(* ** Basic definitions for substiutions *)
(* Version: 19.09. *)
From Undecidability.FOLP Require Export axioms.
(* Notation fin := nat. *)
Definition shift  := S.
Definition scons {X: Type} (x : X) (xi : nat -> X) :=
fun n => match n with
|0 => x
|S n => xi n
end.
Notation "s .: sigma" := (scons s sigma) (at level 70).
Definition var_zero := 0.
Definition funcomp {X Y Z} (g : Y -> Z) (f : X -> Y)  :=
fun x => g (f x).
(* **** Type classes for renamings. *)
Class Ren1 (X1  : Type) (Y Z : Type) :=
ren1 : X1 -> Y -> Z.
Class Ren2 (X1 X2 : Type) (Y Z : Type) :=
ren2 : X1 -> X2 -> Y -> Z.
Class Ren3 (X1 X2 X3 : Type) (Y Z : Type) :=
ren3 : X1 -> X2 -> X3 -> Y -> Z.
Class Ren4 (X1 X2 X3 X4 : Type) (Y Z : Type) :=
ren4 : X1 -> X2 -> X3 -> X4 -> Y -> Z.
Class Ren5 (X1 X2 X3 X4 X5 : Type) (Y Z : Type) :=
ren5 : X1 -> X2 -> X3 -> X4 -> X5 -> Y -> Z.
Declare Scope subst_scope.
Notation "s ⟨ xi1 ⟩" := (ren1 xi1 s) (at level 7, left associativity, format "s  ⟨ xi1 ⟩") : subst_scope.
Notation "s ⟨ xi1 ; xi2 ⟩" := (ren2 xi1 xi2 s) (at level 7, left associativity, format "s  ⟨ xi1 ; xi2 ⟩") : subst_scope.
Notation "s ⟨ xi1 ; xi2 ; xi3 ⟩" := (ren3 xi1 xi2 xi3 s) (at level 7, left associativity, format "s  ⟨ xi1 ; xi2 ; xi3 ⟩") : subst_scope.
Notation "s ⟨ xi1 ; xi2 ; xi3 ; xi4 ⟩" := (ren4  xi1 xi2 xi3 xi4 s) (at level 7, left associativity, format "s  ⟨ xi1 ; xi2 ; xi3 ; xi4 ⟩") : subst_scope.
Notation "s ⟨ xi1 ; xi2 ; xi3 ; xi4 ; xi5 ⟩" := (ren5  xi1 xi2 xi3 xi4 xi5 s) (at level 7, left associativity, format "s  ⟨ xi1 ; xi2 ; xi3 ; xi4 ; xi5 ⟩") : subst_scope.
Declare Scope fscope.
Notation "⟨ xi ⟩" := (ren1 xi) (at level 1, left associativity, format "⟨ xi ⟩") : fscope.
Notation "⟨ xi1 ; xi2 ⟩" := (ren2 xi1 xi2) (at level 1, left associativity, format "⟨ xi1 ; xi2 ⟩") : fscope.
(* **** Type Classes for Substiution *)
Class Subst1 (X1 : Type) (Y Z: Type) :=
subst1 : X1 -> Y -> Z.
Class Subst2 (X1 X2 : Type) (Y Z: Type) :=
subst2 : X1 -> X2 -> Y  -> Z.
Class Subst3 (X1 X2 X3 : Type) (Y Z: Type) :=
subst3 : X1 -> X2 -> X3 ->  Y  -> Z.
Class Subst4 (X1 X2 X3 X4: Type) (Y Z: Type) :=
subst4 : X1 -> X2 -> X3 -> X4 -> Y  -> Z.
Class Subst5 (X1 X2 X3 X4 X5 : Type) (Y Z: Type) :=
subst5 : X1 -> X2 -> X3 -> X4 -> X5  -> Y  -> Z.
Notation "s [ sigma ]" := (subst1 sigma s) (at level 7, left associativity, format "s '/' [ sigma ]") : subst_scope.
Notation "s [ sigma ; tau ]" := (subst2 sigma tau s) (at level 7, left associativity, format "s '/' [ sigma ; '/'  tau ]") : subst_scope.
Class Var X Y :=
ids : X -> Y.
Arguments funcomp {X Y Z} (g)%fscope (f)%fscope.
Notation "f >> g" := (funcomp g f) (*fun x => g (f x)*) (at level 50) : subst_scope.
Open Scope subst_scope.
Definition up_ren (xi : nat -> nat) :=
0 .: (xi >> S).
Definition ap {X Y} (f : X -> Y) {x y : X} (p : x = y) : f x = f y :=
match p with eq_refl => eq_refl end.
Definition apc {X Y} {f g : X -> Y} {x y : X} (p : f = g) (q : x = y) : f x = g y :=
match q with eq_refl => match p with eq_refl => eq_refl end end.
Definition id {X} (x: X) := x.

Lemma scons_eta {T} (f : nat -> T) :
f var_zero .: shift >> f = f.