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diff --git a/mathcomp/basic/fintype.v b/mathcomp/basic/fintype.v new file mode 100644 index 0000000..66ac880 --- /dev/null +++ b/mathcomp/basic/fintype.v @@ -0,0 +1,2056 @@ +(* (c) Copyright Microsoft Corporation and Inria. All rights reserved. *) +Require Import mathcomp.ssreflect.ssreflect. +From mathcomp +Require Import ssrfun ssrbool eqtype ssrnat seq choice. + +(******************************************************************************) +(* The Finite interface describes Types with finitely many elements, *) +(* supplying a duplicate-free sequence of all the elements. It is a subclass *) +(* of Countable and thus of Choice and Equality. As with Countable, the *) +(* interface explicitly includes these somewhat redundant superclasses to *) +(* ensure that Canonical instance inference remains consistent. Finiteness *) +(* could be stated more simply by bounding the range of the pickle function *) +(* supplied by the Countable interface, but this would yield a useless *) +(* computational interpretation due to the wasteful Peano integer encodings. *) +(* Because the Countable interface is closely tied to the Finite interface *) +(* and is not much used on its own, the Countable mixin is included inside *) +(* the Finite mixin; this makes it much easier to derive Finite variants of *) +(* interfaces, in this file for subFinType, and in the finalg library. *) +(* We define the following interfaces and structures: *) +(* finType == the packed class type of the Finite interface. *) +(* FinType m == the packed class for the Finite mixin m. *) +(* Finite.axiom e <-> every x : T occurs exactly once in e : seq T. *) +(* FinMixin ax_e == the Finite mixin for T, encapsulating *) +(* ax_e : Finite.axiom e for some e : seq T. *) +(* UniqFinMixin uniq_e total_e == an alternative mixin constructor that uses *) +(* uniq_e : uniq e and total_e : e =i xpredT. *) +(* PcanFinMixin fK == the Finite mixin for T, given f : T -> fT and g with fT *) +(* a finType and fK : pcancel f g. *) +(* CanFinMixin fK == the Finite mixin for T, given f : T -> fT and g with fT *) +(* a finType and fK : cancel f g. *) +(* subFinType == the join interface type for subType and finType. *) +(* [finType of T for fT] == clone for T of the finType fT. *) +(* [finType of T] == clone for T of the finType inferred for T. *) +(* [subFinType of T] == a subFinType structure for T, when T already has both *) +(* finType and subType structures. *) +(* [finMixin of T by <:] == a finType structure for T, when T has a subType *) +(* structure over an existing finType. *) +(* We define or propagate the finType structure appropriately for all basic *) +(* types : unit, bool, option, prod, sum, sig and sigT. We also define a *) +(* generic type constructor for finite subtypes based on an explicit *) +(* enumeration: *) +(* seq_sub s == the subType of all x \in s, where s : seq T for some *) +(* eqType T; seq_sub s has a canonical finType instance *) +(* when T is a choiceType. *) +(* adhoc_seq_sub_choiceType s, adhoc_seq_sub_finType s == *) +(* non-canonical instances for seq_sub s, s : seq T, *) +(* which can be used when T is not a choiceType. *) +(* Bounded integers are supported by the following type and operations: *) +(* 'I_n, ordinal n == the finite subType of integers i < n, whose *) +(* enumeration is {0, ..., n.-1}. 'I_n coerces to nat, *) +(* so all the integer arithmetic functions can be used *) +(* with 'I_n. *) +(* Ordinal lt_i_n == the element of 'I_n with (nat) value i, given *) +(* lt_i_n : i < n. *) +(* nat_of_ord i == the nat value of i : 'I_n (this function is a *) +(* coercion so it is not usually displayed). *) +(* ord_enum n == the explicit increasing sequence of the i : 'I_n. *) +(* cast_ord eq_n_m i == the element j : 'I_m with the same value as i : 'I_n *) +(* given eq_n_m : n = m (indeed, i : nat and j : nat *) +(* are convertible). *) +(* widen_ord le_n_m i == a j : 'I_m with the same value as i : 'I_n, given *) +(* le_n_m : n <= m. *) +(* rev_ord i == the complement to n.-1 of i : 'I_n, such that *) +(* i + rev_ord i = n.-1. *) +(* inord k == the i : 'I_n.+1 with value k (n is inferred from the *) +(* context). *) +(* sub_ord k == the i : 'I_n.+1 with value n - k (n is inferred from *) +(* the context). *) +(* ord0 == the i : 'I_n.+1 with value 0 (n is inferred from the *) +(* context). *) +(* ord_max == the i : 'I_n.+1 with value n (n is inferred from the *) +(* context). *) +(* bump h k == k.+1 if k >= h, else k (this is a nat function). *) +(* unbump h k == k.-1 if k > h, else k (this is a nat function). *) +(* lift i j == the j' : 'I_n with value bump i j, where i : 'I_n *) +(* and j : 'I_n.-1. *) +(* unlift i j == None if i = j, else Some j', where j' : 'I_n.-1 has *) +(* value unbump i j, given i, j : 'I_n. *) +(* lshift n j == the i : 'I_(m + n) with value j : 'I_m. *) +(* rshift m k == the i : 'I_(m + n) with value m + k, k : 'I_n. *) +(* unsplit u == either lshift n j or rshift m k, depending on *) +(* whether if u : 'I_m + 'I_n is inl j or inr k. *) +(* split i == the u : 'I_m + 'I_n such that i = unsplit u; the *) +(* type 'I_(m + n) of i determines the split. *) +(* Finally, every type T with a finType structure supports the following *) +(* operations: *) +(* enum A == a duplicate-free list of all the x \in A, where A is a *) +(* collective predicate over T. *) +(* #|A| == the cardinal of A, i.e., the number of x \in A. *) +(* enum_val i == the i'th item of enum A, where i : 'I_(#|A|). *) +(* enum_rank x == the i : 'I_(#|T|) such that enum_val i = x. *) +(* enum_rank_in Ax0 x == some i : 'I_(#|A|) such that enum_val i = x if *) +(* x \in A, given Ax0 : x0 \in A. *) +(* A \subset B == all x \in A satisfy x \in B. *) +(* A \proper B == all x \in A satisfy x \in B but not the converse. *) +(* [disjoint A & B] == no x \in A satisfies x \in B. *) +(* image f A == the sequence of f x for all x : T such that x \in A *) +(* (where a is an applicative predicate), of length #|P|. *) +(* The codomain of F can be any type, but image f A can *) +(* only be used as a collective predicate is it is an *) +(* eqType. *) +(* codom f == a sequence spanning the codomain of f (:= image f T). *) +(* [seq F | x : T in A] := image (fun x : T => F) A. *) +(* [seq F | x : T] := [seq F | x <- {: T}]. *) +(* [seq F | x in A], [seq F | x] == variants without casts. *) +(* iinv im_y == some x such that P x holds and f x = y, given *) +(* im_y : y \in image f P. *) +(* invF inj_f y == the x such that f x = y, for inj_j : injective f with *) +(* f : T -> T. *) +(* dinjectiveb A f == the restriction of f : T -> R to A is injective *) +(* (this is a bolean predicate, R must be an eqType). *) +(* injectiveb f == f : T -> R is injective (boolean predicate). *) +(* pred0b A == no x : T satisfies x \in A. *) +(* [forall x, P] == P (in which x can appear) is true for all values of x; *) +(* x must range over a finType. *) +(* [exists x, P] == P is true for some value of x. *) +(* [forall (x | C), P] := [forall x, C ==> P]. *) +(* [forall x in A, P] := [forall (x | x \in A), P]. *) +(* [exists (x | C), P] := [exists x, C && P]. *) +(* [exists x in A, P] := [exists (x | x \in A), P]. *) +(* and typed variants [forall x : T, P], [forall (x : T | C), P], *) +(* [exists x : T, P], [exists x : T in A, P], etc. *) +(* -> The outer brackets can be omitted when nesting finitary quantifiers, *) +(* e.g., [forall i in I, forall j in J, exists a, f i j == a]. *) +(* 'forall_pP == view for [forall x, p _], for pP : reflect .. (p _). *) +(* 'exists_pP == view for [exists x, p _], for pP : reflect .. (p _). *) +(* [pick x | P] == Some x, for an x such that P holds, or None if there *) +(* is no such x. *) +(* [pick x : T] == Some x with x : T, provided T is nonempty, else None. *) +(* [pick x in A] == Some x, with x \in A, or None if A is empty. *) +(* [pick x in A | P] == Some x, with x \in A s.t. P holds, else None. *) +(* [pick x | P & Q] := [pick x | P & Q]. *) +(* [pick x in A | P & Q] := [pick x | P & Q]. *) +(* and (un)typed variants [pick x : T | P], [pick x : T in A], [pick x], etc. *) +(* [arg min_(i < i0 | P) M] == a value of i : T minimizing M : nat, subject *) +(* to the condition P (i may appear in P and M), and *) +(* provided P holds for i0. *) +(* [arg max_(i > i0 | P) M] == a value of i maximizing M subject to P and *) +(* provided P holds for i0. *) +(* [arg min_(i < i0 in A) M] == an i \in A minimizing M if i0 \in A. *) +(* [arg max_(i > i0 in A) M] == an i \in A maximizing M if i0 \in A. *) +(* [arg min_(i < i0) M] == an i : T minimizing M, given i0 : T. *) +(* [arg max_(i > i0) M] == an i : T maximizing M, given i0 : T. *) +(******************************************************************************) + +Set Implicit Arguments. +Unset Strict Implicit. +Unset Printing Implicit Defensive. + +Module Finite. + +Section RawMixin. + +Variable T : eqType. + +Definition axiom e := forall x : T, count_mem x e = 1. + +Lemma uniq_enumP e : uniq e -> e =i T -> axiom e. +Proof. by move=> Ue sT x; rewrite count_uniq_mem ?sT. Qed. + +Record mixin_of := Mixin { + mixin_base : Countable.mixin_of T; + mixin_enum : seq T; + _ : axiom mixin_enum +}. + +End RawMixin. + +Section Mixins. + +Variable T : countType. + +Definition EnumMixin := + let: Countable.Pack _ (Countable.Class _ m) _ as cT := T + return forall e : seq cT, axiom e -> mixin_of cT in + @Mixin (EqType _ _) m. + +Definition UniqMixin e Ue eT := @EnumMixin e (uniq_enumP Ue eT). + +Variable n : nat. + +Definition count_enum := pmap (@pickle_inv T) (iota 0 n). + +Hypothesis ubT : forall x : T, pickle x < n. + +Lemma count_enumP : axiom count_enum. +Proof. +apply: uniq_enumP (pmap_uniq (@pickle_invK T) (iota_uniq _ _)) _ => x. +by rewrite mem_pmap -pickleK_inv map_f // mem_iota ubT. +Qed. + +Definition CountMixin := EnumMixin count_enumP. + +End Mixins. + +Section ClassDef. + +Record class_of T := Class { + base : Choice.class_of T; + mixin : mixin_of (Equality.Pack base T) +}. +Definition base2 T c := Countable.Class (@base T c) (mixin_base (mixin c)). +Local Coercion base : class_of >-> Choice.class_of. + +Structure type : Type := Pack {sort; _ : class_of sort; _ : Type}. +Local Coercion sort : type >-> Sortclass. +Variables (T : Type) (cT : type). +Definition class := let: Pack _ c _ as cT' := cT return class_of cT' in c. +Definition clone c of phant_id class c := @Pack T c T. +Let xT := let: Pack T _ _ := cT in T. +Notation xclass := (class : class_of xT). + +Definition pack b0 (m0 : mixin_of (EqType T b0)) := + fun bT b & phant_id (Choice.class bT) b => + fun m & phant_id m0 m => Pack (@Class T b m) T. + +Definition eqType := @Equality.Pack cT xclass xT. +Definition choiceType := @Choice.Pack cT xclass xT. +Definition countType := @Countable.Pack cT (base2 xclass) xT. + +End ClassDef. + +Module Import Exports. +Coercion mixin_base : mixin_of >-> Countable.mixin_of. +Coercion base : class_of >-> Choice.class_of. +Coercion mixin : class_of >-> mixin_of. +Coercion base2 : class_of >-> Countable.class_of. +Coercion sort : type >-> Sortclass. +Coercion eqType : type >-> Equality.type. +Canonical eqType. +Coercion choiceType : type >-> Choice.type. +Canonical choiceType. +Coercion countType : type >-> Countable.type. +Canonical countType. +Notation finType := type. +Notation FinType T m := (@pack T _ m _ _ id _ id). +Notation FinMixin := EnumMixin. +Notation UniqFinMixin := UniqMixin. +Notation "[ 'finType' 'of' T 'for' cT ]" := (@clone T cT _ idfun) + (at level 0, format "[ 'finType' 'of' T 'for' cT ]") : form_scope. +Notation "[ 'finType' 'of' T ]" := (@clone T _ _ id) + (at level 0, format "[ 'finType' 'of' T ]") : form_scope. +End Exports. + +Module Type EnumSig. +Parameter enum : forall cT : type, seq cT. +Axiom enumDef : enum = fun cT => mixin_enum (class cT). +End EnumSig. + +Module EnumDef : EnumSig. +Definition enum cT := mixin_enum (class cT). +Definition enumDef := erefl enum. +End EnumDef. + +Notation enum := EnumDef.enum. + +End Finite. +Export Finite.Exports. + +Canonical finEnum_unlock := Unlockable Finite.EnumDef.enumDef. + +(* Workaround for the silly syntactic uniformity restriction on coercions; *) +(* this avoids a cross-dependency between finset.v and prime.v for the *) +(* definition of the \pi(A) notation. *) +Definition fin_pred_sort (T : finType) (pT : predType T) := pred_sort pT. +Identity Coercion pred_sort_of_fin : fin_pred_sort >-> pred_sort. + +Definition enum_mem T (mA : mem_pred _) := filter mA (Finite.enum T). +Notation enum A := (enum_mem (mem A)). +Definition pick (T : finType) (P : pred T) := ohead (enum P). + +Notation "[ 'pick' x | P ]" := (pick (fun x => P%B)) + (at level 0, x ident, format "[ 'pick' x | P ]") : form_scope. +Notation "[ 'pick' x : T | P ]" := (pick (fun x : T => P%B)) + (at level 0, x ident, only parsing) : form_scope. +Definition pick_true T (x : T) := true. +Notation "[ 'pick' x : T ]" := [pick x : T | pick_true x] + (at level 0, x ident, only parsing). +Notation "[ 'pick' x ]" := [pick x : _] + (at level 0, x ident, only parsing) : form_scope. +Notation "[ 'pic' 'k' x : T ]" := [pick x : T | pick_true _] + (at level 0, x ident, format "[ 'pic' 'k' x : T ]") : form_scope. +Notation "[ 'pick' x | P & Q ]" := [pick x | P && Q ] + (at level 0, x ident, + format "[ '[hv ' 'pick' x | P '/ ' & Q ] ']'") : form_scope. +Notation "[ 'pick' x : T | P & Q ]" := [pick x : T | P && Q ] + (at level 0, x ident, only parsing) : form_scope. +Notation "[ 'pick' x 'in' A ]" := [pick x | x \in A] + (at level 0, x ident, format "[ 'pick' x 'in' A ]") : form_scope. +Notation "[ 'pick' x : T 'in' A ]" := [pick x : T | x \in A] + (at level 0, x ident, only parsing) : form_scope. +Notation "[ 'pick' x 'in' A | P ]" := [pick x | x \in A & P ] + (at level 0, x ident, + format "[ '[hv ' 'pick' x 'in' A '/ ' | P ] ']'") : form_scope. +Notation "[ 'pick' x : T 'in' A | P ]" := [pick x : T | x \in A & P ] + (at level 0, x ident, only parsing) : form_scope. +Notation "[ 'pick' x 'in' A | P & Q ]" := [pick x in A | P && Q] + (at level 0, x ident, format + "[ '[hv ' 'pick' x 'in' A '/ ' | P '/ ' & Q ] ']'") : form_scope. +Notation "[ 'pick' x : T 'in' A | P & Q ]" := [pick x : T in A | P && Q] + (at level 0, x ident, only parsing) : form_scope. + +(* We lock the definitions of card and subset to mitigate divergence of the *) +(* Coq term comparison algorithm. *) + +Local Notation card_type := (forall T : finType, mem_pred T -> nat). +Local Notation card_def := (fun T mA => size (enum_mem mA)). +Module Type CardDefSig. +Parameter card : card_type. Axiom cardEdef : card = card_def. +End CardDefSig. +Module CardDef : CardDefSig. +Definition card : card_type := card_def. Definition cardEdef := erefl card. +End CardDef. +(* Should be Include, but for a silly restriction: can't Include at toplevel! *) +Export CardDef. + +Canonical card_unlock := Unlockable cardEdef. +(* A is at level 99 to allow the notation #|G : H| in groups. *) +Notation "#| A |" := (card (mem A)) + (at level 0, A at level 99, format "#| A |") : nat_scope. + +Definition pred0b (T : finType) (P : pred T) := #|P| == 0. +Prenex Implicits pred0b. + +Module FiniteQuant. + +CoInductive quantified := Quantified of bool. + +Delimit Scope fin_quant_scope with Q. (* Bogus, only used to declare scope. *) +Bind Scope fin_quant_scope with quantified. + +Notation "F ^*" := (Quantified F) (at level 2). +Notation "F ^~" := (~~ F) (at level 2). + +Section Definitions. + +Variable T : finType. +Implicit Types (B : quantified) (x y : T). + +Definition quant0b Bp := pred0b [pred x : T | let: F^* := Bp x x in F]. +(* The first redundant argument protects the notation from Coq's K-term *) +(* display kludge; the second protects it from simpl and /=. *) +Definition ex B x y := B. +(* Binding the predicate value rather than projecting it prevents spurious *) +(* unfolding of the boolean connectives by unification. *) +Definition all B x y := let: F^* := B in F^~^*. +Definition all_in C B x y := let: F^* := B in (C ==> F)^~^*. +Definition ex_in C B x y := let: F^* := B in (C && F)^*. + +End Definitions. + +Notation "[ x | B ]" := (quant0b (fun x => B x)) (at level 0, x ident). +Notation "[ x : T | B ]" := (quant0b (fun x : T => B x)) (at level 0, x ident). + +Module Exports. + +Notation ", F" := F^* (at level 200, format ", '/ ' F") : fin_quant_scope. + +Notation "[ 'forall' x B ]" := [x | all B] + (at level 0, x at level 99, B at level 200, + format "[ '[hv' 'forall' x B ] ']'") : bool_scope. + +Notation "[ 'forall' x : T B ]" := [x : T | all B] + (at level 0, x at level 99, B at level 200, only parsing) : bool_scope. +Notation "[ 'forall' ( x | C ) B ]" := [x | all_in C B] + (at level 0, x at level 99, B at level 200, + format "[ '[hv' '[' 'forall' ( x '/ ' | C ) ']' B ] ']'") : bool_scope. +Notation "[ 'forall' ( x : T | C ) B ]" := [x : T | all_in C B] + (at level 0, x at level 99, B at level 200, only parsing) : bool_scope. +Notation "[ 'forall' x 'in' A B ]" := [x | all_in (x \in A) B] + (at level 0, x at level 99, B at level 200, + format "[ '[hv' '[' 'forall' x '/ ' 'in' A ']' B ] ']'") : bool_scope. +Notation "[ 'forall' x : T 'in' A B ]" := [x : T | all_in (x \in A) B] + (at level 0, x at level 99, B at level 200, only parsing) : bool_scope. +Notation ", 'forall' x B" := [x | all B]^* + (at level 200, x at level 99, B at level 200, + format ", '/ ' 'forall' x B") : fin_quant_scope. +Notation ", 'forall' x : T B" := [x : T | all B]^* + (at level 200, x at level 99, B at level 200, only parsing) : fin_quant_scope. +Notation ", 'forall' ( x | C ) B" := [x | all_in C B]^* + (at level 200, x at level 99, B at level 200, + format ", '/ ' '[' 'forall' ( x '/ ' | C ) ']' B") : fin_quant_scope. +Notation ", 'forall' ( x : T | C ) B" := [x : T | all_in C B]^* + (at level 200, x at level 99, B at level 200, only parsing) : fin_quant_scope. +Notation ", 'forall' x 'in' A B" := [x | all_in (x \in A) B]^* + (at level 200, x at level 99, B at level 200, + format ", '/ ' '[' 'forall' x '/ ' 'in' A ']' B") : bool_scope. +Notation ", 'forall' x : T 'in' A B" := [x : T | all_in (x \in A) B]^* + (at level 200, x at level 99, B at level 200, only parsing) : bool_scope. + +Notation "[ 'exists' x B ]" := [x | ex B]^~ + (at level 0, x at level 99, B at level 200, + format "[ '[hv' 'exists' x B ] ']'") : bool_scope. +Notation "[ 'exists' x : T B ]" := [x : T | ex B]^~ + (at level 0, x at level 99, B at level 200, only parsing) : bool_scope. +Notation "[ 'exists' ( x | C ) B ]" := [x | ex_in C B]^~ + (at level 0, x at level 99, B at level 200, + format "[ '[hv' '[' 'exists' ( x '/ ' | C ) ']' B ] ']'") : bool_scope. +Notation "[ 'exists' ( x : T | C ) B ]" := [x : T | ex_in C B]^~ + (at level 0, x at level 99, B at level 200, only parsing) : bool_scope. +Notation "[ 'exists' x 'in' A B ]" := [x | ex_in (x \in A) B]^~ + (at level 0, x at level 99, B at level 200, + format "[ '[hv' '[' 'exists' x '/ ' 'in' A ']' B ] ']'") : bool_scope. +Notation "[ 'exists' x : T 'in' A B ]" := [x : T | ex_in (x \in A) B]^~ + (at level 0, x at level 99, B at level 200, only parsing) : bool_scope. +Notation ", 'exists' x B" := [x | ex B]^~^* + (at level 200, x at level 99, B at level 200, + format ", '/ ' 'exists' x B") : fin_quant_scope. +Notation ", 'exists' x : T B" := [x : T | ex B]^~^* + (at level 200, x at level 99, B at level 200, only parsing) : fin_quant_scope. +Notation ", 'exists' ( x | C ) B" := [x | ex_in C B]^~^* + (at level 200, x at level 99, B at level 200, + format ", '/ ' '[' 'exists' ( x '/ ' | C ) ']' B") : fin_quant_scope. +Notation ", 'exists' ( x : T | C ) B" := [x : T | ex_in C B]^~^* + (at level 200, x at level 99, B at level 200, only parsing) : fin_quant_scope. +Notation ", 'exists' x 'in' A B" := [x | ex_in (x \in A) B]^~^* + (at level 200, x at level 99, B at level 200, + format ", '/ ' '[' 'exists' x '/ ' 'in' A ']' B") : bool_scope. +Notation ", 'exists' x : T 'in' A B" := [x : T | ex_in (x \in A) B]^~^* + (at level 200, x at level 99, B at level 200, only parsing) : bool_scope. + +End Exports. + +End FiniteQuant. +Export FiniteQuant.Exports. + +Definition disjoint T (A B : mem_pred _) := @pred0b T (predI A B). +Notation "[ 'disjoint' A & B ]" := (disjoint (mem A) (mem B)) + (at level 0, + format "'[hv' [ 'disjoint' '/ ' A '/' & B ] ']'") : bool_scope. + +Notation Local subset_type := (forall (T : finType) (A B : mem_pred T), bool). +Notation Local subset_def := (fun T A B => pred0b (predD A B)). +Module Type SubsetDefSig. +Parameter subset : subset_type. Axiom subsetEdef : subset = subset_def. +End SubsetDefSig. +Module Export SubsetDef : SubsetDefSig. +Definition subset : subset_type := subset_def. +Definition subsetEdef := erefl subset. +End SubsetDef. +Canonical subset_unlock := Unlockable subsetEdef. +Notation "A \subset B" := (subset (mem A) (mem B)) + (at level 70, no associativity) : bool_scope. + +Definition proper T A B := @subset T A B && ~~ subset B A. +Notation "A \proper B" := (proper (mem A) (mem B)) + (at level 70, no associativity) : bool_scope. + +(* image, xinv, inv, and ordinal operations will be defined later. *) + +Section OpsTheory. + +Variable T : finType. + +Implicit Types A B C P Q : pred T. +Implicit Types x y : T. +Implicit Type s : seq T. + +Lemma enumP : Finite.axiom (Finite.enum T). +Proof. by rewrite unlock; case T => ? [? []]. Qed. + +Section EnumPick. + +Variable P : pred T. + +Lemma enumT : enum T = Finite.enum T. +Proof. exact: filter_predT. Qed. + +Lemma mem_enum A : enum A =i A. +Proof. by move=> x; rewrite mem_filter andbC -has_pred1 has_count enumP. Qed. + +Lemma enum_uniq : uniq (enum P). +Proof. +by apply/filter_uniq/count_mem_uniq => x; rewrite enumP -enumT mem_enum. +Qed. + +Lemma enum0 : enum pred0 = Nil T. Proof. exact: filter_pred0. Qed. + +Lemma enum1 x : enum (pred1 x) = [:: x]. +Proof. +rewrite [enum _](all_pred1P x _ _); first by rewrite size_filter enumP. +by apply/allP=> y; rewrite mem_enum. +Qed. + +CoInductive pick_spec : option T -> Type := + | Pick x of P x : pick_spec (Some x) + | Nopick of P =1 xpred0 : pick_spec None. + +Lemma pickP : pick_spec (pick P). +Proof. +rewrite /pick; case: (enum _) (mem_enum P) => [|x s] Pxs /=. + by right; apply: fsym. +by left; rewrite -[P _]Pxs mem_head. +Qed. + +End EnumPick. + +Lemma eq_enum P Q : P =i Q -> enum P = enum Q. +Proof. by move=> eqPQ; apply: eq_filter. Qed. + +Lemma eq_pick P Q : P =1 Q -> pick P = pick Q. +Proof. by move=> eqPQ; rewrite /pick (eq_enum eqPQ). Qed. + +Lemma cardE A : #|A| = size (enum A). +Proof. by rewrite unlock. Qed. + +Lemma eq_card A B : A =i B -> #|A| = #|B|. +Proof. by move=>eqAB; rewrite !cardE (eq_enum eqAB). Qed. + +Lemma eq_card_trans A B n : #|A| = n -> B =i A -> #|B| = n. +Proof. by move <-; apply: eq_card. Qed. + +Lemma card0 : #|@pred0 T| = 0. Proof. by rewrite cardE enum0. Qed. + +Lemma cardT : #|T| = size (enum T). Proof. by rewrite cardE. Qed. + +Lemma card1 x : #|pred1 x| = 1. +Proof. by rewrite cardE enum1. Qed. + +Lemma eq_card0 A : A =i pred0 -> #|A| = 0. +Proof. exact: eq_card_trans card0. Qed. + +Lemma eq_cardT A : A =i predT -> #|A| = size (enum T). +Proof. exact: eq_card_trans cardT. Qed. + +Lemma eq_card1 x A : A =i pred1 x -> #|A| = 1. +Proof. exact: eq_card_trans (card1 x). Qed. + +Lemma cardUI A B : #|[predU A & B]| + #|[predI A & B]| = #|A| + #|B|. +Proof. by rewrite !cardE !size_filter count_predUI. Qed. + +Lemma cardID B A : #|[predI A & B]| + #|[predD A & B]| = #|A|. +Proof. +rewrite -cardUI addnC [#|predI _ _|]eq_card0 => [|x] /=. + by apply: eq_card => x; rewrite !inE andbC -andb_orl orbN. +by rewrite !inE -!andbA andbC andbA andbN. +Qed. + +Lemma cardC A : #|A| + #|[predC A]| = #|T|. +Proof. by rewrite !cardE !size_filter count_predC. Qed. + +Lemma cardU1 x A : #|[predU1 x & A]| = (x \notin A) + #|A|. +Proof. +case Ax: (x \in A). + by apply: eq_card => y; rewrite inE /=; case: eqP => // ->. +rewrite /= -(card1 x) -cardUI addnC. +rewrite [#|predI _ _|]eq_card0 => [|y /=]; first exact: eq_card. +by rewrite !inE; case: eqP => // ->. +Qed. + +Lemma card2 x y : #|pred2 x y| = (x != y).+1. +Proof. by rewrite cardU1 card1 addn1. Qed. + +Lemma cardC1 x : #|predC1 x| = #|T|.-1. +Proof. by rewrite -(cardC (pred1 x)) card1. Qed. + +Lemma cardD1 x A : #|A| = (x \in A) + #|[predD1 A & x]|. +Proof. +case Ax: (x \in A); last first. + by apply: eq_card => y; rewrite !inE /=; case: eqP => // ->. +rewrite /= -(card1 x) -cardUI addnC /=. +rewrite [#|predI _ _|]eq_card0 => [|y]; last by rewrite !inE; case: eqP. +by apply: eq_card => y; rewrite !inE; case: eqP => // ->. +Qed. + +Lemma max_card A : #|A| <= #|T|. +Proof. by rewrite -(cardC A) leq_addr. Qed. + +Lemma card_size s : #|s| <= size s. +Proof. +elim: s => [|x s IHs] /=; first by rewrite card0. +by rewrite cardU1 /=; case: (~~ _) => //; apply: leqW. +Qed. + +Lemma card_uniqP s : reflect (#|s| = size s) (uniq s). +Proof. +elim: s => [|x s IHs]; first by left; apply: card0. +rewrite cardU1 /= /addn; case: {+}(x \in s) => /=. + by right=> card_Ssz; have:= card_size s; rewrite card_Ssz ltnn. +by apply: (iffP IHs) => [<-| [<-]]. +Qed. + +Lemma card0_eq A : #|A| = 0 -> A =i pred0. +Proof. by move=> A0 x; apply/idP => Ax; rewrite (cardD1 x) Ax in A0. Qed. + +Lemma pred0P P : reflect (P =1 pred0) (pred0b P). +Proof. by apply: (iffP eqP); [apply: card0_eq | apply: eq_card0]. Qed. + +Lemma pred0Pn P : reflect (exists x, P x) (~~ pred0b P). +Proof. +case: (pickP P) => [x Px | P0]. + by rewrite (introN (pred0P P)) => [|P0]; [left; exists x | rewrite P0 in Px]. +by rewrite -lt0n eq_card0 //; right=> [[x]]; rewrite P0. +Qed. + +Lemma card_gt0P A : reflect (exists i, i \in A) (#|A| > 0). +Proof. by rewrite lt0n; apply: pred0Pn. Qed. + +Lemma subsetE A B : (A \subset B) = pred0b [predD A & B]. +Proof. by rewrite unlock. Qed. + +Lemma subsetP A B : reflect {subset A <= B} (A \subset B). +Proof. +rewrite unlock; apply: (iffP (pred0P _)) => [AB0 x | sAB x /=]. + by apply/implyP; apply/idPn; rewrite negb_imply andbC [_ && _]AB0. +by rewrite andbC -negb_imply; apply/negbF/implyP; apply: sAB. +Qed. + +Lemma subsetPn A B : + reflect (exists2 x, x \in A & x \notin B) (~~ (A \subset B)). +Proof. +rewrite unlock; apply: (iffP (pred0Pn _)) => [[x] | [x Ax nBx]]. + by case/andP; exists x. +by exists x; rewrite /= nBx. +Qed. + +Lemma subset_leq_card A B : A \subset B -> #|A| <= #|B|. +Proof. +move=> sAB. +rewrite -(cardID A B) [#|predI _ _|](@eq_card _ A) ?leq_addr //= => x. +by rewrite !inE andbC; case Ax: (x \in A) => //; apply: subsetP Ax. +Qed. + +Lemma subxx_hint (mA : mem_pred T) : subset mA mA. +Proof. +by case: mA => A; have:= introT (subsetP A A); rewrite !unlock => ->. +Qed. +Hint Resolve subxx_hint. + +(* The parametrization by predType makes it easier to apply subxx. *) +Lemma subxx (pT : predType T) (pA : pT) : pA \subset pA. +Proof. by []. Qed. + +Lemma eq_subset A1 A2 : A1 =i A2 -> subset (mem A1) =1 subset (mem A2). +Proof. +move=> eqA12 [B]; rewrite !unlock; congr (_ == 0). +by apply: eq_card => x; rewrite inE /= eqA12. +Qed. + +Lemma eq_subset_r B1 B2 : B1 =i B2 -> + (@subset T)^~ (mem B1) =1 (@subset T)^~ (mem B2). +Proof. +move=> eqB12 [A]; rewrite !unlock; congr (_ == 0). +by apply: eq_card => x; rewrite !inE /= eqB12. +Qed. + +Lemma eq_subxx A B : A =i B -> A \subset B. +Proof. by move/eq_subset->. Qed. + +Lemma subset_predT A : A \subset T. +Proof. by apply/subsetP. Qed. + +Lemma predT_subset A : T \subset A -> forall x, x \in A. +Proof. by move/subsetP=> allA x; apply: allA. Qed. + +Lemma subset_pred1 A x : (pred1 x \subset A) = (x \in A). +Proof. by apply/subsetP/idP=> [-> // | Ax y /eqP-> //]; apply: eqxx. Qed. + +Lemma subset_eqP A B : reflect (A =i B) ((A \subset B) && (B \subset A)). +Proof. +apply: (iffP andP) => [[sAB sBA] x| eqAB]; last by rewrite !eq_subxx. +by apply/idP/idP; apply: subsetP. +Qed. + +Lemma subset_cardP A B : #|A| = #|B| -> reflect (A =i B) (A \subset B). +Proof. +move=> eqcAB; case: (subsetP A B) (subset_eqP A B) => //= sAB. +case: (subsetP B A) => [//|[]] x Bx; apply/idPn => Ax. +case/idP: (ltnn #|A|); rewrite {2}eqcAB (cardD1 x B) Bx /=. +apply: subset_leq_card; apply/subsetP=> y Ay; rewrite inE /= andbC. +by rewrite sAB //; apply/eqP => eqyx; rewrite -eqyx Ay in Ax. +Qed. + +Lemma subset_leqif_card A B : A \subset B -> #|A| <= #|B| ?