105 lines
2.5 KiB
Coq
105 lines
2.5 KiB
Coq
Require Import Autosubst2.core Autosubst2.fintype Autosubst2.syntax common typing.
|
|
From Hammer Require Import Tactics.
|
|
Require Import ssreflect ssrbool.
|
|
Require Import Psatz.
|
|
From stdpp Require Import relations (rtc(..)).
|
|
Require Import Coq.Logic.FunctionalExtensionality.
|
|
|
|
Module HRed.
|
|
Inductive R {n} : PTm n -> PTm n -> Prop :=
|
|
(****************** Beta ***********************)
|
|
| AppAbs a b :
|
|
R (PApp (PAbs a) b) (subst_PTm (scons b VarPTm) a)
|
|
|
|
| ProjPair p a b :
|
|
R (PProj p (PPair a b)) (if p is PL then a else b)
|
|
|
|
(*************** Congruence ********************)
|
|
| AppCong a0 a1 b :
|
|
R a0 a1 ->
|
|
R (PApp a0 b) (PApp a1 b)
|
|
| ProjCong p a0 a1 :
|
|
R a0 a1 ->
|
|
R (PProj p a0) (PProj p a1).
|
|
End HRed.
|
|
|
|
(* Coquand's algorithm with subtyping *)
|
|
Reserved Notation "a ⇔ b" (at level 70).
|
|
Reserved Notation "a ↔ b" (at level 70).
|
|
Reserved Notation "a ≪ b" (at level 70).
|
|
Reserved Notation "a ⋖ b" (at level 70).
|
|
Inductive CoqEq {n} : PTm n -> PTm n -> Prop :=
|
|
| CE_AbsAbs a b :
|
|
a ↔ b ->
|
|
(* --------------------- *)
|
|
PAbs a ⇔ PAbs b
|
|
|
|
| CE_AbsNeu a u :
|
|
ishne u ->
|
|
a ↔ PApp (ren_PTm shift u) (VarPTm var_zero) ->
|
|
(* --------------------- *)
|
|
PAbs a ⇔ u
|
|
|
|
| CE_NeuAbs a u :
|
|
ishne u ->
|
|
PApp (ren_PTm shift u) (VarPTm var_zero) ↔ a ->
|
|
(* --------------------- *)
|
|
u ⇔ PAbs a
|
|
|
|
| CE_PairPair a0 a1 b0 b1 :
|
|
a0 ↔ a1 ->
|
|
b0 ↔ b1 ->
|
|
(* ---------------------------- *)
|
|
PPair a0 b0 ⇔ PPair a1 b1
|
|
|
|
| CE_PairNeu a0 a1 u :
|
|
ishne u ->
|
|
a0 ↔ PProj PL u ->
|
|
a1 ↔ PProj PR u ->
|
|
(* ----------------------- *)
|
|
PPair a0 a1 ⇔ u
|
|
|
|
| CE_NeuPair a0 a1 u :
|
|
ishne u ->
|
|
PProj PL u ↔ a0 ->
|
|
PProj PR u ↔ a1 ->
|
|
(* ----------------------- *)
|
|
u ⇔ PPair a0 a1
|
|
|
|
| CE_ProjCong p u0 u1 :
|
|
ishne u0 ->
|
|
ishne u1 ->
|
|
u0 ⇔ u1 ->
|
|
(* --------------------- *)
|
|
PProj p u0 ⇔ PProj p u1
|
|
|
|
| CE_AppCong u0 u1 a0 a1 :
|
|
ishne u0 ->
|
|
ishne u1 ->
|
|
u0 ⇔ u1 ->
|
|
a0 ↔ a1 ->
|
|
(* ------------------------- *)
|
|
PApp u0 a0 ⇔ PApp u1 a1
|
|
|
|
| CE_VarCong i :
|
|
(* -------------------------- *)
|
|
VarPTm i ⇔ VarPTm i
|
|
|
|
| CE_UnivCong i :
|
|
(* -------------------------- *)
|
|
PUniv i ⇔ PUniv i
|
|
|
|
| CE_BindCong p A0 A1 B0 B1 :
|
|
A0 ↔ A1 ->
|
|
B0 ↔ B1 ->
|
|
(* ---------------------------- *)
|
|
PBind p A0 B0 ⇔ PBind p A1 B1
|
|
|
|
with CoqEq_R {n} : PTm n -> PTm n -> Prop :=
|
|
| CE_HRed a a' b b' :
|
|
rtc HRed.R a a' ->
|
|
rtc HRed.R b b' ->
|
|
a' ⇔ b' ->
|
|
(* ----------------------- *)
|
|
a ↔ b
|
|
where "a ⇔ b" := (CoqEq a b) and "a ↔ b" := (CoqEq_R a b).
|