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Diffstat (limited to 'mathcomp/discrete/finfun.v')
| -rw-r--r-- | mathcomp/discrete/finfun.v | 302 |
1 files changed, 302 insertions, 0 deletions
diff --git a/mathcomp/discrete/finfun.v b/mathcomp/discrete/finfun.v new file mode 100644 index 0000000..f880260 --- /dev/null +++ b/mathcomp/discrete/finfun.v @@ -0,0 +1,302 @@ +(* (c) Copyright Microsoft Corporation and Inria. All rights reserved. *) +Require Import ssreflect 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. *) +(* 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 *) +(* ot n.-tuple T. *) +(* For f : {ffun aT -> rT}, 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) *) +(* f \in ffun_on R == the range of f is a subset of R *) +(* f \in family F == f belongs to the family F (f x \in F x for all x) *) +(* 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. *) +(* f \in pffun_on y D R == f is a y-partial function from D to R: *) +(* f has y-support D and f x \in R for all x \in D. *) +(* f \in pfamily y D F == f belongs to the y-partial family from D to F: *) +(* f has y-support D and f x \in F x for all x \in D. *) +(******************************************************************************) + +Set Implicit Arguments. +Unset Strict Implicit. +Unset Printing Implicit Defensive. + +Section Def. + +Variables (aT : finType) (rT : Type). + +Inductive finfun_type : predArgType := Finfun of #|aT|.-tuple rT. + +Definition finfun_of of phant (aT -> rT) := finfun_type. + +Identity Coercion type_of_finfun : finfun_of >-> finfun_type. + +Definition fgraph f := let: Finfun t := f in t. + +Canonical finfun_subType := Eval hnf in [newType for fgraph]. + +End Def. + +Notation "{ 'ffun' fT }" := (finfun_of (Phant fT)) + (at level 0, format "{ 'ffun' '[hv' fT ']' }") : type_scope. +Definition finexp_domFinType n := ordinal_finType n. +Notation "T ^ n" := (@finfun_of (finexp_domFinType n) T (Phant _)) : type_scope. + +Notation Local fun_of_fin_def := + (fun aT rT f x => tnth (@fgraph aT rT f) (enum_rank x)). + +Notation Local finfun_def := (fun aT rT f => @Finfun aT rT (codom_tuple f)). + +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. +Axiom finfunE : finfun = finfun_def. +End FunFinfunSig. + +Module FunFinfun : FunFinfunSig. +Definition fun_of_fin := fun_of_fin_def. +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. + +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 "[ '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] + (at level 0, format "[ 'ffun' => F ]") : fun_scope. + +(* Helper for defining notation for function families. *) +Definition fmem aT rT (pT : predType rT) (f : aT -> pT) := [fun x => mem (f x)]. + +(* Lemmas on the correspondance between finfun_type and CiC functions. *) +Section PlainTheory. + +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. + +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 fgraph_codom f : fgraph f = codom_tuple f. +Proof. +apply: eq_from_tnth => i; rewrite [@fun_of_fin]unlock tnth_map. +by congr tnth; rewrite -[tnth _ _]enum_val_nth enum_valK. +Qed. + +Lemma codom_ffun f : codom f = val f. +Proof. by rewrite /= fgraph_codom. 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 ffunK : cancel (@fun_of_fin aT rT) (@finfun aT rT). +Proof. by move=> f; apply/ffunP/ffunE. Qed. + +Definition family_mem mF := [pred f : fT | [forall x, in_mem (f x) (mF x)]]. + +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. + +Definition ffun_on_mem mR := family_mem (fun _ => 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. + +Notation family F := (family_mem (fun_of_simpl (fmem F))). +Notation ffun_on R := (ffun_on_mem _ (mem R)). + +Implicit Arguments familyP [aT rT pT F f]. +Implicit Arguments ffun_onP [aT rT R f]. + +(*****************************************************************************) + +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. +Qed. + +Section Support. + +Variables (aT : Type) (rT : eqType). + +Definition support_for y (f : aT -> rT) := [pred x | f x != y]. + +Lemma supportE x y f : (x \in support_for y f) = (f x != y). Proof. by []. Qed. + +End Support. + +Notation "y .-support" := (support_for y) + (at level 2, format "y .-support") : fun_scope. + +Section EqTheory. + +Variables (aT : finType) (rT : eqType). +Notation fT := {ffun aT -> rT}. +Implicit Types (y : rT) (D : pred aT) (R : pred rT) (f : fT). + +Lemma supportP y D g : + reflect (forall x, x \notin D -> g x = y) (y.-support g \subset D). +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). + +Lemma pfamilyP (pT : predType rT) y D (F : aT -> pT) f : + reflect (y.-support f \subset D /\ {in D, forall x, f x \in F x}) + (f \in pfamily_mem y (mem D) (fmem F)). +Proof. +apply: (iffP familyP) => [/= f_pfam | [/supportP f_supp f_fam] x]. + split=> [|x Ax]; last by have:= f_pfam x; rewrite Ax. + by apply/subsetP=> x; case: ifP (f_pfam x) => //= _ fx0 /negP[]. +by case: ifPn => Ax /=; rewrite inE /= (f_fam, f_supp). +Qed. + +Definition pffun_on_mem y mD mR := pfamily_mem y mD (fun _ => mR). + +Lemma pffun_onP y D R f : + reflect (y.-support f \subset D /\ {subset image f D <= R}) + (f \in pffun_on_mem y (mem D) (mem R)). +Proof. +apply: (iffP (pfamilyP y D (fun _ => R) f)) => [] [-> f_fam]; split=> //. + by move=> _ /imageP[x Ax ->]; exact: f_fam. +by move=> x Ax; apply: f_fam; apply/imageP; exists x. +Qed. + +End EqTheory. +Canonical exp_eqType (T : eqType) n := [eqType of T ^ n]. + +Implicit 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)). + +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}]. +Canonical exp_choiceType (T : choiceType) n := [choiceType of T ^ n]. + +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}]. + +(*****************************************************************************) + +Section FinTheory. + +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). + +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]. +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]. +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. +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. +Qed. + +Lemma card_family F : #|family F| = foldr muln 1 [seq #|F x| | x : aT]. +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. +Qed. + +Lemma card_pffun_on y0 D R : #|pffun_on y0 D R| = #|R| ^ #|D|. +Proof. +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|. +Proof. +rewrite card_family /image_mem cardT. +by elim: (enum aT) => //= _ e ->; rewrite expnS. +Qed. + +Lemma card_ffun : #|fT| = #|rT| ^ #|aT|. +Proof. by rewrite -card_ffun_on; apply/esym/eq_card=> f; apply/forallP. Qed. + +End FinTheory. +Canonical exp_finType (T : finType) n := [finType of T ^ n]. + |