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(* (c) Copyright 2006-2016 Microsoft Corporation and Inria. *)
(* Distributed under the terms of CeCILL-B. *)
Require Import mathcomp.ssreflect.ssreflect.
From mathcomp
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. *)
(* 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 *)
(* 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 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)).
Local Notation 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)).
Arguments familyP [aT rT pT F f].
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 ->]; apply: f_fam.
by move=> x Ax; apply: f_fam; apply/imageP; exists x.
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)).
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}].
(*****************************************************************************)
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.
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