Library mathcomp.ssreflect.finfun
Require Import mathcomp.ssreflect.ssreflect.
From mathcomp
Require Import ssrfun ssrbool eqtype ssrnat seq choice fintype tuple.
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.
Definition fmem aT rT (pT : predType rT) (f : aT -> pT) := [fun x => mem (f x)].
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.
From mathcomp
Require Import ssrfun ssrbool eqtype ssrnat seq choice fintype tuple.
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.
Definition fmem aT rT (pT : predType rT) (f : aT -> pT) := [fun x => mem (f x)].
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.