diff options
Diffstat (limited to 'mathcomp')
| -rw-r--r-- | mathcomp/algebra/matrix.v | 5 | ||||
| -rw-r--r-- | mathcomp/solvable/burnside_app.v | 6 | ||||
| -rw-r--r-- | mathcomp/ssreflect/binomial.v | 8 | ||||
| -rw-r--r-- | mathcomp/ssreflect/finfun.v | 400 |
4 files changed, 264 insertions, 155 deletions
diff --git a/mathcomp/algebra/matrix.v b/mathcomp/algebra/matrix.v index 2580741..f87fb78 100644 --- a/mathcomp/algebra/matrix.v +++ b/mathcomp/algebra/matrix.v @@ -215,7 +215,7 @@ Definition mx_val A := let: Matrix g := A in g. Canonical matrix_subType := Eval hnf in [newType for mx_val]. Fact matrix_key : unit. Proof. by []. Qed. -Definition matrix_of_fun_def F := Matrix [ffun ij => F ij.1 ij.2]. +Definition matrix_of_fun_def F := Matrix [ffun ij => F ij.1 ij.2 : R]. Definition matrix_of_fun k := locked_with k matrix_of_fun_def. Canonical matrix_unlockable k := [unlockable fun matrix_of_fun k]. @@ -277,7 +277,8 @@ Notation "\row_ ( j < n ) E" := (@matrix_of_fun _ 1 n matrix_key (fun _ j => E)) Notation "\row_ j E" := (\row_(j < _) E) : ring_scope. Definition matrix_eqMixin (R : eqType) m n := - Eval hnf in [eqMixin of 'M[R]_(m, n) by <:]. + @SubEqMixin (@finfun_eqType _ (fun=> _)) _ (matrix_subType R m n). +(* Eval hnf in [eqMixin of 'M[R]_(m, n) by <:]. *) Canonical matrix_eqType (R : eqType) m n:= Eval hnf in EqType 'M[R]_(m, n) (matrix_eqMixin R m n). Definition matrix_choiceMixin (R : choiceType) m n := diff --git a/mathcomp/solvable/burnside_app.v b/mathcomp/solvable/burnside_app.v index 10cbbaa..34c1c7c 100644 --- a/mathcomp/solvable/burnside_app.v +++ b/mathcomp/solvable/burnside_app.v @@ -686,10 +686,10 @@ move=> p; rewrite inE. by do ?case/orP; move/eqP=> <- a; rewrite !(permM, permE); case: a; do 6?case. Qed. -Definition sop (p : {perm cube}) : seq cube := val (val (val p)). +Definition sop (p : {perm cube}) : seq cube := fgraph (val p). Lemma sop_inj : injective sop. -Proof. by do 2!apply: (inj_comp val_inj); apply: val_inj. Qed. +Proof. by move=> p1 p2 /val_inj/(can_inj fgraphK)/val_inj. Qed. Definition prod_tuple (t1 t2 : seq cube) := map (fun n : 'I_6 => nth F0 t2 n) t1. @@ -700,7 +700,7 @@ Proof. by move=> x n0; rewrite -pvalE unlock enum_rank_ord (tnth_nth F0). Qed. Lemma prod_t_correct : forall (x y : {perm cube}) (i : cube), (x * y) i = nth F0 (prod_tuple (sop x) (sop y)) i. Proof. -move=> x y i; rewrite permM -!sop_spec (nth_map F0) // size_tuple /=. +move=> x y i; rewrite permM -!sop_spec [RHS](nth_map F0) // size_tuple /=. by rewrite card_ord ltn_ord. Qed. diff --git a/mathcomp/ssreflect/binomial.v b/mathcomp/ssreflect/binomial.v index 1f0ae16..c99e296 100644 --- a/mathcomp/ssreflect/binomial.v +++ b/mathcomp/ssreflect/binomial.v @@ -365,10 +365,10 @@ Lemma card_inj_ffuns_on D T (R : pred T) : Proof. rewrite -card_uniq_tuples. have bijFF: {on (_ : pred _), bijective (@Finfun D T)}. - by exists val => // x _; apply: val_inj. -rewrite -(on_card_preimset (bijFF _)); apply: eq_card => t. -rewrite !inE -(codom_ffun (Finfun t)); congr (_ && _); apply: negb_inj. -by rewrite -has_predC has_map enumT has_filter -size_eq0 -cardE. + by exists fgraph => x _; [apply: FinfunK | apply: fgraphK]. +rewrite -(on_card_preimset (bijFF _)); apply: eq_card => /= t. +rewrite !inE -(big_andE predT) -big_filter big_all -all_map. +by rewrite -[injectiveb _]/(uniq _) [map _ _]codom_ffun FinfunK. Qed. Lemma card_inj_ffuns D T : diff --git a/mathcomp/ssreflect/finfun.v b/mathcomp/ssreflect/finfun.v index f8b732a..59b85fe 100644 --- a/mathcomp/ssreflect/finfun.v +++ b/mathcomp/ssreflect/finfun.v @@ -6,22 +6,40 @@ Require Import ssrfun ssrbool eqtype ssrnat seq choice fintype tuple. (******************************************************************************) (* This file implements a type for functions with a finite domain: *) -(* {ffun aT -> rT} where aT should have a finType structure. *) +(* {ffun aT -> rT} where aT should have a finType structure, *) +(* {ffun forall x : aT, rT} for dependent functions over a finType aT, *) +(* and {ffun funT} where funT expands to a product over a finType. *) (* Any eqType, choiceType, countType and finType structures on rT extend to *) (* {ffun aT -> rT} as Leibnitz equality and extensional equalities coincide. *) (* (T ^ n)%type is notation for {ffun 'I_n -> T}, which is isomorphic *) -(* to n.-tuple T. *) -(* For f : {ffun aT -> rT}, we define *) +(* to n.-tuple T, but is structurally positive and thus can be used to *) +(* define inductive types, e.g., Inductive tree := node n of tree ^ n. *) +(* --> More generally, {ffun fT} is always structurally positive. *) +(* For f : {ffun fT} with fT := forall x : aT -> rT x, we define *) (* f x == the image of x under f (f coerces to a CiC function) *) -(* fgraph f == the graph of f, i.e., the #|aT|.-tuple rT of the *) -(* values of f over enum aT. *) -(* finfun lam == the f such that f =1 lam; this is the RECOMMENDED *) -(* interface to build an element of {ffun aT -> rT}. *) -(* [ffun x => expr] == finfun (fun x => expr) *) -(* [ffun=> expr] == finfun (fun _ => expr) *) -(* ffun0 aT0 == the trivial finfun, from a proof aT0 that #|aT| = 0 *) -(* f \in ffun_on R == the range of f is a subset of R *) +(* --> The coercion is structurally decreasing, e.g., Coq will accept *) +(* Fixpoint size t := let: node n f := t in sumn (codom (size \o f)) + 1. *) +(* as structurally decreasing on t of the inductive tree type above. *) +(* total_fun g == the function induced by a dependent function g of type *) +(* forall x, rT on the total space {x : aT & rT x}. *) +(* := fun x => Tagged (fun x => rT) (g x). *) +(* tfgraph f == the total function graph of f, i.e., the #|aT|.-tuple *) +(* of all the (dependent pair) values of total_fun f. *) +(* finfun g == the f extensionally equal to g, and the RECOMMENDED *) +(* interface for building elements of {ffun fT}. *) +(* [ffun x => expr] := finfun (fun x => expr). *) +(* [ffun=> expr] := [ffun _ => expr]. *) +(* [ffun x : aT => expr] := finfun (fun x : aT => expr) (only parsing) *) +(* ffun0 aT0 == the trivial finfun, from a proof aT0 that #|aT| = 0. *) (* f \in family F == f belongs to the family F (f x \in F x for all x) *) +(* Also, {ffun fT} inherits combinatorial structures of rT, i.e., eqType, *) +(* choiceType, countType, and finType. *) +(* There are addidional operations for non-dependent finite functions, *) +(* i.e., f in {ffun aT -> rT}. *) +(* fgraph f == the function graph of f, i.e., the #|aT|.