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diff --git a/docs/htmldoc/mathcomp.ssreflect.seq.html b/docs/htmldoc/mathcomp.ssreflect.seq.html deleted file mode 100644 index 6709b82..0000000 --- a/docs/htmldoc/mathcomp.ssreflect.seq.html +++ /dev/null @@ -1,2987 +0,0 @@ -<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" -"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> -<html xmlns="http://www.w3.org/1999/xhtml"> -<head> -<meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> -<link href="coqdoc.css" rel="stylesheet" type="text/css" /> -<title>mathcomp.ssreflect.seq</title> -</head> - -<body> - -<div id="page"> - -<div id="header"> -</div> - -<div id="main"> - -<h1 class="libtitle">Library mathcomp.ssreflect.seq</h1> - -<div class="code"> -<span class="comment">(* (c) Copyright 2006-2016 Microsoft Corporation and Inria. <br/> - Distributed under the terms of CeCILL-B. *)</span><br/> - -<br/> -</div> - -<div class="doc"> - The seq type is the ssreflect type for sequences; it is an alias for the - standard Coq list type. The ssreflect library equips it with many - operations, as well as eqType and predType (and, later, choiceType) - structures. The operations are geared towards reflection: they generally - expect and provide boolean predicates, e.g., the membership predicate - expects an eqType. To avoid any confusion we do not Import the Coq List - module. - As there is no true subtyping in Coq, we don't use a type for non-empty - sequences; rather, we pass explicitly the head and tail of the sequence. - The empty sequence is especially bothersome for subscripting, since it - forces us to pass a default value. This default value can often be hidden - by a notation. - Here is the list of seq operations: -<a name="lab35"></a><h2 class="section">Constructors:</h2> - - seq T == the type of sequences of items of type T. - bitseq == seq bool. - [:: ], nil, Nil T == the empty sequence (of type T). - x :: s, cons x s, Cons T x s == the sequence x followed by s (of type T). - [:: x] == the singleton sequence. - [:: x_0; ...; x_n] == the explicit sequence of the x_i. - [:: x_0, ..., x_n & s] == the sequence of the x_i, followed by s. - rcons s x == the sequence s, followed by x. - All of the above, except rcons, can be used in patterns. We define a view - lastP and an induction principle last_ind that can be used to decompose - or traverse a sequence in a right to left order. The view lemma lastP has - a dependent family type, so the ssreflect tactic case/lastP: p => [|p' x] - will generate two subgoals in which p has been replaced by [:: ] and by - rcons p' x, respectively. -<a name="lab36"></a><h2 class="section">Factories:</h2> - - nseq n x == a sequence of n x's. - ncons n x s == a sequence of n x's, followed by s. - seqn n x_0 ... x_n-1 == the sequence of the x_i; can be partially applied. - iota m n == the sequence m, m + 1, ..., m + n - 1. - mkseq f n == the sequence f 0, f 1, ..., f (n - 1). -<a name="lab37"></a><h2 class="section">Sequential access:</h2> - - head x0 s == the head (zero'th item) of s if s is non-empty, else x0. - ohead s == None if s is empty, else Some x when the head of s is x. - behead s == s minus its head, i.e., s' if s = x :: s', else [:: ]. - last x s == the last element of x :: s (which is non-empty). - belast x s == x :: s minus its last item. -<a name="lab38"></a><h2 class="section">Dimensions:</h2> - - size s == the number of items (length) in s. - shape ss == the sequence of sizes of the items of the sequence of - sequences ss. -<a name="lab39"></a><h2 class="section">Random access:</h2> - - nth x0 s i == the item i of s (numbered from 0), or x0 if s does - not have at least i+1 items (i.e., size x <= i) - s`<i>i == standard notation for nth x0 s i for a default x0, - e.g., 0 for rings. - set_nth x0 s i y == s where item i has been changed to y; if s does not - have an item i, it is first padded with copies of x0 - to size i+1. - incr_nth s i == the nat sequence s with item i incremented (s is - first padded with 0's to size i+1, if needed). -<a name="lab40"></a><h2 class="section">Predicates:</h2> - - nilp s <=> s is [:: ]. - := (size s == 0). - x \in s == x appears in s (this requires an eqType for T). - index x s == the first index at which x appears in s, or size s if - x \notin s. - has a s <=> a holds for some item in s, where a is an applicative - bool predicate. - all a s <=> a holds for all items in s. - 'has_aP <-> the view reflect (exists2 x, x \in s & A x) (has a s), - where aP x : reflect (A x) (a x). - 'all_aP <=> the view for reflect {in s, forall x, A x} (all a s). - all2 r s t <=> the (bool) relation r holds for all <i>respective</i> items - in s and t, which must also have the same size, i.e., - for s := [:: x1; ...; x_m] and t := [:: y1; ...; y_n], - the condition [&& r x_1 y_1, ..., r x_n y_n & m == n]. - find p s == the index of the first item in s for which p holds, - or size s if no such item is found. - count p s == the number of items of s for which p holds. - count_mem x s == the multiplicity of x in s, i.e., count (pred1 x) s. - tally s == a tally of s, i.e., a sequence of (item, multiplicity) - pairs for all items in sequence s (without duplicates). - incr_tally bs x == increment the multiplicity of x in the tally bs, or add - x with multiplicity 1 at then end if x is not in bs. - bs \is a wf_tally <=> bs is well-formed tally, with no duplicate items or - null multiplicities. - tally_seq bs == the expansion of a tally bs into a sequence where each - (x, n) pair expands into a sequence of n x's. - constant s <=> all items in s are identical (trivial if s = [:: ]). - uniq s <=> all the items in s are pairwise different. - subseq s1 s2 <=> s1 is a subsequence of s2, i.e., s1 = mask m s2 for - some m : bitseq (see below). - perm_eq s1 s2 <=> s2 is a permutation of s1, i.e., s1 and s2 have the - items (with the same repetitions), but possibly in a - different order. - perm_eql s1 s2 <-> s1 and s2 behave identically on the left of perm_eq. - perm_eqr s1 s2 <-> s1 and s2 behave identically on the right of perm_eq. -> These left/right transitive versions of perm_eq make it easier to - chain a sequence of equivalences. - permutations s == a duplicate-free list of all permutations of s. -<a name="lab41"></a><h2 class="section">Filtering:</h2> - - filter p s == the subsequence of s consisting of all the items - for which the (boolean) predicate p holds. - rem x s == the subsequence of s, where the first occurrence - of x has been removed (compare filter (predC1 x) s - where ALL occurrences of x are removed). - undup s == the subsequence of s containing only the first - occurrence of each item in s, i.e., s with all - duplicates removed. - mask m s == the subsequence of s selected by m : bitseq, with - item i of s selected by bit i in m (extra items or - bits are ignored. -<a name="lab42"></a><h2 class="section">Surgery:</h2> - - s1 ++ s2, cat s1 s2 == the concatenation of s1 and s2. - take n s == the sequence containing only the first n items of s - (or all of s if size s <= n). - drop n s == s minus its first n items ( [:: ] if size s <= n) - rot n s == s rotated left n times (or s if size s <= n). - := drop n s ++ take n s - rotr n s == s rotated right n times (or s if size s <= n). - rev s == the (linear time) reversal of s. - catrev s1 s2 == the reversal of s1 followed by s2 (this is the - recursive form of rev). -<a name="lab43"></a><h2 class="section">Dependent iterator: for s : seq S and t : S -> seq T</h2> - - [seq E | x <- s, y <- t] := flatten [seq [seq E | x <- t] | y <- s] - == the sequence of all the f x y, with x and y drawn from - s and t, respectively, in row-major order, - and where t is possibly dependent in elements of s - allpairs_dep f s t := self expanding definition for - [seq f x y | x <- s, y <- t i] -<a name="lab44"></a><h2 class="section">Iterators: for s == [:: x_1, ..., x_n], t == [:: y_1, ..., y_m],</h2> - - allpairs f s t := same as allpairs_dep but where t is non dependent, - i.e. self expanding definition for - [seq f x y | x <- s, y <- t] - := [:: f x_1 y_1; ...; f x_1 y_m; f x_2 y_1; ...; f x_n y_m] - map f s == the sequence [:: f x_1, ..., f x_n]. - pmap pf s == the sequence [:: y_i1, ..., y_ik] where i1 < ... < ik, - pf x_i = Some y_i, and pf x_j = None iff j is not in - {i1, ..., ik}. - foldr f a s == the right fold of s by f (i.e., the natural iterator). - := f x_1 (f x_2 ... (f x_n a)) - sumn s == x_1 + (x_2 + ... + (x_n + 0)) (when s : seq nat). - foldl f a s == the left fold of s by f. - := f (f ... (f a x_1) ... x_n-1) x_n - scanl f a s == the sequence of partial accumulators of foldl f a s. - := [:: f a x_1; ...; foldl f a s] - pairmap f a s == the sequence of f applied to consecutive items in a :: s. - := [:: f a x_1; f x_1 x_2; ...; f x_n-1 x_n] - zip s t == itemwise pairing of s and t (dropping any extra items). - := [:: (x_1, y_1); ...; (x_mn, y_mn) ] with mn = minn n m. - unzip1 s == [:: (x_1).1; ...; (x_n).1] when s : seq (S * T). - unzip2 s == [:: (x_1).2; ...; (x_n).2] when s : seq (S * T). - flatten s == x_1 ++ ... ++ x_n ++ [:: ] when s : seq (seq T). - reshape r s == s reshaped into a sequence of sequences whose sizes are - given by r (truncating if s is too long or too short). - := [:: [:: x_1; ...; x_r1]; - [:: x(r1 + 1); ...; x(r0 + r1) ]; - ...; - [:: x(r1 + ... + r(k-1) + 1); ...; x(r0 + ... rk) ]#] - flatten_index sh r c == the index, in flatten ss, of the item of indexes - (r, c) in any sequence of sequences ss of shape sh - := sh_1 + sh_2 + ... + sh_r + c - reshape_index sh i == the index, in reshape sh s, of the sequence - containing the i-th item of s. - reshape_offset sh i == the offset, in the (reshape_index sh i)-th - sequence of reshape sh s of the i-th item of s -<a name="lab45"></a><h2 class="section">Notation for manifest comprehensions:</h2> - - [seq x <- s | C] := filter (fun x => C) s. - [seq E | x <- s] := map (fun x => E) s. - [seq x <- s | C1 & C2] := [seq x <- s | C1 && C2]. - [seq E | x <- s & C] := [seq E | x <- [seq x | C] ]. -> The above allow optional type casts on the eigenvariables, as in - [seq x : T <- s | C] or [seq E | x : T <- s, y : U <- t]. The cast may be - needed as type inference considers E or C before s. - We are quite systematic in providing lemmas to rewrite any composition - of two operations. "rev", whose simplifications are not natural, is - protected with nosimpl. -<a name="lab46"></a><h2 class="section">The following are equivalent:</h2> - - [<-> P0; P1; ..; Pn] <-> P0, P1, ..., Pn are all equivalent. - := P0 -> P1 -> ... -> Pn -> P0 - if T : [<-> P0; P1; ..; Pn] is such an equivalence, and i, j are in nat - then T i j is a proof of the equivalence Pi <-> Pj between Pi and Pj; - when i (resp. j) is out of bounds, Pi (resp. Pj) defaults to P0. -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Set Implicit Arguments</span>.<br/> - -<br/> -<span class="id" title="keyword">Delimit</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">seq_scope</span> <span class="id" title="keyword">with</span> <span class="id" title="var">SEQ</span>.<br/> -<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">seq_scope</span>.<br/> - -<br/> -</div> - -<div class="doc"> - Inductive seq (T : Type) : Type := Nil | Cons of T & seq T. -</div> -<div class="code"> -<span class="id" title="keyword">Notation</span> <a name="seq"><span class="id" title="abbreviation">seq</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a>.<br/> -<span class="id" title="keyword">Notation</span> <a name="Cons"><span class="id" title="abbreviation">Cons</span></a> <span class="id" title="var">T</span> := (@<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#cons"><span class="id" title="constructor">cons</span></a> <span class="id" title="var">T</span>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/> -<span class="id" title="keyword">Notation</span> <a name="Nil"><span class="id" title="abbreviation">Nil</span></a> <span class="id" title="var">T</span> := (@<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <span class="id" title="var">T</span>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/> - -<br/> -</div> - -<div class="doc"> - As :: and ++ are (improperly) declared in Init.datatypes, we only rebind - them here. -</div> -<div class="code"> -<span class="id" title="keyword">Infix</span> <a name="407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">"</span></a>::" := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#cons"><span class="id" title="constructor">cons</span></a> : <span class="id" title="var">seq_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="0a934e621391740b862762275992e626"><span class="id" title="notation">"</span></a>[ :: ]" := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ :: ]") : <span class="id" title="var">seq_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="506674b18256ef8f50efed43fa1dfd7d"><span class="id" title="notation">"</span></a>[ :: x1 ]" := (<span class="id" title="var">x1</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a>)<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ :: x1 ]") : <span class="id" title="var">seq_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="72c69ac2151566663801757b15a67c9e"><span class="id" title="notation">"</span></a>[ :: x & s ]" := (<span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">s</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">seq_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="ab09fa4bd99b3931b6a319926b973947"><span class="id" title="notation">"</span></a>[ :: x1 , x2 , .. , xn & s ]" := (<span class="id" title="var">x1</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">x2</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> .. (<span class="id" title="var">xn</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">s</span>) ..)<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span><br/> - "'[hv' [ :: '[' x1 , '/' x2 , '/' .. , '/' xn ']' '/ ' & s ] ']'"<br/> - ) : <span class="id" title="var">seq_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="9e9281397dab83046645f1b62dbb2487"><span class="id" title="notation">"</span></a>[ :: x1 ; x2 ; .. ; xn ]" := (<span class="id" title="var">x1</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">x2</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> .. <a class="idref" href="mathcomp.ssreflect.seq.html#506674b18256ef8f50efed43fa1dfd7d"><span class="id" title="notation">[::</span></a> <span class="id" title="var">xn</span><a class="idref" href="mathcomp.ssreflect.seq.html#506674b18256ef8f50efed43fa1dfd7d"><span class="id" title="notation">]</span></a> ..)<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ :: '[' x1 ; '/' x2 ; '/' .. ; '/' xn ']' ]"<br/> - ) : <span class="id" title="var">seq_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Sequences"><span class="id" title="section">Sequences</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="Sequences.n0"><span class="id" title="variable">n0</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>. <span class="comment">(* numerical parameter for take, drop et al *)</span><br/> -<span class="id" title="keyword">Variable</span> <a name="Sequences.T"><span class="id" title="variable">T</span></a> : <span class="id" title="keyword">Type</span>. <span class="comment">(* must come before the implicit Type *)</span><br/> -<span class="id" title="keyword">Variable</span> <a name="Sequences.x0"><span class="id" title="variable">x0</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.T"><span class="id" title="variable">T</span></a>. <span class="comment">(* default for head/nth *)</span><br/> - -<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.T"><span class="id" title="variable">T</span></a>.<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.T"><span class="id" title="variable">T</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="size"><span class="id" title="definition">size</span></a> <span class="id" title="var">s</span> := <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">_</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">s'</span> <span class="id" title="keyword">then</span> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <span class="id" title="var">s'</span><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">).+1</span></a> <span class="id" title="keyword">else</span> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="size0nil"><span class="id" title="lemma">size0nil</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a>. <br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="nilp"><span class="id" title="definition">nilp</span></a> <span class="id" title="var">s</span> := <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nilP"><span class="id" title="lemma">nilP</span></a> <span class="id" title="var">s</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#nilp"><span class="id" title="definition">nilp</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="ohead"><span class="id" title="definition">ohead</span></a> <span class="id" title="var">s</span> := <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">_</span> <span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">x</span> <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a>.<br/> -<span class="id" title="keyword">Definition</span> <a name="head"><span class="id" title="definition">head</span></a> <span class="id" title="var">s</span> := <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">_</span> <span class="id" title="keyword">then</span> <span class="id" title="var">x</span> <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.x0"><span class="id" title="variable">x0</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="behead"><span class="id" title="definition">behead</span></a> <span class="id" title="var">s</span> := <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">_</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">s'</span> <span class="id" title="keyword">then</span> <span class="id" title="var">s'</span> <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="size_behead"><span class="id" title="lemma">size_behead</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#behead"><span class="id" title="definition">behead</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#f953bf7095e0da1cb644443fd0e17d6d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#f953bf7095e0da1cb644443fd0e17d6d"><span class="id" title="notation">).-1</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - Factories -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="ncons"><span class="id" title="definition">ncons</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> := <a class="idref" href="mathcomp.ssreflect.ssrnat.html#iter"><span class="id" title="definition">iter</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#cons"><span class="id" title="constructor">cons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>).<br/> -<span class="id" title="keyword">Definition</span> <a name="nseq"><span class="id" title="definition">nseq</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> := <a class="idref" href="mathcomp.ssreflect.seq.html#ncons"><span class="id" title="definition">ncons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="size_ncons"><span class="id" title="lemma">size_ncons</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#ncons"><span class="id" title="definition">ncons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="size_nseq"><span class="id" title="lemma">size_nseq</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#nseq"><span class="id" title="definition">nseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - n-ary, dependently typed constructor. -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="seqn_type"><span class="id" title="definition">seqn_type</span></a> <span class="id" title="var">n</span> := <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">n'</span><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <span class="id" title="keyword">then</span> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.T"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#seqn_type"><span class="id" title="definition">seqn_type</span></a> <span class="id" title="var">n'</span> <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.T"><span class="id" title="variable">T</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="seqn_rec"><span class="id" title="definition">seqn_rec</span></a> <span class="id" title="var">f</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seqn_type"><span class="id" title="definition">seqn_type</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> :=<br/> - <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">n'</span><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <span class="id" title="keyword">return</span> <a class="idref" href="mathcomp.ssreflect.seq.html#seqn_type"><span class="id" title="definition">seqn_type</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">then</span><br/> - <span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> ⇒ <a class="idref" href="mathcomp.ssreflect.seq.html#seqn_rec"><span class="id" title="definition">seqn_rec</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">s</span> ⇒ <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>)) <span class="id" title="var">n'</span><br/> - <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a>.<br/> -<span class="id" title="keyword">Definition</span> <a name="seqn"><span class="id" title="definition">seqn</span></a> := <a class="idref" href="mathcomp.ssreflect.seq.html#seqn_rec"><span class="id" title="definition">seqn_rec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - Sequence catenation "cat". -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="cat"><span class="id" title="definition">cat</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> := <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">s1'</span> <span class="id" title="keyword">then</span> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">s1'</span> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a><br/> -<span class="id" title="keyword">where</span> <a name="cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">"</span></a>s1 ++ s2" := (<a class="idref" href="mathcomp.ssreflect.seq.html#cat"><span class="id" title="definition">cat</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span>) : <span class="id" title="var">seq_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cat0s"><span class="id" title="lemma">cat0s</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>. <br/> -<span class="id" title="keyword">Lemma</span> <a name="cat1s"><span class="id" title="lemma">cat1s</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#506674b18256ef8f50efed43fa1dfd7d"><span class="id" title="notation">[::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#506674b18256ef8f50efed43fa1dfd7d"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>. <br/> -<span class="id" title="keyword">Lemma</span> <a name="cat_cons"><span class="id" title="lemma">cat_cons</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>. <br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cat_nseq"><span class="id" title="lemma">cat_nseq</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#nseq"><span class="id" title="definition">nseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ncons"><span class="id" title="definition">ncons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cats0"><span class="id" title="lemma">cats0</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="catA"><span class="id" title="lemma">catA</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> <span class="id" title="var">s3</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s3"><span class="id" title="variable">s3</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s3"><span class="id" title="variable">s3</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="size_cat"><span class="id" title="lemma">size_cat</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - last, belast, rcons, and last induction. -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="rcons"><span class="id" title="definition">rcons</span></a> <span class="id" title="var">s</span> <span class="id" title="var">z</span> := <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">s'</span> <span class="id" title="keyword">then</span> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <span class="id" title="var">s'</span> <a class="idref" href="mathcomp.ssreflect.seq.html#z"><span class="id" title="variable">z</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.seq.html#506674b18256ef8f50efed43fa1dfd7d"><span class="id" title="notation">[::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#506674b18256ef8f50efed43fa1dfd7d"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rcons_cons"><span class="id" title="lemma">rcons_cons</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#z"><span class="id" title="variable">z</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cats1"><span class="id" title="lemma">cats1</span></a> <span class="id" title="var">s</span> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#506674b18256ef8f50efed43fa1dfd7d"><span class="id" title="notation">[::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#506674b18256ef8f50efed43fa1dfd7d"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#z"><span class="id" title="variable">z</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="last"><span class="id" title="definition">last</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> := <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">x'</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">s'</span> <span class="id" title="keyword">then</span> <a class="idref" href="mathcomp.ssreflect.seq.html#last"><span class="id" title="definition">last</span></a> <span class="id" title="var">x'</span> <span class="id" title="var">s'</span> <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>.<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="belast"><span class="id" title="definition">belast</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> := <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">x'</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">s'</span> <span class="id" title="keyword">then</span> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#belast"><span class="id" title="definition">belast</span></a> <span class="id" title="var">x'</span> <span class="id" title="var">s'</span><a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">)</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lastI"><span class="id" title="lemma">lastI</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#belast"><span class="id" title="definition">belast</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#last"><span class="id" title="definition">last</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="last_cons"><span class="id" title="lemma">last_cons</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#last"><span class="id" title="definition">last</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#last"><span class="id" title="definition">last</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="size_rcons"><span class="id" title="lemma">size_rcons</span></a> <span class="id" title="var">s</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">).+1</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="size_belast"><span class="id" title="lemma">size_belast</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#belast"><span class="id" title="definition">belast</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="last_cat"><span class="id" title="lemma">last_cat</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#last"><span class="id" title="definition">last</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#last"><span class="id" title="definition">last</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#last"><span class="id" title="definition">last</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="last_rcons"><span class="id" title="lemma">last_rcons</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#last"><span class="id" title="definition">last</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#z"><span class="id" title="variable">z</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#z"><span class="id" title="variable">z</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="belast_cat"><span class="id" title="lemma">belast_cat</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#belast"><span class="id" title="definition">belast</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#belast"><span class="id" title="definition">belast</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#belast"><span class="id" title="definition">belast</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#last"><span class="id" title="definition">last</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="belast_rcons"><span class="id" title="lemma">belast_rcons</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#belast"><span class="id" title="definition">belast</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#z"><span class="id" title="variable">z</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cat_rcons"><span class="id" title="lemma">cat_rcons</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rcons_cat"><span class="id" title="lemma">rcons_cat</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variant</span> <a name="last_spec"><span class="id" title="inductive">last_spec</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.T"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span> :=<br/> - | <a name="LastNil"><span class="id" title="constructor">LastNil</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#last_spec"><span class="id" title="inductive">last_spec</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a><br/> - | <a name="LastRcons"><span class="id" title="constructor">LastRcons</span></a> <span class="id" title="var">s</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#last_spec"><span class="id" title="inductive">last_spec</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="lastP"><span class="id" title="lemma">lastP</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#last_spec"><span class="id" title="inductive">last_spec</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="last_ind"><span class="id" title="lemma">last_ind</span></a> <span class="id" title="var">P</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">s</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.ssreflect.seq.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#P"><span class="id" title="variable">P</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">s</span>, <a class="idref" href="mathcomp.ssreflect.seq.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - Sequence indexing. -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="nth"><span class="id" title="definition">nth</span></a> <span class="id" title="var">s</span> <span class="id" title="var">n</span> {<span class="id" title="keyword">struct</span> <span class="id" title="var">n</span>} :=<br/> - <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">s'</span> <span class="id" title="keyword">then</span> <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">n'</span><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <span class="id" title="keyword">then</span> @<a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <span class="id" title="var">s'</span> <span class="id" title="var">n'</span> <span class="id" title="keyword">else</span> <span class="id" title="var">x</span> <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.x0"><span class="id" title="variable">x0</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="set_nth"><span class="id" title="definition">set_nth</span></a> <span class="id" title="var">s</span> <span class="id" title="var">n</span> <span class="id" title="var">y</span> {<span class="id" title="keyword">struct</span> <span class="id" title="var">n</span>} :=<br/> - <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">s'</span> <span class="id" title="keyword">then</span> <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">n'</span><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <span class="id" title="keyword">then</span> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> @<a class="idref" href="mathcomp.ssreflect.seq.html#set_nth"><span class="id" title="definition">set_nth</span></a> <span class="id" title="var">s'</span> <span class="id" title="var">n'</span> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">s'</span><br/> - <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.seq.html#ncons"><span class="id" title="definition">ncons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.x0"><span class="id" title="variable">x0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#506674b18256ef8f50efed43fa1dfd7d"><span class="id" title="notation">[::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#506674b18256ef8f50efed43fa1dfd7d"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nth0"><span class="id" title="lemma">nth0</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#head"><span class="id" title="definition">head</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>. <br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nth_default"><span class="id" title="lemma">nth_default</span></a> <span class="id" title="var">s</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.x0"><span class="id" title="variable">x0</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nth_nil"><span class="id" title="lemma">nth_nil</span></a> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.x0"><span class="id" title="variable">x0</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="last_nth"><span class="id" title="lemma">last_nth</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#last"><span class="id" title="definition">last</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nth_last"><span class="id" title="lemma">nth_last</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#f953bf7095e0da1cb644443fd0e17d6d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#f953bf7095e0da1cb644443fd0e17d6d"><span class="id" title="notation">).-1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#last"><span class="id" title="definition">last</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.x0"><span class="id" title="variable">x0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nth_behead"><span class="id" title="lemma">nth_behead</span></a> <span class="id" title="var">s</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#behead"><span class="id" title="definition">behead</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nth_cat"><span class="id" title="lemma">nth_cat</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> <span class="id" title="var">n</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#7825ccc99f23b0d30c9d40c317ba7af0"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nth_rcons"><span class="id" title="lemma">nth_rcons</span></a> <span class="id" title="var">s</span> <span class="id" title="var">x</span> <span class="id" title="var">n</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.x0"><span class="id" title="variable">x0</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nth_ncons"><span class="id" title="lemma">nth_ncons</span></a> <span class="id" title="var">m</span> <span class="id" title="var">x</span> <span class="id" title="var">s</span> <span class="id" title="var">n</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#ncons"><span class="id" title="definition">ncons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#7825ccc99f23b0d30c9d40c317ba7af0"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nth_nseq"><span class="id" title="lemma">nth_nseq</span></a> <span class="id" title="var">m</span> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#nseq"><span class="id" title="definition">nseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.x0"><span class="id" title="variable">x0</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_from_nth"><span class="id" title="lemma">eq_from_nth</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="size_set_nth"><span class="id" title="lemma">size_set_nth</span></a> <span class="id" title="var">s</span> <span class="id" title="var">n</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#set_nth"><span class="id" title="definition">set_nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#maxn"><span class="id" title="definition">maxn</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="set_nth_nil"><span class="id" title="lemma">set_nth_nil</span></a> <span class="id" title="var">n</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#set_nth"><span class="id" title="definition">set_nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ncons"><span class="id" title="definition">ncons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.x0"><span class="id" title="variable">x0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#506674b18256ef8f50efed43fa1dfd7d"><span class="id" title="notation">[::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#506674b18256ef8f50efed43fa1dfd7d"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nth_set_nth"><span class="id" title="lemma">nth_set_nth</span></a> <span class="id" title="var">s</span> <span class="id" title="var">n</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#set_nth"><span class="id" title="definition">set_nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#1e16e052495f43108296b1e975fd3236"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.eqtype.html#1e16e052495f43108296b1e975fd3236"><span class="id" title="notation">eta</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#1e16e052495f43108296b1e975fd3236"><span class="id" title="notation">with</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#d74b61a39ba99967624d24970637896f"><span class="id" title="notation">|-></span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.ssreflect.eqtype.html#1e16e052495f43108296b1e975fd3236"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="set_set_nth"><span class="id" title="lemma">set_set_nth</span></a> <span class="id" title="var">s</span> <span class="id" title="var">n1</span> <span class="id" title="var">y1</span> <span class="id" title="var">n2</span> <span class="id" title="var">y2</span> (<span class="id" title="var">s2</span> := <a class="idref" href="mathcomp.ssreflect.seq.html#set_nth"><span class="id" title="definition">set_nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n2"><span class="id" title="variable">n2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y2"><span class="id" title="variable">y2</span></a>) :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#set_nth"><span class="id" title="definition">set_nth</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#set_nth"><span class="id" title="definition">set_nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n1"><span class="id" title="variable">n1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y1"><span class="id" title="variable">y1</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#n2"><span class="id" title="variable">n2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y2"><span class="id" title="variable">y2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n1"><span class="id" title="variable">n1</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n2"><span class="id" title="variable">n2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#set_nth"><span class="id" title="definition">set_nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n1"><span class="id" title="variable">n1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y1"><span class="id" title="variable">y1</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - find, count, has, all. -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Sequences.SeqFind"><span class="id" title="section">SeqFind</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="Sequences.SeqFind.a"><span class="id" title="variable">a</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.T"><span class="id" title="variable">T</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="find"><span class="id" title="definition">find</span></a> <span class="id" title="var">s</span> := <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">s'</span> <span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.SeqFind.a"><span class="id" title="variable">a</span></a> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#find"><span class="id" title="definition">find</span></a> <span class="id" title="var">s'</span><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">).+1</span></a> <span class="id" title="keyword">else</span> 0.<br/> - -<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="filter"><span class="id" title="definition">filter</span></a> <span class="id" title="var">s</span> :=<br/> - <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">s'</span> <span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.SeqFind.a"><span class="id" title="variable">a</span></a> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <span class="id" title="var">s'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <span class="id" title="var">s'</span> <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="count"><span class="id" title="definition">count</span></a> <span class="id" title="var">s</span> := <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">s'</span> <span class="id" title="keyword">then</span> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.SeqFind.a"><span class="id" title="variable">a</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#count"><span class="id" title="definition">count</span></a> <span class="id" title="var">s'</span> <span class="id" title="keyword">else</span> 0.