= iff (B \subset A). +Proof. +move=> sAB; split; [exact: subset_leq_card | apply/eqP/idP]. + by move/subset_cardP=> sABP; rewrite (eq_subset_r (sABP sAB)). +by move=> sBA; apply: eq_card; apply/subset_eqP; rewrite sAB. +Qed. + +Lemma subset_trans A B C : A \subset B -> B \subset C -> A \subset C. +Proof. +by move/subsetP=> sAB /subsetP=> sBC; apply/subsetP=> x /sAB; apply: sBC. +Qed. + +Lemma subset_all s A : (s \subset A) = all (mem A) s. +Proof. exact: (sameP (subsetP _ _) allP). Qed. + +Lemma properE A B : A \proper B = (A \subset B) && ~~(B \subset A). +Proof. by []. Qed. + +Lemma properP A B : + reflect (A \subset B /\ (exists2 x, x \in B & x \notin A)) (A \proper B). +Proof. +by rewrite properE; apply: (iffP andP) => [] [-> /subsetPn]. +Qed. + +Lemma proper_sub A B : A \proper B -> A \subset B. +Proof. by case/andP. Qed. + +Lemma proper_subn A B : A \proper B -> ~~ (B \subset A). +Proof. by case/andP. Qed. + +Lemma proper_trans A B C : A \proper B -> B \proper C -> A \proper C. +Proof. +case/properP=> sAB [x Bx nAx] /properP[sBC [y Cy nBy]]. +rewrite properE (subset_trans sAB) //=; apply/subsetPn; exists y => //. +by apply: contra nBy; apply: subsetP. +Qed. + +Lemma proper_sub_trans A B C : A \proper B -> B \subset C -> A \proper C. +Proof. +case/properP=> sAB [x Bx nAx] sBC; rewrite properE (subset_trans sAB) //. +by apply/subsetPn; exists x; rewrite ?(subsetP _ _ sBC). +Qed. + +Lemma sub_proper_trans A B C : A \subset B -> B \proper C -> A \proper C. +Proof. +move=> sAB /properP[sBC [x Cx nBx]]; rewrite properE (subset_trans sAB) //. +by apply/subsetPn; exists x => //; apply: contra nBx; apply: subsetP. +Qed. + +Lemma proper_card A B : A \proper B -> #|A| < #|B|. +Proof. +by case/andP=> sAB nsBA; rewrite ltn_neqAle !(subset_leqif_card sAB) andbT. +Qed. + +Lemma proper_irrefl A : ~~ (A \proper A). +Proof. by rewrite properE subxx. Qed. + +Lemma properxx A : (A \proper A) = false. +Proof. by rewrite properE subxx. Qed. + +Lemma eq_proper A B : A =i B -> proper (mem A) =1 proper (mem B). +Proof. +move=> eAB [C]; congr (_ && _); first exact: (eq_subset eAB). +by rewrite (eq_subset_r eAB). +Qed. + +Lemma eq_proper_r A B : A =i B -> + (@proper T)^~ (mem A) =1 (@proper T)^~ (mem B). +Proof. +move=> eAB [C]; congr (_ && _); first exact: (eq_subset_r eAB). +by rewrite (eq_subset eAB). +Qed. + +Lemma disjoint_sym A B : [disjoint A & B] = [disjoint B & A]. +Proof. by congr (_ == 0); apply: eq_card => x; apply: andbC. Qed. + +Lemma eq_disjoint A1 A2 : A1 =i A2 -> disjoint (mem A1) =1 disjoint (mem A2). +Proof. +by move=> eqA12 [B]; congr (_ == 0); apply: eq_card => x; rewrite !inE eqA12. +Qed. + +Lemma eq_disjoint_r B1 B2 : B1 =i B2 -> + (@disjoint T)^~ (mem B1) =1 (@disjoint T)^~ (mem B2). +Proof. +by move=> eqB12 [A]; congr (_ == 0); apply: eq_card => x; rewrite !inE eqB12. +Qed. + +Lemma subset_disjoint A B : (A \subset B) = [disjoint A & [predC B]]. +Proof. by rewrite disjoint_sym unlock. Qed. + +Lemma disjoint_subset A B : [disjoint A & B] = (A \subset [predC B]). +Proof. +by rewrite subset_disjoint; apply: eq_disjoint_r => x; rewrite !inE /= negbK. +Qed. + +Lemma disjoint_trans A B C : + A \subset B -> [disjoint B & C] -> [disjoint A & C]. +Proof. by rewrite 2!disjoint_subset; apply: subset_trans. Qed. + +Lemma disjoint0 A : [disjoint pred0 & A]. +Proof. exact/pred0P. Qed. + +Lemma eq_disjoint0 A B : A =i pred0 -> [disjoint A & B]. +Proof. by move/eq_disjoint->; apply: disjoint0. Qed. + +Lemma disjoint1 x A : [disjoint pred1 x & A] = (x \notin A). +Proof. +apply/negbRL/(sameP (pred0Pn _)). +apply: introP => [Ax | notAx [_ /andP[/eqP->]]]; last exact: negP. +by exists x; rewrite !inE eqxx. +Qed. + +Lemma eq_disjoint1 x A B : + A =i pred1 x -> [disjoint A & B] = (x \notin B). +Proof. by move/eq_disjoint->; apply: disjoint1. Qed. + +Lemma disjointU A B C : + [disjoint predU A B & C] = [disjoint A & C] && [disjoint B & C]. +Proof. +case: [disjoint A & C] / (pred0P (xpredI A C)) => [A0 | nA0] /=. + by congr (_ == 0); apply: eq_card => x; rewrite [x \in _]andb_orl A0. +apply/pred0P=> nABC; case: nA0 => x; apply/idPn=> /=; move/(_ x): nABC. +by rewrite [_ x]andb_orl; case/norP. +Qed. + +Lemma disjointU1 x A B : + [disjoint predU1 x A & B] = (x \notin B) && [disjoint A & B]. +Proof. by rewrite disjointU disjoint1. Qed. + +Lemma disjoint_cons x s B : + [disjoint x :: s & B] = (x \notin B) && [disjoint s & B]. +Proof. exact: disjointU1. Qed. + +Lemma disjoint_has s A : [disjoint s & A] = ~~ has (mem A) s. +Proof. +rewrite -(@eq_has _ (mem (enum A))) => [|x]; last exact: mem_enum. +rewrite has_sym has_filter -filter_predI -has_filter has_count -eqn0Ngt. +by rewrite -size_filter /disjoint /pred0b unlock. +Qed. + +Lemma disjoint_cat s1 s2 A : + [disjoint s1 ++ s2 & A] = [disjoint s1 & A] && [disjoint s2 & A]. +Proof. by rewrite !disjoint_has has_cat negb_or. Qed. + +End OpsTheory. + +Hint Resolve subxx_hint. + +Implicit Arguments pred0P [T P]. +Implicit Arguments pred0Pn [T P]. +Implicit Arguments subsetP [T A B]. +Implicit Arguments subsetPn [T A B]. +Implicit Arguments subset_eqP [T A B]. +Implicit Arguments card_uniqP [T s]. +Implicit Arguments properP [T A B]. +Prenex Implicits pred0P pred0Pn subsetP subsetPn subset_eqP card_uniqP. + +(**********************************************************************) +(* *) +(* Boolean quantifiers for finType *) +(* *) +(**********************************************************************) + +Section QuantifierCombinators. + +Variables (T : finType) (P : pred T) (PP : T -> Prop). +Hypothesis viewP : forall x, reflect (PP x) (P x). + +Lemma existsPP : reflect (exists x, PP x) [exists x, P x]. +Proof. by apply: (iffP pred0Pn) => -[x /viewP]; exists x. Qed. + +Lemma forallPP : reflect (forall x, PP x) [forall x, P x]. +Proof. by apply: (iffP pred0P) => /= allP x; have /viewP//=-> := allP x. Qed. + +End QuantifierCombinators. + +Notation "'exists_ view" := (existsPP (fun _ => view)) + (at level 4, right associativity, format "''exists_' view"). +Notation "'forall_ view" := (forallPP (fun _ => view)) + (at level 4, right associativity, format "''forall_' view"). + +Section Quantifiers. + +Variables (T : finType) (rT : T -> eqType). +Implicit Type (D P : pred T) (f : forall x, rT x). + +Lemma forallP P : reflect (forall x, P x) [forall x, P x]. +Proof. exact: 'forall_idP. Qed. + +Lemma eqfunP f1 f2 : reflect (forall x, f1 x = f2 x) [forall x, f1 x == f2 x]. +Proof. exact: 'forall_eqP. Qed. + +Lemma forall_inP D P : reflect (forall x, D x -> P x) [forall (x | D x), P x]. +Proof. exact: 'forall_implyP. Qed. + +Lemma eqfun_inP D f1 f2 : + reflect {in D, forall x, f1 x = f2 x} [forall (x | x \in D), f1 x == f2 x]. +Proof. by apply: (iffP 'forall_implyP) => eq_f12 x Dx; apply/eqP/eq_f12. Qed. + +Lemma existsP P : reflect (exists x, P x) [exists x, P x]. +Proof. exact: 'exists_idP. Qed. + +Lemma exists_eqP f1 f2 : + reflect (exists x, f1 x = f2 x) [exists x, f1 x == f2 x]. +Proof. exact: 'exists_eqP. Qed. + +Lemma exists_inP D P : reflect (exists2 x, D x & P x) [exists (x | D x), P x]. +Proof. by apply: (iffP 'exists_andP) => [[x []] | [x]]; exists x. Qed. + +Lemma exists_eq_inP D f1 f2 : + reflect (exists2 x, D x & f1 x = f2 x) [exists (x | D x), f1 x == f2 x]. +Proof. by apply: (iffP (exists_inP _ _)) => [] [x Dx /eqP]; exists x. Qed. + +Lemma eq_existsb P1 P2 : P1 =1 P2 -> [exists x, P1 x] = [exists x, P2 x]. +Proof. by move=> eqP12; congr (_ != 0); apply: eq_card. Qed. + +Lemma eq_existsb_in D P1 P2 : + (forall x, D x -> P1 x = P2 x) -> + [exists (x | D x), P1 x] = [exists (x | D x), P2 x]. +Proof. by move=> eqP12; apply: eq_existsb => x; apply: andb_id2l => /eqP12. Qed. + +Lemma eq_forallb P1 P2 : P1 =1 P2 -> [forall x, P1 x] = [forall x, P2 x]. +Proof. by move=> eqP12; apply/negb_inj/eq_existsb=> /= x; rewrite eqP12. Qed. + +Lemma eq_forallb_in D P1 P2 : + (forall x, D x -> P1 x = P2 x) -> + [forall (x | D x), P1 x] = [forall (x | D x), P2 x]. +Proof. +by move=> eqP12; apply: eq_forallb => i; case Di: (D i); rewrite // eqP12. +Qed. + +Lemma negb_forall P : ~~ [forall x, P x] = [exists x, ~~ P x]. +Proof. by []. Qed. + +Lemma negb_forall_in D P : + ~~ [forall (x | D x), P x] = [exists (x | D x), ~~ P x]. +Proof. by apply: eq_existsb => x; rewrite negb_imply. Qed. + +Lemma negb_exists P : ~~ [exists x, P x] = [forall x, ~~ P x]. +Proof. by apply/negbLR/esym/eq_existsb=> x; apply: negbK. Qed. + +Lemma negb_exists_in D P : + ~~ [exists (x | D x), P x] = [forall (x | D x), ~~ P x]. +Proof. by rewrite negb_exists; apply/eq_forallb => x; rewrite [~~ _]fun_if. Qed. + +End Quantifiers. + +Implicit Arguments forallP [T P]. +Implicit Arguments eqfunP [T rT f1 f2]. +Implicit Arguments forall_inP [T D P]. +Implicit Arguments eqfun_inP [T rT D f1 f2]. +Implicit Arguments existsP [T P]. +Implicit Arguments exists_eqP [T rT f1 f2]. +Implicit Arguments exists_inP [T D P]. +Implicit Arguments exists_eq_inP [T rT D f1 f2]. + +Section Extrema. + +Variables (I : finType) (i0 : I) (P : pred I) (F : I -> nat). + +Let arg_pred ord := [pred i | P i & [forall (j | P j), ord (F i) (F j)]]. + +Definition arg_min := odflt i0 (pick (arg_pred leq)). + +Definition arg_max := odflt i0 (pick (arg_pred geq)). + +CoInductive extremum_spec (ord : rel nat) : I -> Type := + ExtremumSpec i of P i & (forall j, P j -> ord (F i) (F j)) + : extremum_spec ord i. + +Hypothesis Pi0 : P i0. + +Let FP n := [exists (i | P i), F i == n]. +Let FP_F i : P i -> FP (F i). +Proof. by move=> Pi; apply/existsP; exists i; rewrite Pi /=. Qed. +Let exFP : exists n, FP n. Proof. by exists (F i0); apply: FP_F. Qed. + +Lemma arg_minP : extremum_spec leq arg_min. +Proof. +rewrite /arg_min; case: pickP => [i /andP[Pi /forallP/= min_i] | no_i]. + by split=> // j; apply/implyP. +case/ex_minnP: exFP => n ex_i min_i; case/pred0P: ex_i => i /=. +apply: contraFF (no_i i) => /andP[Pi /eqP def_n]; rewrite /= Pi. +by apply/forall_inP=> j Pj; rewrite def_n min_i ?FP_F. +Qed. + +Lemma arg_maxP : extremum_spec geq arg_max. +Proof. +rewrite /arg_max; case: pickP => [i /andP[Pi /forall_inP/= max_i] | no_i]. + by split=> // j; apply/implyP. +have (n): FP n -> n <= foldr maxn 0 (map F (enum P)). + case/existsP=> i; rewrite -[P i]mem_enum andbC /= => /andP[/eqP <-]. + elim: (enum P) => //= j e IHe; rewrite leq_max orbC !inE. + by case/predU1P=> [-> | /IHe-> //]; rewrite leqnn orbT. +case/ex_maxnP=> // n ex_i max_i; case/pred0P: ex_i => i /=. +apply: contraFF (no_i i) => /andP[Pi def_n]; rewrite /= Pi. +by apply/forall_inP=> j Pj; rewrite (eqP def_n) max_i ?FP_F. +Qed. + +End Extrema. + +Notation "[ 'arg' 'min_' ( i < i0 | P ) F ]" := + (arg_min i0 (fun i => P%B) (fun i => F)) + (at level 0, i, i0 at level 10, + format "[ 'arg' 'min_' ( i < i0 | P ) F ]") : form_scope. + +Notation "[ 'arg' 'min_' ( i < i0 'in' A ) F ]" := + [arg min_(i < i0 | i \in A) F] + (at level 0, i, i0 at level 10, + format "[ 'arg' 'min_' ( i < i0 'in' A ) F ]") : form_scope. + +Notation "[ 'arg' 'min_' ( i < i0 ) F ]" := [arg min_(i < i0 | true) F] + (at level 0, i, i0 at level 10, + format "[ 'arg' 'min_' ( i < i0 ) F ]") : form_scope. + +Notation "[ 'arg' 'max_' ( i > i0 | P ) F ]" := + (arg_max i0 (fun i => P%B) (fun i => F)) + (at level 0, i, i0 at level 10, + format "[ 'arg' 'max_' ( i > i0 | P ) F ]") : form_scope. + +Notation "[ 'arg' 'max_' ( i > i0 'in' A ) F ]" := + [arg max_(i > i0 | i \in A) F] + (at level 0, i, i0 at level 10, + format "[ 'arg' 'max_' ( i > i0 'in' A ) F ]") : form_scope. + +Notation "[ 'arg' 'max_' ( i > i0 ) F ]" := [arg max_(i > i0 | true) F] + (at level 0, i, i0 at level 10, + format "[ 'arg' 'max_' ( i > i0 ) F ]") : form_scope. + +(**********************************************************************) +(* *) +(* Boolean injectivity test for functions with a finType domain *) +(* *) +(**********************************************************************) + +Section Injectiveb. + +Variables (aT : finType) (rT : eqType) (f : aT -> rT). +Implicit Type D : pred aT. + +Definition dinjectiveb D := uniq (map f (enum D)). + +Definition injectiveb := dinjectiveb aT. + +Lemma dinjectivePn D : + reflect (exists2 x, x \in D & exists2 y, y \in [predD1 D & x] & f x = f y) + (~~ dinjectiveb D). +Proof. +apply: (iffP idP) => [injf | [x Dx [y Dxy eqfxy]]]; last first. + move: Dx; rewrite -(mem_enum D) => /rot_to[i E defE]. + rewrite /dinjectiveb -(rot_uniq i) -map_rot defE /=; apply/nandP; left. + rewrite inE /= -(mem_enum D) -(mem_rot i) defE inE in Dxy. + rewrite andb_orr andbC andbN in Dxy. + by rewrite eqfxy map_f //; case/andP: Dxy. +pose p := [pred x in D | [exists (y | y \in [predD1 D & x]), f x == f y]]. +case: (pickP p) => [x /= /andP[Dx /exists_inP[y Dxy /eqP eqfxy]] | no_p]. + by exists x; last exists y. +rewrite /dinjectiveb map_inj_in_uniq ?enum_uniq // in injf => x y Dx Dy eqfxy. +apply: contraNeq (negbT (no_p x)) => ne_xy /=; rewrite -mem_enum Dx. +by apply/existsP; exists y; rewrite /= !inE eq_sym ne_xy -mem_enum Dy eqfxy /=. +Qed. + +Lemma dinjectiveP D : reflect {in D &, injective f} (dinjectiveb D). +Proof. +rewrite -[dinjectiveb D]negbK. +case: dinjectivePn=> [noinjf | injf]; constructor. + case: noinjf => x Dx [y /andP[neqxy /= Dy] eqfxy] injf. + by case/eqP: neqxy; apply: injf. +move=> x y Dx Dy /= eqfxy; apply/eqP; apply/idPn=> nxy; case: injf. +by exists x => //; exists y => //=; rewrite inE /= eq_sym nxy. +Qed. + +Lemma injectivePn : + reflect (exists x, exists2 y, x != y & f x = f y) (~~ injectiveb). +Proof. +apply: (iffP (dinjectivePn _)) => [[x _ [y nxy eqfxy]] | [x [y nxy eqfxy]]]; + by exists x => //; exists y => //; rewrite inE /= andbT eq_sym in nxy *. +Qed. + +Lemma injectiveP : reflect (injective f) injectiveb. +Proof. by apply: (iffP (dinjectiveP _)) => injf x y => [|_ _]; apply: injf. Qed. + +End Injectiveb. + +Definition image_mem T T' f mA : seq T' := map f (@enum_mem T mA). +Notation image f A := (image_mem f (mem A)). +Notation "[ 'seq' F | x 'in' A ]" := (image (fun x => F) A) + (at level 0, F at level 99, x ident, + format "'[hv' [ 'seq' F '/ ' | x 'in' A ] ']'") : seq_scope. +Notation "[ 'seq' F | x : T 'in' A ]" := (image (fun x : T => F) A) + (at level 0, F at level 99, x ident, only parsing) : seq_scope. +Notation "[ 'seq' F | x : T ]" := + [seq F | x : T in sort_of_simpl_pred (@pred_of_argType T)] + (at level 0, F at level 99, x ident, + format "'[hv' [ 'seq' F '/ ' | x : T ] ']'") : seq_scope. +Notation "[ 'seq' F , x ]" := [seq F | x : _ ] + (at level 0, F at level 99, x ident, only parsing) : seq_scope. + +Definition codom T T' f := @image_mem T T' f (mem T). + +Section Image. + +Variable T : finType. +Implicit Type A : pred T. + +Section SizeImage. + +Variables (T' : Type) (f : T -> T'). + +Lemma size_image A : size (image f A) = #|A|. +Proof. by rewrite size_map -cardE. Qed. + +Lemma size_codom : size (codom f) = #|T|. +Proof. exact: size_image. Qed. + +Lemma codomE : codom f = map f (enum T). +Proof. by []. Qed. + +End SizeImage. + +Variables (T' : eqType) (f : T -> T'). + +Lemma imageP A y : reflect (exists2 x, x \in A & y = f x) (y \in image f A). +Proof. +by apply: (iffP mapP) => [] [x Ax y_fx]; exists x; rewrite // mem_enum in Ax *. +Qed. + +Lemma codomP y : reflect (exists x, y = f x) (y \in codom f). +Proof. by apply: (iffP (imageP _ y)) => [][x]; exists x. Qed. + +Remark iinv_proof A y : y \in image f A -> {x | x \in A & f x = y}. +Proof. +move=> fy; pose b x := A x && (f x == y). +case: (pickP b) => [x /andP[Ax /eqP] | nfy]; first by exists x. +by case/negP: fy => /imageP[x Ax fx_y]; case/andP: (nfy x); rewrite fx_y. +Qed. + +Definition iinv A y fAy := s2val (@iinv_proof A y fAy). + +Lemma f_iinv A y fAy : f (@iinv A y fAy) = y. +Proof. exact: s2valP' (iinv_proof fAy). Qed. + +Lemma mem_iinv A y fAy : @iinv A y fAy \in A. +Proof. exact: s2valP (iinv_proof fAy). Qed. + +Lemma in_iinv_f A : {in A &, injective f} -> + forall x fAfx, x \in A -> @iinv A (f x) fAfx = x. +Proof. +by move=> injf x fAfx Ax; apply: injf => //; [apply: mem_iinv | apply: f_iinv]. +Qed. + +Lemma preim_iinv A B y fAy : preim f B (@iinv A y fAy) = B y. +Proof. by rewrite /= f_iinv. Qed. + +Lemma image_f A x : x \in A -> f x \in image f A. +Proof. by move=> Ax; apply/imageP; exists x. Qed. + +Lemma codom_f x : f x \in codom f. +Proof. by apply: image_f. Qed. + +Lemma image_codom A : {subset image f A <= codom f}. +Proof. by move=> _ /imageP[x _ ->]; apply: codom_f. Qed. + +Lemma image_pred0 : image f pred0 =i pred0. +Proof. by move=> x; rewrite /image_mem /= enum0. Qed. + +Section Injective. + +Hypothesis injf : injective f. + +Lemma mem_image A x : (f x \in image f A) = (x \in A). +Proof. by rewrite mem_map ?mem_enum. Qed. + +Lemma pre_image A : [preim f of image f A] =i A. +Proof. by move=> x; rewrite inE /= mem_image. Qed. + +Lemma image_iinv A y (fTy : y \in codom f) : + (y \in image f A) = (iinv fTy \in A). +Proof. by rewrite -mem_image ?f_iinv. Qed. + +Lemma iinv_f x fTfx : @iinv T (f x) fTfx = x. +Proof. by apply: in_iinv_f; first apply: in2W. Qed. + +Lemma image_pre (B : pred T') : image f [preim f of B] =i [predI B & codom f]. +Proof. by move=> y; rewrite /image_mem -filter_map /= mem_filter -enumT. Qed. + +Lemma bij_on_codom (x0 : T) : {on [pred y in codom f], bijective f}. +Proof. +pose g y := iinv (valP (insigd (codom_f x0) y)). +by exists g => [x fAfx | y fAy]; first apply: injf; rewrite f_iinv insubdK. +Qed. + +Lemma bij_on_image A (x0 : T) : {on [pred y in image f A], bijective f}. +Proof. exact: subon_bij (@image_codom A) (bij_on_codom x0). Qed. + +End Injective. + +Fixpoint preim_seq s := + if s is y :: s' then + (if pick (preim f (pred1 y)) is Some x then cons x else id) (preim_seq s') + else [::]. + +Lemma map_preim (s : seq T') : {subset s <= codom f} -> map f (preim_seq s) = s. +Proof. +elim: s => //= y s IHs; case: pickP => [x /eqP fx_y | nfTy] fTs. + by rewrite /= fx_y IHs // => z s_z; apply: fTs; apply: predU1r. +by case/imageP: (fTs y (mem_head y s)) => x _ fx_y; case/eqP: (nfTy x). +Qed. + +End Image. + +Prenex Implicits codom iinv. +Implicit Arguments imageP [T T' f A y]. +Implicit Arguments codomP [T T' f y]. + +Lemma flatten_imageP (aT : finType) (rT : eqType) A (P : pred aT) (y : rT) : + reflect (exists2 x, x \in P & y \in A x) (y \in flatten [seq A x | x in P]). +Proof. +by apply: (iffP flatten_mapP) => [][x Px]; exists x; rewrite ?mem_enum in Px *. +Qed. +Implicit Arguments flatten_imageP [aT rT A P y]. + +Section CardFunImage. + +Variables (T T' : finType) (f : T -> T'). +Implicit Type A : pred T. + +Lemma leq_image_card A : #|image f A| <= #|A|. +Proof. by rewrite (cardE A) -(size_map f) card_size. Qed. + +Lemma card_in_image A : {in A &, injective f} -> #|image f A| = #|A|. +Proof. +move=> injf; rewrite (cardE A) -(size_map f); apply/card_uniqP. +by rewrite map_inj_in_uniq ?enum_uniq // => x y; rewrite !mem_enum; apply: injf. +Qed. + +Lemma image_injP A : reflect {in A &, injective f} (#|image f A| == #|A|). +Proof. +apply: (iffP eqP) => [eqfA |]; last exact: card_in_image. +by apply/dinjectiveP; apply/card_uniqP; rewrite size_map -cardE. +Qed. + +Hypothesis injf : injective f. + +Lemma card_image A : #|image f A| = #|A|. +Proof. by apply: card_in_image; apply: in2W. Qed. + +Lemma card_codom : #|codom f| = #|T|. +Proof. exact: card_image. Qed. + +Lemma card_preim (B : pred T') : #|[preim f of B]| = #|[predI codom f & B]|. +Proof. +rewrite -card_image /=; apply: eq_card => y. +by rewrite [y \in _]image_pre !inE andbC. +Qed. + +Hypothesis card_range : #|T| = #|T'|. + +Lemma inj_card_onto y : y \in codom f. +Proof. by move: y; apply/subset_cardP; rewrite ?card_codom ?subset_predT. Qed. + +Lemma inj_card_bij : bijective f. +Proof. +by exists (fun y => iinv (inj_card_onto y)) => y; rewrite ?iinv_f ?f_iinv. +Qed. + +End CardFunImage. + +Implicit Arguments image_injP [T T' f A]. + +Section FinCancel. + +Variables (T : finType) (f g : T -> T). + +Section Inv. + +Hypothesis injf : injective f. + +Lemma injF_onto y : y \in codom f. Proof. exact: inj_card_onto. Qed. +Definition invF y := iinv (injF_onto y). +Lemma invF_f : cancel f invF. Proof. by move=> x; apply: iinv_f. Qed. +Lemma f_invF : cancel invF f. Proof. by move=> y; apply: f_iinv. Qed. +Lemma injF_bij : bijective f. Proof. exact: inj_card_bij. Qed. + +End Inv. + +Hypothesis fK : cancel f g. + +Lemma canF_sym : cancel g f. +Proof. exact/(bij_can_sym (injF_bij (can_inj fK))). Qed. + +Lemma canF_LR x y : x = g y -> f x = y. +Proof. exact: canLR canF_sym. Qed. + +Lemma canF_RL x y : g x = y -> x = f y. +Proof. exact: canRL canF_sym. Qed. + +Lemma canF_eq x y : (f x == y) = (x == g y). +Proof. exact: (can2_eq fK canF_sym). Qed. + +Lemma canF_invF : g =1 invF (can_inj fK). +Proof. by move=> y; apply: (canLR fK); rewrite f_invF. Qed. + +End FinCancel. + +Section EqImage. + +Variables (T : finType) (T' : Type). + +Lemma eq_image (A B : pred T) (f g : T -> T') : + A =i B -> f =1 g -> image f A = image g B. +Proof. +by move=> eqAB eqfg; rewrite /image_mem (eq_enum eqAB) (eq_map eqfg). +Qed. + +Lemma eq_codom (f g : T -> T') : f =1 g -> codom f = codom g. +Proof. exact: eq_image. Qed. + +Lemma eq_invF f g injf injg : f =1 g -> @invF T f injf =1 @invF T g injg. +Proof. +by move=> eq_fg x; apply: (canLR (invF_f injf)); rewrite eq_fg f_invF. +Qed. + +End EqImage. + +(* Standard finTypes *) + +Lemma unit_enumP : Finite.axiom [::tt]. Proof. by case. Qed. +Definition unit_finMixin := Eval hnf in FinMixin unit_enumP. +Canonical unit_finType := Eval hnf in FinType unit unit_finMixin. +Lemma card_unit : #|{: unit}| = 1. Proof. by rewrite cardT enumT unlock. Qed. + +Lemma bool_enumP : Finite.axiom [:: true; false]. Proof. by case. Qed. +Definition bool_finMixin := Eval hnf in FinMixin bool_enumP. +Canonical bool_finType := Eval hnf in FinType bool bool_finMixin. +Lemma card_bool : #|{: bool}| = 2. Proof. by rewrite cardT enumT unlock. Qed. + +Local Notation enumF T := (Finite.enum T). + +Section OptionFinType. + +Variable T : finType. + +Definition option_enum := None :: map some (enumF T). + +Lemma option_enumP : Finite.axiom option_enum. +Proof. by case=> [x|]; rewrite /= count_map (count_pred0, enumP). Qed. + +Definition option_finMixin := Eval hnf in FinMixin option_enumP. +Canonical option_finType := Eval hnf in FinType (option T) option_finMixin. + +Lemma card_option : #|{: option T}| = #|T|.+1. +Proof. by rewrite !cardT !enumT {1}unlock /= !size_map. Qed. + +End OptionFinType. + +Section TransferFinType. + +Variables (eT : countType) (fT : finType) (f : eT -> fT). + +Lemma pcan_enumP g : pcancel f g -> Finite.axiom (undup (pmap g (enumF fT))). +Proof. +move=> fK x; rewrite count_uniq_mem ?undup_uniq // mem_undup. +by rewrite mem_pmap -fK map_f // -enumT mem_enum. +Qed. + +Definition PcanFinMixin g fK := FinMixin (@pcan_enumP g fK). + +Definition CanFinMixin g (fK : cancel f g) := PcanFinMixin (can_pcan fK). + +End TransferFinType. + +Section SubFinType. + +Variables (T : choiceType) (P : pred T). +Import Finite. + +Structure subFinType := SubFinType { + subFin_sort :> subType P; + _ : mixin_of (sub_eqType subFin_sort) +}. + +Definition pack_subFinType U := + fun cT b m & phant_id (class cT) (@Class U b m) => + fun sT m' & phant_id m' m => @SubFinType sT m'. + +Implicit Type sT : subFinType. + +Definition subFin_mixin sT := + let: SubFinType _ m := sT return mixin_of (sub_eqType sT) in m. + +Coercion subFinType_subCountType sT := @SubCountType _ _ sT (subFin_mixin sT). +Canonical subFinType_subCountType. + +Coercion subFinType_finType sT := + Pack (@Class sT (sub_choiceClass sT) (subFin_mixin sT)) sT. +Canonical subFinType_finType. + +Lemma codom_val sT x : (x \in codom (val : sT -> T)) = P x. +Proof. +by apply/codomP/idP=> [[u ->]|Px]; last exists (Sub x Px); rewrite ?valP ?SubK. +Qed. + +End SubFinType. + +(* This assumes that T has both finType and subCountType structures. *) +Notation "[ 'subFinType' 'of' T ]" := (@pack_subFinType _ _ T _ _ _ id _ _ id) + (at level 0, format "[ 'subFinType' 'of' T ]") : form_scope. + +Section FinTypeForSub. + +Variables (T : finType) (P : pred T) (sT : subCountType P). + +Definition sub_enum : seq sT := pmap insub (enumF T). + +Lemma mem_sub_enum u : u \in sub_enum. +Proof. by rewrite mem_pmap_sub -enumT mem_enum. Qed. + +Lemma sub_enum_uniq : uniq sub_enum. +Proof. by rewrite pmap_sub_uniq // -enumT enum_uniq. Qed. + +Lemma val_sub_enum : map val sub_enum = enum P. +Proof. +rewrite pmap_filter; last exact: insubK. +by apply: eq_filter => x; apply: isSome_insub. +Qed. + +(* We can't declare a canonical structure here because we've already *) +(* stated that subType_sort and FinType.sort unify via to the *) +(* subType_finType structure. *) + +Definition SubFinMixin := UniqFinMixin sub_enum_uniq mem_sub_enum. +Definition SubFinMixin_for (eT : eqType) of phant eT := + eq_rect _ Finite.mixin_of SubFinMixin eT. + +Variable sfT : subFinType P. + +Lemma card_sub : #|sfT| = #|[pred x | P x]|. +Proof. by rewrite -(eq_card (codom_val sfT)) (card_image val_inj). Qed. + +Lemma eq_card_sub (A : pred sfT) : A =i predT -> #|A| = #|[pred x | P x]|. +Proof. exact: eq_card_trans card_sub. Qed. + +End FinTypeForSub. + +(* This assumes that T has a subCountType structure over a type that *) +(* has a finType structure. *) +Notation "[ 'finMixin' 'of' T 'by' <: ]" := + (SubFinMixin_for (Phant T) (erefl _)) + (at level 0, format "[ 'finMixin' 'of' T 'by' <: ]") : form_scope. + +(* Regression for the subFinType stack +Record myb : Type := MyB {myv : bool; _ : ~~ myv}. +Canonical myb_sub := Eval hnf in [subType for myv]. +Definition myb_eqm := Eval hnf in [eqMixin of myb by <:]. +Canonical myb_eq := Eval hnf in EqType myb myb_eqm. +Definition myb_chm := [choiceMixin of myb by <:]. +Canonical myb_ch := Eval hnf in ChoiceType myb myb_chm. +Definition myb_cntm := [countMixin of myb by <:]. +Canonical myb_cnt := Eval hnf in CountType myb myb_cntm. +Canonical myb_scnt := Eval hnf in [subCountType of myb]. +Definition myb_finm := [finMixin of myb by <:]. +Canonical myb_fin := Eval hnf in FinType myb myb_finm. +Canonical myb_sfin := Eval hnf in [subFinType of myb]. +Print Canonical Projections. +Print myb_finm. +Print myb_cntm. +*) + +Section CardSig. + +Variables (T : finType) (P : pred T). + +Definition sig_finMixin := [finMixin of {x | P x} by <:]. +Canonical sig_finType := Eval hnf in FinType {x | P x} sig_finMixin. +Canonical sig_subFinType := Eval hnf in [subFinType of {x | P x}]. + +Lemma card_sig : #|{: {x | P x}}| = #|[pred x | P x]|. +Proof. exact: card_sub. Qed. + +End CardSig. + +(* Subtype for an explicit enumeration. *) +Section SeqSubType. + +Variables (T : eqType) (s : seq T). + +Record seq_sub : Type := SeqSub {ssval : T; ssvalP : in_mem ssval (@mem T _ s)}. + +Canonical seq_sub_subType := Eval hnf in [subType for ssval]. +Definition seq_sub_eqMixin := Eval hnf in [eqMixin of seq_sub by <:]. +Canonical seq_sub_eqType := Eval hnf in EqType seq_sub seq_sub_eqMixin. + +Definition seq_sub_enum : seq seq_sub := undup (pmap insub s). + +Lemma mem_seq_sub_enum x : x \in seq_sub_enum. +Proof. by rewrite mem_undup mem_pmap -valK map_f ?ssvalP. Qed. + +Lemma val_seq_sub_enum : uniq s -> map val seq_sub_enum = s. +Proof. +move=> Us; rewrite /seq_sub_enum undup_id ?pmap_sub_uniq //. +rewrite (pmap_filter (@insubK _ _ _)); apply/all_filterP. +by apply/allP => x; rewrite isSome_insub. +Qed. + +Definition seq_sub_pickle x := index x seq_sub_enum. +Definition seq_sub_unpickle n := nth None (map some seq_sub_enum) n. +Lemma seq_sub_pickleK : pcancel seq_sub_pickle seq_sub_unpickle. +Proof. +rewrite /seq_sub_unpickle => x. +by rewrite (nth_map x) ?nth_index ?index_mem ?mem_seq_sub_enum. +Qed. + +Definition seq_sub_countMixin := CountMixin seq_sub_pickleK. +Fact seq_sub_axiom : Finite.axiom seq_sub_enum. +Proof. exact: Finite.uniq_enumP (undup_uniq _) mem_seq_sub_enum. Qed. +Definition seq_sub_finMixin := Finite.Mixin seq_sub_countMixin seq_sub_axiom. + +(* Beware: these are not the canonical instances, as they are not consistent *) +(* the generic sub_choiceType canonical instance. *) +Definition adhoc_seq_sub_choiceMixin := PcanChoiceMixin seq_sub_pickleK. +Definition adhoc_seq_sub_choiceType := + Eval hnf in ChoiceType seq_sub adhoc_seq_sub_choiceMixin. +Definition adhoc_seq_sub_finType := + [finType of seq_sub for FinType adhoc_seq_sub_choiceType