55 lines
1.5 KiB
Coq
55 lines
1.5 KiB
Coq
(* This module defines the typing relation
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* where each expression is assigned a type.
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*)
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From Coq Require Import Strings.String.
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Require Import terms.
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Require Import subst.
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Include Terms.
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Include Subst.
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Module Typing.
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Inductive context : Type :=
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| ctx_assign : string -> type_term -> context -> context
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| ctx_empty : context
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.
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Inductive context_contains : context -> string -> type_term -> Prop :=
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| C_take : forall (x:string) (X:type_term) (Γ:context),
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(context_contains (ctx_assign x X Γ) x X)
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| C_shuffle : forall x X y Y Γ,
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(context_contains Γ x X) ->
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(context_contains (ctx_assign y Y Γ) x X).
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Reserved Notation "Gamma '|-' x '\in' X" (at level 101, x at next level, X at level 0).
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Inductive expr_type : context -> expr -> ladder_type -> Prop :=
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| T_Var : forall Γ x X,
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(context_contains Γ x X) ->
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Γ |- x \in X
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| T_Let : forall Γ s (σ:ladder_type) t τ x,
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Γ |- s \in σ ->
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Γ |- t \in τ ->
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Γ |- (expr_let x σ s t) \in τ
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| T_Abs : forall (Γ:context) (x:string) (X:ladder_type) (t:expr) (T:ladder_type),
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Γ |- t \in T ->
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Γ |- (expr_tm_abs x X t) \in (type_fun X T)
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| T_App : forall (Γ:context) (f:expr) (a:expr) (S:ladder_type) (T:ladder_type),
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Γ |- f \in (type_fun S T) ->
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Γ |- a \in S ->
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Γ |- (expr_tm_app f a) \in T
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where "Γ '|-' x '\in' X" := (expr_type Γ x X).
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Example typing1 :
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ctx_empty |-
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(expr_ty_abs "T" (expr_tm_abs "x" (type_var "T") (expr_var "x"))) \in
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(type_univ "T" (type_fun (type_var "T") (type_var "T"))).
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Proof.
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Admitted.
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End Typing.
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