<br/> - -<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="has"><span class="id" title="definition">has</span></a> <span class="id" title="var">s</span> := <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">s'</span> <span class="id" title="keyword">then</span> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.SeqFind.a"><span class="id" title="variable">a</span></a> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <span class="id" title="var">s'</span> <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="all"><span class="id" title="definition">all</span></a> <span class="id" title="var">s</span> := <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">s'</span> <span class="id" title="keyword">then</span> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.SeqFind.a"><span class="id" title="variable">a</span></a> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> <span class="id" title="var">s'</span> <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="size_filter"><span class="id" title="lemma">size_filter</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#count"><span class="id" title="definition">count</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="has_count"><span class="id" title="lemma">has_count</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#count"><span class="id" title="definition">count</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="count_size"><span class="id" title="lemma">count_size</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#count"><span class="id" title="definition">count</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="all_count"><span class="id" title="lemma">all_count</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#count"><span class="id" title="definition">count</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="filter_all"><span class="id" title="lemma">filter_all</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="all_filterP"><span class="id" title="lemma">all_filterP</span></a> <span class="id" title="var">s</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="filter_id"><span class="id" title="lemma">filter_id</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="has_find"><span class="id" title="lemma">has_find</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#find"><span class="id" title="definition">find</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="find_size"><span class="id" title="lemma">find_size</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#find"><span class="id" title="definition">find</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="find_cat"><span class="id" title="lemma">find_cat</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#find"><span class="id" title="definition">find</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#find"><span class="id" title="definition">find</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#find"><span class="id" title="definition">find</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="has_nil"><span class="id" title="lemma">has_nil</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>. <br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="has_seq1"><span class="id" title="lemma">has_seq1</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#506674b18256ef8f50efed43fa1dfd7d"><span class="id" title="notation">[::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#506674b18256ef8f50efed43fa1dfd7d"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.SeqFind.a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="has_nseq"><span class="id" title="lemma">has_nseq</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#nseq"><span class="id" title="definition">nseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.SeqFind.a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="has_seqb"><span class="id" title="lemma">has_seqb</span></a> (<span class="id" title="var">b</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#nseq"><span class="id" title="definition">nseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.SeqFind.a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="all_nil"><span class="id" title="lemma">all_nil</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>. <br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="all_seq1"><span class="id" title="lemma">all_seq1</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#506674b18256ef8f50efed43fa1dfd7d"><span class="id" title="notation">[::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#506674b18256ef8f50efed43fa1dfd7d"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.SeqFind.a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="all_nseq"><span class="id" title="lemma">all_nseq</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#nseq"><span class="id" title="definition">nseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.SeqFind.a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="all_nseqb"><span class="id" title="lemma">all_nseqb</span></a> (<span class="id" title="var">b</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#nseq"><span class="id" title="definition">nseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#a133e82bab56729f895f9b2b31e837cd"><span class="id" title="notation">==></span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.SeqFind.a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="filter_nseq"><span class="id" title="lemma">filter_nseq</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#nseq"><span class="id" title="definition">nseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nseq"><span class="id" title="definition">nseq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.SeqFind.a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="count_nseq"><span class="id" title="lemma">count_nseq</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#count"><span class="id" title="definition">count</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#nseq"><span class="id" title="definition">nseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.SeqFind.a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="find_nseq"><span class="id" title="lemma">find_nseq</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#find"><span class="id" title="definition">find</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#nseq"><span class="id" title="definition">nseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.SeqFind.a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nth_find"><span class="id" title="lemma">nth_find</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.SeqFind.a"><span class="id" title="variable">a</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#find"><span class="id" title="definition">find</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>)).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="before_find"><span class="id" title="lemma">before_find</span></a> <span class="id" title="var">s</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#find"><span class="id" title="definition">find</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.SeqFind.a"><span class="id" title="variable">a</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="filter_cat"><span class="id" title="lemma">filter_cat</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="filter_rcons"><span class="id" title="lemma">filter_rcons</span></a> <span class="id" title="var">s</span> <span class="id" title="var">x</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.SeqFind.a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="count_cat"><span class="id" title="lemma">count_cat</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#count"><span class="id" title="definition">count</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#count"><span class="id" title="definition">count</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#count"><span class="id" title="definition">count</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="has_cat"><span class="id" title="lemma">has_cat</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="has_rcons"><span class="id" title="lemma">has_rcons</span></a> <span class="id" title="var">s</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.SeqFind.a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="all_cat"><span class="id" title="lemma">all_cat</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="all_rcons"><span class="id" title="lemma">all_rcons</span></a> <span class="id" title="var">s</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.SeqFind.a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.SeqFind"><span class="id" title="section">SeqFind</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_find"><span class="id" title="lemma">eq_find</span></a> <span class="id" title="var">a1</span> <span class="id" title="var">a2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a2"><span class="id" title="variable">a2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#find"><span class="id" title="definition">find</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#find"><span class="id" title="definition">find</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a2"><span class="id" title="variable">a2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_filter"><span class="id" title="lemma">eq_filter</span></a> <span class="id" title="var">a1</span> <span class="id" title="var">a2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a2"><span class="id" title="variable">a2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a2"><span class="id" title="variable">a2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_count"><span class="id" title="lemma">eq_count</span></a> <span class="id" title="var">a1</span> <span class="id" title="var">a2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a2"><span class="id" title="variable">a2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#count"><span class="id" title="definition">count</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#count"><span class="id" title="definition">count</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a2"><span class="id" title="variable">a2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_has"><span class="id" title="lemma">eq_has</span></a> <span class="id" title="var">a1</span> <span class="id" title="var">a2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a2"><span class="id" title="variable">a2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a2"><span class="id" title="variable">a2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_all"><span class="id" title="lemma">eq_all</span></a> <span class="id" title="var">a1</span> <span class="id" title="var">a2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a2"><span class="id" title="variable">a2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a2"><span class="id" title="variable">a2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Sequences.SubPred"><span class="id" title="section">SubPred</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> (<a name="Sequences.SubPred.a1"><span class="id" title="variable">a1</span></a> <a name="Sequences.SubPred.a2"><span class="id" title="variable">a2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.T"><span class="id" title="variable">T</span></a>).<br/> -<span class="id" title="keyword">Hypothesis</span> <a name="Sequences.SubPred.s12"><span class="id" title="variable">s12</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#subpred"><span class="id" title="definition">subpred</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.SubPred.a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.SubPred.a2"><span class="id" title="variable">a2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sub_find"><span class="id" title="lemma">sub_find</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#find"><span class="id" title="definition">find</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.SubPred.a2"><span class="id" title="variable">a2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#find"><span class="id" title="definition">find</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.SubPred.a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sub_has"><span class="id" title="lemma">sub_has</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.SubPred.a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.SubPred.a2"><span class="id" title="variable">a2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sub_count"><span class="id" title="lemma">sub_count</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#count"><span class="id" title="definition">count</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.SubPred.a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#count"><span class="id" title="definition">count</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.SubPred.a2"><span class="id" title="variable">a2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sub_all"><span class="id" title="lemma">sub_all</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.SubPred.a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.SubPred.a2"><span class="id" title="variable">a2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.SubPred"><span class="id" title="section">SubPred</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="filter_pred0"><span class="id" title="lemma">filter_pred0</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred0"><span class="id" title="definition">pred0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a>. <br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="filter_predT"><span class="id" title="lemma">filter_predT</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#predT"><span class="id" title="definition">predT</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="filter_predI"><span class="id" title="lemma">filter_predI</span></a> <span class="id" title="var">a1</span> <span class="id" title="var">a2</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#predI"><span class="id" title="definition">predI</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a2"><span class="id" title="variable">a2</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a1"><span class="id" title="variable">a1</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a2"><span class="id" title="variable">a2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="count_pred0"><span class="id" title="lemma">count_pred0</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#count"><span class="id" title="definition">count</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred0"><span class="id" title="definition">pred0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="count_predT"><span class="id" title="lemma">count_predT</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#count"><span class="id" title="definition">count</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#predT"><span class="id" title="definition">predT</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="count_predUI"><span class="id" title="lemma">count_predUI</span></a> <span class="id" title="var">a1</span> <span class="id" title="var">a2</span> <span class="id" title="var">s</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#count"><span class="id" title="definition">count</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#predU"><span class="id" title="definition">predU</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a2"><span class="id" title="variable">a2</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#count"><span class="id" title="definition">count</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#predI"><span class="id" title="definition">predI</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a2"><span class="id" title="variable">a2</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#count"><span class="id" title="definition">count</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#count"><span class="id" title="definition">count</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a2"><span class="id" title="variable">a2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="count_predC"><span class="id" title="lemma">count_predC</span></a> <span class="id" title="var">a</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#count"><span class="id" title="definition">count</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#count"><span class="id" title="definition">count</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#predC"><span class="id" title="definition">predC</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="count_filter"><span class="id" title="lemma">count_filter</span></a> <span class="id" title="var">a1</span> <span class="id" title="var">a2</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#count"><span class="id" title="definition">count</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a1"><span class="id" title="variable">a1</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a2"><span class="id" title="variable">a2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#count"><span class="id" title="definition">count</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#predI"><span class="id" title="definition">predI</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a2"><span class="id" title="variable">a2</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="has_pred0"><span class="id" title="lemma">has_pred0</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred0"><span class="id" title="definition">pred0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="has_predT"><span class="id" title="lemma">has_predT</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#predT"><span class="id" title="definition">predT</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="has_predC"><span class="id" title="lemma">has_predC</span></a> <span class="id" title="var">a</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#predC"><span class="id" title="definition">predC</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="has_predU"><span class="id" title="lemma">has_predU</span></a> <span class="id" title="var">a1</span> <span class="id" title="var">a2</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#predU"><span class="id" title="definition">predU</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a2"><span class="id" title="variable">a2</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a2"><span class="id" title="variable">a2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="all_pred0"><span class="id" title="lemma">all_pred0</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred0"><span class="id" title="definition">pred0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="all_predT"><span class="id" title="lemma">all_predT</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#predT"><span class="id" title="definition">predT</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="all_predC"><span class="id" title="lemma">all_predC</span></a> <span class="id" title="var">a</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#predC"><span class="id" title="definition">predC</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="all_predI"><span class="id" title="lemma">all_predI</span></a> <span class="id" title="var">a1</span> <span class="id" title="var">a2</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#predI"><span class="id" title="definition">predI</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a2"><span class="id" title="variable">a2</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a2"><span class="id" title="variable">a2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - Surgery: drop, take, rot, rotr. -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="drop"><span class="id" title="definition">drop</span></a> <span class="id" title="var">n</span> <span class="id" title="var">s</span> {<span class="id" title="keyword">struct</span> <span class="id" title="var">s</span>} :=<br/> - <span class="id" title="keyword">match</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>, <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span><br/> - | <span class="id" title="var">_</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">s'</span>, <span class="id" title="var">n'</span><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> ⇒ <a class="idref" href="mathcomp.ssreflect.seq.html#drop"><span class="id" title="definition">drop</span></a> <span class="id" title="var">n'</span> <span class="id" title="var">s'</span><br/> - | <span class="id" title="var">_</span>, <span class="id" title="var">_</span> ⇒ <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><br/> - <span class="id" title="keyword">end</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="drop_behead"><span class="id" title="lemma">drop_behead</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#drop"><span class="id" title="definition">drop</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#iter"><span class="id" title="definition">iter</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#behead"><span class="id" title="definition">behead</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="drop0"><span class="id" title="lemma">drop0</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#drop"><span class="id" title="definition">drop</span></a> 0 <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>. <br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="drop1"><span class="id" title="lemma">drop1</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#drop"><span class="id" title="definition">drop</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#behead"><span class="id" title="definition">behead</span></a>. <br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="drop_oversize"><span class="id" title="lemma">drop_oversize</span></a> <span class="id" title="var">n</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#drop"><span class="id" title="definition">drop</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="drop_size"><span class="id" title="lemma">drop_size</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#drop"><span class="id" title="definition">drop</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="drop_cons"><span class="id" title="lemma">drop_cons</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#drop"><span class="id" title="definition">drop</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.n0"><span class="id" title="variable">n0</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.n0"><span class="id" title="variable">n0</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">n</span><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <span class="id" title="keyword">then</span> <a class="idref" href="mathcomp.ssreflect.seq.html#drop"><span class="id" title="definition">drop</span></a> <span class="id" title="var">n</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="size_drop"><span class="id" title="lemma">size_drop</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#drop"><span class="id" title="definition">drop</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#7825ccc99f23b0d30c9d40c317ba7af0"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.n0"><span class="id" title="variable">n0</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="drop_cat"><span class="id" title="lemma">drop_cat</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#drop"><span class="id" title="definition">drop</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.n0"><span class="id" title="variable">n0</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#drop"><span class="id" title="definition">drop</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#drop"><span class="id" title="definition">drop</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#7825ccc99f23b0d30c9d40c317ba7af0"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="drop_size_cat"><span class="id" title="lemma">drop_size_cat</span></a> <span class="id" title="var">n</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#drop"><span class="id" title="definition">drop</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nconsK"><span class="id" title="lemma">nconsK</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#ncons"><span class="id" title="definition">ncons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#drop"><span class="id" title="definition">drop</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="drop_drop"><span class="id" title="lemma">drop_drop</span></a> <span class="id" title="var">s</span> <span class="id" title="var">n1</span> <span class="id" title="var">n2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#drop"><span class="id" title="definition">drop</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n1"><span class="id" title="variable">n1</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#drop"><span class="id" title="definition">drop</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n2"><span class="id" title="variable">n2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#drop"><span class="id" title="definition">drop</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#n1"><span class="id" title="variable">n1</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n2"><span class="id" title="variable">n2</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="take"><span class="id" title="definition">take</span></a> <span class="id" title="var">n</span> <span class="id" title="var">s</span> {<span class="id" title="keyword">struct</span> <span class="id" title="var">s</span>} :=<br/> - <span class="id" title="keyword">match</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>, <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">with</span><br/> - | <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">s'</span>, <span class="id" title="var">n'</span><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> ⇒ <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#take"><span class="id" title="definition">take</span></a> <span class="id" title="var">n'</span> <span class="id" title="var">s'</span><br/> - | <span class="id" title="var">_</span>, <span class="id" title="var">_</span> ⇒ <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a><br/> - <span class="id" title="keyword">end</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="take0"><span class="id" title="lemma">take0</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#take"><span class="id" title="definition">take</span></a> 0 <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a>. <br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="take_oversize"><span class="id" title="lemma">take_oversize</span></a> <span class="id" title="var">n</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#take"><span class="id" title="definition">take</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="take_size"><span class="id" title="lemma">take_size</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#take"><span class="id" title="definition">take</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="take_cons"><span class="id" title="lemma">take_cons</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#take"><span class="id" title="definition">take</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.n0"><span class="id" title="variable">n0</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.n0"><span class="id" title="variable">n0</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">n</span><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <span class="id" title="keyword">then</span> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#take"><span class="id" title="definition">take</span></a> <span class="id" title="var">n</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">)</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="drop_rcons"><span class="id" title="lemma">drop_rcons</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.ssreflect.seq.html#drop"><span class="id" title="definition">drop</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.n0"><span class="id" title="variable">n0</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#drop"><span class="id" title="definition">drop</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cat_take_drop"><span class="id" title="lemma">cat_take_drop</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#take"><span class="id" title="definition">take</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#drop"><span class="id" title="definition">drop</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="size_takel"><span class="id" title="lemma">size_takel</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#take"><span class="id" title="definition">take</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.n0"><span class="id" title="variable">n0</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="size_take"><span class="id" title="lemma">size_take</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#take"><span class="id" title="definition">take</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="take_cat"><span class="id" title="lemma">take_cat</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#take"><span class="id" title="definition">take</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.n0"><span class="id" title="variable">n0</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#take"><span class="id" title="definition">take</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#take"><span class="id" title="definition">take</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#7825ccc99f23b0d30c9d40c317ba7af0"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="take_size_cat"><span class="id" title="lemma">take_size_cat</span></a> <span class="id" title="var">n</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#take"><span class="id" title="definition">take</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="takel_cat"><span class="id" title="lemma">takel_cat</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#take"><span class="id" title="definition">take</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.n0"><span class="id" title="variable">n0</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#take"><span class="id" title="definition">take</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nth_drop"><span class="id" title="lemma">nth_drop</span></a> <span class="id" title="var">s</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#drop"><span class="id" title="definition">drop</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nth_take"><span class="id" title="lemma">nth_take</span></a> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">s</span>, <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#take"><span class="id" title="definition">take</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - drop_nth and take_nth below do NOT use the default n0, because the "n" - can be inferred from the condition, whereas the nth default value x0 - will have to be given explicitly (and this will provide "d" as well). -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="drop_nth"><span class="id" title="lemma">drop_nth</span></a> <span class="id" title="var">n</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#drop"><span class="id" title="definition">drop</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#drop"><span class="id" title="definition">drop</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="take_nth"><span class="id" title="lemma">take_nth</span></a> <span class="id" title="var">n</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#take"><span class="id" title="definition">take</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#take"><span class="id" title="definition">take</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a>).<br/> - -<br/> -</div> - -<div class="doc"> - Rotation -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="rot"><span class="id" title="definition">rot</span></a> <span class="id" title="var">n</span> <span class="id" title="var">s</span> := <a class="idref" href="mathcomp.ssreflect.seq.html#drop"><span class="id" title="definition">drop</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#take"><span class="id" title="definition">take</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rot0"><span class="id" title="lemma">rot0</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> 0 <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="size_rot"><span class="id" title="lemma">size_rot</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rot_oversize"><span class="id" title="lemma">rot_oversize</span></a> <span class="id" title="var">n</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rot_size"><span class="id" title="lemma">rot_size</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="has_rot"><span class="id" title="lemma">has_rot</span></a> <span class="id" title="var">s</span> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rot_size_cat"><span class="id" title="lemma">rot_size_cat</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="rotr"><span class="id" title="definition">rotr</span></a> <span class="id" title="var">n</span> <span class="id" title="var">s</span> := <a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#7825ccc99f23b0d30c9d40c317ba7af0"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rotK"><span class="id" title="lemma">rotK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.n0"><span class="id" title="variable">n0</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#rotr"><span class="id" title="definition">rotr</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.n0"><span class="id" title="variable">n0</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rot_inj"><span class="id" title="lemma">rot_inj</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences.n0"><span class="id" title="variable">n0</span></a>). <br/> - -<br/> -</div> - -<div class="doc"> - (efficient) reversal -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="catrev"><span class="id" title="definition">catrev</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> := <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">s1'</span> <span class="id" title="keyword">then</span> <a class="idref" href="mathcomp.ssreflect.seq.html#catrev"><span class="id" title="definition">catrev</span></a> <span class="id" title="var">s1'</span> (<span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="rev"><span class="id" title="definition">rev</span></a> <span class="id" title="var">s</span> := <a class="idref" href="mathcomp.ssreflect.seq.html#catrev"><span class="id" title="definition">catrev</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="catrev_catl"><span class="id" title="lemma">catrev_catl</span></a> <span class="id" title="var">s</span> <span class="id" title="var">t</span> <span class="id" title="var">u</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#catrev"><span class="id" title="definition">catrev</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#catrev"><span class="id" title="definition">catrev</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#catrev"><span class="id" title="definition">catrev</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#u"><span class="id" title="variable">u</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="catrev_catr"><span class="id" title="lemma">catrev_catr</span></a> <span class="id" title="var">s</span> <span class="id" title="var">t</span> <span class="id" title="var">u</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#catrev"><span class="id" title="definition">catrev</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#u"><span class="id" title="variable">u</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#catrev"><span class="id" title="definition">catrev</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#u"><span class="id" title="variable">u</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="catrevE"><span class="id" title="lemma">catrevE</span></a> <span class="id" title="var">s</span> <span class="id" title="var">t</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#catrev"><span class="id" title="definition">catrev</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rev_cons"><span class="id" title="lemma">rev_cons</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="size_rev"><span class="id" title="lemma">size_rev</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rev_cat"><span class="id" title="lemma">rev_cat</span></a> <span class="id" title="var">s</span> <span class="id" title="var">t</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cfcff6535a1677da2547b032de3120ce"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rev_rcons"><span class="id" title="lemma">rev_rcons</span></a> <span class="id" title="var">s</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="revK"><span class="id" title="lemma">revK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#involutive"><span class="id" title="definition">involutive</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nth_rev"><span class="id" title="lemma">nth_rev</span></a> <span class="id" title="var">n</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#7825ccc99f23b0d30c9d40c317ba7af0"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="filter_rev"><span class="id" title="lemma">filter_rev</span></a> <span class="id" title="var">a</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="count_rev"><span class="id" title="lemma">count_rev</span></a> <span class="id" title="var">a</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#count"><span class="id" title="definition">count</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#count"><span class="id" title="definition">count</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="has_rev"><span class="id" title="lemma">has_rev</span></a> <span class="id" title="var">a</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="all_rev"><span class="id" title="lemma">all_rev</span></a> <span class="id" title="var">a</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.seq.html#Sequences"><span class="id" title="section">Sequences</span></a>.<br/> - -<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="count_mem"><span class="id" title="abbreviation">count_mem</span></a> <span class="id" title="var">x</span> := (<a class="idref" href="mathcomp.ssreflect.seq.html#count"><span class="id" title="definition">count</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred_of_simpl"><span class="id" title="definition">pred_of_simpl</span></a> (<a class="idref" href="mathcomp.ssreflect.eqtype.html#pred1"><span class="id" title="definition">pred1</span></a> <span class="id" title="var">x</span>))).<br/> - -<br/> -<span class="id" title="keyword">Infix</span> <a name="93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">"</span></a>++" := <a class="idref" href="mathcomp.ssreflect.seq.html#cat"><span class="id" title="definition">cat</span></a> : <span class="id" title="var">seq_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="3de67eb42556f00623e74d6756c5ab67"><span class="id" title="notation">"</span></a>[ 'seq' x <- s | C ]" := (<a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> ⇒ <span class="id" title="var">C</span>%<span class="id" title="var">B</span>) <span class="id" title="var">s</span>)<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">x</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 99,<br/> - <span class="id" title="var">format</span> "[ '[hv' 'seq' x <- s '/ ' | C ] ']'") : <span class="id" title="var">seq_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="021ca27646f2093224bc04273af48564"><span class="id" title="notation">"</span></a>[ 'seq' x <- s | C1 & C2 ]" := <a class="idref" href="mathcomp.ssreflect.seq.html#3de67eb42556f00623e74d6756c5ab67"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#3de67eb42556f00623e74d6756c5ab67"><span class="id" title="notation">seq</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#3de67eb42556f00623e74d6756c5ab67"><span class="id" title="notation"><-</span></a> <span class="id" title="var">s</span> <a class="idref" href="mathcomp.ssreflect.seq.html#3de67eb42556f00623e74d6756c5ab67"><span class="id" title="notation">|</span></a> <span class="id" title="var">C1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <span class="id" title="var">C2</span><a class="idref" href="mathcomp.ssreflect.seq.html#3de67eb42556f00623e74d6756c5ab67"><span class="id" title="notation">]</span></a><br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">x</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 99,<br/> - <span class="id" title="var">format</span> "[ '[hv' 'seq' x <- s '/ ' | C1 '/ ' & C2 ] ']'") : <span class="id" title="var">seq_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="3d61eb9bcecd9ede478c4672178648de"><span class="id" title="notation">"</span></a>[ 'seq' x : T <- s | C ]" := (<a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> : <span class="id" title="var">T</span> ⇒ <span class="id" title="var">C</span>%<span class="id" title="var">B</span>) <span class="id" title="var">s</span>)<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">x</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 99, <span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/> -<span class="id" title="keyword">Notation</span> <a name="97ed8600c5e4ffe49097c0e9374942e0"><span class="id" title="notation">"</span></a>[ 'seq' x : T <- s | C1 & C2 ]" := <a class="idref" href="mathcomp.ssreflect.seq.html#3d61eb9bcecd9ede478c4672178648de"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#3d61eb9bcecd9ede478c4672178648de"><span class="id" title="notation">seq</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#3d61eb9bcecd9ede478c4672178648de"><span class="id" title="notation">:</span></a> <span class="id" title="var">T</span> <a class="idref" href="mathcomp.ssreflect.seq.html#3d61eb9bcecd9ede478c4672178648de"><span class="id" title="notation"><-</span></a> <span class="id" title="var">s</span> <a class="idref" href="mathcomp.ssreflect.seq.html#3d61eb9bcecd9ede478c4672178648de"><span class="id" title="notation">|</span></a> <span class="id" title="var">C1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <span class="id" title="var">C2</span><a class="idref" href="mathcomp.ssreflect.seq.html#3d61eb9bcecd9ede478c4672178648de"><span class="id" title="notation">]</span></a><br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">x</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 99, <span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/> - -<br/> -</div> - -<div class="doc"> - Double induction/recursion. -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="seq_ind2"><span class="id" title="lemma">seq_ind2</span></a> {<span class="id" title="var">S</span> <span class="id" title="var">T</span>} (<span class="id" title="var">P</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#T"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span>) :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">s</span> <span class="id" title="var">t</span>, <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#P"><span class="id" title="variable">P</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <span class="id" title="keyword">∀</span> <span class="id" title="var">s</span> <span class="id" title="var">t</span>, <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="RotRcons"><span class="id" title="section">RotRcons</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="RotRcons.T"><span class="id" title="variable">T</span></a> : <span class="id" title="keyword">Type</span>.<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#RotRcons.T"><span class="id" title="variable">T</span></a>) (<span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#RotRcons.T"><span class="id" title="variable">T</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rot1_cons"><span class="id" title="lemma">rot1_cons</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> 1 (<a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rcons_inj"><span class="id" title="lemma">rcons_inj</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> <span class="id" title="var">x1</span> <span class="id" title="var">x2</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x1"><span class="id" title="variable">x1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#b8b2ebc8e1a8b9aa935c0702efb5dccf"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x2"><span class="id" title="variable">x2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#b8b2ebc8e1a8b9aa935c0702efb5dccf"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#RotRcons.T"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x1"><span class="id" title="variable">x1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x2"><span class="id" title="variable">x2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rcons_injl"><span class="id" title="lemma">rcons_injl</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#d89396f990d6b54d736cfe259e498cf4"><span class="id" title="notation">^~</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rcons_injr"><span class="id" title="lemma">rcons_injr</span></a> <span class="id" title="var">s</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.seq.html#RotRcons"><span class="id" title="section">RotRcons</span></a>.<br/> - -<br/> - -<br/> -</div> - -<div class="doc"> - Equality and eqType for seq. -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Section</span> <a name="EqSeq"><span class="id" title="section">EqSeq</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="EqSeq.n0"><span class="id" title="variable">n0</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a name="EqSeq.T"><span class="id" title="variable">T</span></a> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>) (<a name="EqSeq.x0"><span class="id" title="variable">x0</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#T"><span class="id" title="variable">T</span></a>).