-tuple *) +(* listing the values of f x, for x ranging over enum aT. *) +(* Finfun G == the finfun f whose (simple) function graph is G. *) +(* f \in ffun_on R == the range of f is a subset of R. *) (* y.-support f == the y-support of f, i.e., [pred x | f x != y]. *) (* Thus, y.-support f \subset D means f has y-support D. *) (* We will put Notation support := 0.-support in ssralg. *) @@ -37,136 +55,235 @@ Unset Printing Implicit Defensive. Section Def. -Variables (aT : finType) (rT : Type). +Variables (aT : finType) (rT : aT -> Type). -Inductive finfun_type : predArgType := Finfun of #|aT|.-tuple rT. +Inductive finfun_on : seq aT -> Type := +| finfun_nil : finfun_on [::] +| finfun_cons x s of rT x & finfun_on s : finfun_on (x :: s). -Definition finfun_of of phant (aT -> rT) := finfun_type. +Local Fixpoint finfun_rec (g : forall x, rT x) s : finfun_on s := + if s is x1 :: s1 then finfun_cons (g x1) (finfun_rec g s1) else finfun_nil. -Identity Coercion type_of_finfun : finfun_of >-> finfun_type. +Local Fixpoint fun_of_fin_rec x s (f_s : finfun_on s) : x \in s -> rT x := + if f_s is finfun_cons x1 s1 y1 f_s1 then + if eqP is ReflectT Dx in reflect _ Dxb return Dxb || (x \in s1) -> rT x then + fun=> ecast x (rT x) (esym Dx) y1 + else fun_of_fin_rec f_s1 + else fun isF => False_rect (rT x) (notF isF). -Definition fgraph f := let: Finfun t := f in t. +Variant finfun_of (ph : phant (forall x, rT x)) : predArgType := + FinfunOf of finfun_on (enum aT). -Canonical finfun_subType := Eval hnf in [newType for fgraph]. +Definition fun_of_fin ph (f : finfun_of ph) x := + let: FinfunOf f_aT := f in fun_of_fin_rec f_aT (mem_enum aT x). End Def. +Coercion fun_of_fin : finfun_of >-> Funclass. +Arguments fun_of_fin {aT rT ph} f x. + Notation "{ 'ffun' fT }" := (finfun_of (Phant fT)) (at level 0, format "{ 'ffun' '[hv' fT ']' }") : type_scope. -Definition exp_finIndexType n := ordinal_finType n. -Notation "T ^ n" := (@finfun_of (exp_finIndexType n) T (Phant _)) : type_scope. - -Local Notation fun_of_fin_def := - (fun aT rT f x => tnth (@fgraph aT rT f) (enum_rank x)). +Definition exp_finIndexType := ordinal_finType. +Notation "T ^ n" := + (@finfun_of (exp_finIndexType n) (fun=> T) (Phant _)) : type_scope. -Local Notation finfun_def := (fun aT rT f => @Finfun aT rT (codom_tuple f)). +Local Notation finPi aT rT := (forall x : Finite.sort aT, rT x) (only parsing). +Local Notation finfun_def := + (fun aT rT g => FinfunOf (Phant (finPi aT rT)) (finfun_rec g (enum aT))). -Module Type FunFinfunSig. -Parameter fun_of_fin : forall aT rT, finfun_type aT rT -> aT -> rT. -Parameter finfun : forall (aT : finType) rT, (aT -> rT) -> {ffun aT -> rT}. -Axiom fun_of_finE : fun_of_fin = fun_of_fin_def. +Module Type FinfunSig. +Parameter finfun : forall aT rT, finPi aT rT -> {ffun finPi aT rT}. Axiom finfunE : finfun = finfun_def. -End FunFinfunSig. +End FinfunSig. -Module FunFinfun : FunFinfunSig. -Definition fun_of_fin := fun_of_fin_def. +Module Finfun : FinfunSig. Definition finfun := finfun_def. -Lemma fun_of_finE : fun_of_fin = fun_of_fin_def. Proof. by []. Qed. Lemma finfunE : finfun = finfun_def. Proof. by []. Qed. -End