Library ModelSyntax
Require Import Term Conv.
Require Import Models.
Module SyntacticModel <: CC_Model.
Definition X := term.
Definition eqX : X -> X -> Prop := conv.
Instance eqX_equiv : Equivalence eqX.
Qed.
Notation "x == y" := (eqX x y) (at level 70).
Inductive mem : term -> term -> Prop :=
| MemProp :
(forall t, mem (subst t M) (Srt prop)) ->
mem (Abs T M) (Srt prop).
Parameter inX : X -> X -> Prop.
Notation "x \in y" := (inX x y) (at level 60).
Parameter in_ext: Proper (eqX ==> eqX ==> iff) inX.
Parameter props : X.
Parameter app : X -> X -> X.
Parameter lam : X -> (X -> X) -> X.
Parameter prod : X -> (X -> X) -> X.
Definition eq_fun (x:X) (f1 f2:X->X) :=
forall y1 y2, y1 \in x -> y1 == y2 -> f1 y1 == f2 y2.
Parameter lam_ext :
forall x1 x2 f1 f2,
x1 == x2 ->
eq_fun x1 f1 f2 ->
lam x1 f1 == lam x2 f2.
Parameter app_ext: Proper (eqX ==> eqX ==> eqX) app.
Parameter prod_ext :
forall x1 x2 f1 f2,
x1 == x2 ->
eq_fun x1 f1 f2 ->
prod x1 f1 == prod x2 f2.
Parameter prod_intro : forall dom f F,
eq_fun dom f f ->
eq_fun dom F F ->
(forall x, x \in dom -> f x \in F x) ->
lam dom f \in prod dom F.
Parameter prod_elim : forall dom f x F,
eq_fun dom F F ->
f \in prod dom F ->
x \in dom ->
app f x \in F x.
Parameter impredicative_prod : forall dom F,
eq_fun dom F F ->
(forall x, x \in dom -> F x \in props) ->
prod dom F \in props.
Parameter beta_eq:
forall dom F x,
eq_fun dom F F ->
x \in dom ->
app (lam dom F) x == F x.