<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.T"><span class="id" title="variable">T</span></a>) (<span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.T"><span class="id" title="variable">T</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="eqseq"><span class="id" title="definition">eqseq</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> {<span class="id" title="keyword">struct</span> <span class="id" title="var">s2</span>} :=<br/> - <span class="id" title="keyword">match</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a>, <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <span class="id" title="keyword">with</span><br/> - | <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a>, <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/> - | <span class="id" title="var">x1</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">s1'</span>, <span class="id" title="var">x2</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">s2'</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><span class="id" title="var">x1</span> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <span class="id" title="var">x2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#eqseq"><span class="id" title="definition">eqseq</span></a> <span class="id" title="var">s1'</span> <span class="id" title="var">s2'</span><br/> - | <span class="id" title="var">_</span>, <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/> - <span class="id" title="keyword">end</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eqseqP"><span class="id" title="lemma">eqseqP</span></a> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.axiom"><span class="id" title="definition">Equality.axiom</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#eqseq"><span class="id" title="definition">eqseq</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">seq_eqMixin</span> := <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.EqMixin"><span class="id" title="abbreviation">EqMixin</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#eqseqP"><span class="id" title="lemma">eqseqP</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">seq_eqType</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.EqType"><span class="id" title="abbreviation">EqType</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.T"><span class="id" title="variable">T</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#seq_eqMixin"><span class="id" title="definition">seq_eqMixin</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eqseqE"><span class="id" title="lemma">eqseqE</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#eqseq"><span class="id" title="definition">eqseq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#eq_op"><span class="id" title="definition">eq_op</span></a>. <br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eqseq_cons"><span class="id" title="lemma">eqseq_cons</span></a> <span class="id" title="var">x1</span> <span class="id" title="var">x2</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#x1"><span class="id" title="variable">x1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x2"><span class="id" title="variable">x2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#x1"><span class="id" title="variable">x1</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x2"><span class="id" title="variable">x2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eqseq_cat"><span class="id" title="lemma">eqseq_cat</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> <span class="id" title="var">s3</span> <span class="id" title="var">s4</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s3"><span class="id" title="variable">s3</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s4"><span class="id" title="variable">s4</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#s3"><span class="id" title="variable">s3</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s4"><span class="id" title="variable">s4</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eqseq_rcons"><span class="id" title="lemma">eqseq_rcons</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> <span class="id" title="var">x1</span> <span class="id" title="var">x2</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x1"><span class="id" title="variable">x1</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x2"><span class="id" title="variable">x2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#x1"><span class="id" title="variable">x1</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x2"><span class="id" title="variable">x2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="size_eq0"><span class="id" title="lemma">size_eq0</span></a> <span class="id" title="var">s</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="has_filter"><span class="id" title="lemma">has_filter</span></a> <span class="id" title="var">a</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - mem_seq and index. - mem_seq defines a predType for seq. -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="mem_seq"><span class="id" title="definition">mem_seq</span></a> (<span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.T"><span class="id" title="variable">T</span></a>) :=<br/> - <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">s'</span> <span class="id" title="keyword">then</span> <a class="idref" href="mathcomp.ssreflect.eqtype.html#xpredU1"><span class="id" title="abbreviation">xpredU1</span></a> <span class="id" title="var">y</span> (<a class="idref" href="mathcomp.ssreflect.seq.html#mem_seq"><span class="id" title="definition">mem_seq</span></a> <span class="id" title="var">s'</span>) <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#xpred0"><span class="id" title="abbreviation">xpred0</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="seq_eqclass"><span class="id" title="definition">seq_eqclass</span></a> := <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.T"><span class="id" title="variable">T</span></a>.<br/> -<span class="id" title="keyword">Identity</span> <span class="id" title="keyword">Coercion</span> <span class="id" title="var">seq_of_eqclass</span> : <span class="id" title="var">seq_eqclass</span> >-> <span class="id" title="var">seq</span>.<br/> -<span class="id" title="keyword">Coercion</span> <span class="id" title="var">pred_of_seq</span> (<span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq_eqclass"><span class="id" title="definition">seq_eqclass</span></a>) : <a class="idref" href="mathcomp.ssreflect.ssrbool.html#64f8873130736b599801d4930af00e74"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.ssrbool.html#64f8873130736b599801d4930af00e74"><span class="id" title="notation">pred</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.T"><span class="id" title="variable">T</span></a><a class="idref" href="mathcomp.ssreflect.ssrbool.html#64f8873130736b599801d4930af00e74"><span class="id" title="notation">}</span></a> := <a class="idref" href="mathcomp.ssreflect.seq.html#mem_seq"><span class="id" title="definition">mem_seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">seq_predType</span> := <a class="idref" href="mathcomp.ssreflect.ssrbool.html#PredType"><span class="id" title="definition">PredType</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#pred_of_seq"><span class="id" title="definition">pred_of_seq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.T"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.T"><span class="id" title="variable">T</span></a>).<br/> -</div> - -<div class="doc"> - The line below makes mem_seq a canonical instance of topred. -</div> -<div class="code"> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">mem_seq_predType</span> := <a class="idref" href="mathcomp.ssreflect.ssrbool.html#PredType"><span class="id" title="definition">PredType</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#mem_seq"><span class="id" title="definition">mem_seq</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="in_cons"><span class="id" title="lemma">in_cons</span></a> <span class="id" title="var">y</span> <span class="id" title="var">s</span> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="in_nil"><span class="id" title="lemma">in_nil</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem_seq1"><span class="id" title="lemma">mem_seq1</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#506674b18256ef8f50efed43fa1dfd7d"><span class="id" title="notation">[::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#506674b18256ef8f50efed43fa1dfd7d"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> - <span class="comment">(* to be repeated after the Section discharge. *)</span><br/> -<span class="id" title="keyword">Let</span> <a name="EqSeq.inE"><span class="id" title="variable">inE</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#mem_seq1"><span class="id" title="lemma">mem_seq1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#in_cons"><span class="id" title="lemma">in_cons</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.ssrbool.html#inE"><span class="id" title="definition">inE</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem_seq2"><span class="id" title="lemma">mem_seq2</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#9e9281397dab83046645f1b62dbb2487"><span class="id" title="notation">[::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#9e9281397dab83046645f1b62dbb2487"><span class="id" title="notation">;</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#9e9281397dab83046645f1b62dbb2487"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#xpred2"><span class="id" title="abbreviation">xpred2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem_seq3"><span class="id" title="lemma">mem_seq3</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> <span class="id" title="var">t</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#9e9281397dab83046645f1b62dbb2487"><span class="id" title="notation">[::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#9e9281397dab83046645f1b62dbb2487"><span class="id" title="notation">;</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#9e9281397dab83046645f1b62dbb2487"><span class="id" title="notation">;</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#9e9281397dab83046645f1b62dbb2487"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#xpred3"><span class="id" title="abbreviation">xpred3</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem_seq4"><span class="id" title="lemma">mem_seq4</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> <span class="id" title="var">t</span> <span class="id" title="var">u</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#9e9281397dab83046645f1b62dbb2487"><span class="id" title="notation">[::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#9e9281397dab83046645f1b62dbb2487"><span class="id" title="notation">;</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#9e9281397dab83046645f1b62dbb2487"><span class="id" title="notation">;</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#9e9281397dab83046645f1b62dbb2487"><span class="id" title="notation">;</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#u"><span class="id" title="variable">u</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#9e9281397dab83046645f1b62dbb2487"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#xpred4"><span class="id" title="abbreviation">xpred4</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem_cat"><span class="id" title="lemma">mem_cat</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem_rcons"><span class="id" title="lemma">mem_rcons</span></a> <span class="id" title="var">s</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem_head"><span class="id" title="lemma">mem_head</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem_last"><span class="id" title="lemma">mem_last</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#last"><span class="id" title="definition">last</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem_behead"><span class="id" title="lemma">mem_behead</span></a> <span class="id" title="var">s</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#behead"><span class="id" title="definition">behead</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem_belast"><span class="id" title="lemma">mem_belast</span></a> <span class="id" title="var">s</span> <span class="id" title="var">y</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#belast"><span class="id" title="definition">belast</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem_nth"><span class="id" title="lemma">mem_nth</span></a> <span class="id" title="var">s</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="abbreviation">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem_take"><span class="id" title="lemma">mem_take</span></a> <span class="id" title="var">s</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#take"><span class="id" title="definition">take</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem_drop"><span class="id" title="lemma">mem_drop</span></a> <span class="id" title="var">s</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#drop"><span class="id" title="definition">drop</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="last_eq"><span class="id" title="lemma">last_eq</span></a> <span class="id" title="var">s</span> <span class="id" title="var">z</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#c385a484ee9d1b4e0615924561a9b75e"><span class="id" title="notation">!=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#last"><span class="id" title="definition">last</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#last"><span class="id" title="definition">last</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="EqSeq.Filters"><span class="id" title="section">Filters</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">a</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.T"><span class="id" title="variable">T</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="hasP"><span class="id" title="lemma">hasP</span></a> {<span class="id" title="var">a</span> <span class="id" title="var">s</span>} : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="allP"><span class="id" title="lemma">allP</span></a> {<span class="id" title="var">a</span> <span class="id" title="var">s</span>} : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">}</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="hasPn"><span class="id" title="lemma">hasPn</span></a> <span class="id" title="var">a</span> <span class="id" title="var">s</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">}</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="allPn"><span class="id" title="lemma">allPn</span></a> <span class="id" title="var">a</span> <span class="id" title="var">s</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem_filter"><span class="id" title="lemma">mem_filter</span></a> <span class="id" title="var">a</span> <span class="id" title="var">x</span> <span class="id" title="var">s</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="EqSeq.Filters.a"><span class="id" title="variable">a</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.T"><span class="id" title="variable">T</span></a>) (<a name="EqSeq.Filters.s"><span class="id" title="variable">s</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.T"><span class="id" title="variable">T</span></a>) (<a name="EqSeq.Filters.A"><span class="id" title="variable">A</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.T"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>).<br/> -<span class="id" title="keyword">Hypothesis</span> <a name="EqSeq.Filters.aP"><span class="id" title="variable">aP</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.Filters.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.Filters.a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="hasPP"><span class="id" title="lemma">hasPP</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.Filters.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.Filters.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.Filters.a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.Filters.s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="allPP"><span class="id" title="lemma">allPP</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.Filters.s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.Filters.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">}</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.Filters.a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.Filters.s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.Filters"><span class="id" title="section">Filters</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="c7a8000a6cd687fa747a1b027fdad1c9"><span class="id" title="notation">"</span></a>'has_ view" := (<a class="idref" href="mathcomp.ssreflect.seq.html#hasPP"><span class="id" title="lemma">hasPP</span></a> <span class="id" title="var">_</span> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> ⇒ <span class="id" title="var">view</span>))<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 4, <span class="id" title="tactic">right</span> <span class="id" title="keyword">associativity</span>, <span class="id" title="var">format</span> "''has_' view").<br/> -<span class="id" title="keyword">Notation</span> <a name="ab51245ad31bdb97452f83c8b446e9f3"><span class="id" title="notation">"</span></a>'all_ view" := (<a class="idref" href="mathcomp.ssreflect.seq.html#allPP"><span class="id" title="lemma">allPP</span></a> <span class="id" title="var">_</span> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> ⇒ <span class="id" title="var">view</span>))<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 4, <span class="id" title="tactic">right</span> <span class="id" title="keyword">associativity</span>, <span class="id" title="var">format</span> "''all_' view").<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="EqSeq.EqIn"><span class="id" title="section">EqIn</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> <a name="EqSeq.EqIn.a1"><span class="id" title="variable">a1</span></a> <a name="EqSeq.EqIn.a2"><span class="id" title="variable">a2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.T"><span class="id" title="variable">T</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_in_filter"><span class="id" title="lemma">eq_in_filter</span></a> <span class="id" title="var">s</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.EqIn.a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.EqIn.a2"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.EqIn.a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.EqIn.a2"><span class="id" title="variable">a2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_in_find"><span class="id" title="lemma">eq_in_find</span></a> <span class="id" title="var">s</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.EqIn.a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.EqIn.a2"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#find"><span class="id" title="definition">find</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.EqIn.a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#find"><span class="id" title="definition">find</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.EqIn.a2"><span class="id" title="variable">a2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_in_count"><span class="id" title="lemma">eq_in_count</span></a> <span class="id" title="var">s</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.EqIn.a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.EqIn.a2"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#count"><span class="id" title="definition">count</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.EqIn.a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#count"><span class="id" title="definition">count</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.EqIn.a2"><span class="id" title="variable">a2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_in_all"><span class="id" title="lemma">eq_in_all</span></a> <span class="id" title="var">s</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.EqIn.a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.EqIn.a2"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.EqIn.a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.EqIn.a2"><span class="id" title="variable">a2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_in_has"><span class="id" title="lemma">eq_in_has</span></a> <span class="id" title="var">s</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.EqIn.a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.EqIn.a2"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.EqIn.a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.EqIn.a2"><span class="id" title="variable">a2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.EqIn"><span class="id" title="section">EqIn</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_has_r"><span class="id" title="lemma">eq_has_r</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#d89396f990d6b54d736cfe259e498cf4"><span class="id" title="notation">^~</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#d89396f990d6b54d736cfe259e498cf4"><span class="id" title="notation">^~</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_all_r"><span class="id" title="lemma">eq_all_r</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#d89396f990d6b54d736cfe259e498cf4"><span class="id" title="notation">^~</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#d89396f990d6b54d736cfe259e498cf4"><span class="id" title="notation">^~</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="has_sym"><span class="id" title="lemma">has_sym</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="has_pred1"><span class="id" title="lemma">has_pred1</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> (<a class="idref" href="mathcomp.ssreflect.eqtype.html#pred1"><span class="id" title="definition">pred1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem_rev"><span class="id" title="lemma">mem_rev</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - Constant sequences, i.e., the image of nseq. -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="constant"><span class="id" title="definition">constant</span></a> <span class="id" title="var">s</span> := <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">s'</span> <span class="id" title="keyword">then</span> <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> (<a class="idref" href="mathcomp.ssreflect.eqtype.html#pred1"><span class="id" title="definition">pred1</span></a> <span class="id" title="var">x</span>) <span class="id" title="var">s'</span> <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="all_pred1P"><span class="id" title="lemma">all_pred1P</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nseq"><span class="id" title="definition">nseq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> (<a class="idref" href="mathcomp.ssreflect.eqtype.html#pred1"><span class="id" title="definition">pred1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="all_pred1_constant"><span class="id" title="lemma">all_pred1_constant</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> (<a class="idref" href="mathcomp.ssreflect.eqtype.html#pred1"><span class="id" title="definition">pred1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#constant"><span class="id" title="definition">constant</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="all_pred1_nseq"><span class="id" title="lemma">all_pred1_nseq</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> (<a class="idref" href="mathcomp.ssreflect.eqtype.html#pred1"><span class="id" title="definition">pred1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#nseq"><span class="id" title="definition">nseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem_nseq"><span class="id" title="lemma">mem_nseq</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nseq"><span class="id" title="definition">nseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a>0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nseqP"><span class="id" title="lemma">nseqP</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#7f2a7ef2c63af7359b22787a9daf336e"><span class="id" title="notation">></span></a> 0) (<a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nseq"><span class="id" title="definition">nseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="constant_nseq"><span class="id" title="lemma">constant_nseq</span></a> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#constant"><span class="id" title="definition">constant</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#nseq"><span class="id" title="definition">nseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>).<br/> - -<br/> -</div> - -<div class="doc"> - Uses x0 -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="constantP"><span class="id" title="lemma">constantP</span></a> <span class="id" title="var">s</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nseq"><span class="id" title="definition">nseq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#constant"><span class="id" title="definition">constant</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -</div> - -<div class="doc"> - Duplicate-freenes. -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="uniq"><span class="id" title="definition">uniq</span></a> <span class="id" title="var">s</span> := <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">s'</span> <span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">notin</span></a> <span class="id" title="var">s'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <span class="id" title="var">s'</span> <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cons_uniq"><span class="id" title="lemma">cons_uniq</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">notin</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cat_uniq"><span class="id" title="lemma">cat_uniq</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">[&&</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#5a7d806905be2a0d04047156433535f1"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="uniq_catC"><span class="id" title="lemma">uniq_catC</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="uniq_catCA"><span class="id" title="lemma">uniq_catCA</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> <span class="id" title="var">s3</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s3"><span class="id" title="variable">s3</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s3"><span class="id" title="variable">s3</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rcons_uniq"><span class="id" title="lemma">rcons_uniq</span></a> <span class="id" title="var">s</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">notin</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="filter_uniq"><span class="id" title="lemma">filter_uniq</span></a> <span class="id" title="var">s</span> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rot_uniq"><span class="id" title="lemma">rot_uniq</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rev_uniq"><span class="id" title="lemma">rev_uniq</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="count_memPn"><span class="id" title="lemma">count_memPn</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#count_mem"><span class="id" title="abbreviation">count_mem</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0) (<a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">notin</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="count_uniq_mem"><span class="id" title="lemma">count_uniq_mem</span></a> <span class="id" title="var">s</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#count_mem"><span class="id" title="abbreviation">count_mem</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="filter_pred1_uniq"><span class="id" title="lemma">filter_pred1_uniq</span></a> <span class="id" title="var">s</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> (<a class="idref" href="mathcomp.ssreflect.eqtype.html#pred1"><span class="id" title="definition">pred1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#506674b18256ef8f50efed43fa1dfd7d"><span class="id" title="notation">[::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#506674b18256ef8f50efed43fa1dfd7d"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - Removing duplicates -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="undup"><span class="id" title="definition">undup</span></a> <span class="id" title="var">s</span> :=<br/> - <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">s'</span> <span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <span class="id" title="var">s'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#undup"><span class="id" title="definition">undup</span></a> <span class="id" title="var">s'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#undup"><span class="id" title="definition">undup</span></a> <span class="id" title="var">s'</span> <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="size_undup"><span class="id" title="lemma">size_undup</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#undup"><span class="id" title="definition">undup</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem_undup"><span class="id" title="lemma">mem_undup</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#undup"><span class="id" title="definition">undup</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="undup_uniq"><span class="id" title="lemma">undup_uniq</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#undup"><span class="id" title="definition">undup</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="undup_id"><span class="id" title="lemma">undup_id</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#undup"><span class="id" title="definition">undup</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="ltn_size_undup"><span class="id" title="lemma">ltn_size_undup</span></a> <span class="id" title="var">s</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#undup"><span class="id" title="definition">undup</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="filter_undup"><span class="id" title="lemma">filter_undup</span></a> <span class="id" title="var">p</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#p"><span class="id" title="variable">p</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#undup"><span class="id" title="definition">undup</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#undup"><span class="id" title="definition">undup</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="undup_nil"><span class="id" title="lemma">undup_nil</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#undup"><span class="id" title="definition">undup</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - Lookup -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="index"><span class="id" title="definition">index</span></a> <span class="id" title="var">x</span> := <a class="idref" href="mathcomp.ssreflect.seq.html#find"><span class="id" title="definition">find</span></a> (<a class="idref" href="mathcomp.ssreflect.eqtype.html#pred1"><span class="id" title="definition">pred1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="index_size"><span class="id" title="lemma">index_size</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#index"><span class="id" title="definition">index</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="index_mem"><span class="id" title="lemma">index_mem</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#index"><span class="id" title="definition">index</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nth_index"><span class="id" title="lemma">nth_index</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="abbreviation">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#index"><span class="id" title="definition">index</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="index_cat"><span class="id" title="lemma">index_cat</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#index"><span class="id" title="definition">index</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#index"><span class="id" title="definition">index</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#index"><span class="id" title="definition">index</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nthK"><span class="id" title="lemma">nthK</span></a> <span class="id" title="var">s</span>: <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#gtn"><span class="id" title="definition">gtn</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="abbreviation">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#index"><span class="id" title="definition">index</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#d89396f990d6b54d736cfe259e498cf4"><span class="id" title="notation">^~</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="index_uniq"><span class="id" title="lemma">index_uniq</span></a> <span class="id" title="var">i</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#index"><span class="id" title="definition">index</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="abbreviation">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="index_head"><span class="id" title="lemma">index_head</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#index"><span class="id" title="definition">index</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="index_last"><span class="id" title="lemma">index_last</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#index"><span class="id" title="definition">index</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#last"><span class="id" title="definition">last</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nth_uniq"><span class="id" title="lemma">nth_uniq</span></a> <span class="id" title="var">s</span> <span class="id" title="var">i</span> <span class="id" title="var">j</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="abbreviation">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="abbreviation">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#j"><span class="id" title="variable">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#j"><span class="id" title="variable">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="uniqPn"><span class="id" title="lemma">uniqPn</span></a> <span class="id" title="var">s</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <span class="id" title="var">i</span> <span class="id" title="var">j</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">[/\</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#j"><span class="id" title="variable">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="abbreviation">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="abbreviation">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#j"><span class="id" title="variable">j</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">]</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="uniqP"><span class="id" title="lemma">uniqP</span></a> <span class="id" title="var">s</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#gtn"><span class="id" title="definition">gtn</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="abbreviation">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">}</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem_rot"><span class="id" title="lemma">mem_rot</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eqseq_rot"><span class="id" title="lemma">eqseq_rot</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variant</span> <a name="rot_to_spec"><span class="id" title="inductive">rot_to_spec</span></a> <span class="id" title="var">s</span> <span class="id" title="var">x</span> := <a name="RotToSpec"><span class="id" title="constructor">RotToSpec</span></a> <span class="id" title="var">i</span> <span class="id" title="var">s'</span> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s'"><span class="id" title="variable">s'</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rot_to"><span class="id" title="lemma">rot_to</span></a> <span class="id" title="var">s</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rot_to_spec"><span class="id" title="inductive">rot_to_spec</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.seq.html#EqSeq"><span class="id" title="section">EqSeq</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="inE"><span class="id" title="definition">inE</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#mem_seq1"><span class="id" title="lemma">mem_seq1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#in_cons"><span class="id" title="lemma">in_cons</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.ssrbool.html#inE"><span class="id" title="definition">inE</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/> - -<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="NthTheory"><span class="id" title="section">NthTheory</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nthP"><span class="id" title="lemma">nthP</span></a> (<span class="id" title="var">T</span> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>) (<span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#T"><span class="id" title="variable">T</span></a>) <span class="id" title="var">x</span> <span class="id" title="var">x0</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">i</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x0"><span class="id" title="variable">x0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="NthTheory.T"><span class="id" title="variable">T</span></a> : <span class="id" title="keyword">Type</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="has_nthP"><span class="id" title="lemma">has_nthP</span></a> (<span class="id" title="var">a</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#NthTheory.T"><span class="id" title="variable">T</span></a>) <span class="id" title="var">s</span> <span class="id" title="var">x0</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">i</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x0"><span class="id" title="variable">x0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a>)) (<a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="all_nthP"><span class="id" title="lemma">all_nthP</span></a> (<span class="id" title="var">a</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#NthTheory.T"><span class="id" title="variable">T</span></a>) <span class="id" title="var">s</span> <span class="id" title="var">x0</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x0"><span class="id" title="variable">x0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a>)) (<a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.seq.html#NthTheory"><span class="id" title="section">NthTheory</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="set_nth_default"><span class="id" title="lemma">set_nth_default</span></a> <span class="id" title="var">T</span> <span class="id" title="var">s</span> (<span class="id" title="var">y0</span> <span class="id" title="var">x0</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#T"><span class="id" title="variable">T</span></a>) <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x0"><span class="id" title="variable">x0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y0"><span class="id" title="variable">y0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="headI"><span class="id" title="lemma">headI</span></a> <span class="id" title="var">T</span> <span class="id" title="var">s</span> (<span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#T"><span class="id" title="variable">T</span></a>) : <a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#head"><span class="id" title="definition">head</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#behead"><span class="id" title="definition">behead</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>).<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="bitseq"><span class="id" title="definition">bitseq</span></a> := <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">bitseq_eqType</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.eqtype.html#2b9222c46a529018a8ebb5be6355801c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.eqtype.html#2b9222c46a529018a8ebb5be6355801c"><span class="id" title="notation">eqType</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#2b9222c46a529018a8ebb5be6355801c"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#bitseq"><span class="id" title="definition">bitseq</span></a><a class="idref" href="mathcomp.ssreflect.eqtype.html#2b9222c46a529018a8ebb5be6355801c"><span class="id" title="notation">]</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">bitseq_predType</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#7b8433622bdffcf2887bf528bd2481ba"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#7b8433622bdffcf2887bf528bd2481ba"><span class="id" title="notation">predType</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#7b8433622bdffcf2887bf528bd2481ba"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#bitseq"><span class="id" title="definition">bitseq</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#7b8433622bdffcf2887bf528bd2481ba"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - Incrementing the ith nat in a seq nat, padding with 0's if needed. This - allows us to use nat seqs as bags of nats. -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="incr_nth"><span class="id" title="definition">incr_nth</span></a> <span class="id" title="var">v</span> <span class="id" title="var">i</span> {<span class="id" title="keyword">struct</span> <span class="id" title="var">i</span>} :=<br/> - <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#v"><span class="id" title="variable">v</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">n</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">v'</span> <span class="id" title="keyword">then</span> <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">i'</span><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <span class="id" title="keyword">then</span> <span class="id" title="var">n</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#incr_nth"><span class="id" title="definition">incr_nth</span></a> <span class="id" title="var">v'</span> <span class="id" title="var">i'</span> <span class="id" title="keyword">else</span> <span class="id" title="var">n</span><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">v'</span><br/> - <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.seq.html#ncons"><span class="id" title="definition">ncons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> 0 <a class="idref" href="mathcomp.ssreflect.seq.html#506674b18256ef8f50efed43fa1dfd7d"><span class="id" title="notation">[::</span></a> 1<a class="idref" href="mathcomp.ssreflect.seq.html#506674b18256ef8f50efed43fa1dfd7d"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nth_incr_nth"><span class="id" title="lemma">nth_incr_nth</span></a> <span class="id" title="var">v</span> <span class="id" title="var">i</span> <span class="id" title="var">j</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> 0 (<a class="idref" href="mathcomp.ssreflect.seq.html#incr_nth"><span class="id" title="definition">incr_nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#v"><span class="id" title="variable">v</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#j"><span class="id" title="variable">j</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> 0 <a class="idref" href="mathcomp.ssreflect.seq.html#v"><span class="id" title="variable">v</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#j"><span class="id" title="variable">j</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="size_incr_nth"><span class="id" title="lemma">size_incr_nth</span></a> <span class="id" title="var">v</span> <span class="id" title="var">i</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#incr_nth"><span class="id" title="definition">incr_nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#v"><span class="id" title="variable">v</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#v"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#v"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="incr_nth_inj"><span class="id" title="lemma">incr_nth_inj</span></a> <span class="id" title="var">v</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#incr_nth"><span class="id" title="definition">incr_nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#v"><span class="id" title="variable">v</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="incr_nthC"><span class="id" title="lemma">incr_nthC</span></a> <span class="id" title="var">v</span> <span class="id" title="var">i</span> <span class="id" title="var">j</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#incr_nth"><span class="id" title="definition">incr_nth</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#incr_nth"><span class="id" title="definition">incr_nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#v"><span class="id" title="variable">v</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#incr_nth"><span class="id" title="definition">incr_nth</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#incr_nth"><span class="id" title="definition">incr_nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#v"><span class="id" title="variable">v</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#j"><span class="id" title="variable">j</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - Equality up to permutation -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Section</span> <a name="PermSeq"><span class="id" title="section">PermSeq</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="PermSeq.T"><span class="id" title="variable">T</span></a> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>.