FunFinfun. +End Finfun. -Notation fun_of_fin := FunFinfun.fun_of_fin. -Notation finfun := FunFinfun.finfun. -Coercion fun_of_fin : finfun_type >-> Funclass. -Canonical fun_of_fin_unlock := Unlockable FunFinfun.fun_of_finE. -Canonical finfun_unlock := Unlockable FunFinfun.finfunE. +Notation finfun := Finfun.finfun. +Canonical finfun_unlock := Unlockable Finfun.finfunE. +Arguments finfun {aT rT} g. Notation "[ 'ffun' x : aT => F ]" := (finfun (fun x : aT => F)) (at level 0, x ident, only parsing) : fun_scope. -Notation "[ 'ffun' : aT => F ]" := (finfun (fun _ : aT => F)) - (at level 0, only parsing) : fun_scope. - Notation "[ 'ffun' x => F ]" := [ffun x : _ => F] (at level 0, x ident, format "[ 'ffun' x => F ]") : fun_scope. -Notation "[ 'ffun' => F ]" := [ffun : _ => F] +Notation "[ 'ffun' => F ]" := [ffun _ => F] (at level 0, format "[ 'ffun' => F ]") : fun_scope. +(* Test that the type and accessor are structurally positive and decreasing, + respectively. +Inductive tree := node n of tree ^ n. +Fixpoint size t := let: node n f := t in sumn (codom (size \o f)) + 1. +Inductive tri_tree n := tri_row of {ffun forall i : 'I_n, tri_tree i}. +Fixpoint tri_size n (t : tri_tree n) := + let: tri_row f := t in sumn [seq tri_size (f i) | i : 'I_n] + 1. +*) + +(* Lemmas on the correspondance between finfun_of and CiC dependent functions. *) +Section DepPlainTheory. +Variables (aT : finType) (rT : aT -> Type). +Notation fT := {ffun finPi aT rT}. +Implicit Type f : fT. + +Fact ffun0 (aT0 : #|aT| = 0) : fT. +Proof. by apply/finfun=> x; have:= card0_eq aT0 x. Qed. + +Lemma ffunE g x : (finfun g : fT) x = g x. +Proof. +rewrite unlock /=; set s := enum aT; set s_x : mem_seq s x := mem_enum _ _. +by elim: s s_x => //= x1 s IHs; case: eqP => [|_]; [case: x1 / | apply: IHs]. +Qed. + +Lemma ffunP (f1 f2 : fT) : (forall x, f1 x = f2 x) <-> f1 = f2. +Proof. +suffices ffunK f g: (forall x, f x = g x) -> f = finfun g. + by split=> [/ffunK|] -> //; apply/esym/ffunK. +case: f => f Dg; rewrite unlock; congr FinfunOf. +have{Dg} Dg x (aTx : mem_seq (enum aT) x): g x = fun_of_fin_rec f aTx. + by rewrite -Dg /= (bool_irrelevance (mem_enum _ _) aTx). +elim: (enum aT) / f (enum_uniq aT) => //= x1 s y f IHf /andP[s'x1 Us] in Dg *. +rewrite Dg ?eqxx //=; case: eqP => // /eq_axiomK-> /= _. +rewrite {}IHf // => x s_x; rewrite Dg ?s_x ?orbT //. +by case: eqP (memPn s'x1 x s_x) => // _ _ /(bool_irrelevance s_x) <-. +Qed. + +Lemma ffunK : @cancel (finPi aT rT) fT fun_of_fin finfun. +Proof. by move=> f; apply/ffunP=> x; rewrite ffunE. Qed. + +Lemma eq_dffun (g1 g2 : forall x, rT x) : + (forall x, g1 x = g2 x) -> finfun g1 = finfun g2. +Proof. by move=> eq_g; apply/ffunP => x; rewrite !ffunE eq_g. Qed. + +Definition total_fun g x := Tagged rT (g x : rT x). + +Definition tfgraph f := codom_tuple (total_fun f). + +Lemma codom_tffun f : codom (total_fun f) = tfgraph f. Proof. by []. Qed. + +Local Definition tfgraph_inv (G : #|aT|.