seq_sub_finMixin]. + +End SeqSubType. + +Section SeqFinType. + +Variables (T : choiceType) (s : seq T). +Local Notation sT := (seq_sub s). + +Definition seq_sub_choiceMixin := [choiceMixin of sT by <:]. +Canonical seq_sub_choiceType := Eval hnf in ChoiceType sT seq_sub_choiceMixin. + +Canonical seq_sub_countType := Eval hnf in CountType sT (seq_sub_countMixin s). +Canonical seq_sub_subCountType := Eval hnf in [subCountType of sT]. +Canonical seq_sub_finType := Eval hnf in FinType sT (seq_sub_finMixin s). +Canonical seq_sub_subFinType := Eval hnf in [subFinType of sT]. + +Lemma card_seq_sub : uniq s -> #|{:sT}| = size s. +Proof. +by move=> Us; rewrite cardE enumT -(size_map val) unlock val_seq_sub_enum. +Qed. + +End SeqFinType. + + +(**********************************************************************) +(* *) +(* Ordinal finType : {0, ... , n-1} *) +(* *) +(**********************************************************************) + +Section OrdinalSub. + +Variable n : nat. + +Inductive ordinal : predArgType := Ordinal m of m < n. + +Coercion nat_of_ord i := let: Ordinal m _ := i in m. + +Canonical ordinal_subType := [subType for nat_of_ord]. +Definition ordinal_eqMixin := Eval hnf in [eqMixin of ordinal by <:]. +Canonical ordinal_eqType := Eval hnf in EqType ordinal ordinal_eqMixin. +Definition ordinal_choiceMixin := [choiceMixin of ordinal by <:]. +Canonical ordinal_choiceType := + Eval hnf in ChoiceType ordinal ordinal_choiceMixin. +Definition ordinal_countMixin := [countMixin of ordinal by <:]. +Canonical ordinal_countType := Eval hnf in CountType ordinal ordinal_countMixin. +Canonical ordinal_subCountType := [subCountType of ordinal]. + +Lemma ltn_ord (i : ordinal) : i < n. Proof. exact: valP i. Qed. + +Lemma ord_inj : injective nat_of_ord. Proof. exact: val_inj. Qed. + +Definition ord_enum : seq ordinal := pmap insub (iota 0 n). + +Lemma val_ord_enum : map val ord_enum = iota 0 n. +Proof. +rewrite pmap_filter; last exact: insubK. +by apply/all_filterP; apply/allP=> i; rewrite mem_iota isSome_insub. +Qed. + +Lemma ord_enum_uniq : uniq ord_enum. +Proof. by rewrite pmap_sub_uniq ?iota_uniq. Qed. + +Lemma mem_ord_enum i : i \in ord_enum. +Proof. by rewrite -(mem_map ord_inj) val_ord_enum mem_iota ltn_ord. Qed. + +Definition ordinal_finMixin := + Eval hnf in UniqFinMixin ord_enum_uniq mem_ord_enum. +Canonical ordinal_finType := Eval hnf in FinType ordinal ordinal_finMixin. +Canonical ordinal_subFinType := Eval hnf in [subFinType of ordinal]. + +End OrdinalSub. + +Notation "''I_' n" := (ordinal n) + (at level 8, n at level 2, format "''I_' n"). + +Hint Resolve ltn_ord. + +Section OrdinalEnum. + +Variable n : nat. + +Lemma val_enum_ord : map val (enum 'I_n) = iota 0 n. +Proof. by rewrite enumT unlock val_ord_enum. Qed. + +Lemma size_enum_ord : size (enum 'I_n) = n. +Proof. by rewrite -(size_map val) val_enum_ord size_iota. Qed. + +Lemma card_ord : #|'I_n| = n. +Proof. by rewrite cardE size_enum_ord. Qed. + +Lemma nth_enum_ord i0 m : m < n -> nth i0 (enum 'I_n) m = m :> nat. +Proof. +by move=> ?; rewrite -(nth_map _ 0) (size_enum_ord, val_enum_ord) // nth_iota. +Qed. + +Lemma nth_ord_enum (i0 i : 'I_n) : nth i0 (enum 'I_n) i = i. +Proof. by apply: val_inj; apply: nth_enum_ord. Qed. + +Lemma index_enum_ord (i : 'I_n) : index i (enum 'I_n) = i. +Proof. +by rewrite -{1}(nth_ord_enum i i) index_uniq ?(enum_uniq, size_enum_ord). +Qed. + +End OrdinalEnum. + +Lemma widen_ord_proof n m (i : 'I_n) : n <= m -> i < m. +Proof. exact: leq_trans. Qed. +Definition widen_ord n m le_n_m i := Ordinal (@widen_ord_proof n m i le_n_m). + +Lemma cast_ord_proof n m (i : 'I_n) : n = m -> i < m. +Proof. by move <-. Qed. +Definition cast_ord n m eq_n_m i := Ordinal (@cast_ord_proof n m i eq_n_m). + +Lemma cast_ord_id n eq_n i : cast_ord eq_n i = i :> 'I_n. +Proof. exact: val_inj. Qed. + +Lemma cast_ord_comp n1 n2 n3 eq_n2 eq_n3 i : + @cast_ord n2 n3 eq_n3 (@cast_ord n1 n2 eq_n2 i) = + cast_ord (etrans eq_n2 eq_n3) i. +Proof. exact: val_inj. Qed. + +Lemma cast_ordK n1 n2 eq_n : + cancel (@cast_ord n1 n2 eq_n) (cast_ord (esym eq_n)). +Proof. by move=> i; apply: val_inj. Qed. + +Lemma cast_ordKV n1 n2 eq_n : + cancel (cast_ord (esym eq_n)) (@cast_ord n1 n2 eq_n). +Proof. by move=> i; apply: val_inj. Qed. + +Lemma cast_ord_inj n1 n2 eq_n : injective (@cast_ord n1 n2 eq_n). +Proof. exact: can_inj (cast_ordK eq_n). Qed. + +Lemma rev_ord_proof n (i : 'I_n) : n - i.+1 < n. +Proof. by case: n i => [|n] [i lt_i_n] //; rewrite ltnS subSS leq_subr. Qed. +Definition rev_ord n i := Ordinal (@rev_ord_proof n i). + +Lemma rev_ordK n : involutive (@rev_ord n). +Proof. +by case: n => [|n] [i lti] //; apply: val_inj; rewrite /= !subSS subKn. +Qed. + +Lemma rev_ord_inj {n} : injective (@rev_ord n). +Proof. exact: inv_inj (@rev_ordK n). Qed. + +(* bijection between any finType T and the Ordinal finType of its cardinal *) +Section EnumRank. + +Variable T : finType. +Implicit Type A : pred T. + +Lemma enum_rank_subproof x0 A : x0 \in A -> 0 < #|A|. +Proof. by move=> Ax0; rewrite (cardD1 x0) Ax0. Qed. + +Definition enum_rank_in x0 A (Ax0 : x0 \in A) x := + insubd (Ordinal (@enum_rank_subproof x0 [eta A] Ax0)) (index x (enum A)). + +Definition enum_rank x := @enum_rank_in x T (erefl true) x. + +Lemma enum_default A : 'I_(#|A|) -> T. +Proof. by rewrite cardE; case: (enum A) => [|//] []. Qed. + +Definition enum_val A i := nth (@enum_default [eta A] i) (enum A) i. +Prenex Implicits enum_val. + +Lemma enum_valP A i : @enum_val A i \in A. +Proof. by rewrite -mem_enum mem_nth -?cardE. Qed. + +Lemma enum_val_nth A x i : @enum_val A i = nth x (enum A) i. +Proof. by apply: set_nth_default; rewrite cardE in i *; apply: ltn_ord. Qed. + +Lemma nth_image T' y0 (f : T -> T') A (i : 'I_#|A|) : + nth y0 (image f A) i = f (enum_val i). +Proof. by rewrite -(nth_map _ y0) // -cardE. Qed. + +Lemma nth_codom T' y0 (f : T -> T') (i : 'I_#|T|) : + nth y0 (codom f) i = f (enum_val i). +Proof. exact: nth_image. Qed. + +Lemma nth_enum_rank_in x00 x0 A Ax0 : + {in A, cancel (@enum_rank_in x0 A Ax0) (nth x00 (enum A))}. +Proof. +move=> x Ax; rewrite /= insubdK ?nth_index ?mem_enum //. +by rewrite cardE [_ \in _]index_mem mem_enum. +Qed. + +Lemma nth_enum_rank x0 : cancel enum_rank (nth x0 (enum T)). +Proof. by move=> x; apply: nth_enum_rank_in. Qed. + +Lemma enum_rankK_in x0 A Ax0 : + {in A, cancel (@enum_rank_in x0 A Ax0) enum_val}. +Proof. by move=> x; apply: nth_enum_rank_in. Qed. + +Lemma enum_rankK : cancel enum_rank enum_val. +Proof. by move=> x; apply: enum_rankK_in. Qed. + +Lemma enum_valK_in x0 A Ax0 : cancel enum_val (@enum_rank_in x0 A Ax0). +Proof. +move=> x; apply: ord_inj; rewrite insubdK; last first. + by rewrite cardE [_ \in _]index_mem mem_nth // -cardE. +by rewrite index_uniq ?enum_uniq // -cardE. +Qed. + +Lemma enum_valK : cancel enum_val enum_rank. +Proof. by move=> x; apply: enum_valK_in. Qed. + +Lemma enum_rank_inj : injective enum_rank. +Proof. exact: can_inj enum_rankK. Qed. + +Lemma enum_val_inj A : injective (@enum_val A). +Proof. by move=> i; apply: can_inj (enum_valK_in (enum_valP i)) (i). Qed. + +Lemma enum_val_bij_in x0 A : x0 \in A -> {on A, bijective (@enum_val A)}. +Proof. +move=> Ax0; exists (enum_rank_in Ax0) => [i _|]; last exact: enum_rankK_in. +exact: enum_valK_in. +Qed. + +Lemma enum_rank_bij : bijective enum_rank. +Proof. by move: enum_rankK enum_valK; exists (@enum_val T). Qed. + +Lemma enum_val_bij : bijective (@enum_val T). +Proof. by move: enum_rankK enum_valK; exists enum_rank. Qed. + +(* Due to the limitations of the Coq unification patterns, P can only be *) +(* inferred from the premise of this lemma, not its conclusion. As a result *) +(* this lemma will only be usable in forward chaining style. *) +Lemma fin_all_exists U (P : forall x : T, U x -> Prop) : + (forall x, exists u, P x u) -> (exists u, forall x, P x (u x)). +Proof. +move=> ex_u; pose Q m x := enum_rank x < m -> {ux | P x ux}. +suffices: forall m, m <= #|T| -> exists w : forall x, Q m x, True. + case/(_ #|T|)=> // w _; pose u x := sval (w x (ltn_ord _)). + by exists u => x; rewrite {}/u; case: (w x _). +elim=> [|m IHm] ltmX; first by have w x: Q 0 x by []; exists w. +have{IHm} [w _] := IHm (ltnW ltmX); pose i := Ordinal ltmX. +have [u Pu] := ex_u (enum_val i); suffices w' x: Q m.+1 x by exists w'. +rewrite /Q ltnS leq_eqVlt (val_eqE _ i); case: eqP => [def_i _ | _ /w //]. +by rewrite -def_i enum_rankK in u Pu; exists u. +Qed. + +Lemma fin_all_exists2 U (P Q : forall x : T, U x -> Prop) : + (forall x, exists2 u, P x u & Q x u) -> + (exists2 u, forall x, P x (u x) & forall x, Q x (u x)). +Proof. +move=> ex_u; have (x): exists u, P x u /\ Q x u by have [u] := ex_u x; exists u. +by case/fin_all_exists=> u /all_and2[]; exists u. +Qed. + +End EnumRank. + +Implicit Arguments enum_val_inj [[T] [A] x1 x2]. +Implicit Arguments enum_rank_inj [[T] x1 x2]. +Prenex Implicits enum_val enum_rank. + +Lemma enum_rank_ord n i : enum_rank i = cast_ord (esym (card_ord n)) i. +Proof. +by apply: val_inj; rewrite insubdK ?index_enum_ord // card_ord [_ \in _]ltn_ord. +Qed. + +Lemma enum_val_ord n i : enum_val i = cast_ord (card_ord n) i. +Proof. +by apply: canLR (@enum_rankK _) _; apply: val_inj; rewrite enum_rank_ord. +Qed. + +(* The integer bump / unbump operations. *) + +Definition bump h i := (h <= i) + i. +Definition unbump h i := i - (h < i). + +Lemma bumpK h : cancel (bump h) (unbump h). +Proof. +rewrite /bump /unbump => i. +have [le_hi | lt_ih] := leqP h i; first by rewrite ltnS le_hi subn1. +by rewrite ltnNge ltnW ?subn0. +Qed. + +Lemma neq_bump h i : h != bump h i. +Proof. +rewrite /bump eqn_leq; have [le_hi | lt_ih] := leqP h i. + by rewrite ltnNge le_hi andbF. +by rewrite leqNgt lt_ih. +Qed. + +Lemma unbumpKcond h i : bump h (unbump h i) = (i == h) + i. +Proof. +rewrite /bump /unbump leqNgt -subSKn. +case: (ltngtP i h) => /= [-> | ltih | ->] //; last by rewrite ltnn. +by rewrite subn1 /= leqNgt !