<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#PermSeq.T"><span class="id" title="variable">T</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="perm_eq"><span class="id" title="definition">perm_eq</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> :=<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#27dabc72ea2c2c768f2db80a79f42524"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#27dabc72ea2c2c768f2db80a79f42524"><span class="id" title="notation">pred</span></a> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#27dabc72ea2c2c768f2db80a79f42524"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#count_mem"><span class="id" title="abbreviation">count_mem</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#count_mem"><span class="id" title="abbreviation">count_mem</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#27dabc72ea2c2c768f2db80a79f42524"><span class="id" title="notation">]</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="permP"><span class="id" title="lemma">permP</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#count"><span class="id" title="definition">count</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#d89396f990d6b54d736cfe259e498cf4"><span class="id" title="notation">^~</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#count"><span class="id" title="definition">count</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#d89396f990d6b54d736cfe259e498cf4"><span class="id" title="notation">^~</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="perm_refl"><span class="id" title="lemma">perm_refl</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - <span class="id" title="keyword">Hint Resolve</span> <span class="id" title="var">perm_refl</span> : <span class="id" title="var">core</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="perm_sym"><span class="id" title="lemma">perm_sym</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#symmetric"><span class="id" title="definition">symmetric</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="perm_trans"><span class="id" title="lemma">perm_trans</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#transitive"><span class="id" title="definition">transitive</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="perm_eql"><span class="id" title="abbreviation">perm_eql</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> := (<a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <span class="id" title="var">s1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <span class="id" title="var">s2</span>).<br/> -<span class="id" title="keyword">Notation</span> <a name="perm_eqr"><span class="id" title="abbreviation">perm_eqr</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> := (<a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#d89396f990d6b54d736cfe259e498cf4"><span class="id" title="notation">^~</span></a> <span class="id" title="var">s1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#d89396f990d6b54d736cfe259e498cf4"><span class="id" title="notation">^~</span></a> <span class="id" title="var">s2</span>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="permEl"><span class="id" title="lemma">permEl</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eql"><span class="id" title="abbreviation">perm_eql</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>. <br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="permPl"><span class="id" title="lemma">permPl</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#perm_eql"><span class="id" title="abbreviation">perm_eql</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="permPr"><span class="id" title="lemma">permPr</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#perm_eqr"><span class="id" title="abbreviation">perm_eqr</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="perm_catC"><span class="id" title="lemma">perm_catC</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eql"><span class="id" title="abbreviation">perm_eql</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="perm_cat2l"><span class="id" title="lemma">perm_cat2l</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> <span class="id" title="var">s3</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s3"><span class="id" title="variable">s3</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s3"><span class="id" title="variable">s3</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="perm_catl"><span class="id" title="lemma">perm_catl</span></a> <span class="id" title="var">s</span> <span class="id" title="var">t1</span> <span class="id" title="var">t2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t1"><span class="id" title="variable">t1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t2"><span class="id" title="variable">t2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eql"><span class="id" title="abbreviation">perm_eql</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t1"><span class="id" title="variable">t1</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t2"><span class="id" title="variable">t2</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="perm_cons"><span class="id" title="lemma">perm_cons</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="perm_cat2r"><span class="id" title="lemma">perm_cat2r</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> <span class="id" title="var">s3</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#s3"><span class="id" title="variable">s3</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s3"><span class="id" title="variable">s3</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="perm_catr"><span class="id" title="lemma">perm_catr</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> <span class="id" title="var">t</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eql"><span class="id" title="abbreviation">perm_eql</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="perm_cat"><span class="id" title="lemma">perm_cat</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> <span class="id" title="var">t1</span> <span class="id" title="var">t2</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t1"><span class="id" title="variable">t1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t2"><span class="id" title="variable">t2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t1"><span class="id" title="variable">t1</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t2"><span class="id" title="variable">t2</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="perm_catAC"><span class="id" title="lemma">perm_catAC</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> <span class="id" title="var">s3</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eql"><span class="id" title="abbreviation">perm_eql</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s3"><span class="id" title="variable">s3</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s3"><span class="id" title="variable">s3</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="perm_catCA"><span class="id" title="lemma">perm_catCA</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> <span class="id" title="var">s3</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eql"><span class="id" title="abbreviation">perm_eql</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s3"><span class="id" title="variable">s3</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s3"><span class="id" title="variable">s3</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="perm_rcons"><span class="id" title="lemma">perm_rcons</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eql"><span class="id" title="abbreviation">perm_eql</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="perm_rot"><span class="id" title="lemma">perm_rot</span></a> <span class="id" title="var">n</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eql"><span class="id" title="abbreviation">perm_eql</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="perm_rotr"><span class="id" title="lemma">perm_rotr</span></a> <span class="id" title="var">n</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eql"><span class="id" title="abbreviation">perm_eql</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rotr"><span class="id" title="definition">rotr</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="perm_rev"><span class="id" title="lemma">perm_rev</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eql"><span class="id" title="abbreviation">perm_eql</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="perm_filter"><span class="id" title="lemma">perm_filter</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> <span class="id" title="var">a</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="perm_filterC"><span class="id" title="lemma">perm_filterC</span></a> <span class="id" title="var">a</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eql"><span class="id" title="abbreviation">perm_eql</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#predC"><span class="id" title="definition">predC</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="perm_size"><span class="id" title="lemma">perm_size</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="perm_mem"><span class="id" title="lemma">perm_mem</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="perm_nilP"><span class="id" title="lemma">perm_nilP</span></a> <span class="id" title="var">s</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="perm_consP"><span class="id" title="lemma">perm_consP</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> <span class="id" title="var">t</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <span class="id" title="var">i</span> <span class="id" title="var">u</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>)<br/> - (<a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>)).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="perm_has"><span class="id" title="lemma">perm_has</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="perm_all"><span class="id" title="lemma">perm_all</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="perm_small_eq"><span class="id" title="lemma">perm_small_eq</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="uniq_leq_size"><span class="id" title="lemma">uniq_leq_size</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="leq_size_uniq"><span class="id" title="lemma">leq_size_uniq</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="uniq_size_uniq"><span class="id" title="lemma">uniq_size_uniq</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="uniq_min_size"><span class="id" title="lemma">uniq_min_size</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_uniq"><span class="id" title="lemma">eq_uniq</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="perm_uniq"><span class="id" title="lemma">perm_uniq</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="uniq_perm"><span class="id" title="lemma">uniq_perm</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="perm_undup"><span class="id" title="lemma">perm_undup</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#undup"><span class="id" title="definition">undup</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#undup"><span class="id" title="definition">undup</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="count_mem_uniq"><span class="id" title="lemma">count_mem_uniq</span></a> <span class="id" title="var">s</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.ssreflect.seq.html#count_mem"><span class="id" title="abbreviation">count_mem</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="catCA_perm_ind"><span class="id" title="lemma">catCA_perm_ind</span></a> <span class="id" title="var">P</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> <span class="id" title="var">s3</span>, <a class="idref" href="mathcomp.ssreflect.seq.html#P"><span class="id" title="variable">P</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s3"><span class="id" title="variable">s3</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#P"><span class="id" title="variable">P</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s3"><span class="id" title="variable">s3</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span>, <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="catCA_perm_subst"><span class="id" title="lemma">catCA_perm_subst</span></a> <span class="id" title="var">R</span> <span class="id" title="var">F</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> <span class="id" title="var">s3</span>, <a class="idref" href="mathcomp.ssreflect.seq.html#F"><span class="id" title="variable">F</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s3"><span class="id" title="variable">s3</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#b8b2ebc8e1a8b9aa935c0702efb5dccf"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#F"><span class="id" title="variable">F</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s3"><span class="id" title="variable">s3</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#b8b2ebc8e1a8b9aa935c0702efb5dccf"><span class="id" title="notation">:></span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span>, <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.seq.html#PermSeq"><span class="id" title="section">PermSeq</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="perm_eql"><span class="id" title="abbreviation">perm_eql</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> := (<a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <span class="id" title="var">s1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <span class="id" title="var">s2</span>).<br/> -<span class="id" title="keyword">Notation</span> <a name="perm_eqr"><span class="id" title="abbreviation">perm_eqr</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> := (<a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#d89396f990d6b54d736cfe259e498cf4"><span class="id" title="notation">^~</span></a> <span class="id" title="var">s1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#d89396f990d6b54d736cfe259e498cf4"><span class="id" title="notation">^~</span></a> <span class="id" title="var">s2</span>).<br/> - -<br/> -<span class="id" title="keyword">Hint Resolve</span> <span class="id" title="var">perm_refl</span> : <span class="id" title="var">core</span>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="RotrLemmas"><span class="id" title="section">RotrLemmas</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="RotrLemmas.n0"><span class="id" title="variable">n0</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a name="RotrLemmas.T"><span class="id" title="variable">T</span></a> : <span class="id" title="keyword">Type</span>) (<a name="RotrLemmas.T'"><span class="id" title="variable">T'</span></a> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>).<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#RotrLemmas.T"><span class="id" title="variable">T</span></a>) (<span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#RotrLemmas.T"><span class="id" title="variable">T</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="size_rotr"><span class="id" title="lemma">size_rotr</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rotr"><span class="id" title="definition">rotr</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#RotrLemmas.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem_rotr"><span class="id" title="lemma">mem_rotr</span></a> (<span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#RotrLemmas.T'"><span class="id" title="variable">T'</span></a>) : <a class="idref" href="mathcomp.ssreflect.seq.html#rotr"><span class="id" title="definition">rotr</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#RotrLemmas.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rotr_size_cat"><span class="id" title="lemma">rotr_size_cat</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#rotr"><span class="id" title="definition">rotr</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rotr1_rcons"><span class="id" title="lemma">rotr1_rcons</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#rotr"><span class="id" title="definition">rotr</span></a> 1 (<a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="has_rotr"><span class="id" title="lemma">has_rotr</span></a> <span class="id" title="var">a</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rotr"><span class="id" title="definition">rotr</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#RotrLemmas.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rotr_uniq"><span class="id" title="lemma">rotr_uniq</span></a> (<span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#RotrLemmas.T'"><span class="id" title="variable">T'</span></a>) : <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rotr"><span class="id" title="definition">rotr</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#RotrLemmas.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rotrK"><span class="id" title="lemma">rotrK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> (@<a class="idref" href="mathcomp.ssreflect.seq.html#rotr"><span class="id" title="definition">rotr</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#RotrLemmas.T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#RotrLemmas.n0"><span class="id" title="variable">n0</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#RotrLemmas.n0"><span class="id" title="variable">n0</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rotr_inj"><span class="id" title="lemma">rotr_inj</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> (@<a class="idref" href="mathcomp.ssreflect.seq.html#rotr"><span class="id" title="definition">rotr</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#RotrLemmas.T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#RotrLemmas.n0"><span class="id" title="variable">n0</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="take_rev"><span class="id" title="lemma">take_rev</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#take"><span class="id" title="definition">take</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#RotrLemmas.n0"><span class="id" title="variable">n0</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#drop"><span class="id" title="definition">drop</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#7825ccc99f23b0d30c9d40c317ba7af0"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#RotrLemmas.n0"><span class="id" title="variable">n0</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="drop_rev"><span class="id" title="lemma">drop_rev</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#drop"><span class="id" title="definition">drop</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#RotrLemmas.n0"><span class="id" title="variable">n0</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#take"><span class="id" title="definition">take</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#7825ccc99f23b0d30c9d40c317ba7af0"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#RotrLemmas.n0"><span class="id" title="variable">n0</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rev_rotr"><span class="id" title="lemma">rev_rotr</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rotr"><span class="id" title="definition">rotr</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#RotrLemmas.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#RotrLemmas.n0"><span class="id" title="variable">n0</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rev_rot"><span class="id" title="lemma">rev_rot</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#RotrLemmas.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rotr"><span class="id" title="definition">rotr</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#RotrLemmas.n0"><span class="id" title="variable">n0</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.seq.html#RotrLemmas"><span class="id" title="section">RotrLemmas</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="RotCompLemmas"><span class="id" title="section">RotCompLemmas</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="RotCompLemmas.T"><span class="id" title="variable">T</span></a> : <span class="id" title="keyword">Type</span>.<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#RotCompLemmas.T"><span class="id" title="variable">T</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rot_addn"><span class="id" title="lemma">rot_addn</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rotS"><span class="id" title="lemma">rotS</span></a> <span class="id" title="var">n</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> 1 (<a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rot_add_mod"><span class="id" title="lemma">rot_add_mod</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#7825ccc99f23b0d30c9d40c317ba7af0"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rot_rot"><span class="id" title="lemma">rot_rot</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rot_rotr"><span class="id" title="lemma">rot_rotr</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rotr"><span class="id" title="definition">rotr</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rotr"><span class="id" title="definition">rotr</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rotr_rotr"><span class="id" title="lemma">rotr_rotr</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#rotr"><span class="id" title="definition">rotr</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rotr"><span class="id" title="definition">rotr</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rotr"><span class="id" title="definition">rotr</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rotr"><span class="id" title="definition">rotr</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.seq.html#RotCompLemmas"><span class="id" title="section">RotCompLemmas</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Mask"><span class="id" title="section">Mask</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="Mask.n0"><span class="id" title="variable">n0</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a name="Mask.T"><span class="id" title="variable">T</span></a> : <span class="id" title="keyword">Type</span>).<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">m</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#bitseq"><span class="id" title="definition">bitseq</span></a>) (<span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Mask.T"><span class="id" title="variable">T</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="mask"><span class="id" title="definition">mask</span></a> <span class="id" title="var">m</span> <span class="id" title="var">s</span> {<span class="id" title="keyword">struct</span> <span class="id" title="var">m</span>} :=<br/> - <span class="id" title="keyword">match</span> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a>, <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/> - | <span class="id" title="var">b</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">m'</span>, <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">s'</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <span class="id" title="var">b</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#mask"><span class="id" title="definition">mask</span></a> <span class="id" title="var">m'</span> <span class="id" title="var">s'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#mask"><span class="id" title="definition">mask</span></a> <span class="id" title="var">m'</span> <span class="id" title="var">s'</span><br/> - | <span class="id" title="var">_</span>, <span class="id" title="var">_</span> ⇒ <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a><br/> - <span class="id" title="keyword">end</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mask_false"><span class="id" title="lemma">mask_false</span></a> <span class="id" title="var">s</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#mask"><span class="id" title="definition">mask</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#nseq"><span class="id" title="definition">nseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mask_true"><span class="id" title="lemma">mask_true</span></a> <span class="id" title="var">s</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#mask"><span class="id" title="definition">mask</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#nseq"><span class="id" title="definition">nseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mask0"><span class="id" title="lemma">mask0</span></a> <span class="id" title="var">m</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#mask"><span class="id" title="definition">mask</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mask1"><span class="id" title="lemma">mask1</span></a> <span class="id" title="var">b</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#mask"><span class="id" title="definition">mask</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#506674b18256ef8f50efed43fa1dfd7d"><span class="id" title="notation">[::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#506674b18256ef8f50efed43fa1dfd7d"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#506674b18256ef8f50efed43fa1dfd7d"><span class="id" title="notation">[::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#506674b18256ef8f50efed43fa1dfd7d"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nseq"><span class="id" title="definition">nseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mask_cons"><span class="id" title="lemma">mask_cons</span></a> <span class="id" title="var">b</span> <span class="id" title="var">m</span> <span class="id" title="var">x</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#mask"><span class="id" title="definition">mask</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nseq"><span class="id" title="definition">nseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#mask"><span class="id" title="definition">mask</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="size_mask"><span class="id" title="lemma">size_mask</span></a> <span class="id" title="var">m</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#mask"><span class="id" title="definition">mask</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#count"><span class="id" title="definition">count</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mask_cat"><span class="id" title="lemma">mask_cat</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m1"><span class="id" title="variable">m1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#mask"><span class="id" title="definition">mask</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#m1"><span class="id" title="variable">m1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m2"><span class="id" title="variable">m2</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#mask"><span class="id" title="definition">mask</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m1"><span class="id" title="variable">m1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#mask"><span class="id" title="definition">mask</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m2"><span class="id" title="variable">m2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="has_mask_cons"><span class="id" title="lemma">has_mask_cons</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">m</span> <span class="id" title="var">x</span> <span class="id" title="var">s</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#mask"><span class="id" title="definition">mask</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#mask"><span class="id" title="definition">mask</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="has_mask"><span class="id" title="lemma">has_mask</span></a> <span class="id" title="var">a</span> <span class="id" title="var">m</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#mask"><span class="id" title="definition">mask</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mask_rot"><span class="id" title="lemma">mask_rot</span></a> <span class="id" title="var">m</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#mask"><span class="id" title="definition">mask</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Mask.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Mask.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#count"><span class="id" title="definition">count</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#take"><span class="id" title="definition">take</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Mask.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a>)) (<a class="idref" href="mathcomp.ssreflect.seq.html#mask"><span class="id" title="definition">mask</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="resize_mask"><span class="id" title="lemma">resize_mask</span></a> <span class="id" title="var">m</span> <span class="id" title="var">s</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#c0bbd202248f4def7aaf0c316cf2c29e"><span class="id" title="notation">{</span></a><span class="id" title="var">m1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#c0bbd202248f4def7aaf0c316cf2c29e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m1"><span class="id" title="variable">m1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#c0bbd202248f4def7aaf0c316cf2c29e"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#mask"><span class="id" title="definition">mask</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#mask"><span class="id" title="definition">mask</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m1"><span class="id" title="variable">m1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#c0bbd202248f4def7aaf0c316cf2c29e"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.seq.html#Mask"><span class="id" title="section">Mask</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="EqMask"><span class="id" title="section">EqMask</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="EqMask.n0"><span class="id" title="variable">n0</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a name="EqMask.T"><span class="id" title="variable">T</span></a> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>).<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqMask.T"><span class="id" title="variable">T</span></a>) (<span class="id" title="var">m</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#bitseq"><span class="id" title="definition">bitseq</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem_mask_cons"><span class="id" title="lemma">mem_mask_cons</span></a> <span class="id" title="var">x</span> <span class="id" title="var">b</span> <span class="id" title="var">m</span> <span class="id" title="var">y</span> <span class="id" title="var">s</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#mask"><span class="id" title="definition">mask</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#mask"><span class="id" title="definition">mask</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem_mask"><span class="id" title="lemma">mem_mask</span></a> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#mask"><span class="id" title="definition">mask</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mask_uniq"><span class="id" title="lemma">mask_uniq</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">m</span>, <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#mask"><span class="id" title="definition">mask</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem_mask_rot"><span class="id" title="lemma">mem_mask_rot</span></a> <span class="id" title="var">m</span> <span class="id" title="var">s</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#mask"><span class="id" title="definition">mask</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqMask.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqMask.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#mask"><span class="id" title="definition">mask</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.seq.html#EqMask"><span class="id" title="section">EqMask</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Subseq"><span class="id" title="section">Subseq</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="Subseq.T"><span class="id" title="variable">T</span></a> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>.<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Subseq.T"><span class="id" title="variable">T</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="subseq"><span class="id" title="definition">subseq</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> :=<br/> - <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">s2'</span> <span class="id" title="keyword">then</span><br/> - <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">s1'</span> <span class="id" title="keyword">then</span> <a class="idref" href="mathcomp.ssreflect.seq.html#subseq"><span class="id" title="definition">subseq</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <span class="id" title="var">s1'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a>) <span class="id" title="var">s2'</span> <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/> - <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sub0seq"><span class="id" title="lemma">sub0seq</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#subseq"><span class="id" title="definition">subseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>. <br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="subseq0"><span class="id" title="lemma">subseq0</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#subseq"><span class="id" title="definition">subseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>. <br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="subseq_refl"><span class="id" title="lemma">subseq_refl</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#subseq"><span class="id" title="definition">subseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - <span class="id" title="keyword">Hint Resolve</span> <span class="id" title="var">subseq_refl</span> : <span class="id" title="var">core</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="subseqP"><span class="id" title="lemma">subseqP</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">m</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#mask"><span class="id" title="definition">mask</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#subseq"><span class="id" title="definition">subseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mask_subseq"><span class="id" title="lemma">mask_subseq</span></a> <span class="id" title="var">m</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#subseq"><span class="id" title="definition">subseq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#mask"><span class="id" title="definition">mask</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="subseq_trans"><span class="id" title="lemma">subseq_trans</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#transitive"><span class="id" title="definition">transitive</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#subseq"><span class="id" title="definition">subseq</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cat_subseq"><span class="id" title="lemma">cat_subseq</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> <span class="id" title="var">s3</span> <span class="id" title="var">s4</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#subseq"><span class="id" title="definition">subseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s3"><span class="id" title="variable">s3</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#subseq"><span class="id" title="definition">subseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s4"><span class="id" title="variable">s4</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#subseq"><span class="id" title="definition">subseq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#s3"><span class="id" title="variable">s3</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s4"><span class="id" title="variable">s4</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="prefix_subseq"><span class="id" title="lemma">prefix_subseq</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#subseq"><span class="id" title="definition">subseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="suffix_subseq"><span class="id" title="lemma">suffix_subseq</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#subseq"><span class="id" title="definition">subseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="take_subseq"><span class="id" title="lemma">take_subseq</span></a> <span class="id" title="var">s</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#subseq"><span class="id" title="definition">subseq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#take"><span class="id" title="definition">take</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="drop_subseq"><span class="id" title="lemma">drop_subseq</span></a> <span class="id" title="var">s</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#subseq"><span class="id" title="definition">subseq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#drop"><span class="id" title="definition">drop</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem_subseq"><span class="id" title="lemma">mem_subseq</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#subseq"><span class="id" title="definition">subseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sub1seq"><span class="id" title="lemma">sub1seq</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#subseq"><span class="id" title="definition">subseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#506674b18256ef8f50efed43fa1dfd7d"><span class="id" title="notation">[::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#506674b18256ef8f50efed43fa1dfd7d"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="size_subseq"><span class="id" title="lemma">size_subseq</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#subseq"><span class="id" title="definition">subseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="size_subseq_leqif"><span class="id" title="lemma">size_subseq_leqif</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#subseq"><span class="id" title="definition">subseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#22d09a36997010daec8f30c044c9e5d4"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#22d09a36997010daec8f30c044c9e5d4"><span class="id" title="notation">?=</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#22d09a36997010daec8f30c044c9e5d4"><span class="id" title="notation">iff</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#22d09a36997010daec8f30c044c9e5d4"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#22d09a36997010daec8f30c044c9e5d4"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="subseq_cons"><span class="id" title="lemma">subseq_cons</span></a> <span class="id" title="var">s</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#subseq"><span class="id" title="definition">subseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="subseq_rcons"><span class="id" title="lemma">subseq_rcons</span></a> <span class="id" title="var">s</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#subseq"><span class="id" title="definition">subseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="subseq_uniq"><span class="id" title="lemma">subseq_uniq</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#subseq"><span class="id" title="definition">subseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.seq.html#Subseq"><span class="id" title="section">Subseq</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Hint Resolve</span> <span class="id" title="var">subseq_refl</span> : <span class="id" title="var">core</span>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Rem"><span class="id" title="section">Rem</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="Rem.T"><span class="id" title="variable">T</span></a> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>) (<a name="Rem.x"><span class="id" title="variable">x</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#T"><span class="id" title="variable">T</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="rem"><span class="id" title="definition">rem</span></a> <span class="id" title="var">s</span> := <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">t</span> <span class="id" title="keyword">then</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Rem.x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <span class="id" title="var">t</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rem"><span class="id" title="definition">rem</span></a> <span class="id" title="var">t</span>) <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rem_id"><span class="id" title="lemma">rem_id</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#Rem.x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">notin</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rem"><span class="id" title="definition">rem</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="perm_to_rem"><span class="id" title="lemma">perm_to_rem</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#Rem.x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#Rem.x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rem"><span class="id" title="definition">rem</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="size_rem"><span class="id" title="lemma">size_rem</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#Rem.x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rem"><span class="id" title="definition">rem</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#f953bf7095e0da1cb644443fd0e17d6d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#f953bf7095e0da1cb644443fd0e17d6d"><span class="id" title="notation">).-1</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rem_subseq"><span class="id" title="lemma">rem_subseq</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#subseq"><span class="id" title="definition">subseq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rem"><span class="id" title="definition">rem</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rem_uniq"><span class="id" title="lemma">rem_uniq</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rem"><span class="id" title="definition">rem</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem_rem"><span class="id" title="lemma">mem_rem</span></a> <span class="id" title="var">s</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rem"><span class="id" title="definition">rem</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#ca592708f529c7c7ee5f3dbd6cf93463"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rem_filter"><span class="id" title="lemma">rem_filter</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rem"><span class="id" title="definition">rem</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> (<a class="idref" href="mathcomp.ssreflect.eqtype.html#predC1"><span class="id" title="definition">predC1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Rem.x"><span class="id" title="variable">x</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem_rem_uniq"><span class="id" title="lemma">mem_rem_uniq</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rem"><span class="id" title="definition">rem</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#bcb59f838ed564b993945f7efc641d66"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.eqtype.html#bcb59f838ed564b993945f7efc641d66"><span class="id" title="notation">predD1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#bcb59f838ed564b993945f7efc641d66"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Rem.x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.eqtype.html#bcb59f838ed564b993945f7efc641d66"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem_rem_uniqF"><span class="id" title="lemma">mem_rem_uniqF</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Rem.x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rem"><span class="id" title="definition">rem</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.seq.html#Rem"><span class="id" title="section">Rem</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Map"><span class="id" title="section">Map</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="Map.n0"><span class="id" title="variable">n0</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a name="Map.T1"><span class="id" title="variable">T1</span></a> : <span class="id" title="keyword">Type</span>) (<a name="Map.x1"><span class="id" title="variable">x1</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#T1"><span class="id" title="variable">T1</span></a>).<br/> -<span class="id" title="keyword">Variables</span> (<a name="Map.T2"><span class="id" title="variable">T2</span></a> : <span class="id" title="keyword">Type</span>) (<a name="Map.x2"><span class="id" title="variable">x2</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#T2"><span class="id" title="variable">T2</span></a>) (<a name="Map.