-tuple {x : aT & rT x}) : option fT := + if eqfunP isn't ReflectT Dtg then None else + Some [ffun x => ecast x (rT x) (Dtg x) (tagged (tnth G (enum_rank x)))]. + +Local Lemma tfgraphK : pcancel tfgraph tfgraph_inv. +Proof. +move=> f; have Dg x: tnth (tfgraph f) (enum_rank x) = total_fun f x. + by rewrite tnth_map -[tnth _ _]enum_val_nth enum_rankK. +rewrite /tfgraph_inv; case: eqfunP => /= [Dtg | [] x]; last by rewrite Dg. +congr (Some _); apply/ffunP=> x; rewrite ffunE. +by rewrite Dg in (Dx := Dtg x) *; rewrite eq_axiomK. +Qed. + +Lemma tfgraph_inj : injective tfgraph. Proof. exact: pcan_inj tfgraphK. Qed. + +Definition family_mem mF := [pred f : fT | [forall x, in_mem (f x) (mF x)]]. + +Variables (pT : forall x, predType (rT x)) (F : forall x, pT x). + (* Helper for defining notation for function families. *) -Definition fmem aT rT (pT : predType rT) (f : aT -> pT) := [fun x => mem (f x)]. +Local Definition fmem F x := mem (F x : pT x). + +Lemma familyP f : reflect (forall x, f x \in F x) (f \in family_mem (fmem F)). +Proof. exact: forallP. Qed. + +End DepPlainTheory. + +Arguments ffunK {aT rT} f : rename. +Arguments ffun0 {aT rT} aT0. +Arguments eq_dffun {aT rT} [g1] g2 eq_g12. +Arguments total_fun {aT rT} g x. +Arguments tfgraph {aT rT} f. +Arguments tfgraph_inj {aT rT} [f1 f2] : rename. -(* Lemmas on the correspondance between finfun_type and CiC functions. *) -Section PlainTheory. +Arguments fmem {aT rT pT} F x /. +Arguments familyP {aT rT pT F f}. +Notation family F := (family_mem (fmem F)). + +Definition finfun_eqMixin aT (rT : _ -> eqType) := + PcanEqMixin (@tfgraphK aT rT). +Canonical finfun_eqType aT rT := + EqType {ffun finPi aT _} (finfun_eqMixin rT). + +Definition finfun_choiceMixin aT (rT : _ -> choiceType) := + PcanChoiceMixin (@tfgraphK aT rT). +Canonical finfun_choiceType aT rT := + ChoiceType {ffun finPi aT _} (finfun_choiceMixin rT). + +Definition finfun_countMixin aT (rT : _ -> countType) := + PcanCountMixin (@tfgraphK aT rT). +Canonical finfun_countType aT rT := + CountType {ffun finPi aT _} (finfun_countMixin rT). + +Definition finfun_finMixin aT (rT : _ -> finType) := + PcanFinMixin (@tfgraphK aT rT). +Canonical finfun_finType aT rT := + FinType {ffun finPi aT _} (finfun_finMixin rT). + +Section FunPlainTheory. Variables (aT : finType) (rT : Type). Notation fT := {ffun aT -> rT}. Implicit Types (f : fT) (R : pred rT). -Canonical finfun_of_subType := Eval hnf in [subType of fT]. - -Lemma tnth_fgraph f i : tnth (fgraph f) i = f (enum_val i). -Proof. by rewrite [@fun_of_fin]unlock enum_valK. Qed. +Definition fgraph f := codom_tuple f. -Definition ffun0 (aT0 : #|aT| = 0) : {ffun aT -> rT} := - Finfun (ecast _ _ (esym aT0) (nil_tuple _)). +Definition Finfun (G : #|aT|.-tuple rT) := [ffun x => tnth G (enum_rank x)]. -Lemma ffunE (g : aT -> rT) : finfun g =1 g. -Proof. -move=> x; rewrite [@finfun]unlock unlock tnth_map. -by rewrite -[tnth _ _]enum_val_nth enum_rankK. -Qed. +Lemma tnth_fgraph f i : tnth (fgraph f) i = f (enum_val i). +Proof. by rewrite tnth_map /tnth -enum_val_nth. Qed. -Lemma fgraph_codom f : fgraph f = codom_tuple f. +Lemma FinfunK : cancel Finfun fgraph. Proof. -apply: eq_from_tnth => i; rewrite [@fun_of_fin]unlock tnth_map. -by congr tnth; rewrite -[tnth _ _]enum_val_nth enum_valK. +by move=> G; apply/eq_from_tnth=> i; rewrite tnth_fgraph ffunE enum_valK. Qed. -Lemma codom_ffun f : codom f = val f. -Proof. by rewrite /= fgraph_codom. Qed. +Lemma fgraphK : cancel