(ltn_predK ltih, ltih, add1n). +Qed. + +Lemma unbumpK h : {in predC1 h, cancel (unbump h) (bump h)}. +Proof. by move=> i; move/negbTE=> neq_h_i; rewrite unbumpKcond neq_h_i. Qed. + +Lemma bump_addl h i k : bump (k + h) (k + i) = k + bump h i. +Proof. by rewrite /bump leq_add2l addnCA. Qed. + +Lemma bumpS h i : bump h.+1 i.+1 = (bump h i).+1. +Proof. exact: addnS. Qed. + +Lemma unbump_addl h i k : unbump (k + h) (k + i) = k + unbump h i. +Proof. +apply: (can_inj (bumpK (k + h))). +by rewrite bump_addl !unbumpKcond eqn_add2l addnCA. +Qed. + +Lemma unbumpS h i : unbump h.+1 i.+1 = (unbump h i).+1. +Proof. exact: unbump_addl 1. Qed. + +Lemma leq_bump h i j : (i <= bump h j) = (unbump h i <= j). +Proof. +rewrite /bump leq_subLR. +case: (leqP i h) (leqP h j) => [le_i_h | lt_h_i] [le_h_j | lt_j_h] //. + by rewrite leqW (leq_trans le_i_h). +by rewrite !(leqNgt i) ltnW (leq_trans _ lt_h_i). +Qed. + +Lemma leq_bump2 h i j : (bump h i <= bump h j) = (i <= j). +Proof. by rewrite leq_bump bumpK. Qed. + +Lemma bumpC h1 h2 i : + bump h1 (bump h2 i) = bump (bump h1 h2) (bump (unbump h2 h1) i). +Proof. +rewrite {1 5}/bump -leq_bump addnCA; congr (_ + (_ + _)). +rewrite 2!leq_bump /unbump /bump; case: (leqP h1 h2) => [le_h12 | lt_h21]. + by rewrite subn0 ltnS le_h12 subn1. +by rewrite subn1 (ltn_predK lt_h21) (leqNgt h1) lt_h21 subn0. +Qed. + +(* The lift operations on ordinals; to avoid a messy dependent type, *) +(* unlift is a partial operation (returns an option). *) + +Lemma lift_subproof n h (i : 'I_n.-1) : bump h i < n. +Proof. by case: n i => [[]|n] //= i; rewrite -addnS (leq_add (leq_b1 _)). Qed. + +Definition lift n (h : 'I_n) (i : 'I_n.-1) := Ordinal (lift_subproof h i). + +Lemma unlift_subproof n (h : 'I_n) (u : {j | j != h}) : unbump h (val u) < n.-1. +Proof. +case: n h u => [|n h] [] //= j ne_jh. +rewrite -(leq_bump2 h.+1) bumpS unbumpK // /bump. +case: (ltngtP n h) => [|_|eq_nh]; rewrite ?(leqNgt _ h) ?ltn_ord //. +by rewrite ltn_neqAle [j <= _](valP j) {2}eq_nh andbT. +Qed. + +Definition unlift n (h i : 'I_n) := + omap (fun u : {j | j != h} => Ordinal (unlift_subproof u)) (insub i). + +CoInductive unlift_spec n h i : option 'I_n.-1 -> Type := + | UnliftSome j of i = lift h j : unlift_spec h i (Some j) + | UnliftNone of i = h : unlift_spec h i None. + +Lemma unliftP n (h i : 'I_n) : unlift_spec h i (unlift h i). +Proof. +rewrite /unlift; case: insubP => [u nhi | ] def_i /=; constructor. + by apply: val_inj; rewrite /= def_i unbumpK. +by rewrite negbK in def_i; apply/eqP. +Qed. + +Lemma neq_lift n (h : 'I_n) i : h != lift h i. +Proof. exact: neq_bump. Qed. + +Lemma unlift_none n (h : 'I_n) : unlift h h = None. +Proof. by case: unliftP => // j Dh; case/eqP: (neq_lift h j). Qed. + +Lemma unlift_some n (h i : 'I_n) : + h != i -> {j | i = lift h j & unlift h i = Some j}. +Proof. +rewrite eq_sym => /eqP neq_ih. +by case Dui: (unlift h i) / (unliftP h i) => [j Dh|//]; exists j. +Qed. + +Lemma lift_inj n (h : 'I_n) : injective (lift h). +Proof. +move=> i1 i2; move/eqP; rewrite [_ == _](can_eq (@bumpK _)) => eq_i12. +exact/eqP. +Qed. + +Lemma liftK n (h : 'I_n) : pcancel (lift h) (unlift h). +Proof. +by move=> i; case: (unlift_some (neq_lift h i)) => j; move/lift_inj->. +Qed. + +(* Shifting and splitting indices, for cutting and pasting arrays *) + +Lemma lshift_subproof m n (i : 'I_m) : i < m + n. +Proof. by apply: leq_trans (valP i) _; apply: leq_addr. Qed. + +Lemma rshift_subproof m n (i : 'I_n) : m + i < m + n. +Proof. by rewrite ltn_add2l. Qed. + +Definition lshift m n (i : 'I_m) := Ordinal (lshift_subproof n i). +Definition rshift m n (i : 'I_n) := Ordinal (rshift_subproof m i). + +Lemma split_subproof m n (i : 'I_(m + n)) : i >= m -> i - m < n. +Proof. by move/subSn <-; rewrite leq_subLR. Qed. + +Definition split m n (i : 'I_(m + n)) : 'I_m + 'I_n := + match ltnP (i) m with + | LtnNotGeq lt_i_m => inl _ (Ordinal lt_i_m) + | GeqNotLtn ge_i_m => inr _ (Ordinal (split_subproof ge_i_m)) + end. + +CoInductive split_spec m n (i : 'I_(m + n)) : 'I_m + 'I_n -> bool -> Type := + | SplitLo (j : 'I_m) of i = j :> nat : split_spec i (inl _ j) true + | SplitHi (k : 'I_n) of i = m + k :> nat : split_spec i (inr _ k) false. + +Lemma splitP m n (i : 'I_(m + n)) : split_spec i (split i) (i < m). +Proof. +rewrite /split {-3}/leq. +by case: (@ltnP i m) => cmp_i_m //=; constructor; rewrite ?subnKC. +Qed. + +Definition unsplit m n (jk : 'I_m + 'I_n) := + match jk with inl j => lshift n j | inr k => rshift m k end. + +Lemma ltn_unsplit m n (jk : 'I_m + 'I_n) : (unsplit jk < m) = jk. +Proof. by case: jk => [j|k]; rewrite /= ?ltn_ord // ltnNge leq_addr. Qed. + +Lemma splitK m n : cancel (@split m n) (@unsplit m n). +Proof. by move=> i; apply: val_inj; case: splitP. Qed. + +Lemma unsplitK m n : cancel (@unsplit m n) (@split m n). +Proof. +move=> jk; have:= ltn_unsplit jk. +by do [case: splitP; case: jk => //= i j] => [|/addnI] => /ord_inj->. +Qed. + +Section OrdinalPos. + +Variable n' : nat. +Local Notation n := n'.+1. + +Definition ord0 := Ordinal (ltn0Sn n'). +Definition ord_max := Ordinal (ltnSn n'). + +Lemma leq_ord (i : 'I_n) : i <= n'. Proof. exact: valP i. Qed. + +Lemma sub_ord_proof m : n' - m < n. +Proof. by rewrite ltnS leq_subr. Qed. +Definition sub_ord m := Ordinal (sub_ord_proof m). + +Lemma sub_ordK (i : 'I_n) : n' - (n' - i) = i. +Proof. by rewrite subKn ?leq_ord. Qed. + +Definition inord m : 'I_n := insubd ord0 m. + +Lemma inordK m : m < n -> inord m = m :> nat. +Proof. by move=> lt_m; rewrite val_insubd lt_m. Qed. + +Lemma inord_val (i : 'I_n) : inord i = i. +Proof. by rewrite /inord /insubd valK. Qed. + +Lemma enum_ordS : enum 'I_n = ord0 :: map (lift ord0) (enum 'I_n'). +Proof. +apply: (inj_map val_inj); rewrite val_enum_ord /= -map_comp. +by rewrite (map_comp (addn 1)) val_enum_ord -iota_addl. +Qed. + +Lemma lift_max (i : 'I_n') : lift ord_max i = i :> nat. +Proof. by rewrite /= /bump leqNgt ltn_ord. Qed. + +Lemma lift0 (i : 'I_n') : lift ord0 i = i.+1 :> nat. Proof. by []. Qed. + +End OrdinalPos. + +Implicit Arguments ord0 [[n']]. +Implicit Arguments ord_max [[n']]. +Implicit Arguments inord [[n']]. +Implicit Arguments sub_ord [[n']]. + +(* Product of two fintypes which is a fintype *) +Section ProdFinType. + +Variable T1 T2 : finType. + +Definition prod_enum := [seq (x1, x2) | x1 <- enum T1, x2 <- enum T2]. + +Lemma predX_prod_enum (A1 : pred T1) (A2 : pred T2) : + count [predX A1 & A2] prod_enum = #|A1| * #|A2|. +Proof. +rewrite !cardE !size_filter -!enumT /prod_enum. +elim: (enum T1) => //= x1 s1 IHs; rewrite count_cat {}IHs count_map /preim /=. +by case: (x1 \in A1); rewrite ?count_pred0. +Qed. + +Lemma prod_enumP : Finite.axiom prod_enum. +Proof. +by case=> x1 x2; rewrite (predX_prod_enum (pred1 x1) (pred1 x2)) !card1. +Qed. + +Definition prod_finMixin := Eval hnf in FinMixin prod_enumP. +Canonical prod_finType := Eval hnf in FinType (T1 * T2) prod_finMixin. + +Lemma cardX (A1 : pred T1) (A2 : pred T2) : #|[predX A1 & A2]| = #|A1| * #|A2|. +Proof. by rewrite -predX_prod_enum unlock size_filter unlock. Qed. + +Lemma card_prod : #|{: T1 * T2}| = #|T1| * #|T2|. +Proof. by rewrite -cardX; apply: eq_card; case. Qed. + +Lemma eq_card_prod (A : pred (T1 * T2)) : A =i predT -> #|A| = #|T1| * #|T2|. +Proof. exact: eq_card_trans card_prod. Qed. + +End ProdFinType. + +Section TagFinType. + +Variables (I : finType) (T_ : I -> finType). + +Definition tag_enum := + flatten [seq [seq Tagged T_ x | x <- enumF (T_ i)] | i <- enumF I]. + +Lemma tag_enumP : Finite.axiom tag_enum. +Proof. +case=> i x; rewrite -(enumP i) /tag_enum -enumT. +elim: (enum I) => //= j e IHe. +rewrite count_cat count_map {}IHe; congr (_ + _). +rewrite -size_filter -cardE /=; case: eqP => [-> | ne_j_i]. + by apply: (@eq_card1 _ x) => y; rewrite -topredE /= tagged_asE ?eqxx. +by apply: eq_card0 => y. +Qed. + +Definition tag_finMixin := Eval hnf in FinMixin tag_enumP. +Canonical tag_finType := Eval hnf in FinType {i : I & T_ i} tag_finMixin. + +Lemma card_tagged : + #|{: {i : I & T_ i}}| = sumn (map (fun i => #|T_ i|) (enum I)). +Proof. +rewrite cardE !enumT {1}unlock size_flatten /shape -map_comp. +by congr (sumn _); apply: eq_map => i; rewrite /= size_map -enumT -cardE. +Qed. + +End TagFinType. + +Section SumFinType. + +Variables T1 T2 : finType. + +Definition sum_enum := + [seq inl _ x | x <- enumF T1] ++ [seq inr _ y | y <- enumF T2]. + +Lemma sum_enum_uniq : uniq sum_enum. +Proof. +rewrite cat_uniq -!enumT !(enum_uniq, map_inj_uniq); try by move=> ? ? []. +by rewrite andbT; apply/hasP=> [[_ /mapP[x _ ->] /mapP[]]]. +Qed. + +Lemma mem_sum_enum u : u \in sum_enum. +Proof. by case: u => x; rewrite mem_cat -!enumT map_f ?mem_enum ?orbT. Qed. + +Definition sum_finMixin := Eval hnf in UniqFinMixin sum_enum_uniq mem_sum_enum. +Canonical sum_finType := Eval hnf in FinType (T1 + T2) sum_finMixin. + +Lemma card_sum : #|{: T1 + T2}| = #|T1| + #|T2|. +Proof. by rewrite !cardT !enumT {1}unlock size_cat !size_map. Qed. + + +End SumFinType. |