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coq: wip typing

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Michael Sippel 2024-07-27 13:30:12 +02:00
parent d880e07d57
commit eebb096f8a
Signed by: senvas
GPG key ID: F96CF119C34B64A6
1 changed files with 53 additions and 19 deletions
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72
coq/typing.v
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@ -21,35 +21,69 @@ Inductive context_contains : context -> string -> type_term -> Prop :=
(context_contains Γ x X) -> (context_contains Γ x X) ->
(context_contains (ctx_assign y Y Γ) x X). (context_contains (ctx_assign y Y Γ) x X).
Reserved Notation "Gamma '|-' x '\in' X" (at level 101, x at next level, X at level 0). Reserved Notation "Gamma '|-' x '\is' X" (at level 101, x at next level, X at level 0).
Reserved Notation "Gamma '|-' x '\compatible' X" (at level 101, x at next level, X at level 0).
Inductive expr_type : context -> expr -> ladder_type -> Prop := Inductive expr_type : context -> expr_term -> type_term -> Prop :=
| T_Var : forall Γ x X, | T_Var : forall Γ x τ,
(context_contains Γ x X) -> (context_contains Γ x τ) ->
Γ |- x \in X (Γ |- x \is τ)
| T_Let : forall Γ s (σ:ladder_type) t τ x, | T_Let : forall Γ s (σ:type_term) t τ x,
Γ |- s \in σ -> (Γ |- s \is σ) ->
Γ |- t \in τ -> (Γ |- t \is τ) ->
Γ |- (expr_let x σ s t) \in τ (Γ |- (expr_let x σ s t) \is τ)
| T_Abs : forall (Γ:context) (x:string) (X:ladder_type) (t:expr) (T:ladder_type), | T_TypeAbs : forall Γ (e:expr_term) (τ:type_term) α,
Γ |- t \in T -> Γ |- e \is τ ->
Γ |- (expr_tm_abs x X t) \in (type_fun X T) Γ |- (expr_ty_abs α e) \is (type_univ α τ)
| T_App : forall (Γ:context) (f:expr) (a:expr) (S:ladder_type) (T:ladder_type), | T_TypeApp : forall Γ α (e:expr_term) (σ:type_term) (τ:type_term),
Γ |- f \in (type_fun S T) -> Γ |- e \is (type_univ α τ) ->
Γ |- a \in S -> Γ |- (expr_ty_app e σ) \is (type_subst α σ τ)
Γ |- (expr_tm_app f a) \in T
where "Γ '|-' x '\in' X" := (expr_type Γ x X). | T_Abs : forall (Γ:context) (x:string) (σ:type_term) (t:expr_term) (τ:type_term),
(context_contains Γ x σ) ->
Γ |- t \is τ ->
Γ |- (expr_tm_abs x σ t) \is (type_fun σ τ)
| T_App : forall (Γ:context) (f:expr_term) (a:expr_term) (σ:type_term) (τ:type_term),
Γ |- f \is (type_fun σ τ) ->
Γ |- a \is σ ->
Γ |- (expr_tm_app f a) \is τ
where "Γ '|-' x '\is' τ" := (expr_type Γ x τ).
Inductive expr_type_compatible : context -> expr_term -> type_term -> Prop :=
| T_Compatible : forall Γ x τ,
(Γ |- x \is τ) ->
(Γ |- x \compatible τ)
where "Γ '|-' x '\compatible' τ" := (expr_type_compatible Γ x τ).
Example typing1 : Example typing1 :
ctx_empty |- forall Γ,
(expr_ty_abs "T" (expr_tm_abs "x" (type_var "T") (expr_var "x"))) \in (context_contains Γ "x" (type_var "T")) ->
Γ |- (expr_ty_abs "T" (expr_tm_abs "x" (type_var "T") (expr_var "x"))) \is
(type_univ "T" (type_fun (type_var "T") (type_var "T"))). (type_univ "T" (type_fun (type_var "T") (type_var "T"))).
Proof. Proof.
intros.
apply T_TypeAbs.
apply T_Abs.
apply H.
apply T_Var.
apply H.
Admitted.
Example typing2 :
ctx_empty |- (expr_ty_abs "T" (expr_tm_abs "x" (type_var "T") (expr_var "x"))) \is
(type_univ "T" (type_fun (type_var "T") (type_var "T"))).
Proof.
apply T_TypeAbs.
apply T_Abs.
Admitted. Admitted.
End Typing. End Typing.