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#Map.T1"><span class="id" title="variable">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#T2"><span class="id" title="variable">T2</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="map"><span class="id" title="definition">map</span></a> <span class="id" title="var">s</span> := <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">s'</span> <span class="id" title="keyword">then</span> <a class="idref" href="mathcomp.ssreflect.seq.html#Map.f"><span class="id" title="variable">f</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <span class="id" title="var">s'</span> <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_cons"><span class="id" title="lemma">map_cons</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Map.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_nseq"><span class="id" title="lemma">map_nseq</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#nseq"><span class="id" title="definition">nseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Map.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nseq"><span class="id" title="definition">nseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Map.n0"><span class="id" title="variable">n0</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#Map.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_cat"><span class="id" title="lemma">map_cat</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="size_map"><span class="id" title="lemma">size_map</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="behead_map"><span class="id" title="lemma">behead_map</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#behead"><span class="id" title="definition">behead</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#behead"><span class="id" title="definition">behead</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nth_map"><span class="id" title="lemma">nth_map</span></a> <span class="id" title="var">n</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Map.x2"><span class="id" title="variable">x2</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Map.f"><span class="id" title="variable">f</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Map.x1"><span class="id" title="variable">x1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_rcons"><span class="id" title="lemma">map_rcons</span></a> <span class="id" title="var">s</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#Map.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="last_map"><span class="id" title="lemma">last_map</span></a> <span class="id" title="var">s</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#last"><span class="id" title="definition">last</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#Map.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Map.f"><span class="id" title="variable">f</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#last"><span class="id" title="definition">last</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="belast_map"><span class="id" title="lemma">belast_map</span></a> <span class="id" title="var">s</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#belast"><span class="id" title="definition">belast</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#Map.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#belast"><span class="id" title="definition">belast</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="filter_map"><span class="id" title="lemma">filter_map</span></a> <span class="id" title="var">a</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#preim"><span class="id" title="definition">preim</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Map.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="find_map"><span class="id" title="lemma">find_map</span></a> <span class="id" title="var">a</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#find"><span class="id" title="definition">find</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#find"><span class="id" title="definition">find</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#preim"><span class="id" title="definition">preim</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Map.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="has_map"><span class="id" title="lemma">has_map</span></a> <span class="id" title="var">a</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#has"><span class="id" title="definition">has</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#preim"><span class="id" title="definition">preim</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Map.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="all_map"><span class="id" title="lemma">all_map</span></a> <span class="id" title="var">a</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#preim"><span class="id" title="definition">preim</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Map.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="count_map"><span class="id" title="lemma">count_map</span></a> <span class="id" title="var">a</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#count"><span class="id" title="definition">count</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#count"><span class="id" title="definition">count</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#preim"><span class="id" title="definition">preim</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Map.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_take"><span class="id" title="lemma">map_take</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#take"><span class="id" title="definition">take</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Map.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#take"><span class="id" title="definition">take</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Map.n0"><span class="id" title="variable">n0</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_drop"><span class="id" title="lemma">map_drop</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#drop"><span class="id" title="definition">drop</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Map.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#drop"><span class="id" title="definition">drop</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Map.n0"><span class="id" title="variable">n0</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_rot"><span class="id" title="lemma">map_rot</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Map.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Map.n0"><span class="id" title="variable">n0</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_rotr"><span class="id" title="lemma">map_rotr</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rotr"><span class="id" title="definition">rotr</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Map.n0"><span class="id" title="variable">n0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rotr"><span class="id" title="definition">rotr</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Map.n0"><span class="id" title="variable">n0</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_rev"><span class="id" title="lemma">map_rev</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_mask"><span class="id" title="lemma">map_mask</span></a> <span class="id" title="var">m</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#mask"><span class="id" title="definition">mask</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#mask"><span class="id" title="definition">mask</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="inj_map"><span class="id" title="lemma">inj_map</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Map.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.seq.html#Map"><span class="id" title="section">Map</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">"</span></a>[ 'seq' E | i <- s ]" := (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">i</span> ⇒ <span class="id" title="var">E</span>) <span class="id" title="var">s</span>)<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">E</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 99, <span class="id" title="var">i</span> <span class="id" title="var">ident</span>,<br/> - <span class="id" title="var">format</span> "[ '[hv' 'seq' E '/ ' | i <- s ] ']'") : <span class="id" title="var">seq_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="3f8af8f1c43d737adb935ffe24912c2b"><span class="id" title="notation">"</span></a>[ 'seq' E | i <- s & C ]" := <a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">seq</span></a> <span class="id" title="var">E</span> <a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">|</span></a> <span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#3de67eb42556f00623e74d6756c5ab67"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#3de67eb42556f00623e74d6756c5ab67"><span class="id" title="notation">seq</span></a> <span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.seq.html#3de67eb42556f00623e74d6756c5ab67"><span class="id" title="notation"><-</span></a> <span class="id" title="var">s</span> <a class="idref" href="mathcomp.ssreflect.seq.html#3de67eb42556f00623e74d6756c5ab67"><span class="id" title="notation">|</span></a> <span class="id" title="var">C</span><a class="idref" href="mathcomp.ssreflect.seq.html#3de67eb42556f00623e74d6756c5ab67"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">]</span></a><br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">E</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 99, <span class="id" title="var">i</span> <span class="id" title="var">ident</span>,<br/> - <span class="id" title="var">format</span> "[ '[hv' 'seq' E '/ ' | i <- s '/ ' & C ] ']'") : <span class="id" title="var">seq_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="035c9924d34358d2b2816d54c1fb0043"><span class="id" title="notation">"</span></a>[ 'seq' E | i : T <- s ]" := (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">i</span> : <span class="id" title="var">T</span> ⇒ <span class="id" title="var">E</span>) <span class="id" title="var">s</span>)<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">E</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 99, <span class="id" title="var">i</span> <span class="id" title="var">ident</span>, <span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">seq_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="c9c10f3f61c4e97c645239bfa50a8854"><span class="id" title="notation">"</span></a>[ 'seq' E | i : T <- s & C ]" :=<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#035c9924d34358d2b2816d54c1fb0043"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#035c9924d34358d2b2816d54c1fb0043"><span class="id" title="notation">seq</span></a> <span class="id" title="var">E</span> <a class="idref" href="mathcomp.ssreflect.seq.html#035c9924d34358d2b2816d54c1fb0043"><span class="id" title="notation">|</span></a> <span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.seq.html#035c9924d34358d2b2816d54c1fb0043"><span class="id" title="notation">:</span></a> <span class="id" title="var">T</span> <a class="idref" href="mathcomp.ssreflect.seq.html#035c9924d34358d2b2816d54c1fb0043"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#3d61eb9bcecd9ede478c4672178648de"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#3d61eb9bcecd9ede478c4672178648de"><span class="id" title="notation">seq</span></a> <span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.seq.html#3d61eb9bcecd9ede478c4672178648de"><span class="id" title="notation">:</span></a> <span class="id" title="var">T</span> <a class="idref" href="mathcomp.ssreflect.seq.html#3d61eb9bcecd9ede478c4672178648de"><span class="id" title="notation"><-</span></a> <span class="id" title="var">s</span> <a class="idref" href="mathcomp.ssreflect.seq.html#3d61eb9bcecd9ede478c4672178648de"><span class="id" title="notation">|</span></a> <span class="id" title="var">C</span><a class="idref" href="mathcomp.ssreflect.seq.html#3d61eb9bcecd9ede478c4672178648de"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#035c9924d34358d2b2816d54c1fb0043"><span class="id" title="notation">]</span></a><br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">E</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 99, <span class="id" title="var">i</span> <span class="id" title="var">ident</span>, <span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">seq_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="047e648644be092ebdd09919d3f414d3"><span class="id" title="notation">"</span></a>[ 'seq' E : R | i <- s ]" := (@<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <span class="id" title="var">_</span> <span class="id" title="var">R</span> (<span class="id" title="keyword">fun</span> <span class="id" title="var">i</span> ⇒ <span class="id" title="var">E</span>) <span class="id" title="var">s</span>)<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">E</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 99, <span class="id" title="var">i</span> <span class="id" title="var">ident</span>, <span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">seq_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="470a0f35a6a80dfc540ce4d1bb552af4"><span class="id" title="notation">"</span></a>[ 'seq' E : R | i <- s & C ]" := <a class="idref" href="mathcomp.ssreflect.seq.html#047e648644be092ebdd09919d3f414d3"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#047e648644be092ebdd09919d3f414d3"><span class="id" title="notation">seq</span></a> <span class="id" title="var">E</span> <a class="idref" href="mathcomp.ssreflect.seq.html#047e648644be092ebdd09919d3f414d3"><span class="id" title="notation">:</span></a> <span class="id" title="var">R</span> <a class="idref" href="mathcomp.ssreflect.seq.html#047e648644be092ebdd09919d3f414d3"><span class="id" title="notation">|</span></a> <span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.seq.html#047e648644be092ebdd09919d3f414d3"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#3de67eb42556f00623e74d6756c5ab67"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#3de67eb42556f00623e74d6756c5ab67"><span class="id" title="notation">seq</span></a> <span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.seq.html#3de67eb42556f00623e74d6756c5ab67"><span class="id" title="notation"><-</span></a> <span class="id" title="var">s</span> <a class="idref" href="mathcomp.ssreflect.seq.html#3de67eb42556f00623e74d6756c5ab67"><span class="id" title="notation">|</span></a> <span class="id" title="var">C</span><a class="idref" href="mathcomp.ssreflect.seq.html#3de67eb42556f00623e74d6756c5ab67"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#047e648644be092ebdd09919d3f414d3"><span class="id" title="notation">]</span></a><br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">E</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 99, <span class="id" title="var">i</span> <span class="id" title="var">ident</span>, <span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">seq_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="732c880e3345bf5133e68085a32d1a68"><span class="id" title="notation">"</span></a>[ 'seq' E : R | i : T <- s ]" := (@<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <span class="id" title="var">T</span> <span class="id" title="var">R</span> (<span class="id" title="keyword">fun</span> <span class="id" title="var">i</span> : <span class="id" title="var">T</span> ⇒ <span class="id" title="var">E</span>) <span class="id" title="var">s</span>)<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">E</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 99, <span class="id" title="var">i</span> <span class="id" title="var">ident</span>, <span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">seq_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="769368013541fda2d54cc44e75d7c105"><span class="id" title="notation">"</span></a>[ 'seq' E : R | i : T <- s & C ]" :=<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#732c880e3345bf5133e68085a32d1a68"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#732c880e3345bf5133e68085a32d1a68"><span class="id" title="notation">seq</span></a> <span class="id" title="var">E</span> <a class="idref" href="mathcomp.ssreflect.seq.html#732c880e3345bf5133e68085a32d1a68"><span class="id" title="notation">:</span></a> <span class="id" title="var">R</span> <a class="idref" href="mathcomp.ssreflect.seq.html#732c880e3345bf5133e68085a32d1a68"><span class="id" title="notation">|</span></a> <span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.seq.html#732c880e3345bf5133e68085a32d1a68"><span class="id" title="notation">:</span></a> <span class="id" title="var">T</span> <a class="idref" href="mathcomp.ssreflect.seq.html#732c880e3345bf5133e68085a32d1a68"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#3d61eb9bcecd9ede478c4672178648de"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#3d61eb9bcecd9ede478c4672178648de"><span class="id" title="notation">seq</span></a> <span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.seq.html#3d61eb9bcecd9ede478c4672178648de"><span class="id" title="notation">:</span></a> <span class="id" title="var">T</span> <a class="idref" href="mathcomp.ssreflect.seq.html#3d61eb9bcecd9ede478c4672178648de"><span class="id" title="notation"><-</span></a> <span class="id" title="var">s</span> <a class="idref" href="mathcomp.ssreflect.seq.html#3d61eb9bcecd9ede478c4672178648de"><span class="id" title="notation">|</span></a> <span class="id" title="var">C</span><a class="idref" href="mathcomp.ssreflect.seq.html#3d61eb9bcecd9ede478c4672178648de"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#732c880e3345bf5133e68085a32d1a68"><span class="id" title="notation">]</span></a><br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">E</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 99, <span class="id" title="var">i</span> <span class="id" title="var">ident</span>, <span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">seq_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="filter_mask"><span class="id" title="lemma">filter_mask</span></a> <span class="id" title="var">T</span> <span class="id" title="var">a</span> (<span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#T"><span class="id" title="variable">T</span></a>) : <a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#mask"><span class="id" title="definition">mask</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="FilterSubseq"><span class="id" title="section">FilterSubseq</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="FilterSubseq.T"><span class="id" title="variable">T</span></a> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>.<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#FilterSubseq.T"><span class="id" title="variable">T</span></a>) (<span class="id" title="var">a</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#FilterSubseq.T"><span class="id" title="variable">T</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="filter_subseq"><span class="id" title="lemma">filter_subseq</span></a> <span class="id" title="var">a</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#subseq"><span class="id" title="definition">subseq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="subseq_filter"><span class="id" title="lemma">subseq_filter</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> <span class="id" title="var">a</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#subseq"><span class="id" title="definition">subseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#subseq"><span class="id" title="definition">subseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="subseq_uniqP"><span class="id" title="lemma">subseq_uniqP</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#subseq"><span class="id" title="definition">subseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="perm_to_subseq"><span class="id" title="lemma">perm_to_subseq</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#subseq"><span class="id" title="definition">subseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#bc4528e836ab0e91ea7e942fb09e898f"><span class="id" title="notation">{</span></a><span class="id" title="var">s3</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#bc4528e836ab0e91ea7e942fb09e898f"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s3"><span class="id" title="variable">s3</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#bc4528e836ab0e91ea7e942fb09e898f"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.seq.html#FilterSubseq"><span class="id" title="section">FilterSubseq</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="EqMap"><span class="id" title="section">EqMap</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="EqMap.n0"><span class="id" title="variable">n0</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a name="EqMap.T1"><span class="id" title="variable">T1</span></a> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>) (<a name="EqMap.x1"><span class="id" title="variable">x1</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#T1"><span class="id" title="variable">T1</span></a>).<br/> -<span class="id" title="keyword">Variables</span> (<a name="EqMap.T2"><span class="id" title="variable">T2</span></a> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>) (<a name="EqMap.x2"><span class="id" title="variable">x2</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#T2"><span class="id" title="variable">T2</span></a>) (<a name="EqMap.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#EqMap.T1"><span class="id" title="variable">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#T2"><span class="id" title="variable">T2</span></a>).<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqMap.T1"><span class="id" title="variable">T1</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_f"><span class="id" title="lemma">map_f</span></a> <span class="id" title="var">s</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqMap.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqMap.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mapP"><span class="id" title="lemma">mapP</span></a> <span class="id" title="var">s</span> <span class="id" title="var">y</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqMap.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqMap.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_uniq"><span class="id" title="lemma">map_uniq</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqMap.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_inj_in_uniq"><span class="id" title="lemma">map_inj_in_uniq</span></a> <span class="id" title="var">s</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqMap.f"><span class="id" title="variable">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqMap.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_subseq"><span class="id" title="lemma">map_subseq</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#subseq"><span class="id" title="definition">subseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#subseq"><span class="id" title="definition">subseq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqMap.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqMap.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nth_index_map"><span class="id" title="lemma">nth_index_map</span></a> <span class="id" title="var">s</span> <span class="id" title="var">x0</span> <span class="id" title="var">x</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqMap.f"><span class="id" title="variable">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x0"><span class="id" title="variable">x0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#index"><span class="id" title="definition">index</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#EqMap.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqMap.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="perm_map"><span class="id" title="lemma">perm_map</span></a> <span class="id" title="var">s</span> <span class="id" title="var">t</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqMap.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqMap.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Hypothesis</span> <a name="EqMap.Hf"><span class="id" title="variable">Hf</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqMap.f"><span class="id" title="variable">f</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem_map"><span class="id" title="lemma">mem_map</span></a> <span class="id" title="var">s</span> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#EqMap.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqMap.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="index_map"><span class="id" title="lemma">index_map</span></a> <span class="id" title="var">s</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#index"><span class="id" title="definition">index</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#EqMap.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqMap.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#index"><span class="id" title="definition">index</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_inj_uniq"><span class="id" title="lemma">map_inj_uniq</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqMap.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="perm_map_inj"><span class="id" title="lemma">perm_map_inj</span></a> <span class="id" title="var">s</span> <span class="id" title="var">t</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqMap.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqMap.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.seq.html#EqMap"><span class="id" title="section">EqMap</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_of_seq"><span class="id" title="lemma">map_of_seq</span></a> (<span class="id" title="var">T1</span> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>) <span class="id" title="var">T2</span> (<span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#T1"><span class="id" title="variable">T1</span></a>) (<span class="id" title="var">fs</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#T2"><span class="id" title="variable">T2</span></a>) (<span class="id" title="var">y0</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#T2"><span class="id" title="variable">T2</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#bc4528e836ab0e91ea7e942fb09e898f"><span class="id" title="notation">{</span></a><span class="id" title="var">f</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#bc4528e836ab0e91ea7e942fb09e898f"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#fs"><span class="id" title="variable">fs</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#fs"><span class="id" title="variable">fs</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#bc4528e836ab0e91ea7e942fb09e898f"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="MapComp"><span class="id" title="section">MapComp</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="MapComp.T1"><span class="id" title="variable">T1</span></a> <a name="MapComp.T2"><span class="id" title="variable">T2</span></a> <a name="MapComp.T3"><span class="id" title="variable">T3</span></a> : <span class="id" title="keyword">Type</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_id"><span class="id" title="lemma">map_id</span></a> (<span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#MapComp.T1"><span class="id" title="variable">T1</span></a>) : <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_map"><span class="id" title="lemma">eq_map</span></a> (<span class="id" title="var">f1</span> <span class="id" title="var">f2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#MapComp.T1"><span class="id" title="variable">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#MapComp.T2"><span class="id" title="variable">T2</span></a>) : <a class="idref" href="mathcomp.ssreflect.seq.html#f1"><span class="id" title="variable">f1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f2"><span class="id" title="variable">f2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f1"><span class="id" title="variable">f1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f2"><span class="id" title="variable">f2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_comp"><span class="id" title="lemma">map_comp</span></a> (<span class="id" title="var">f1</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#MapComp.T2"><span class="id" title="variable">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#MapComp.T3"><span class="id" title="variable">T3</span></a>) (<span class="id" title="var">f2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#MapComp.T1"><span class="id" title="variable">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#MapComp.T2"><span class="id" title="variable">T2</span></a>) <span class="id" title="var">s</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#f1"><span class="id" title="variable">f1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#8b4742e3f67816503ce4ab2f3b81c27e"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#8b4742e3f67816503ce4ab2f3b81c27e"><span class="id" title="notation">o</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f2"><span class="id" title="variable">f2</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f1"><span class="id" title="variable">f1</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f2"><span class="id" title="variable">f2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mapK"><span class="id" title="lemma">mapK</span></a> (<span class="id" title="var">f1</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#MapComp.T1"><span class="id" title="variable">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#MapComp.T2"><span class="id" title="variable">T2</span></a>) (<span class="id" title="var">f2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#MapComp.T2"><span class="id" title="variable">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#MapComp.T1"><span class="id" title="variable">T1</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f1"><span class="id" title="variable">f1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f2"><span class="id" title="variable">f2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f1"><span class="id" title="variable">f1</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f2"><span class="id" title="variable">f2</span></a>).<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.seq.html#MapComp"><span class="id" title="section">MapComp</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_in_map"><span class="id" title="lemma">eq_in_map</span></a> (<span class="id" title="var">T1</span> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>) <span class="id" title="var">T2</span> (<span class="id" title="var">f1</span> <span class="id" title="var">f2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#T1"><span class="id" title="variable">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#T2"><span class="id" title="variable">T2</span></a>) (<span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#T1"><span class="id" title="variable">T1</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f1"><span class="id" title="variable">f1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f2"><span class="id" title="variable">f2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#4bfb4f2d0721ba668e3a802ab1b745a1"><span class="id" title="notation">↔</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f1"><span class="id" title="variable">f1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f2"><span class="id" title="variable">f2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_id_in"><span class="id" title="lemma">map_id_in</span></a> (<span class="id" title="var">T</span> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>) <span class="id" title="var">f</span> (<span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#T"><span class="id" title="variable">T</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - Map a partial function -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Pmap"><span class="id" title="section">Pmap</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="Pmap.aT"><span class="id" title="variable">aT</span></a> <a name="Pmap.rT"><span class="id" title="variable">rT</span></a> : <span class="id" title="keyword">Type</span>) (<a name="Pmap.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#aT"><span class="id" title="variable">aT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rT"><span class="id" title="variable">rT</span></a>) (<a name="Pmap.g"><span class="id" title="variable">g</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#rT"><span class="id" title="variable">rT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#aT"><span class="id" title="variable">aT</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="pmap"><span class="id" title="definition">pmap</span></a> <span class="id" title="var">s</span> :=<br/> - <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">s'</span> <span class="id" title="keyword">then</span> <span class="id" title="keyword">let</span> <span class="id" title="var">r</span> := <a class="idref" href="mathcomp.ssreflect.seq.html#pmap"><span class="id" title="definition">pmap</span></a> <span class="id" title="var">s'</span> <span class="id" title="tactic">in</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#oapp"><span class="id" title="abbreviation">oapp</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#cons"><span class="id" title="constructor">cons</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#d89396f990d6b54d736cfe259e498cf4"><span class="id" title="notation">^~</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#r"><span class="id" title="variable">r</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#r"><span class="id" title="variable">r</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#Pmap.f"><span class="id" title="variable">f</span></a> <span class="id" title="var">x</span>) <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_pK"><span class="id" title="lemma">map_pK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#pcancel"><span class="id" title="definition">pcancel</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Pmap.g"><span class="id" title="variable">g</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Pmap.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Pmap.g"><span class="id" title="variable">g</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#pmap"><span class="id" title="definition">pmap</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="size_pmap"><span class="id" title="lemma">size_pmap</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#pmap"><span class="id" title="definition">pmap</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#count"><span class="id" title="definition">count</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#20b75710d0541e9ffd06f8e723fd3daf"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#20b75710d0541e9ffd06f8e723fd3daf"><span class="id" title="notation">eta</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Pmap.f"><span class="id" title="variable">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#20b75710d0541e9ffd06f8e723fd3daf"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pmapS_filter"><span class="id" title="lemma">pmapS_filter</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#some"><span class="id" title="abbreviation">some</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#pmap"><span class="id" title="definition">pmap</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Pmap.f"><span class="id" title="variable">f</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#20b75710d0541e9ffd06f8e723fd3daf"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#20b75710d0541e9ffd06f8e723fd3daf"><span class="id" title="notation">eta</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Pmap.f"><span class="id" title="variable">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#20b75710d0541e9ffd06f8e723fd3daf"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Hypothesis</span> <a name="Pmap.fK"><span class="id" title="variable">fK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#ocancel"><span class="id" title="definition">ocancel</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Pmap.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Pmap.g"><span class="id" title="variable">g</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pmap_filter"><span class="id" title="lemma">pmap_filter</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Pmap.g"><span class="id" title="variable">g</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#pmap"><span class="id" title="definition">pmap</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#20b75710d0541e9ffd06f8e723fd3daf"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#20b75710d0541e9ffd06f8e723fd3daf"><span class="id" title="notation">eta</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Pmap.f"><span class="id" title="variable">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#20b75710d0541e9ffd06f8e723fd3daf"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pmap_cat"><span class="id" title="lemma">pmap_cat</span></a> <span class="id" title="var">s</span> <span class="id" title="var">t</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#pmap"><span class="id" title="definition">pmap</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#pmap"><span class="id" title="definition">pmap</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#pmap"><span class="id" title="definition">pmap</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.seq.html#Pmap"><span class="id" title="section">Pmap</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="EqPmap"><span class="id" title="section">EqPmap</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="EqPmap.aT"><span class="id" title="variable">aT</span></a> <a name="EqPmap.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>) (<a name="EqPmap.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#aT"><span class="id" title="variable">aT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rT"><span class="id" title="variable">rT</span></a>) (<a name="EqPmap.g"><span class="id" title="variable">g</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#rT"><span class="id" title="variable">rT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#aT"><span class="id" title="variable">aT</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_pmap"><span class="id" title="lemma">eq_pmap</span></a> (<span class="id" title="var">f1</span> <span class="id" title="var">f2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#EqPmap.aT"><span class="id" title="variable">aT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqPmap.rT"><span class="id" title="variable">rT</span></a>) : <a class="idref" href="mathcomp.ssreflect.seq.html#f1"><span class="id" title="variable">f1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f2"><span class="id" title="variable">f2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#pmap"><span class="id" title="definition">pmap</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f1"><span class="id" title="variable">f1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#pmap"><span class="id" title="definition">pmap</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f2"><span class="id" title="variable">f2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem_pmap"><span class="id" title="lemma">mem_pmap</span></a> <span class="id" title="var">s</span> <span class="id" title="var">u</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#pmap"><span class="id" title="definition">pmap</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqPmap.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqPmap.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Hypothesis</span> <a name="EqPmap.fK"><span class="id" title="variable">fK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#ocancel"><span class="id" title="definition">ocancel</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqPmap.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqPmap.g"><span class="id" title="variable">g</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="can2_mem_pmap"><span class="id" title="lemma">can2_mem_pmap</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#pcancel"><span class="id" title="definition">pcancel</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqPmap.g"><span class="id" title="variable">g</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqPmap.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">s</span> <span class="id" title="var">u</span>, <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#pmap"><span class="id" title="definition">pmap</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqPmap.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#EqPmap.g"><span class="id" title="variable">g</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pmap_uniq"><span class="id" title="lemma">pmap_uniq</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#pmap"><span class="id" title="definition">pmap</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqPmap.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="perm_pmap"><span class="id" title="lemma">perm_pmap</span></a> <span class="id" title="var">s</span> <span class="id" title="var">t</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#pmap"><span class="id" title="definition">pmap</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqPmap.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#pmap"><span class="id" title="definition">pmap</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqPmap.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a>).