fgraph Finfun. +Proof. by move=> f; apply/ffunP=> x; rewrite ffunE tnth_fgraph enum_rankK. Qed. -Lemma ffunP f1 f2 : f1 =1 f2 <-> f1 = f2. -Proof. -split=> [eq_f12 | -> //]; do 2!apply: val_inj => /=. -by rewrite !fgraph_codom /= (eq_codom eq_f12). -Qed. +Lemma fgraph_ffun0 aT0 : fgraph (ffun0 aT0) = nil :> seq rT. +Proof. by apply/nilP/eqP; rewrite size_tuple. Qed. -Lemma eq_ffun (g1 g2 : aT -> rT) : - g1 =1 g2 -> finfun g1 = finfun g2. -Proof. by move=> eq_g; apply/ffunP => x; rewrite !ffunE eq_g. Qed. +Lemma codom_ffun f : codom f = fgraph f. Proof. by []. Qed. -Lemma ffunK : cancel (@fun_of_fin aT rT) (@finfun aT rT). -Proof. by move=> f; apply/ffunP/ffunE. Qed. +Lemma tagged_tfgraph f : @map _ rT tagged (tfgraph f) = fgraph f. +Proof. by rewrite -map_comp. Qed. -Definition family_mem mF := [pred f : fT | [forall x, in_mem (f x) (mF x)]]. +Lemma eq_ffun (g1 g2 : aT -> rT) : g1 =1 g2 -> finfun g1 = finfun g2. +Proof. exact: eq_dffun. Qed. -Lemma familyP (pT : predType rT) (F : aT -> pT) f : - reflect (forall x, f x \in F x) (f \in family_mem (fmem F)). -Proof. exact: forallP. Qed. +Lemma fgraph_codom f : fgraph f = codom_tuple f. +Proof. exact/esym/val_inj/codom_ffun. Qed. -Definition ffun_on_mem mR := family_mem (fun _ => mR). +Definition ffun_on_mem (mR : mem_pred rT) := family_mem (fun _ : aT => mR). Lemma ffun_onP R f : reflect (forall x, f x \in R) (f \in ffun_on_mem (mem R)). Proof. exact: forallP. Qed. -End PlainTheory. +End FunPlainTheory. -Notation family F := (family_mem (fun_of_simpl (fmem F))). -Notation ffun_on R := (ffun_on_mem _ (mem R)). - -Arguments ffunK {aT rT}. +Arguments Finfun {aT rT} G. +Arguments fgraph {aT rT} f. +Arguments FinfunK {aT rT} G : rename. +Arguments fgraphK {aT rT} f : rename. Arguments eq_ffun {aT rT} [g1] g2 eq_g12. -Arguments familyP {aT rT pT F f}. Arguments ffun_onP {aT rT R f}. -(*****************************************************************************) +Notation ffun_on R := (ffun_on_mem _ (mem R)). +Notation "@ 'ffun_on' aT R" := + (ffun_on R : simpl_pred (finfun_of (Phant (aT -> id _)))) + (at level 10, aT, R at level 9). Lemma nth_fgraph_ord T n (x0 : T) (i : 'I_n) f : nth x0 (fgraph f) i = f i. Proof. -by rewrite -{2}(enum_rankK i) -tnth_fgraph (tnth_nth x0) enum_rank_ord. +by rewrite -[i in RHS]enum_rankK -tnth_fgraph (tnth_nth x0) enum_rank_ord. Qed. +(*****************************************************************************) + Section Support. Variables (aT : Type) (rT : eqType). @@ -192,11 +309,6 @@ Proof. by apply: (iffP subsetP) => Dg x; [apply: contraNeq | apply: contraR] => /Dg->. Qed. -Definition finfun_eqMixin := - Eval hnf in [eqMixin of finfun_type aT rT by <:]. -Canonical finfun_eqType := Eval hnf in EqType _ finfun_eqMixin. -Canonical finfun_of_eqType := Eval hnf in [eqType of fT]. - Definition pfamily_mem y mD (mF : aT -> mem_pred rT) := family (fun i : aT => if in_mem i mD then pred_of_simpl (mF i) else pred1 y). @@ -224,73 +336,70 @@ Qed. End EqTheory. Arguments supportP {aT rT y D g}. -Notation pfamily y D F := (pfamily_mem y (mem D) (fun_of_simpl (fmem F))). -Notation pffun_on y D R := (pffun_on_mem y (mem D) (mem R)). +Arguments pfamilyP {aT rT pT y D F f}. +Arguments