Library mathcomp.ssreflect.finfun
(* (c) Copyright 2006-2015 Microsoft Corporation and Inria.
Distributed under the terms of CeCILL-B. *)
Require Import mathcomp.ssreflect.ssreflect.
Distributed under the terms of CeCILL-B. *)
Require Import mathcomp.ssreflect.ssreflect.
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.
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.
Module Type FunFinfunSig.
Parameter fun_of_fin : ∀ aT rT, finfun_type aT rT → aT → rT.
Parameter finfun : ∀ (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.
Lemma finfunE : finfun = finfun_def.
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.
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).
Lemma ffunE (g : aT → rT) : finfun g =1 g.
Lemma fgraph_codom f : fgraph f = codom_tuple f.
Lemma codom_ffun f : codom f = val f.
Lemma ffunP f1 f2 : f1 =1 f2 ↔ f1 = f2.
Lemma ffunK : cancel (@fun_of_fin aT rT) (@finfun aT rT).
Definition family_mem mF := [pred f : fT | [∀ x, in_mem (f x) (mF x)]].
Lemma familyP (pT : predType rT) (F : aT → pT) f :
reflect (∀ x, f x \in F x) (f \in family_mem (fmem F)).
Definition ffun_on_mem mR := family_mem (fun _ ⇒ mR).
Lemma ffun_onP R f : reflect (∀ x, f x \in R) (f \in ffun_on_mem (mem R)).
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.
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).
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 (∀ x, x \notin D → g x = y) (y.-support g \subset D).
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, ∀ x, f x \in F x})
(f \in pfamily_mem y (mem D) (fmem F)).
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)).
End EqTheory.
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}].
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].
Lemma card_family F : #|family F| = foldr muln 1 [seq #|F x| | x : aT].
Lemma card_pffun_on y0 D R : #|pffun_on y0 D R| = #|R| ^ #|D|.
Lemma card_ffun_on R : #|ffun_on R| = #|R| ^ #|aT|.
Lemma card_ffun : #|fT| = #|rT| ^ #|aT|.
End FinTheory.
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).
Lemma ffunE (g : aT → rT) : finfun g =1 g.
Lemma fgraph_codom f : fgraph f = codom_tuple f.
Lemma codom_ffun f : codom f = val f.
Lemma ffunP f1 f2 : f1 =1 f2 ↔ f1 = f2.
Lemma ffunK : cancel (@fun_of_fin aT rT) (@finfun aT rT).
Definition family_mem mF := [pred f : fT | [∀ x, in_mem (f x) (mF x)]].
Lemma familyP (pT : predType rT) (F : aT → pT) f :
reflect (∀ x, f x \in F x) (f \in family_mem (fmem F)).
Definition ffun_on_mem mR := family_mem (fun _ ⇒ mR).
Lemma ffun_onP R f : reflect (∀ x, f x \in R) (f \in ffun_on_mem (mem R)).
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.
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).
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 (∀ x, x \notin D → g x = y) (y.-support g \subset D).
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, ∀ x, f x \in F x})
(f \in pfamily_mem y (mem D) (fmem F)).
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)).
End EqTheory.
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}].
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].
Lemma card_family F : #|family F| = foldr muln 1 [seq #|F x| | x : aT].
Lemma card_pffun_on y0 D R : #|pffun_on y0 D R| = #|R| ^ #|D|.
Lemma card_ffun_on R : #|ffun_on R| = #|R| ^ #|aT|.
Lemma card_ffun : #|fT| = #|rT| ^ #|aT|.
End FinTheory.