<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.seq.html#EqPmap"><span class="id" title="section">EqPmap</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="PmapSub"><span class="id" title="section">PmapSub</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="PmapSub.T"><span class="id" title="variable">T</span></a> : <span class="id" title="keyword">Type</span>) (<a name="PmapSub.p"><span class="id" title="variable">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#T"><span class="id" title="variable">T</span></a>) (<a name="PmapSub.sT"><span class="id" title="variable">sT</span></a> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#subType"><span class="id" title="record">subType</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#p"><span class="id" title="variable">p</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="size_pmap_sub"><span class="id" title="lemma">size_pmap_sub</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#pmap"><span class="id" title="definition">pmap</span></a> (<a class="idref" href="mathcomp.ssreflect.eqtype.html#insub"><span class="id" title="definition">insub</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#PmapSub.T"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#PmapSub.sT"><span class="id" title="variable">sT</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#count"><span class="id" title="definition">count</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#PmapSub.p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.seq.html#PmapSub"><span class="id" title="section">PmapSub</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="EqPmapSub"><span class="id" title="section">EqPmapSub</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="EqPmapSub.T"><span class="id" title="variable">T</span></a> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>) (<a name="EqPmapSub.p"><span class="id" title="variable">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#T"><span class="id" title="variable">T</span></a>) (<a name="EqPmapSub.sT"><span class="id" title="variable">sT</span></a> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#subType"><span class="id" title="record">subType</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#p"><span class="id" title="variable">p</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Let</span> <a name="EqPmapSub.insT"><span class="id" title="variable">insT</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#EqPmapSub.T"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqPmapSub.sT"><span class="id" title="variable">sT</span></a> := <a class="idref" href="mathcomp.ssreflect.eqtype.html#insub"><span class="id" title="definition">insub</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem_pmap_sub"><span class="id" title="lemma">mem_pmap_sub</span></a> <span class="id" title="var">s</span> <span class="id" title="var">u</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#pmap"><span class="id" title="definition">pmap</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqPmapSub.insT"><span class="id" title="variable">insT</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.eqtype.html#val"><span class="id" title="projection">val</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pmap_sub_uniq"><span class="id" title="lemma">pmap_sub_uniq</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#pmap"><span class="id" title="definition">pmap</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqPmapSub.insT"><span class="id" title="variable">insT</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.seq.html#EqPmapSub"><span class="id" title="section">EqPmapSub</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - Index sequence -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="iota"><span class="id" title="definition">iota</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> := <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">n'</span><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <span class="id" title="keyword">then</span> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <span class="id" title="var">n'</span> <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="size_iota"><span class="id" title="lemma">size_iota</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="iota_add"><span class="id" title="lemma">iota_add</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n1</span> <span class="id" title="var">n2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#n1"><span class="id" title="variable">n1</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n2"><span class="id" title="variable">n2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n1"><span class="id" title="variable">n1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#iota"><span class="id" title="definition">iota</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n1"><span class="id" title="variable">n1</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#n2"><span class="id" title="variable">n2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="iota_addl"><span class="id" title="lemma">iota_addl</span></a> <span class="id" title="var">m1</span> <span class="id" title="var">m2</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#iota"><span class="id" title="definition">iota</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#m1"><span class="id" title="variable">m1</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m2"><span class="id" title="variable">m2</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> (<a class="idref" href="mathcomp.ssreflect.ssrnat.html#addn"><span class="id" title="definition">addn</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m1"><span class="id" title="variable">m1</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m2"><span class="id" title="variable">m2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nth_iota"><span class="id" title="lemma">nth_iota</span></a> <span class="id" title="var">p</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#p"><span class="id" title="variable">p</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem_iota"><span class="id" title="lemma">mem_iota</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> <span class="id" title="var">i</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="iota_uniq"><span class="id" title="lemma">iota_uniq</span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#iota"><span class="id" title="definition">iota</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a>).<br/> - -<br/> -</div> - -<div class="doc"> - Making a sequence of a specific length, using indexes to compute items. -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Section</span> <a name="MakeSeq"><span class="id" title="section">MakeSeq</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="MakeSeq.T"><span class="id" title="variable">T</span></a> : <span class="id" title="keyword">Type</span>) (<a name="MakeSeq.x0"><span class="id" title="variable">x0</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#T"><span class="id" title="variable">T</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="mkseq"><span class="id" title="definition">mkseq</span></a> <span class="id" title="var">f</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#MakeSeq.T"><span class="id" title="variable">T</span></a> := <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#iota"><span class="id" title="definition">iota</span></a> 0 <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="size_mkseq"><span class="id" title="lemma">size_mkseq</span></a> <span class="id" title="var">f</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#mkseq"><span class="id" title="definition">mkseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_mkseq"><span class="id" title="lemma">eq_mkseq</span></a> <span class="id" title="var">f</span> <span class="id" title="var">g</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#g"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#mkseq"><span class="id" title="definition">mkseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#mkseq"><span class="id" title="definition">mkseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#g"><span class="id" title="variable">g</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nth_mkseq"><span class="id" title="lemma">nth_mkseq</span></a> <span class="id" title="var">f</span> <span class="id" title="var">n</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#MakeSeq.x0"><span class="id" title="variable">x0</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#mkseq"><span class="id" title="definition">mkseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mkseq_nth"><span class="id" title="lemma">mkseq_nth</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#mkseq"><span class="id" title="definition">mkseq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#MakeSeq.x0"><span class="id" title="variable">x0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.seq.html#MakeSeq"><span class="id" title="section">MakeSeq</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="MakeEqSeq"><span class="id" title="section">MakeEqSeq</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="MakeEqSeq.T"><span class="id" title="variable">T</span></a> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mkseq_uniq"><span class="id" title="lemma">mkseq_uniq</span></a> (<span class="id" title="var">f</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#MakeEqSeq.T"><span class="id" title="variable">T</span></a>) <span class="id" title="var">n</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#mkseq"><span class="id" title="definition">mkseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="perm_iotaP"><span class="id" title="lemma">perm_iotaP</span></a> {<span class="id" title="var">s</span> <span class="id" title="var">t</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#MakeEqSeq.T"><span class="id" title="variable">T</span></a>} <span class="id" title="var">x0</span> (<span class="id" title="var">It</span> := <a class="idref" href="mathcomp.ssreflect.seq.html#iota"><span class="id" title="definition">iota</span></a> 0 (<a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a>)) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">Is</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Is"><span class="id" title="variable">Is</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#It"><span class="id" title="variable">It</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x0"><span class="id" title="variable">x0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#Is"><span class="id" title="variable">Is</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a>).<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.seq.html#MakeEqSeq"><span class="id" title="section">MakeEqSeq</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="FoldRight"><span class="id" title="section">FoldRight</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="FoldRight.T"><span class="id" title="variable">T</span></a> : <span class="id" title="keyword">Type</span>) (<a name="FoldRight.R"><span class="id" title="variable">R</span></a> : <span class="id" title="keyword">Type</span>) (<a name="FoldRight.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#T"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#R"><span class="id" title="variable">R</span></a>) (<a name="FoldRight.z0"><span class="id" title="variable">z0</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#R"><span class="id" title="variable">R</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="foldr"><span class="id" title="definition">foldr</span></a> <span class="id" title="var">s</span> := <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">s'</span> <span class="id" title="keyword">then</span> <a class="idref" href="mathcomp.ssreflect.seq.html#FoldRight.f"><span class="id" title="variable">f</span></a> <span class="id" title="var">x</span> (<a class="idref" href="mathcomp.ssreflect.seq.html#foldr"><span class="id" title="definition">foldr</span></a> <span class="id" title="var">s'</span>) <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.seq.html#FoldRight.z0"><span class="id" title="variable">z0</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.seq.html#FoldRight"><span class="id" title="section">FoldRight</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="FoldRightComp"><span class="id" title="section">FoldRightComp</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="FoldRightComp.T1"><span class="id" title="variable">T1</span></a> <a name="FoldRightComp.T2"><span class="id" title="variable">T2</span></a> : <span class="id" title="keyword">Type</span>) (<a name="FoldRightComp.h"><span class="id" title="variable">h</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#T1"><span class="id" title="variable">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#T2"><span class="id" title="variable">T2</span></a>).<br/> -<span class="id" title="keyword">Variables</span> (<a name="FoldRightComp.R"><span class="id" title="variable">R</span></a> : <span class="id" title="keyword">Type</span>) (<a name="FoldRightComp.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#FoldRightComp.T2"><span class="id" title="variable">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#R"><span class="id" title="variable">R</span></a>) (<a name="FoldRightComp.z0"><span class="id" title="variable">z0</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#R"><span class="id" title="variable">R</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="foldr_cat"><span class="id" title="lemma">foldr_cat</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#foldr"><span class="id" title="definition">foldr</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#FoldRightComp.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#FoldRightComp.z0"><span class="id" title="variable">z0</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#foldr"><span class="id" title="definition">foldr</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#FoldRightComp.f"><span class="id" title="variable">f</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#foldr"><span class="id" title="definition">foldr</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#FoldRightComp.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#FoldRightComp.z0"><span class="id" title="variable">z0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="foldr_map"><span class="id" title="lemma">foldr_map</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#foldr"><span class="id" title="definition">foldr</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#FoldRightComp.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#FoldRightComp.z0"><span class="id" title="variable">z0</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#FoldRightComp.h"><span class="id" title="variable">h</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#foldr"><span class="id" title="definition">foldr</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> <span class="id" title="var">z</span> ⇒ <a class="idref" href="mathcomp.ssreflect.seq.html#FoldRightComp.f"><span class="id" title="variable">f</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#FoldRightComp.h"><span class="id" title="variable">h</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#z"><span class="id" title="variable">z</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#FoldRightComp.z0"><span class="id" title="variable">z0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.seq.html#FoldRightComp"><span class="id" title="section">FoldRightComp</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - Quick characterization of the null sequence. -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="sumn"><span class="id" title="definition">sumn</span></a> := <a class="idref" href="mathcomp.ssreflect.seq.html#foldr"><span class="id" title="definition">foldr</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#addn"><span class="id" title="definition">addn</span></a> 0.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sumn_nseq"><span class="id" title="lemma">sumn_nseq</span></a> <span class="id" title="var">x</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#sumn"><span class="id" title="definition">sumn</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#nseq"><span class="id" title="definition">nseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sumn_cat"><span class="id" title="lemma">sumn_cat</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#sumn"><span class="id" title="definition">sumn</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sumn"><span class="id" title="definition">sumn</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sumn"><span class="id" title="definition">sumn</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sumn_count"><span class="id" title="lemma">sumn_count</span></a> <span class="id" title="var">T</span> (<span class="id" title="var">a</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#T"><span class="id" title="variable">T</span></a>) <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#sumn"><span class="id" title="definition">sumn</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#047e648644be092ebdd09919d3f414d3"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#047e648644be092ebdd09919d3f414d3"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#047e648644be092ebdd09919d3f414d3"><span class="id" title="notation">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#047e648644be092ebdd09919d3f414d3"><span class="id" title="notation">|</span></a> <span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.seq.html#047e648644be092ebdd09919d3f414d3"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#047e648644be092ebdd09919d3f414d3"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#count"><span class="id" title="definition">count</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sumn_rcons"><span class="id" title="lemma">sumn_rcons</span></a> <span class="id" title="var">s</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#sumn"><span class="id" title="definition">sumn</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sumn"><span class="id" title="definition">sumn</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="perm_sumn"><span class="id" title="lemma">perm_sumn</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sumn"><span class="id" title="definition">sumn</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sumn"><span class="id" title="definition">sumn</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sumn_rot"><span class="id" title="lemma">sumn_rot</span></a> <span class="id" title="var">s</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#sumn"><span class="id" title="definition">sumn</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sumn"><span class="id" title="definition">sumn</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sumn_rev"><span class="id" title="lemma">sumn_rev</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#sumn"><span class="id" title="definition">sumn</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sumn"><span class="id" title="definition">sumn</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="natnseq0P"><span class="id" title="lemma">natnseq0P</span></a> <span class="id" title="var">s</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nseq"><span class="id" title="definition">nseq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) 0) (<a class="idref" href="mathcomp.ssreflect.seq.html#sumn"><span class="id" title="definition">sumn</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 0).<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="FoldLeft"><span class="id" title="section">FoldLeft</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="FoldLeft.T"><span class="id" title="variable">T</span></a> <a name="FoldLeft.R"><span class="id" title="variable">R</span></a> : <span class="id" title="keyword">Type</span>) (<a name="FoldLeft.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#T"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#R"><span class="id" title="variable">R</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="foldl"><span class="id" title="definition">foldl</span></a> <span class="id" title="var">z</span> <span class="id" title="var">s</span> := <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">s'</span> <span class="id" title="keyword">then</span> <a class="idref" href="mathcomp.ssreflect.seq.html#foldl"><span class="id" title="definition">foldl</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#FoldLeft.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#z"><span class="id" title="variable">z</span></a> <span class="id" title="var">x</span>) <span class="id" title="var">s'</span> <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.seq.html#z"><span class="id" title="variable">z</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="foldl_rev"><span class="id" title="lemma">foldl_rev</span></a> <span class="id" title="var">z</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#foldl"><span class="id" title="definition">foldl</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#z"><span class="id" title="variable">z</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#foldr"><span class="id" title="definition">foldr</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> <span class="id" title="var">z</span> ⇒ <a class="idref" href="mathcomp.ssreflect.seq.html#FoldLeft.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="foldl_cat"><span class="id" title="lemma">foldl_cat</span></a> <span class="id" title="var">z</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#foldl"><span class="id" title="definition">foldl</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#z"><span class="id" title="variable">z</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#foldl"><span class="id" title="definition">foldl</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#foldl"><span class="id" title="definition">foldl</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.seq.html#FoldLeft"><span class="id" title="section">FoldLeft</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Scan"><span class="id" title="section">Scan</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="Scan.T1"><span class="id" title="variable">T1</span></a> : <span class="id" title="keyword">Type</span>) (<a name="Scan.x1"><span class="id" title="variable">x1</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#T1"><span class="id" title="variable">T1</span></a>) (<a name="Scan.T2"><span class="id" title="variable">T2</span></a> : <span class="id" title="keyword">Type</span>) (<a name="Scan.x2"><span class="id" title="variable">x2</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#T2"><span class="id" title="variable">T2</span></a>).<br/> -<span class="id" title="keyword">Variables</span> (<a name="Scan.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#Scan.T1"><span class="id" title="variable">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Scan.T1"><span class="id" title="variable">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Scan.T2"><span class="id" title="variable">T2</span></a>) (<a name="Scan.g"><span class="id" title="variable">g</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#Scan.T1"><span class="id" title="variable">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Scan.T2"><span class="id" title="variable">T2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Scan.T1"><span class="id" title="variable">T1</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="pairmap"><span class="id" title="definition">pairmap</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> := <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">s'</span> <span class="id" title="keyword">then</span> <a class="idref" href="mathcomp.ssreflect.seq.html#Scan.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#pairmap"><span class="id" title="definition">pairmap</span></a> <span class="id" title="var">y</span> <span class="id" title="var">s'</span> <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="size_pairmap"><span class="id" title="lemma">size_pairmap</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#pairmap"><span class="id" title="definition">pairmap</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pairmap_cat"><span class="id" title="lemma">pairmap_cat</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#pairmap"><span class="id" title="definition">pairmap</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#pairmap"><span class="id" title="definition">pairmap</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#pairmap"><span class="id" title="definition">pairmap</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#last"><span class="id" title="definition">last</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nth_pairmap"><span class="id" title="lemma">nth_pairmap</span></a> <span class="id" title="var">s</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Scan.x2"><span class="id" title="variable">x2</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#pairmap"><span class="id" title="definition">pairmap</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Scan.f"><span class="id" title="variable">f</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Scan.x1"><span class="id" title="variable">x1</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Scan.x1"><span class="id" title="variable">x1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="scanl"><span class="id" title="definition">scanl</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> :=<br/> - <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">s'</span> <span class="id" title="keyword">then</span> <span class="id" title="keyword">let</span> <span class="id" title="var">x'</span> := <a class="idref" href="mathcomp.ssreflect.seq.html#Scan.g"><span class="id" title="variable">g</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="var">y</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.seq.html#x'"><span class="id" title="variable">x'</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#scanl"><span class="id" title="definition">scanl</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x'"><span class="id" title="variable">x'</span></a> <span class="id" title="var">s'</span> <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="size_scanl"><span class="id" title="lemma">size_scanl</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#scanl"><span class="id" title="definition">scanl</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="scanl_cat"><span class="id" title="lemma">scanl_cat</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#scanl"><span class="id" title="definition">scanl</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#scanl"><span class="id" title="definition">scanl</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#scanl"><span class="id" title="definition">scanl</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#foldl"><span class="id" title="definition">foldl</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Scan.g"><span class="id" title="variable">g</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nth_scanl"><span class="id" title="lemma">nth_scanl</span></a> <span class="id" title="var">s</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Scan.x1"><span class="id" title="variable">x1</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#scanl"><span class="id" title="definition">scanl</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#foldl"><span class="id" title="definition">foldl</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Scan.g"><span class="id" title="variable">g</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#take"><span class="id" title="definition">take</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="scanlK"><span class="id" title="lemma">scanlK</span></a> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#Scan.g"><span class="id" title="variable">g</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#Scan.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#scanl"><span class="id" title="definition">scanl</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#pairmap"><span class="id" title="definition">pairmap</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pairmapK"><span class="id" title="lemma">pairmapK</span></a> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#Scan.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#Scan.g"><span class="id" title="variable">g</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#pairmap"><span class="id" title="definition">pairmap</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#scanl"><span class="id" title="definition">scanl</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>).<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.seq.html#Scan"><span class="id" title="section">Scan</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Zip"><span class="id" title="section">Zip</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> <a name="Zip.S"><span class="id" title="variable">S</span></a> <a name="Zip.T"><span class="id" title="variable">T</span></a> : <span class="id" title="keyword">Type</span>.<br/> - -<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="zip"><span class="id" title="definition">zip</span></a> (<span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Zip.S"><span class="id" title="variable">S</span></a>) (<span class="id" title="var">t</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Zip.T"><span class="id" title="variable">T</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">t</span>} :=<br/> - <span class="id" title="keyword">match</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>, <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span><br/> - | <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">s'</span>, <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">t'</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#zip"><span class="id" title="definition">zip</span></a> <span class="id" title="var">s'</span> <span class="id" title="var">t'</span><br/> - | <span class="id" title="var">_</span>, <span class="id" title="var">_</span> ⇒ <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a><br/> - <span class="id" title="keyword">end</span>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="unzip1"><span class="id" title="definition">unzip1</span></a> := <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> (@<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Zip.S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Zip.T"><span class="id" title="variable">T</span></a>).<br/> -<span class="id" title="keyword">Definition</span> <a name="unzip2"><span class="id" title="definition">unzip2</span></a> := <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> (@<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Zip.S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Zip.T"><span class="id" title="variable">T</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="all2"><span class="id" title="definition">all2</span></a> (<span class="id" title="var">r</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#Zip.S"><span class="id" title="variable">S</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Zip.T"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) <span class="id" title="var">s</span> <span class="id" title="var">t</span> :=<br/> - <span class="id" title="keyword">match</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>, <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span><br/> - | <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a>, <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/> - | <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">s</span>, <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">t</span> ⇒ <a class="idref" href="mathcomp.ssreflect.seq.html#r"><span class="id" title="variable">r</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#all2"><span class="id" title="definition">all2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a><br/> - | <span class="id" title="var">_</span>, <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/> - <span class="id" title="keyword">end</span>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="zip_unzip"><span class="id" title="lemma">zip_unzip</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#zip"><span class="id" title="definition">zip</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#unzip1"><span class="id" title="definition">unzip1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#unzip2"><span class="id" title="definition">unzip2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="unzip1_zip"><span class="id" title="lemma">unzip1_zip</span></a> <span class="id" title="var">s</span> <span class="id" title="var">t</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#unzip1"><span class="id" title="definition">unzip1</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#zip"><span class="id" title="definition">zip</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="unzip2_zip"><span class="id" title="lemma">unzip2_zip</span></a> <span class="id" title="var">s</span> <span class="id" title="var">t</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#unzip2"><span class="id" title="definition">unzip2</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#zip"><span class="id" title="definition">zip</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="size1_zip"><span class="id" title="lemma">size1_zip</span></a> <span class="id" title="var">s</span> <span class="id" title="var">t</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#zip"><span class="id" title="definition">zip</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="size2_zip"><span class="id" title="lemma">size2_zip</span></a> <span class="id" title="var">s</span> <span class="id" title="var">t</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#zip"><span class="id" title="definition">zip</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="size_zip"><span class="id" title="lemma">size_zip</span></a> <span class="id" title="var">s</span> <span class="id" title="var">t</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#zip"><span class="id" title="definition">zip</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#minn"><span class="id" title="definition">minn</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="zip_cat"><span class="id" title="lemma">zip_cat</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> <span class="id" title="var">t1</span> <span class="id" title="var">t2</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t1"><span class="id" title="variable">t1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#zip"><span class="id" title="definition">zip</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#t1"><span class="id" title="variable">t1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t2"><span class="id" title="variable">t2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#zip"><span class="id" title="definition">zip</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t1"><span class="id" title="variable">t1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#zip"><span class="id" title="definition">zip</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t2"><span class="id" title="variable">t2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nth_zip"><span class="id" title="lemma">nth_zip</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">s</span> <span class="id" title="var">t</span> <span class="id" title="var">i</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#zip"><span class="id" title="definition">zip</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nth_zip_cond"><span class="id" title="lemma">nth_zip_cond</span></a> <span class="id" title="var">p</span> <span class="id" title="var">s</span> <span class="id" title="var">t</span> <span class="id" title="var">i</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#p"><span class="id" title="variable">p</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#zip"><span class="id" title="definition">zip</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#zip"><span class="id" title="definition">zip</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">.1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">.2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="zip_rcons"><span class="id" title="lemma">zip_rcons</span></a> <span class="id" title="var">s</span> <span class="id" title="var">t</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#zip"><span class="id" title="definition">zip</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#zip"><span class="id" title="definition">zip</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rev_zip"><span class="id" title="lemma">rev_zip</span></a> <span class="id" title="var">s</span> <span class="id" title="var">t</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#zip"><span class="id" title="definition">zip</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#zip"><span class="id" title="definition">zip</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="all2E"><span class="id" title="lemma">all2E</span></a> <span class="id" title="var">r</span> <span class="id" title="var">s</span> <span class="id" title="var">t</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#all2"><span class="id" title="definition">all2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#27dabc72ea2c2c768f2db80a79f42524"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#27dabc72ea2c2c768f2db80a79f42524"><span class="id" title="notation">pred</span></a> <span class="id" title="var">xy</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#27dabc72ea2c2c768f2db80a79f42524"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#xy"><span class="id" title="variable">xy</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">.1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#xy"><span class="id" title="variable">xy</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">.2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#27dabc72ea2c2c768f2db80a79f42524"><span class="id" title="notation">]</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#zip"><span class="id" title="definition">zip</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a>).<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.seq.html#Zip"><span class="id" title="section">Zip</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Flatten"><span class="id" title="section">Flatten</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="Flatten.T"><span class="id" title="variable">T</span></a> : <span class="id" title="keyword">Type</span>.<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Flatten.T"><span class="id" title="variable">T</span></a>) (<span class="id" title="var">ss</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Flatten.T"><span class="id" title="variable">T</span></a>)).<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="flatten"><span class="id" title="definition">flatten</span></a> := <a class="idref" href="mathcomp.ssreflect.seq.html#foldr"><span class="id" title="definition">foldr</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cat"><span class="id" title="definition">cat</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#Nil"><span class="id" title="abbreviation">Nil</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Flatten.T"><span class="id" title="variable">T</span></a>).<br/> -<span class="id" title="keyword">Definition</span> <a name="shape"><span class="id" title="definition">shape</span></a> := <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> (@<a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Flatten.T"><span class="id" title="variable">T</span></a>).<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="reshape"><span class="id" title="definition">reshape</span></a> <span class="id" title="var">sh</span> <span class="id" title="var">s</span> :=<br/> - <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">n</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">sh'</span> <span class="id" title="keyword">then</span> <a class="idref" href="mathcomp.ssreflect.seq.html#take"><span class="id" title="definition">take</span></a> <span class="id" title="var">n</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#reshape"><span class="id" title="definition">reshape</span></a> <span class="id" title="var">sh'</span> (<a class="idref" href="mathcomp.ssreflect.seq.html#drop"><span class="id" title="definition">drop</span></a> <span class="id" title="var">n</span> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="flatten_index"><span class="id" title="definition">flatten_index</span></a> <span class="id" title="var">sh</span> <span class="id" title="var">r</span> <span class="id" title="var">c</span> := <a class="idref" href="mathcomp.ssreflect.seq.html#sumn"><span class="id" title="definition">sumn</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#take"><span class="id" title="definition">take</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a>) <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#c"><span class="id" title="variable">c</span></a>.<br/> -<span class="id" title="keyword">Definition</span> <a name="reshape_index"><span class="id" title="definition">reshape_index</span></a> <span class="id" title="var">sh</span> <span class="id" title="var">i</span> := <a class="idref" href="mathcomp.ssreflect.seq.html#find"><span class="id" title="definition">find</span></a> (<a class="idref" href="mathcomp.ssreflect.eqtype.html#pred1"><span class="id" title="definition">pred1</span></a> 0) (<a class="idref" href="mathcomp.ssreflect.seq.html#scanl"><span class="id" title="definition">scanl</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#subn"><span class="id" title="definition">subn</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a>).<br/> -<span class="id" title="keyword">Definition</span> <a name="reshape_offset"><span class="id" title="definition">reshape_offset</span></a> <span class="id" title="var">sh</span> <span class="id" title="var">i</span> := <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#7825ccc99f23b0d30c9d40c317ba7af0"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sumn"><span class="id" title="definition">sumn</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#take"><span class="id" title="definition">take</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#reshape_index"><span class="id" title="definition">reshape_index</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="size_flatten"><span class="id" title="lemma">size_flatten</span></a> <span class="id" title="var">ss</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#flatten"><span class="id" title="definition">flatten</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ss"><span class="id" title="variable">ss</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sumn"><span class="id" title="definition">sumn</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#shape"><span class="id" title="definition">shape</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ss"><span class="id" title="variable">ss</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="flatten_cat"><span class="id" title="lemma">flatten_cat</span></a> <span class="id" title="var">ss1</span> <span class="id" title="var">ss2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#flatten"><span class="id" title="definition">flatten</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#ss1"><span class="id" title="variable">ss1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ss2"><span class="id" title="variable">ss2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#flatten"><span class="id" title="definition">flatten</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ss1"><span class="id" title="variable">ss1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#flatten"><span class="id" title="definition">flatten</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ss2"><span class="id" title="variable">ss2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="size_reshape"><span class="id" title="lemma">size_reshape</span></a> <span class="id" title="var">sh</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#reshape"><span class="id" title="definition">reshape</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nth_reshape"><span class="id" title="lemma">nth_reshape</span></a> (<span class="id" title="var">sh</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <span class="id" title="var">l</span> <span class="id" title="var">n</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#reshape"><span class="id" title="definition">reshape</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#l"><span class="id" title="variable">l</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#take"><span class="id" title="definition">take</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> 0 <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#drop"><span class="id" title="definition">drop</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#sumn"><span class="id" title="definition">sumn</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#take"><span class="id" title="definition">take</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a>)) <a class="idref" href="mathcomp.ssreflect.seq.html#l"><span class="id" title="variable">l</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="flattenK"><span class="id" title="lemma">flattenK</span></a> <span class="id" title="var">ss</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#reshape"><span class="id" title="definition">reshape</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#shape"><span class="id" title="definition">shape</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ss"><span class="id" title="variable">ss</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#flatten"><span class="id" title="definition">flatten</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ss"><span class="id" title="variable">ss</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ss"><span class="id" title="variable">ss</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="reshapeKr"><span class="id" title="lemma">reshapeKr</span></a> <span class="id" title="var">sh</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sumn"><span class="id" title="definition">sumn</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#flatten"><span class="id" title="definition">flatten</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#reshape"><span class="id" title="definition">reshape</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="reshapeKl"><span class="id" title="lemma">reshapeKl</span></a> <span class="id" title="var">sh</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#73030c22bc0b1fa771c65aa5414c65f9"><span class="id" title="notation">≥</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sumn"><span class="id" title="definition">sumn</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#shape"><span class="id" title="definition">shape</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#reshape"><span class="id" title="definition">reshape</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="flatten_rcons"><span class="id" title="lemma">flatten_rcons</span></a> <span class="id" title="var">ss</span> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#flatten"><span class="id" title="definition">flatten</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ss"><span class="id" title="variable">ss</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#flatten"><span class="id" title="definition">flatten</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ss"><span class="id" title="variable">ss</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="flatten_seq1"><span class="id" title="lemma">flatten_seq1</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#flatten"><span class="id" title="definition">flatten</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#506674b18256ef8f50efed43fa1dfd7d"><span class="id" title="notation">[::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#506674b18256ef8f50efed43fa1dfd7d"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="count_flatten"><span class="id" title="lemma">count_flatten</span></a> <span class="id" title="var">ss</span> <span class="id" title="var">P</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#count"><span class="id" title="definition">count</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#P"><span class="id" title="variable">P</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#flatten"><span class="id" title="definition">flatten</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ss"><span class="id" title="variable">ss</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sumn"><span class="id" title="definition">sumn</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#count"><span class="id" title="definition">count</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ss"><span class="id" title="variable">ss</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="filter_flatten"><span class="id" title="lemma">filter_flatten</span></a> <span class="id" title="var">ss</span> (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Flatten.T"><span class="id" title="variable">T</span></a>) :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#P"><span class="id" title="variable">P</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#flatten"><span class="id" title="definition">flatten</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ss"><span class="id" title="variable">ss</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#flatten"><span class="id" title="definition">flatten</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">|</span></a> <span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ss"><span class="id" title="variable">ss</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rev_flatten"><span class="id" title="lemma">rev_flatten</span></a> <span class="id" title="var">ss</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#flatten"><span class="id" title="definition">flatten</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ss"><span class="id" title="variable">ss</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#flatten"><span class="id" title="definition">flatten</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ss"><span class="id" title="variable">ss</span></a>)).