pffun_onP {aT rT y D R f}. -Definition finfun_choiceMixin aT (rT : choiceType) := - [choiceMixin of finfun_type aT rT by <:]. -Canonical finfun_choiceType aT rT := - Eval hnf in ChoiceType _ (finfun_choiceMixin aT rT). -Canonical finfun_of_choiceType (aT : finType) (rT : choiceType) := - Eval hnf in [choiceType of {ffun aT -> rT}]. - -Definition finfun_countMixin aT (rT : countType) := - [countMixin of finfun_type aT rT by <:]. -Canonical finfun_countType aT (rT : countType) := - Eval hnf in CountType _ (finfun_countMixin aT rT). -Canonical finfun_of_countType (aT : finType) (rT : countType) := - Eval hnf in [countType of {ffun aT -> rT}]. -Canonical finfun_subCountType aT (rT : countType) := - Eval hnf in [subCountType of finfun_type aT rT]. -Canonical finfun_of_subCountType (aT : finType) (rT : countType) := - Eval hnf in [subCountType of {ffun aT -> rT}]. +Notation pfamily y D F := (pfamily_mem y (mem D) (fmem F)). +Notation pffun_on y D R := (pffun_on_mem y (mem D) (mem R)). (*****************************************************************************) -Section FinTheory. +Section FinDepTheory. -Variables aT rT : finType. -Notation fT := {ffun aT -> rT}. -Notation ffT := (finfun_type aT rT). -Implicit Types (D : pred aT) (R : pred rT) (F : aT -> pred rT). +Variables (aT : finType) (rT : aT -> finType). +Notation fT := {ffun forall x : aT, rT x}. -Definition finfun_finMixin := [finMixin of ffT by <:]. -Canonical finfun_finType := Eval hnf in FinType ffT finfun_finMixin. -Canonical finfun_subFinType := Eval hnf in [subFinType of ffT]. -Canonical finfun_of_finType := Eval hnf in [finType of fT for finfun_finType]. -Canonical finfun_of_subFinType := Eval hnf in [subFinType of fT]. - -Lemma card_pfamily y0 D F : - #|pfamily y0 D F| = foldr muln 1 [seq #|F x| | x in D]. +Lemma card_family (F : forall x, pred (rT x)) : + #|family F| = foldr muln 1 [seq #|F x| | x : aT]. Proof. -rewrite /image_mem; transitivity #|pfamily y0 (enum D) F|. - by apply/eq_card=> f; apply/eq_forallb=> x /=; rewrite mem_enum. -elim: {D}(enum D) (enum_uniq D) => /= [_|x0 s IHs /andP[s'x0 /IHs<-{IHs}]]. - apply: eq_card1 [ffun=> y0] _ _ => f. - apply/familyP/eqP=> [y0_f|-> x]; last by rewrite ffunE inE. - by apply/ffunP=> x; rewrite ffunE (eqP (y0_f x)). -pose g (xf : rT * fT) := finfun [eta xf.2 with x0 |-> xf.1]. +rewrite /image_mem; set E := enum aT in (uniqE := enum_uniq aT) *. +have trivF x: x \notin E -> #|F x| = 1 by rewrite mem_enum. +elim: E uniqE => /= [_ | x0 E IH_E /andP[E'x0 uniqE]] in F trivF *. + have /fin_all_exists[f0 Ff0] x: exists y0, F x =i pred1 y0. + have /pred0Pn[y Fy]: #|F x| != 0 by rewrite trivF. + by exists y; apply/fsym/subset_cardP; rewrite ?subset_pred1 // card1 trivF. + apply: (@eq_card1 _ (finfun f0)) => f; apply/familyP/eqP=> [Ff | {f}-> x]. + by apply/ffunP=> x; have:= Ff x; rewrite Ff0 ffunE => /eqP. + by rewrite ffunE Ff0 inE /=. +have [y0 Fxy0 | Fx00] := pickP (F x0); last first. + by rewrite !eq_card0 // => f; apply: contraFF (Fx00 (f x0))=> /familyP; apply. +pose F1 x := if eqP is ReflectT Dx then xpred1 (ecast x (rT x) Dx y0) else F x. +transitivity (#|[predX F x0 & family F1]|) => [{uniqE IH_E}|]; last first. + rewrite