Library finfun

 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. 
  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)                                
      ffun_on R f == the range of f is a subset of R                       
       family F f == f belongs to the family F (f x \in F x for all x)     
    support y f x == x is in the y-support of f (i.e., f x != y)           
 pffun_on y D R f == f is a (y,D)-partial function with range R, i.e., the 
                     y-support of f is a subset of D, and f x \in R for    
                     all x \in D.                                          
 pfamily y D pF f == f belongs to the partial family <y, D, pF>, i.e., the 
                     y-support of f is a subset of D, and f x \in pF x for 
                     all x \in D.                                          
     powerset D f == f is a sub-predicate of D (when rT == bool)           

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 Structure finfun_subType :=
  Eval hnf in [newType for fgraph by finfun_type_rect].

End Def.

Notation "{ 'ffun' fT }" := (finfun_of (Phant fT))
  (at level 0, format "{ 'ffun' '[hv' fT ']' }") : 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 [tuple of map f (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.
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 Structure fun_of_fin_unlock := Unlockable FunFinfun.fun_of_finE.
Canonical Structure 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.

 Lemma on the correspondance between finfun_type and finite domain function
Section PlainTheory.

Variables (aT : finType) (rT : Type).
Notation fT := {ffun aT -> rT}.

Canonical Structure finfun_of_subType := Eval hnf in [subType of fT].

Lemma tnth_fgraph : forall (f : fT) i, tnth (fgraph f) i = f (enum_val i).

Lemma ffunE : forall f : aT -> rT, finfun f =1 f.

Lemma fgraph_map : forall f : fT, fgraph f = [tuple of map f (enum aT)].

Lemma map_ffun_enum : forall f : fT, map f (enum aT) = val f.

Lemma ffunP : forall f1 f2 : fT, f1 =1 f2 <-> f1 = f2.

Lemma ffunK : cancel (@fun_of_fin aT rT) (@finfun aT rT).

Section Family.

Variable F : aT -> pred rT.

Definition family : pred fT := fun f => forallb x, F x (f x).

Lemma familyP : forall f : fT, reflect (forall x, F x (f x)) (family f).

End Family.

Definition ffun_on r := family (fun _ => r).

Lemma ffun_onP : forall (r : pred rT) (f : fT),
  reflect (forall x, r (f x)) (ffun_on r f).

End PlainTheory.

Lemma nth_fgraph_ord : forall T n (x0 : T) (i : 'I_n) f,
  nth x0 (fgraph f) i = f i.

Section EqTheory.

Variables (aT : finType) (rT : eqType).

Notation fT := {ffun aT -> rT}.

Definition finfun_eqMixin :=
  Eval hnf in [eqMixin of finfun_type aT rT by <:].
Canonical Structure finfun_eqType := Eval hnf in EqType _ finfun_eqMixin.

Canonical Structure finfun_of_eqType := Eval hnf in [eqType of fT].

Section Partial.

Variables (y0 : rT) (d : pred aT).

Definition support T f : pred T := fun x => f x != y0.

Definition pfamily F :=
  family (fun i => if d i then F i else pred1 y0 : pred _).

Lemma pfamilyP : forall (F : aT -> pred rT) (f : fT),
  reflect (subpred (support f) d /\ forall x, d x -> F x (f x)) (pfamily F f).

Definition pffun_on r := pfamily (fun _ => r).

Lemma pffun_onP : forall r (f : fT),
  reflect (subpred (support f) d /\ subpred (image f d) r) (pffun_on r f).

End Partial.

End EqTheory.

Definition finfun_choiceMixin aT (rT : choiceType) :=
  [choiceMixin of finfun_type aT rT by <:].
Canonical Structure finfun_choiceType aT rT :=
  Eval hnf in ChoiceType _ (finfun_choiceMixin aT rT).
Canonical Structure 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 Structure finfun_countType aT (rT : countType) :=
  Eval hnf in CountType _ (finfun_countMixin aT rT).
Canonical Structure finfun_of_countType (aT : finType) (rT : countType) :=
  Eval hnf in [countType of {ffun aT -> rT}].
Canonical Structure finfun_subCountType aT (rT : countType) :=
  Eval hnf in [subCountType of finfun_type aT rT].
Canonical Structure 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).

Definition finfun_finMixin := [finMixin of ffT by <:].
Canonical Structure finfun_finType := Eval hnf in FinType ffT finfun_finMixin.
Canonical Structure finfun_subFinType := Eval hnf in [subFinType of ffT].

Canonical Structure finfun_of_finType := Eval hnf in [finType of fT for finfun_finType].

Canonical Structure finfun_of_subFinType := Eval hnf in [subFinType of fT].

Lemma card_pfamily : forall y0 d (F : aT -> pred rT),
  #|pfamily y0 d F| = foldr (fun x m => #|F x| * m) 1 (enum d).

Lemma card_family : forall F : aT -> pred rT,
  #|family F| = foldr (fun x m => #|F x| * m) 1 (enum aT).

Lemma card_pffun_on : forall y0 d r, #|pffun_on y0 d r| = #|r| ^ #|d|.

Lemma card_ffun_on : forall r, #|ffun_on r| = #|r| ^ #|aT|.

Lemma card_ffun : #|fT| = #|rT| ^ #|aT|.

End FinTheory.

Section FinPowerSet.

Variable eT : finType.
Variable A : pred eT.

Definition powerset := pffun_on false A predT.

Lemma card_powerset : #|powerset| = 2 ^ #|A|.

End FinPowerSet.