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nth_shape"><span class="id" title="lemma">nth_shape</span></a> <span class="id" title="var">ss</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> 0 (<a class="idref" href="mathcomp.ssreflect.seq.html#shape"><span class="id" title="definition">shape</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ss"><span class="id" title="variable">ss</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ss"><span class="id" title="variable">ss</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="shape_rev"><span class="id" title="lemma">shape_rev</span></a> <span class="id" title="var">ss</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#shape"><span class="id" title="definition">shape</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ss"><span class="id" title="variable">ss</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#shape"><span class="id" title="definition">shape</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ss"><span class="id" title="variable">ss</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_from_flatten_shape"><span class="id" title="lemma">eq_from_flatten_shape</span></a> <span class="id" title="var">ss1</span> <span class="id" title="var">ss2</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#flatten"><span class="id" title="definition">flatten</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ss1"><span class="id" title="variable">ss1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#flatten"><span class="id" title="definition">flatten</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ss2"><span class="id" title="variable">ss2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#shape"><span class="id" title="definition">shape</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ss1"><span class="id" title="variable">ss1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#shape"><span class="id" title="definition">shape</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ss2"><span class="id" title="variable">ss2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ss1"><span class="id" title="variable">ss1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ss2"><span class="id" title="variable">ss2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="rev_reshape"><span class="id" title="lemma">rev_reshape</span></a> <span class="id" title="var">sh</span> <span class="id" title="var">s</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sumn"><span class="id" title="definition">sumn</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#reshape"><span class="id" title="definition">reshape</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#reshape"><span class="id" title="definition">reshape</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#rev"><span class="id" title="definition">rev</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>)).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="reshape_rcons"><span class="id" title="lemma">reshape_rcons</span></a> <span class="id" title="var">s</span> <span class="id" title="var">sh</span> <span class="id" title="var">n</span> (<span class="id" title="var">m</span> := <a class="idref" href="mathcomp.ssreflect.seq.html#sumn"><span class="id" title="definition">sumn</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a>) :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#reshape"><span class="id" title="definition">reshape</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#reshape"><span class="id" title="definition">reshape</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#take"><span class="id" title="definition">take</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>)) (<a class="idref" href="mathcomp.ssreflect.seq.html#drop"><span class="id" title="definition">drop</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="flatten_indexP"><span class="id" title="lemma">flatten_indexP</span></a> <span class="id" title="var">sh</span> <span class="id" title="var">r</span> <span class="id" title="var">c</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> 0 <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#flatten_index"><span class="id" title="definition">flatten_index</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sumn"><span class="id" title="definition">sumn</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="reshape_indexP"><span class="id" title="lemma">reshape_indexP</span></a> <span class="id" title="var">sh</span> <span class="id" title="var">i</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sumn"><span class="id" title="definition">sumn</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#reshape_index"><span class="id" title="definition">reshape_index</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="reshape_offsetP"><span class="id" title="lemma">reshape_offsetP</span></a> <span class="id" title="var">sh</span> <span class="id" title="var">i</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sumn"><span class="id" title="definition">sumn</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#reshape_offset"><span class="id" title="definition">reshape_offset</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> 0 <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#reshape_index"><span class="id" title="definition">reshape_index</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="reshape_indexK"><span class="id" title="lemma">reshape_indexK</span></a> <span class="id" title="var">sh</span> <span class="id" title="var">i</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#flatten_index"><span class="id" title="definition">flatten_index</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#reshape_index"><span class="id" title="definition">reshape_index</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#reshape_offset"><span class="id" title="definition">reshape_offset</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="flatten_indexKl"><span class="id" title="lemma">flatten_indexKl</span></a> <span class="id" title="var">sh</span> <span class="id" title="var">r</span> <span class="id" title="var">c</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> 0 <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#reshape_index"><span class="id" title="definition">reshape_index</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#flatten_index"><span class="id" title="definition">flatten_index</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#c"><span class="id" title="variable">c</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#r"><span class="id" title="variable">r</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="flatten_indexKr"><span class="id" title="lemma">flatten_indexKr</span></a> <span class="id" title="var">sh</span> <span class="id" title="var">r</span> <span class="id" title="var">c</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> 0 <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#reshape_offset"><span class="id" title="definition">reshape_offset</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#flatten_index"><span class="id" title="definition">flatten_index</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#c"><span class="id" title="variable">c</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#c"><span class="id" title="variable">c</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nth_flatten"><span class="id" title="lemma">nth_flatten</span></a> <span class="id" title="var">x0</span> <span class="id" title="var">ss</span> <span class="id" title="var">i</span> (<span class="id" title="var">r</span> := <a class="idref" href="mathcomp.ssreflect.seq.html#reshape_index"><span class="id" title="definition">reshape_index</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#shape"><span class="id" title="definition">shape</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ss"><span class="id" title="variable">ss</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a>) :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x0"><span class="id" title="variable">x0</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#flatten"><span class="id" title="definition">flatten</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ss"><span class="id" title="variable">ss</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x0"><span class="id" title="variable">x0</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ss"><span class="id" title="variable">ss</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#r"><span class="id" title="variable">r</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#reshape_offset"><span class="id" title="definition">reshape_offset</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#shape"><span class="id" title="definition">shape</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ss"><span class="id" title="variable">ss</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="reshape_leq"><span class="id" title="lemma">reshape_leq</span></a> <span class="id" title="var">sh</span> <span class="id" title="var">i1</span> <span class="id" title="var">i2</span><br/> - (<span class="id" title="var">r1</span> := <a class="idref" href="mathcomp.ssreflect.seq.html#reshape_index"><span class="id" title="definition">reshape_index</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i1"><span class="id" title="variable">i1</span></a>) (<span class="id" title="var">c1</span> := <a class="idref" href="mathcomp.ssreflect.seq.html#reshape_offset"><span class="id" title="definition">reshape_offset</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i1"><span class="id" title="variable">i1</span></a>)<br/> - (<span class="id" title="var">r2</span> := <a class="idref" href="mathcomp.ssreflect.seq.html#reshape_index"><span class="id" title="definition">reshape_index</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i2"><span class="id" title="variable">i2</span></a>) (<span class="id" title="var">c2</span> := <a class="idref" href="mathcomp.ssreflect.seq.html#reshape_offset"><span class="id" title="definition">reshape_offset</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i2"><span class="id" title="variable">i2</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#i1"><span class="id" title="variable">i1</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i2"><span class="id" title="variable">i2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#r1"><span class="id" title="variable">r1</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#r2"><span class="id" title="variable">r2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">||</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#r1"><span class="id" title="variable">r1</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#r2"><span class="id" title="variable">r2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#c1"><span class="id" title="variable">c1</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#c2"><span class="id" title="variable">c2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#081ff67d3116402bb680e8692aa39185"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.seq.html#Flatten"><span class="id" title="section">Flatten</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_flatten"><span class="id" title="lemma">map_flatten</span></a> <span class="id" title="var">S</span> <span class="id" title="var">T</span> (<span class="id" title="var">f</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#T"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#S"><span class="id" title="variable">S</span></a>) <span class="id" title="var">ss</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#flatten"><span class="id" title="definition">flatten</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ss"><span class="id" title="variable">ss</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#flatten"><span class="id" title="definition">flatten</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#ss"><span class="id" title="variable">ss</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sumn_flatten"><span class="id" title="lemma">sumn_flatten</span></a> (<span class="id" title="var">ss</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)) :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#sumn"><span class="id" title="definition">sumn</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#flatten"><span class="id" title="definition">flatten</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ss"><span class="id" title="variable">ss</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sumn"><span class="id" title="definition">sumn</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sumn"><span class="id" title="definition">sumn</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ss"><span class="id" title="variable">ss</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_reshape"><span class="id" title="lemma">map_reshape</span></a> <span class="id" title="var">T</span> <span class="id" title="var">S</span> (<span class="id" title="var">f</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#T"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#S"><span class="id" title="variable">S</span></a>) <span class="id" title="var">sh</span> <span class="id" title="var">s</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#reshape"><span class="id" title="definition">reshape</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#reshape"><span class="id" title="definition">reshape</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sh"><span class="id" title="variable">sh</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="EqFlatten"><span class="id" title="section">EqFlatten</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> <a name="EqFlatten.S"><span class="id" title="variable">S</span></a> <a name="EqFlatten.T"><span class="id" title="variable">T</span></a> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="flattenP"><span class="id" title="lemma">flattenP</span></a> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqFlatten.T"><span class="id" title="variable">T</span></a>)) <span class="id" title="var">x</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">s</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#flatten"><span class="id" title="definition">flatten</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#A"><span class="id" title="variable">A</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="flatten_mapP"><span class="id" title="lemma">flatten_mapP</span></a> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#EqFlatten.S"><span class="id" title="variable">S</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqFlatten.T"><span class="id" title="variable">T</span></a>) <span class="id" title="var">s</span> <span class="id" title="var">y</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#flatten"><span class="id" title="definition">flatten</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>)).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="perm_flatten"><span class="id" title="lemma">perm_flatten</span></a> (<span class="id" title="var">ss1</span> <span class="id" title="var">ss2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqFlatten.T"><span class="id" title="variable">T</span></a>)) :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ss1"><span class="id" title="variable">ss1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ss2"><span class="id" title="variable">ss2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#flatten"><span class="id" title="definition">flatten</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ss1"><span class="id" title="variable">ss1</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#flatten"><span class="id" title="definition">flatten</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ss2"><span class="id" title="variable">ss2</span></a>).<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.seq.html#EqFlatten"><span class="id" title="section">EqFlatten</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">"</span></a>[ 'seq' E | x <- s , y <- t ]" :=<br/> - (<a class="idref" href="mathcomp.ssreflect.seq.html#flatten"><span class="id" title="definition">flatten</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">seq</span></a> <span class="id" title="var">E</span> <a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">|</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation"><-</span></a> <span class="id" title="var">t</span><a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation"><-</span></a> <span class="id" title="var">s</span><a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">]</span></a>)<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">E</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 99, <span class="id" title="var">x</span> <span class="id" title="var">ident</span>, <span class="id" title="var">y</span> <span class="id" title="var">ident</span>,<br/> - <span class="id" title="var">format</span> "[ '[hv' 'seq' E '/ ' | x <- s , '/ ' y <- t ] ']'")<br/> - : <span class="id" title="var">seq_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="7324da4c7b03b301c0281718dc535c41"><span class="id" title="notation">"</span></a>[ 'seq' E | x : S <- s , y : T <- t ]" :=<br/> - (<a class="idref" href="mathcomp.ssreflect.seq.html#flatten"><span class="id" title="definition">flatten</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#035c9924d34358d2b2816d54c1fb0043"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#035c9924d34358d2b2816d54c1fb0043"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#035c9924d34358d2b2816d54c1fb0043"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#035c9924d34358d2b2816d54c1fb0043"><span class="id" title="notation">seq</span></a> <span class="id" title="var">E</span> <a class="idref" href="mathcomp.ssreflect.seq.html#035c9924d34358d2b2816d54c1fb0043"><span class="id" title="notation">|</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#035c9924d34358d2b2816d54c1fb0043"><span class="id" title="notation">:</span></a> <span class="id" title="var">T</span> <a class="idref" href="mathcomp.ssreflect.seq.html#035c9924d34358d2b2816d54c1fb0043"><span class="id" title="notation"><-</span></a> <span class="id" title="var">t</span><a class="idref" href="mathcomp.ssreflect.seq.html#035c9924d34358d2b2816d54c1fb0043"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#035c9924d34358d2b2816d54c1fb0043"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#035c9924d34358d2b2816d54c1fb0043"><span class="id" title="notation">:</span></a> <span class="id" title="var">S</span> <a class="idref" href="mathcomp.ssreflect.seq.html#035c9924d34358d2b2816d54c1fb0043"><span class="id" title="notation"><-</span></a> <span class="id" title="var">s</span><a class="idref" href="mathcomp.ssreflect.seq.html#035c9924d34358d2b2816d54c1fb0043"><span class="id" title="notation">]</span></a>)<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">E</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 99, <span class="id" title="var">x</span> <span class="id" title="var">ident</span>, <span class="id" title="var">y</span> <span class="id" title="var">ident</span>, <span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">seq_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="1fc9c30bdeb8ed720514ea7776d09231"><span class="id" title="notation">"</span></a>[ 'seq' E : R | x : S <- s , y : T <- t ]" :=<br/> - (<a class="idref" href="mathcomp.ssreflect.seq.html#flatten"><span class="id" title="definition">flatten</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#035c9924d34358d2b2816d54c1fb0043"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#035c9924d34358d2b2816d54c1fb0043"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#732c880e3345bf5133e68085a32d1a68"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#732c880e3345bf5133e68085a32d1a68"><span class="id" title="notation">seq</span></a> <span class="id" title="var">E</span> <a class="idref" href="mathcomp.ssreflect.seq.html#732c880e3345bf5133e68085a32d1a68"><span class="id" title="notation">:</span></a> <span class="id" title="var">R</span> <a class="idref" href="mathcomp.ssreflect.seq.html#732c880e3345bf5133e68085a32d1a68"><span class="id" title="notation">|</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#732c880e3345bf5133e68085a32d1a68"><span class="id" title="notation">:</span></a> <span class="id" title="var">T</span> <a class="idref" href="mathcomp.ssreflect.seq.html#732c880e3345bf5133e68085a32d1a68"><span class="id" title="notation"><-</span></a> <span class="id" title="var">t</span><a class="idref" href="mathcomp.ssreflect.seq.html#732c880e3345bf5133e68085a32d1a68"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#035c9924d34358d2b2816d54c1fb0043"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#035c9924d34358d2b2816d54c1fb0043"><span class="id" title="notation">:</span></a> <span class="id" title="var">S</span> <a class="idref" href="mathcomp.ssreflect.seq.html#035c9924d34358d2b2816d54c1fb0043"><span class="id" title="notation"><-</span></a> <span class="id" title="var">s</span><a class="idref" href="mathcomp.ssreflect.seq.html#035c9924d34358d2b2816d54c1fb0043"><span class="id" title="notation">]</span></a>)<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">E</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 99, <span class="id" title="var">x</span> <span class="id" title="var">ident</span>, <span class="id" title="var">y</span> <span class="id" title="var">ident</span>, <span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">seq_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="00929bf63896446405b6c979ee089e42"><span class="id" title="notation">"</span></a>[ 'seq' E : R | x <- s , y <- t ]" :=<br/> - (<a class="idref" href="mathcomp.ssreflect.seq.html#flatten"><span class="id" title="definition">flatten</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#047e648644be092ebdd09919d3f414d3"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#047e648644be092ebdd09919d3f414d3"><span class="id" title="notation">seq</span></a> <span class="id" title="var">E</span> <a class="idref" href="mathcomp.ssreflect.seq.html#047e648644be092ebdd09919d3f414d3"><span class="id" title="notation">:</span></a> <span class="id" title="var">R</span> <a class="idref" href="mathcomp.ssreflect.seq.html#047e648644be092ebdd09919d3f414d3"><span class="id" title="notation">|</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#047e648644be092ebdd09919d3f414d3"><span class="id" title="notation"><-</span></a> <span class="id" title="var">t</span><a class="idref" href="mathcomp.ssreflect.seq.html#047e648644be092ebdd09919d3f414d3"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation"><-</span></a> <span class="id" title="var">s</span><a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">]</span></a>)<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">E</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 99, <span class="id" title="var">x</span> <span class="id" title="var">ident</span>, <span class="id" title="var">y</span> <span class="id" title="var">ident</span>, <span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">seq_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="AllPairsDep"><span class="id" title="section">AllPairsDep</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="AllPairsDep.S"><span class="id" title="variable">S</span></a> <a name="AllPairsDep.S'"><span class="id" title="variable">S'</span></a> : <span class="id" title="keyword">Type</span>) (<a name="AllPairsDep.T"><span class="id" title="variable">T</span></a> <a name="AllPairsDep.T'"><span class="id" title="variable">T'</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Type</span>) (<a name="AllPairsDep.R"><span class="id" title="variable">R</span></a> : <span class="id" title="keyword">Type</span>).<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">f</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.ssreflect.seq.html#AllPairsDep.T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#AllPairsDep.R"><span class="id" title="variable">R</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="allpairs_dep"><span class="id" title="definition">allpairs_dep</span></a> <span class="id" title="var">f</span> <span class="id" title="var">s</span> <span class="id" title="var">t</span> := <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">,</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="size_allpairs_dep"><span class="id" title="lemma">size_allpairs_dep</span></a> <span class="id" title="var">f</span> <span class="id" title="var">s</span> <span class="id" title="var">t</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">,</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sumn"><span class="id" title="definition">sumn</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_allpairs"><span class="id" title="lemma">eq_allpairs</span></a> (<span class="id" title="var">f1</span> <span class="id" title="var">f2</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.ssreflect.seq.html#AllPairsDep.T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#AllPairsDep.R"><span class="id" title="variable">R</span></a>) <span class="id" title="var">s</span> <span class="id" title="var">t</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.ssreflect.seq.html#f1"><span class="id" title="variable">f1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f2"><span class="id" title="variable">f2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f1"><span class="id" title="variable">f1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">,</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f2"><span class="id" title="variable">f2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">,</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="allpairs_cat"><span class="id" title="lemma">allpairs_cat</span></a> <span class="id" title="var">f</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> <span class="id" title="var">t</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">,</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">,</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">,</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="allpairs_mapl"><span class="id" title="lemma">allpairs_mapl</span></a> <span class="id" title="var">f</span> (<span class="id" title="var">g</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#AllPairsDep.S'"><span class="id" title="variable">S'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#AllPairsDep.S"><span class="id" title="variable">S</span></a>) <span class="id" title="var">s</span> <span class="id" title="var">t</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#g"><span class="id" title="variable">g</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">,</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#g"><span class="id" title="variable">g</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">,</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#g"><span class="id" title="variable">g</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>)<a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="allpairs_mapr"><span class="id" title="lemma">allpairs_mapr</span></a> <span class="id" title="var">f</span> (<span class="id" title="var">g</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.ssreflect.seq.html#AllPairsDep.T'"><span class="id" title="variable">T'</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#AllPairsDep.T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) <span class="id" title="var">s</span> <span class="id" title="var">t</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">,</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#g"><span class="id" title="variable">g</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>)<a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#g"><span class="id" title="variable">g</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">,</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.seq.html#AllPairsDep"><span class="id" title="section">AllPairsDep</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="map_allpairs"><span class="id" title="lemma">map_allpairs</span></a> <span class="id" title="var">S</span> <span class="id" title="var">T</span> <span class="id" title="var">R</span> <span class="id" title="var">R'</span> (<span class="id" title="var">g</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#R'"><span class="id" title="variable">R'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#R"><span class="id" title="variable">R</span></a>) <span class="id" title="var">f</span> <span class="id" title="var">s</span> <span class="id" title="var">t</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#g"><span class="id" title="variable">g</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#7324da4c7b03b301c0281718dc535c41"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#7324da4c7b03b301c0281718dc535c41"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#7324da4c7b03b301c0281718dc535c41"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#7324da4c7b03b301c0281718dc535c41"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#7324da4c7b03b301c0281718dc535c41"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#7324da4c7b03b301c0281718dc535c41"><span class="id" title="notation">,</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#7324da4c7b03b301c0281718dc535c41"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#7324da4c7b03b301c0281718dc535c41"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#7324da4c7b03b301c0281718dc535c41"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#g"><span class="id" title="variable">g</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">,</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="AllPairsNonDep"><span class="id" title="section">AllPairsNonDep</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="AllPairsNonDep.S"><span class="id" title="variable">S</span></a> <a name="AllPairsNonDep.T"><span class="id" title="variable">T</span></a> <a name="AllPairsNonDep.R"><span class="id" title="variable">R</span></a> : <span class="id" title="keyword">Type</span>) (<a name="AllPairsNonDep.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#T"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#R"><span class="id" title="variable">R</span></a>).<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#AllPairsNonDep.S"><span class="id" title="variable">S</span></a>) (<span class="id" title="var">t</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#AllPairsNonDep.T"><span class="id" title="variable">T</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="allpairs"><span class="id" title="definition">allpairs</span></a> <span class="id" title="var">s</span> <span class="id" title="var">t</span> := <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#AllPairsNonDep.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">,</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="size_allpairs"><span class="id" title="lemma">size_allpairs</span></a> <span class="id" title="var">s</span> <span class="id" title="var">t</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#AllPairsNonDep.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">,</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.seq.html#AllPairsNonDep"><span class="id" title="section">AllPairsNonDep</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="EqAllPairsDep"><span class="id" title="section">EqAllPairsDep</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="EqAllPairsDep.S"><span class="id" title="variable">S</span></a> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>) (<a name="EqAllPairsDep.T"><span class="id" title="variable">T</span></a> : <a class="idref" href="mathcomp.ssreflect.seq.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>).<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>) (<span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqAllPairsDep.S"><span class="id" title="variable">S</span></a>) (<span class="id" title="var">t</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#EqAllPairsDep.T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>)).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="allpairsPdep"><span class="id" title="lemma">allpairsPdep</span></a> <span class="id" title="var">R</span> (<span class="id" title="var">f</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.ssreflect.seq.html#EqAllPairsDep.T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#R"><span class="id" title="variable">R</span></a>) <span class="id" title="var">s</span> <span class="id" title="var">t</span> (<span class="id" title="var">z</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#R"><span class="id" title="variable">R</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">[/\</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">]</span></a>)<br/> - (<a class="idref" href="mathcomp.ssreflect.seq.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">,</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">]</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="EqAllPairsDep.R"><span class="id" title="variable">R</span></a> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>.<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">f</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.ssreflect.seq.html#EqAllPairsDep.T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqAllPairsDep.R"><span class="id" title="variable">R</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="allpairs_f_dep"><span class="id" title="lemma">allpairs_f_dep</span></a> <span class="id" title="var">f</span> <span class="id" title="var">s</span> <span class="id" title="var">t</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">,</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_in_allpairs_dep"><span class="id" title="lemma">eq_in_allpairs_dep</span></a> <span class="id" title="var">f1</span> <span class="id" title="var">f2</span> <span class="id" title="var">s</span> <span class="id" title="var">t</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f1"><span class="id" title="variable">f1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#876aa133fb3472bffd492f74ff496035"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f2"><span class="id" title="variable">f2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">}}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#4bfb4f2d0721ba668e3a802ab1b745a1"><span class="id" title="notation">↔</span></a><br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#00929bf63896446405b6c979ee089e42"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#00929bf63896446405b6c979ee089e42"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f1"><span class="id" title="variable">f1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#00929bf63896446405b6c979ee089e42"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqAllPairsDep.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#00929bf63896446405b6c979ee089e42"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#00929bf63896446405b6c979ee089e42"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#00929bf63896446405b6c979ee089e42"><span class="id" title="notation">,</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#00929bf63896446405b6c979ee089e42"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#00929bf63896446405b6c979ee089e42"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f2"><span class="id" title="variable">f2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">,</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem_allpairs_dep"><span class="id" title="lemma">mem_allpairs_dep</span></a> <span class="id" title="var">f</span> <span class="id" title="var">s1</span> <span class="id" title="var">t1</span> <span class="id" title="var">s2</span> <span class="id" title="var">t2</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.ssreflect.seq.html#t1"><span class="id" title="variable">t1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t2"><span class="id" title="variable">t2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">,</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t1"><span class="id" title="variable">t1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">,</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t2"><span class="id" title="variable">t2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="allpairs_catr"><span class="id" title="lemma">allpairs_catr</span></a> <span class="id" title="var">f</span> <span class="id" title="var">s</span> <span class="id" title="var">t1</span> <span class="id" title="var">t2</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">,</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t1"><span class="id" title="variable">t1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t2"><span class="id" title="variable">t2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">i</span></a><br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">,</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t1"><span class="id" title="variable">t1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">,</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t2"><span class="id" title="variable">t2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="allpairs_uniq_dep"><span class="id" title="lemma">allpairs_uniq_dep</span></a> <span class="id" title="var">f</span> <span class="id" title="var">s</span> <span class="id" title="var">t</span> (<span class="id" title="var">st</span> := <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">seq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#Tagged"><span class="id" title="definition">Tagged</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqAllPairsDep.T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">,</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">]</span></a>) :<br/> - <span class="id" title="keyword">let</span> <span class="id" title="var">g</span> (<span class="id" title="var">p</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">{</span></a><span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqAllPairsDep.S"><span class="id" title="variable">S</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqAllPairsDep.T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">}</span></a>) : <a class="idref" href="mathcomp.ssreflect.seq.html#EqAllPairsDep.R"><span class="id" title="variable">R</span></a> := <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#tag"><span class="id" title="definition">tag</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#p"><span class="id" title="variable">p</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#tagged"><span class="id" title="definition">tagged</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#p"><span class="id" title="variable">p</span></a>) <span class="id" title="tactic">in</span><br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#st"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#g"><span class="id" title="variable">g</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">,</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.seq.html#EqAllPairsDep"><span class="id" title="section">EqAllPairsDep</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="EqAllPairs"><span class="id" title="section">EqAllPairs</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> <a name="EqAllPairs.S"><span class="id" title="variable">S</span></a> <a name="EqAllPairs.T"><span class="id" title="variable">T</span></a> <a name="EqAllPairs.R"><span class="id" title="variable">R</span></a> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>.<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">f</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#EqAllPairs.S"><span class="id" title="variable">S</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqAllPairs.T"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqAllPairs.R"><span class="id" title="variable">R</span></a>) (<span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqAllPairs.S"><span class="id" title="variable">S</span></a>) (<span class="id" title="var">t</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqAllPairs.T"><span class="id" title="variable">T</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="allpairsP"><span class="id" title="lemma">allpairsP</span></a> <span class="id" title="var">f</span> <span class="id" title="var">s</span> <span class="id" title="var">t</span> (<span class="id" title="var">z</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#EqAllPairs.R"><span class="id" title="variable">R</span></a>) :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <span class="id" title="var">p</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">[/\</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">.1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">.2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">.1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">.2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">]</span></a>)<br/> - (<a class="idref" href="mathcomp.ssreflect.seq.