cardX {}IH_E {uniqE}// => [|x E'x]; last first. + rewrite /F1; case: eqP => [Dx | /nesym/eqP-x0'x]; first exact: card1. + by rewrite trivF // negb_or x0'x. + congr (_ * foldr _ _ _); apply/eq_in_map=> x Ex. + by rewrite /F1; case: eqP => // Dx0; rewrite Dx0 Ex in E'x0. +pose g yf := let: (y, f) := yf : rT x0 * fT in + [ffun x => if eqP is ReflectT Dx then ecast x (rT x) Dx y else f x]. have gK: cancel (fun f : fT => (f x0, g (y0, f))) g. - by move=> f; apply/ffunP=> x; do !rewrite ffunE /=; case: eqP => // ->. -rewrite -cardX -(card_image (can_inj gK)); apply: eq_card => [] [y f] /=. -apply/imageP/andP=> [[f0 /familyP/=Ff0] [{f}-> ->]| [Fy /familyP/=Ff]]. - split; first by have:= Ff0 x0; rewrite /= mem_head. - apply/familyP=> x; have:= Ff0 x; rewrite ffunE inE /=. - by case: eqP => //= -> _; rewrite ifN ?inE. + by move=> f; apply/ffunP=> x; rewrite !ffunE; case: eqP => //; case: x /. +rewrite -(card_image (can_inj gK)); apply: eq_card => [] [y f] /=. +apply/imageP/andP=> [[f1 /familyP/=Ff1] [-> ->]| [/=Fx0y /familyP/=Ff]]. + split=> //; apply/familyP=> x; rewrite ffunE /F1 /=. + by case: eqP => // Dx; apply: eqxx. exists (g (y, f)). - by apply/familyP=> x; have:= Ff x; rewrite ffunE /= inE; case: eqP => // ->. -congr (_, _); last apply/ffunP=> x; do !rewrite ffunE /= ?eqxx //. -by case: eqP => // ->{x}; apply/eqP; have:= Ff x0; rewrite ifN. + by apply/familyP=> x; have:= Ff x; rewrite ffunE /F1; case: eqP; [case: x /|]. +congr (_, _); first by rewrite /= ffunE; case: eqP => // Dx; rewrite eq_axiomK. +by apply/ffunP=> x; have:= Ff x; rewrite !ffunE /F1; case: eqP => // Dx /eqP. Qed. -Lemma card_family F : #|family F| = foldr muln 1 [seq #|F x| | x : aT]. +Lemma card_dep_ffun : #|fT| = foldr muln 1 [seq #|rT x| | x : aT]. +Proof. by rewrite -card_family; apply/esym/eq_card=> f; apply/familyP. Qed. + +End FinDepTheory. + +Section FinFunTheory. + +Variables aT rT : finType. +Notation fT := {ffun aT -> rT}. +Implicit Types (D : pred aT) (R : pred rT) (F : aT -> pred rT). + +Lemma card_pfamily y0 D F : + #|pfamily y0 D F| = foldr muln 1 [seq #|F x| | x in D]. Proof. -have [y0 _ | rT0] := pickP rT; first exact: (card_pfamily y0 aT). -rewrite /image_mem; case DaT: (enum aT) => [{rT0}|x0 e] /=; last first. - by rewrite !eq_card0 // => [f | y]; [have:= rT0 (f x0) | have:= rT0 y]. -have{DaT} no_aT P (x : aT) : P by have:= mem_enum aT x; rewrite DaT. -apply: eq_card1 [ffun x => no_aT rT x] _ _ => f. -by apply/familyP/eqP=> _; [apply/ffunP | ] => x; apply: no_aT. +rewrite card_family !/(image _ _) /(enum D) -enumT /=. +by elim: (enum aT) => //= x E ->; have [// | D'x] := ifP; rewrite card1 mul1n. Qed. Lemma card_pffun_on y0 D R : #|pffun_on y0 D R| = #|R| ^ #|D|. @@ -299,7 +408,7 @@ rewrite (cardE D) card_pfamily /image_mem. by elim: (enum D) => //= _ e ->; rewrite expnS. Qed. -Lemma card_ffun_on R : #|ffun_on R| = #|R| ^ #|aT|. +Lemma card_ffun_on R : #|@ffun_on aT R| = #|R| ^ #|aT|. Proof. rewrite card_family /image_mem cardT. by elim: (enum aT) => //= _ e ->; rewrite expnS. @@ -308,5 +417,4 @@ Qed. Lemma card_ffun : #|fT| = #|rT| ^ #|aT|. Proof. by rewrite -card_ffun_on; apply/esym/eq_card=> f; apply/forallP. Qed. -End FinTheory. - +End FinFunTheory. |