coq: smallstep: define delta expansion

coq/bbencode.v

@@ -37,7 +37,7 @@ Definition bb_two : expr_term :=
 Definition bb_succ : expr_term :=
   (expr_tm_abs "n" (type_ladder (type_id "ℕ")
                     (type_ladder (type_id "BBNat")
-                     (type_abs "α"
+                     (type_univ "α"
                       (type_fun (type_fun (type_var "α") (type_var "α"))
                          (type_fun (type_var "α") (type_var "α"))))))
 
@@ -54,7 +54,7 @@ Definition bb_succ : expr_term :=
 Definition e1 : expr_term :=
   (expr_let "bb-zero" (type_ladder (type_id "ℕ")
                         (type_ladder (type_id "BBNat")
-                          (type_abs "α"
+                          (type_univ "α"
                             (type_fun (type_fun (type_var "α") (type_var "α"))
                               (type_fun (type_var "α") (type_var "α"))))))
                        bb_zero

coq/smallstep.v

@@ -1,51 +1,76 @@
 From Coq Require Import Strings.String.
 Require Import terms.
 Require Import subst.
+Require Import typing.
 
 Include Terms.
 Include Subst.
+Include Typing.
 
 Module Smallstep.
 
-Reserved Notation " s '-->α' t " (at level 40).
 Reserved Notation " s '-->β' t " (at level 40).
 Reserved Notation " s '-->δ' t " (at level 40).
+Reserved Notation " s '-->eval' t " (at level 40).
 
 Inductive beta_step : expr_term -> expr_term -> Prop :=
-  | E_AppLeft : forall e1 e1' e2,
+  | E_App1 : forall e1 e1' e2,
     e1 -->β e1' ->
     (expr_tm_app e1 e2) -->β (expr_tm_app e1' e2)
 
-  | E_AppRight : forall e1 e2 e2',
+  | E_App2 : forall e1 e2 e2',
     e2 -->β e2' ->
     (expr_tm_app e1 e2) -->β (expr_tm_app e1 e2')
 
-  | E_AppTmAbs : forall x τ e a,
+  | E_TypApp : forall e e' τ,
-    (expr_tm_app (expr_tm_abs x τ e) a) -->β (expr_subst x a e)
+    e -->β e' ->
+    (expr_ty_app e τ) -->β (expr_ty_app e' τ)
 
-  | E_AppTyAbs : forall x e a,
+  | E_TypAppLam : forall x e a,
     (expr_ty_app (expr_ty_abs x e) a) -->β (expr_specialize x a e)
 
+  | E_AppLam : forall x τ e a,
+    (expr_tm_app (expr_tm_abs x τ e) a) -->β (expr_subst x a e)
+
   | E_AppLet : forall x t e a,
     (expr_let x t a e) -->β (expr_subst x a e)
 
 where "s '-->β' t" := (beta_step s t).
 
-(*
+
-Inductive multi {X : Type} (R : relation X) : relation X :=
+
-  | multi_refl : forall (x : X), multi R x x
+Inductive delta_step : expr_term -> expr_term -> Prop :=
-  | multi_step : forall (x y z : X),
+
+  | E_ImplicitCast : forall (Γ:context) (f:expr_term) (h:string) (a:expr_term) (τ:type_term) (s:type_term) (p:type_term),
+
+    (context_contains Γ h (type_morph p s)) ->
+    Γ |- f \is (type_fun s τ) ->
+    Γ |- a \is p ->
+    (expr_tm_app f a) -->δ (expr_tm_app f (expr_tm_app (expr_var h) a))
+
+where "s '-->δ' t" := (delta_step s t).
+
+
+Inductive eval_step : expr_term -> expr_term -> Prop :=
+  | E_Beta : forall s t,
+    (s -->β t) ->
+    (s -->eval t)
+
+  | E_Delta : forall s t,
+    (s -->δ t) ->
+    (s -->eval t)
+
+where "s '-->eval' t" := (eval_step s t).
+
+Inductive multi {X : Type} (R : X -> X -> Prop) : X -> X -> Prop :=
+  | Multi_Refl : forall (x : X), multi R x x
+  | Multi_Step : forall (x y z : X),
                     R x y ->
                     multi R y z ->
                     multi R x z.
 
 Notation " s -->β* t " := (multi beta_step s t) (at level 40).
-*)
+Notation " s -->δ* t " := (multi delta_step s t) (at level 40).
-
+Notation " s -->eval* t " := (multi eval_step s t) (at level 40).
-(*
-Inductive delta_expand : expr_term -> expr_term -> Prop :=
-  | E_ImplicitCast
-  (expr_tm_app e1 e2)
-*)
 
 End Smallstep.

coq/subst.v

@@ -23,6 +23,7 @@ Fixpoint expr_subst (v:string) (n:expr_term) (e0:expr_term) :=
   | expr_ty_abs x e => if (eqb v x) then e0 else expr_ty_abs x (expr_subst v n e)
   | expr_ty_app e t => expr_ty_app (expr_subst v n e) t
   | expr_tm_abs x t e => if (eqb v x) then e0 else expr_tm_abs x t (expr_subst v n e)
+  | expr_tm_abs_morph x t e => if (eqb v x) then e0 else expr_tm_abs_morph x t (expr_subst v n e)
   | expr_tm_app e a => expr_tm_app (expr_subst v n e) (expr_subst v n a)
   | expr_let x t a e => expr_let x t (expr_subst v n a) (expr_subst v n e)
   | expr_ascend t e => expr_ascend t (expr_subst v n e)

coq/terms.v

@@ -15,6 +15,7 @@ Inductive type_term : Type :=
   | type_fun : type_term -> type_term -> type_term
   | type_univ : string -> type_term -> type_term
   | type_spec : type_term -> type_term -> type_term
+  | type_morph : type_term -> type_term -> type_term
   | type_ladder : type_term -> type_term -> type_term
 .
 
@@ -24,6 +25,7 @@ Inductive expr_term : Type :=
   | expr_ty_abs : string -> expr_term -> expr_term
   | expr_ty_app : expr_term -> type_term -> expr_term
   | expr_tm_abs : string -> type_term -> expr_term -> expr_term
+  | expr_tm_abs_morph : string -> type_term -> expr_term -> expr_term
   | expr_tm_app : expr_term -> expr_term -> expr_term
   | expr_let : string -> type_term -> expr_term -> expr_term -> expr_term
   | expr_ascend : type_term -> expr_term -> expr_term