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">,</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">]</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="allpairs_f"><span class="id" title="lemma">allpairs_f</span></a> <span class="id" title="var">f</span> <span class="id" title="var">s</span> <span class="id" title="var">t</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">,</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="eq_in_allpairs"><span class="id" title="lemma">eq_in_allpairs</span></a> <span class="id" title="var">f1</span> <span class="id" title="var">f2</span> <span class="id" title="var">s</span> <span class="id" title="var">t</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#5f76d9959f82823e4253cd67e7dc0e96"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#5f76d9959f82823e4253cd67e7dc0e96"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#5f76d9959f82823e4253cd67e7dc0e96"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#5f76d9959f82823e4253cd67e7dc0e96"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f1"><span class="id" title="variable">f1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#c717e24548d7d4d1086535addc681262"><span class="id" title="notation">=2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f2"><span class="id" title="variable">f2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#5f76d9959f82823e4253cd67e7dc0e96"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#4bfb4f2d0721ba668e3a802ab1b745a1"><span class="id" title="notation">↔</span></a><br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#00929bf63896446405b6c979ee089e42"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#00929bf63896446405b6c979ee089e42"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f1"><span class="id" title="variable">f1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#00929bf63896446405b6c979ee089e42"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#EqAllPairs.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#00929bf63896446405b6c979ee089e42"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#00929bf63896446405b6c979ee089e42"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#00929bf63896446405b6c979ee089e42"><span class="id" title="notation">,</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#00929bf63896446405b6c979ee089e42"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#00929bf63896446405b6c979ee089e42"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f2"><span class="id" title="variable">f2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">,</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem_allpairs"><span class="id" title="lemma">mem_allpairs</span></a> <span class="id" title="var">f</span> <span class="id" title="var">s1</span> <span class="id" title="var">t1</span> <span class="id" title="var">s2</span> <span class="id" title="var">t2</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t1"><span class="id" title="variable">t1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t2"><span class="id" title="variable">t2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">,</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t1"><span class="id" title="variable">t1</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">=</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#1e6a438ff685c38fcd9034a94f271777"><span class="id" title="notation">i</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">,</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t2"><span class="id" title="variable">t2</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="allpairs_uniq"><span class="id" title="lemma">allpairs_uniq</span></a> <span class="id" title="var">f</span> <span class="id" title="var">s</span> <span class="id" title="var">t</span> (<span class="id" title="var">st</span> := <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">seq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">,</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">]</span></a>) :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#st"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#prod_curry"><span class="id" title="definition">prod_curry</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b4f176550f5b849a7fbba2ee164934d3"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">,</span></a> <span class="id" title="var">y</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.seq.html#EqAllPairs"><span class="id" title="section">EqAllPairs</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Permutations"><span class="id" title="section">Permutations</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="Permutations.T"><span class="id" title="variable">T</span></a> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>.<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#Permutations.T"><span class="id" title="variable">T</span></a>) (<span class="id" title="var">s</span> <span class="id" title="var">t</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Permutations.T"><span class="id" title="variable">T</span></a>) (<span class="id" title="var">bs</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#Permutations.T"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>)) (<span class="id" title="var">acc</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Permutations.T"><span class="id" title="variable">T</span></a>)).<br/> - -<br/> -<span class="id" title="keyword">Fixpoint</span> <a name="incr_tally"><span class="id" title="definition">incr_tally</span></a> <span class="id" title="var">bs</span> <span class="id" title="var">x</span> :=<br/> - <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#bs"><span class="id" title="variable">bs</span></a> <span class="id" title="var">isn't</span> <span class="id" title="var">b</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">bs</span> <span class="id" title="keyword">then</span> <a class="idref" href="mathcomp.ssreflect.seq.html#506674b18256ef8f50efed43fa1dfd7d"><span class="id" title="notation">[::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#506674b18256ef8f50efed43fa1dfd7d"><span class="id" title="notation">]</span></a> <span class="id" title="keyword">else</span><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <span class="id" title="var">b</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">.1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">b</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">.2</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#bs"><span class="id" title="variable">bs</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <span class="id" title="var">b</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#incr_tally"><span class="id" title="definition">incr_tally</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#bs"><span class="id" title="variable">bs</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="tally"><span class="id" title="definition">tally</span></a> <span class="id" title="var">s</span> := <a class="idref" href="mathcomp.ssreflect.seq.html#foldl"><span class="id" title="definition">foldl</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#incr_tally"><span class="id" title="definition">incr_tally</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="wf_tally"><span class="id" title="definition">wf_tally</span></a> :=<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#65c8f47ea0daafc83f7bb18bc9eca61f"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#65c8f47ea0daafc83f7bb18bc9eca61f"><span class="id" title="notation">qualify</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#65c8f47ea0daafc83f7bb18bc9eca61f"><span class="id" title="notation">a</span></a> <span class="id" title="var">bs</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#65c8f47ea0daafc83f7bb18bc9eca61f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#Permutations.T"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#65c8f47ea0daafc83f7bb18bc9eca61f"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#unzip1"><span class="id" title="definition">unzip1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#bs"><span class="id" title="variable">bs</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">notin</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#unzip2"><span class="id" title="definition">unzip2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#bs"><span class="id" title="variable">bs</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#65c8f47ea0daafc83f7bb18bc9eca61f"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="tally_seq"><span class="id" title="definition">tally_seq</span></a> <span class="id" title="var">bs</span> := <a class="idref" href="mathcomp.ssreflect.seq.html#flatten"><span class="id" title="definition">flatten</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nseq"><span class="id" title="definition">nseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#675082cc4d4538da052b547bdc6ea4c9"><span class="id" title="notation">.2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#e0817251e7d67ad994b4d9b1aa82a412"><span class="id" title="notation">.1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">|</span></a> <span class="id" title="var">b</span> <a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#bs"><span class="id" title="variable">bs</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="size_tally_seq"><span class="id" title="lemma">size_tally_seq</span></a> <span class="id" title="var">bs</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#tally_seq"><span class="id" title="definition">tally_seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#bs"><span class="id" title="variable">bs</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#sumn"><span class="id" title="definition">sumn</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#unzip2"><span class="id" title="definition">unzip2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#bs"><span class="id" title="variable">bs</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="tally_seqK"><span class="id" title="lemma">tally_seqK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#wf_tally"><span class="id" title="definition">wf_tally</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#tally_seq"><span class="id" title="definition">tally_seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#tally"><span class="id" title="definition">tally</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#8c08d4203604dbed63e7afa9b689d858"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="incr_tallyP"><span class="id" title="lemma">incr_tallyP</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#b05c841abe740b4dbf549b223cd4518f"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#b05c841abe740b4dbf549b223cd4518f"><span class="id" title="notation">homo</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#incr_tally"><span class="id" title="definition">incr_tally</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#d89396f990d6b54d736cfe259e498cf4"><span class="id" title="notation">^~</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#b05c841abe740b4dbf549b223cd4518f"><span class="id" title="notation">:</span></a> <span class="id" title="var">bs</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#b05c841abe740b4dbf549b223cd4518f"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#bs"><span class="id" title="variable">bs</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#wf_tally"><span class="id" title="definition">wf_tally</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrfun.html#b05c841abe740b4dbf549b223cd4518f"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="tallyP"><span class="id" title="lemma">tallyP</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#tally"><span class="id" title="definition">tally</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">is</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#f6c65697fefaf4504de1d4d641cd4409"><span class="id" title="notation">a</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#wf_tally"><span class="id" title="definition">wf_tally</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="tallyK"><span class="id" title="lemma">tallyK</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#tally_seq"><span class="id" title="definition">tally_seq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#tally"><span class="id" title="definition">tally</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>)) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="tallyEl"><span class="id" title="lemma">tallyEl</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#unzip1"><span class="id" title="definition">unzip1</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#tally"><span class="id" title="definition">tally</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>)) (<a class="idref" href="mathcomp.ssreflect.seq.html#undup"><span class="id" title="definition">undup</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="tallyE"><span class="id" title="lemma">tallyE</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#tally"><span class="id" title="definition">tally</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">seq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#count_mem"><span class="id" title="abbreviation">count_mem</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#undup"><span class="id" title="definition">undup</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="perm_tally"><span class="id" title="lemma">perm_tally</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#tally"><span class="id" title="definition">tally</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#tally"><span class="id" title="definition">tally</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="perm_tally_seq"><span class="id" title="lemma">perm_tally_seq</span></a> <span class="id" title="var">bs1</span> <span class="id" title="var">bs2</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#bs1"><span class="id" title="variable">bs1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#bs2"><span class="id" title="variable">bs2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#tally_seq"><span class="id" title="definition">tally_seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#bs1"><span class="id" title="variable">bs1</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#tally_seq"><span class="id" title="definition">tally_seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#bs2"><span class="id" title="variable">bs2</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="perm_count_undup"><span class="id" title="lemma">perm_count_undup</span></a> <span class="id" title="var">s</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#flatten"><span class="id" title="definition">flatten</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nseq"><span class="id" title="definition">nseq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#count_mem"><span class="id" title="abbreviation">count_mem</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#undup"><span class="id" title="definition">undup</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">]</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="permutations"><span class="id" title="definition">permutations</span></a> <span class="id" title="var">s</span> := <a class="idref" href="mathcomp.ssreflect.seq.html#perms_rec"><span class="id" title="definition">perms_rec</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#tally"><span class="id" title="definition">tally</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Let</span> <a name="Permutations.permsP"><span class="id" title="variable">permsP</span></a> <span class="id" title="var">s</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <span class="id" title="var">n</span> <span class="id" title="var">bs</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">[/\</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#permutations"><span class="id" title="definition">permutations</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perms_rec"><span class="id" title="definition">perms_rec</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#bs"><span class="id" title="variable">bs</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">,</span></a><br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#tseq"><span class="id" title="abbreviation">tseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#bs"><span class="id" title="variable">bs</span></a>) <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#tseq"><span class="id" title="abbreviation">tseq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#bs"><span class="id" title="variable">bs</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#unzip1"><span class="id" title="definition">unzip1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#bs"><span class="id" title="variable">bs</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Let</span> <a name="Permutations.cons_permsE"><span class="id" title="variable">cons_permsE</span></a> : <span class="id" title="keyword">∀</span> <span class="id" title="var">n</span> <span class="id" title="var">x</span> <span class="id" title="var">bs</span> <span class="id" title="var">bs1</span> <span class="id" title="var">bs2</span>,<br/> - <span class="id" title="keyword">let</span> <span class="id" title="var">cp</span> := <a class="idref" href="mathcomp.ssreflect.seq.html#cons_perms"><span class="id" title="abbreviation">cons_perms</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#bs"><span class="id" title="variable">bs</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#bs2"><span class="id" title="variable">bs2</span></a> <span class="id" title="tactic">in</span> <span class="id" title="keyword">let</span> <span class="id" title="var">perms</span> <span class="id" title="var">s</span> := <a class="idref" href="mathcomp.ssreflect.seq.html#perms_rec"><span class="id" title="definition">perms_rec</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#bs1"><span class="id" title="variable">bs1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a> <span class="id" title="tactic">in</span><br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#cp"><span class="id" title="variable">cp</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#perms"><span class="id" title="variable">perms</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#506674b18256ef8f50efed43fa1dfd7d"><span class="id" title="notation">[::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#506674b18256ef8f50efed43fa1dfd7d"><span class="id" title="notation">]</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#cp"><span class="id" title="variable">cp</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#93e0a78b945d3f9f22195c004c67aa36"><span class="id" title="notation">++</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">|</span></a> <span class="id" title="var">t</span> <a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perms"><span class="id" title="variable">perms</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#0a934e621391740b862762275992e626"><span class="id" title="notation">[::]</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#dcd18413b33436252c77b6c6465f02bc"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="mem_permutations"><span class="id" title="lemma">mem_permutations</span></a> <span class="id" title="var">s</span> <span class="id" title="var">t</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#permutations"><span class="id" title="definition">permutations</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="permutations_uniq"><span class="id" title="lemma">permutations_uniq</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#permutations"><span class="id" title="definition">permutations</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="perms"><span class="id" title="abbreviation">perms</span></a> := <a class="idref" href="mathcomp.ssreflect.seq.html#permutations"><span class="id" title="definition">permutations</span></a>.<br/> -<span class="id" title="keyword">Lemma</span> <a name="permutationsE"><span class="id" title="lemma">permutationsE</span></a> <span class="id" title="var">s</span> :<br/> - 0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#00fe0eaf5e6949f0a31725357afa4bba"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#perms"><span class="id" title="abbreviation">perms</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#undup"><span class="id" title="definition">undup</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">,</span></a> <span class="id" title="var">t</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perms"><span class="id" title="abbreviation">perms</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#rem"><span class="id" title="definition">rem</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>)<a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="permutationsErot"><span class="id" title="lemma">permutationsErot</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> (<span class="id" title="var">le_x</span> := <span class="id" title="keyword">fun</span> <span class="id" title="var">t</span> ⇒ <a class="idref" href="mathcomp.ssreflect.seq.html#iota"><span class="id" title="definition">iota</span></a> 0 (<a class="idref" href="mathcomp.ssreflect.seq.html#index"><span class="id" title="definition">index</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> 1)) :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#perms"><span class="id" title="abbreviation">perms</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>)) <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rot"><span class="id" title="definition">rot</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#i"><span class="id" title="variable">i</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">|</span></a> <span class="id" title="var">t</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perms"><span class="id" title="abbreviation">perms</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">,</span></a> <span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#le_x"><span class="id" title="variable">le_x</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#623af7e6d063f6c25f88725170ed3a4c"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="size_permutations"><span class="id" title="lemma">size_permutations</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#permutations"><span class="id" title="definition">permutations</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#fb85b56582e20c3ab47a4e55b1f6f111"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#fb85b56582e20c3ab47a4e55b1f6f111"><span class="id" title="notation">)`!</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="permutations_all_uniq"><span class="id" title="lemma">permutations_all_uniq</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq"><span class="id" title="definition">uniq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#permutations"><span class="id" title="definition">permutations</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="perm_permutations"><span class="id" title="lemma">perm_permutations</span></a> <span class="id" title="var">s</span> <span class="id" title="var">t</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#permutations"><span class="id" title="definition">permutations</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s"><span class="id" title="variable">s</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#permutations"><span class="id" title="definition">permutations</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#t"><span class="id" title="variable">t</span></a>).<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.seq.html#Permutations"><span class="id" title="section">Permutations</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="AllIff"><span class="id" title="section">AllIff</span></a>.<br/> -</div> - -<div class="doc"> - The Following Are Equivalent -<div class="paragraph"> </div> - - We introduce a specific conjunction, used to chain the consecutive - items in a circular list of implications -</div> -<div class="code"> -<span class="id" title="keyword">Inductive</span> <a name="all_iff_and"><span class="id" title="inductive">all_iff_and</span></a> (<span class="id" title="var">P</span> <span class="id" title="var">Q</span> : <span class="id" title="keyword">Prop</span>) : <span class="id" title="keyword">Prop</span> := <a name="AllIffConj"><span class="id" title="constructor">AllIffConj</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.ssreflect.seq.html#P"><span class="id" title="variable">P</span></a> & <a class="idref" href="mathcomp.ssreflect.seq.html#Q"><span class="id" title="variable">Q</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="all_iff"><span class="id" title="definition">all_iff</span></a> (<span class="id" title="var">P0</span> : <span class="id" title="keyword">Prop</span>) (<span class="id" title="var">Ps</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <span class="id" title="keyword">Prop</span>) : <span class="id" title="keyword">Prop</span> :=<br/> - <span class="id" title="keyword">let</span> <span class="id" title="keyword">fix</span> <span class="id" title="var">loop</span> (<span class="id" title="var">P</span> : <span class="id" title="keyword">Prop</span>) (<span class="id" title="var">Qs</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <span class="id" title="keyword">Prop</span>) : <span class="id" title="keyword">Prop</span> :=<br/> - <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.seq.html#Qs"><span class="id" title="variable">Qs</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">Q</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <span class="id" title="var">Qs</span> <span class="id" title="keyword">then</span> <a class="idref" href="mathcomp.ssreflect.seq.html#all_iff_and"><span class="id" title="inductive">all_iff_and</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <span class="id" title="var">Q</span>) (<a class="idref" href="mathcomp.ssreflect.seq.html#loop"><span class="id" title="variable">loop</span></a> <span class="id" title="var">Q</span> <a class="idref" href="mathcomp.ssreflect.seq.html#Qs"><span class="id" title="variable">Qs</span></a>) <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.seq.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#P0"><span class="id" title="variable">P0</span></a> <span class="id" title="tactic">in</span><br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#loop"><span class="id" title="variable">loop</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#P0"><span class="id" title="variable">P0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Ps"><span class="id" title="variable">Ps</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="all_iffLR"><span class="id" title="lemma">all_iffLR</span></a> <span class="id" title="var">P0</span> <span class="id" title="var">Ps</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#all_iff"><span class="id" title="definition">all_iff</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#P0"><span class="id" title="variable">P0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Ps"><span class="id" title="variable">Ps</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span>, <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#P0"><span class="id" title="variable">P0</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#P0"><span class="id" title="variable">P0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Ps"><span class="id" title="variable">Ps</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#P0"><span class="id" title="variable">P0</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#P0"><span class="id" title="variable">P0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Ps"><span class="id" title="variable">Ps</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="all_iffP"><span class="id" title="lemma">all_iffP</span></a> <span class="id" title="var">P0</span> <span class="id" title="var">Ps</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.seq.html#all_iff"><span class="id" title="definition">all_iff</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#P0"><span class="id" title="variable">P0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Ps"><span class="id" title="variable">Ps</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span>, <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#P0"><span class="id" title="variable">P0</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#P0"><span class="id" title="variable">P0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Ps"><span class="id" title="variable">Ps</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#4bfb4f2d0721ba668e3a802ab1b745a1"><span class="id" title="notation">↔</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#nth"><span class="id" title="definition">nth</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#P0"><span class="id" title="variable">P0</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#P0"><span class="id" title="variable">P0</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Ps"><span class="id" title="variable">Ps</span></a>) <a class="idref" href="mathcomp.ssreflect.seq.html#n"><span class="id" title="variable">n</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.seq.html#AllIff"><span class="id" title="section">AllIff</span></a>.<br/> -<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.ssreflect.seq.html#all_iffP"><span class="id" title="lemma">all_iffP</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#all_iffP"><span class="id" title="lemma">:</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#all_iffP"><span class="id" title="lemma">all_iff</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#all_iffP"><span class="id" title="lemma">>-></span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#all_iffP"><span class="id" title="lemma">Funclass</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - This means "the following are all equivalent: P0, ... Pn" -</div> -<div class="code"> -<span class="id" title="keyword">Notation</span> <a name="a3088764e59abfd113df52de1a8d59b1"><span class="id" title="notation">"</span></a>[ '<->' P0 ; P1 ; .. ; Pn ]" := (<a class="idref" href="mathcomp.ssreflect.seq.html#all_iff"><span class="id" title="definition">all_iff</span></a> <span class="id" title="var">P0</span> (<span class="id" title="var">P1</span> <a class="idref" href="mathcomp.ssreflect.seq.html#407cde5b61fbf27196d1a7c5a475e083"><span class="id" title="notation">::</span></a> .. <a class="idref" href="mathcomp.ssreflect.seq.html#506674b18256ef8f50efed43fa1dfd7d"><span class="id" title="notation">[::</span></a> <span class="id" title="var">Pn</span><a class="idref" href="mathcomp.ssreflect.seq.html#506674b18256ef8f50efed43fa1dfd7d"><span class="id" title="notation">]</span></a> ..))<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ '<->' '[' P0 ; '/' P1 ; '/' .. ; '/' Pn ']' ]")<br/> - : <span class="id" title="var">form_scope</span>.<br/> - -<br/> -</div> - -<div class="doc"> - Temporary backward compatibility. -</div> -<div class="code"> -<span class="id" title="keyword">Notation</span> <a name="perm_eqP"><span class="id" title="abbreviation">perm_eqP</span></a> := (<a class="idref" href="mathcomp.ssreflect.ssreflect.html#Deprecation.Exports.deprecate"><span class="id" title="abbreviation">deprecate</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eqP"><span class="id" title="variable">perm_eqP</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#permP"><span class="id" title="variable">permP</span></a>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/> -<span class="id" title="keyword">Notation</span> <a name="perm_eqlP"><span class="id" title="abbreviation">perm_eqlP</span></a> := (<a class="idref" href="mathcomp.ssreflect.ssreflect.html#Deprecation.Exports.deprecate"><span class="id" title="abbreviation">deprecate</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eqlP"><span class="id" title="variable">perm_eqlP</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#permPl"><span class="id" title="variable">permPl</span></a>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/> -<span class="id" title="keyword">Notation</span> <a name="perm_eqrP"><span class="id" title="abbreviation">perm_eqrP</span></a> := (<a class="idref" href="mathcomp.ssreflect.ssreflect.html#Deprecation.Exports.deprecate"><span class="id" title="abbreviation">deprecate</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eqrP"><span class="id" title="variable">perm_eqrP</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#permPr"><span class="id" title="variable">permPr</span></a>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/> -<span class="id" title="keyword">Notation</span> <a name="perm_eqlE"><span class="id" title="abbreviation">perm_eqlE</span></a> := (<a class="idref" href="mathcomp.ssreflect.ssreflect.html#Deprecation.Exports.deprecate"><span class="id" title="abbreviation">deprecate</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eqlE"><span class="id" title="variable">perm_eqlE</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#permEl"><span class="id" title="variable">permEl</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/> -<span class="id" title="keyword">Notation</span> <a name="perm_eq_refl"><span class="id" title="abbreviation">perm_eq_refl</span></a> := (<a class="idref" href="mathcomp.ssreflect.ssreflect.html#Deprecation.Exports.deprecate"><span class="id" title="abbreviation">deprecate</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq_refl"><span class="id" title="variable">perm_eq_refl</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_refl"><span class="id" title="variable">perm_refl</span></a> <span class="id" title="var">_</span>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/> -<span class="id" title="keyword">Notation</span> <a name="perm_eq_sym"><span class="id" title="abbreviation">perm_eq_sym</span></a> := (<a class="idref" href="mathcomp.ssreflect.ssreflect.html#Deprecation.Exports.deprecate"><span class="id" title="abbreviation">deprecate</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq_sym"><span class="id" title="variable">perm_eq_sym</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_sym"><span class="id" title="variable">perm_sym</span></a> <span class="id" title="var">_</span>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/> -<span class="id" title="keyword">Notation</span> <a name="30471ea39317493f33418c522b69149e"><span class="id" title="notation">"</span></a>@ 'perm_eq_trans'" := (<a class="idref" href="mathcomp.ssreflect.ssreflect.html#Deprecation.Exports.deprecate"><span class="id" title="abbreviation">deprecate</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq_trans"><span class="id" title="variable">perm_eq_trans</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_trans"><span class="id" title="variable">perm_trans</span></a>)<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 10, <span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/> -<span class="id" title="keyword">Notation</span> <a name="perm_eq_trans"><span class="id" title="abbreviation">perm_eq_trans</span></a> := (<a class="idref" href="mathcomp.ssreflect.seq.html#30471ea39317493f33418c522b69149e"><span class="id" title="notation">@</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#30471ea39317493f33418c522b69149e"><span class="id" title="notation">perm_eq_trans</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#30471ea39317493f33418c522b69149e"><span class="id" title="notation">_</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#30471ea39317493f33418c522b69149e"><span class="id" title="notation">_</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#30471ea39317493f33418c522b69149e"><span class="id" title="notation">_</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#30471ea39317493f33418c522b69149e"><span class="id" title="notation">_</span></a>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/> -<span class="id" title="keyword">Notation</span> <a name="perm_eq_size"><span class="id" title="abbreviation">perm_eq_size</span></a> := (<a class="idref" href="mathcomp.ssreflect.ssreflect.html#Deprecation.Exports.deprecate"><span class="id" title="abbreviation">deprecate</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq_size"><span class="id" title="variable">perm_eq_size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_size"><span class="id" title="variable">perm_size</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>)<br/> - (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/> -<span class="id" title="keyword">Notation</span> <a name="perm_eq_mem"><span class="id" title="abbreviation">perm_eq_mem</span></a> := (<a class="idref" href="mathcomp.ssreflect.ssreflect.html#Deprecation.Exports.deprecate"><span class="id" title="abbreviation">deprecate</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq_mem"><span class="id" title="variable">perm_eq_mem</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_mem"><span class="id" title="variable">perm_mem</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>)<br/> - (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/> -<span class="id" title="keyword">Notation</span> <a name="perm_eq_uniq"><span class="id" title="abbreviation">perm_eq_uniq</span></a> := (<a class="idref" href="mathcomp.ssreflect.ssreflect.html#Deprecation.Exports.deprecate"><span class="id" title="abbreviation">deprecate</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq_uniq"><span class="id" title="variable">perm_eq_uniq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_uniq"><span class="id" title="variable">perm_uniq</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>)<br/> - (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/> -<span class="id" title="keyword">Notation</span> <a name="perm_eq_rev"><span class="id" title="abbreviation">perm_eq_rev</span></a> := (<a class="idref" href="mathcomp.ssreflect.ssreflect.html#Deprecation.Exports.deprecate"><span class="id" title="abbreviation">deprecate</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq_rev"><span class="id" title="variable">perm_eq_rev</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_rev"><span class="id" title="variable">perm_rev</span></a> <span class="id" title="var">_</span>)<br/> - (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/> -<span class="id" title="keyword">Notation</span> <a name="perm_eq_flatten"><span class="id" title="abbreviation">perm_eq_flatten</span></a> := (<a class="idref" href="mathcomp.ssreflect.ssreflect.html#Deprecation.Exports.deprecate"><span class="id" title="abbreviation">deprecate</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq_flatten"><span class="id" title="variable">perm_eq_flatten</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_flatten"><span class="id" title="variable">perm_flatten</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>)<br/> - (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/> -<span class="id" title="keyword">Notation</span> <a name="perm_eq_all"><span class="id" title="abbreviation">perm_eq_all</span></a> := (<a class="idref" href="mathcomp.ssreflect.ssreflect.html#Deprecation.Exports.deprecate"><span class="id" title="abbreviation">deprecate</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq_all"><span class="id" title="variable">perm_eq_all</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_all"><span class="id" title="variable">perm_all</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>)<br/> - (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/> -<span class="id" title="keyword">Notation</span> <a name="perm_eq_small"><span class="id" title="abbreviation">perm_eq_small</span></a> := (<a class="idref" href="mathcomp.ssreflect.ssreflect.html#Deprecation.Exports.deprecate"><span class="id" title="abbreviation">deprecate</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq_small"><span class="id" title="variable">perm_eq_small</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_small_eq"><span class="id" title="variable">perm_small_eq</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>)<br/> - (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/> -<span class="id" title="keyword">Notation</span> <a name="perm_eq_nilP"><span class="id" title="abbreviation">perm_eq_nilP</span></a> := (<a class="idref" href="mathcomp.ssreflect.ssreflect.html#Deprecation.Exports.deprecate"><span class="id" title="abbreviation">deprecate</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq_nilP"><span class="id" title="variable">perm_eq_nilP</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_nilP"><span class="id" title="variable">perm_nilP</span></a>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/> -<span class="id" title="keyword">Notation</span> <a name="perm_eq_consP"><span class="id" title="abbreviation">perm_eq_consP</span></a> := (<a class="idref" href="mathcomp.ssreflect.ssreflect.html#Deprecation.Exports.deprecate"><span class="id" title="abbreviation">deprecate</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq_consP"><span class="id" title="variable">perm_eq_consP</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_consP"><span class="id" title="variable">perm_consP</span></a>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/> -<span class="id" title="keyword">Notation</span> <a name="leq_size_perm"><span class="id" title="abbreviation">leq_size_perm</span></a> :=<br/> - ((<span class="id" title="keyword">fun</span> <span class="id" title="var">T</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> <span class="id" title="var">Us1</span> <span class="id" title="var">ss12</span> <span class="id" title="var">les21</span> ⇒<br/> - <span class="id" title="keyword">let</span>: <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">Esz12</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">Es12</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> :=<br/> - <a class="idref" href="mathcomp.ssreflect.ssreflect.html#Deprecation.Exports.deprecate"><span class="id" title="abbreviation">deprecate</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#leq_size_perm"><span class="id" title="variable">leq_size_perm</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq_min_size"><span class="id" title="variable">uniq_min_size</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#Us1"><span class="id" title="variable">Us1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ss12"><span class="id" title="variable">ss12</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#les21"><span class="id" title="variable">les21</span></a><br/> - <span class="id" title="tactic">in</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#conj"><span class="id" title="constructor">conj</span></a> <span class="id" title="var">Es12</span> <span class="id" title="var">Esz12</span>) <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>)<br/> - (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/> -<span class="id" title="keyword">Notation</span> <a name="uniq_perm_eq"><span class="id" title="abbreviation">uniq_perm_eq</span></a> := (<a class="idref" href="mathcomp.ssreflect.ssreflect.html#Deprecation.Exports.deprecate"><span class="id" title="abbreviation">deprecate</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq_perm_eq"><span class="id" title="variable">uniq_perm_eq</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#uniq_perm"><span class="id" title="variable">uniq_perm</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>)<br/> - (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/> -<span class="id" title="keyword">Notation</span> <a name="perm_eq_iotaP"><span class="id" title="abbreviation">perm_eq_iotaP</span></a> := (<a class="idref" href="mathcomp.ssreflect.ssreflect.html#Deprecation.Exports.deprecate"><span class="id" title="abbreviation">deprecate</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq_iotaP"><span class="id" title="variable">perm_eq_iotaP</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_iotaP"><span class="id" title="variable">perm_iotaP</span></a>) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/> -<span class="id" title="keyword">Notation</span> <a name="perm_undup_count"><span class="id" title="abbreviation">perm_undup_count</span></a> := (<a class="idref" href="mathcomp.ssreflect.ssreflect.html#Deprecation.Exports.deprecate"><span class="id" title="abbreviation">deprecate</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_undup_count"><span class="id" title="variable">perm_undup_count</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#perm_count_undup"><span class="id" title="variable">perm_count_undup</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span>)<br/> - (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/> -</div> -</div> - -<div id="footer"> -<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a> -</div> - -</div> - -</body> -</html>
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