146 lines
3.8 KiB
Coq
146 lines
3.8 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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Require Import equiv.
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Include Terms.
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Include Subst.
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Include Equiv.
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Module Typing.
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(** Subtyping *)
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Reserved Notation "s ':<=' t" (at level 50).
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Reserved Notation "s '~=~' t" (at level 50).
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Inductive is_syntactic_subtype : type_term -> type_term -> Prop :=
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| S_Refl : forall t t', (t === t') -> (t :<= t')
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| S_Trans : forall x y z, (x :<= y) -> (y :<= z) -> (x :<= z)
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| S_SynRepr : forall x' x y, (x :<= y) -> ((type_ladder x' x) :<= y)
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where "s ':<=' t" := (is_syntactic_subtype s t).
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Inductive is_semantic_subtype : type_term -> type_term -> Prop :=
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| S_Synt : forall x y,
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(x :<= y) -> (x ~=~ y)
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| S_SemRepr : forall x y y',
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(type_ladder x y) ~=~ (type_ladder x y')
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where "s '~=~' t" := (is_semantic_subtype s t).
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Open Scope ladder_type_scope.
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Example sub0 :
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[ < $"Seq"$ < $"Digit"$ $"10"$ > >
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~ < $"Seq"$ $"Char"$ > ]
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:<=
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[ < $"Seq"$ $"Char"$ > ]
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.
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Proof.
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apply S_SynRepr.
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apply S_Refl.
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apply L_Refl.
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Qed.
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Example sub1 :
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[ < $"Seq"$ < $"Digit"$ $"10"$ > ~ $"Char"$ > ]
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:<= [ < $"Seq"$ $"Char"$ > ]
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.
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Proof.
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set [ < $"Seq"$ < $"Digit"$ $"10"$ > > ~ < $"Seq"$ $"Char"$ > ].
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set [ < $"Seq"$ < $"Digit"$ $"10"$ > ~ $"Char"$ > ].
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set [ < $"Seq"$ $"Char"$ > ].
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set (t0 === t).
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set (t :<= t0).
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set (t :<= t2).
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apply S_Trans with t1.
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apply S_Refl.
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Qed.
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(** Typing Derivation *)
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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:context),
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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 '\is' X" (at level 101, x at next level, X at level 0).
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Reserved Notation "Gamma '|-' x '\compatible' X" (at level 101, x at next level, X at level 0).
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Inductive expr_type : context -> expr_term -> type_term -> Prop :=
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| T_Var : forall Γ x τ,
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(context_contains Γ x τ) ->
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(Γ |- x \is τ)
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| T_Let : forall Γ s (σ:type_term) t τ x,
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(Γ |- s \is σ) ->
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(Γ |- t \is τ) ->
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(Γ |- (expr_let x σ s t) \is τ)
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| T_TypeAbs : forall Γ (e:expr_term) (τ:type_term) α,
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Γ |- e \is τ ->
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Γ |- (expr_ty_abs α e) \is (type_univ α τ)
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| T_TypeApp : forall Γ α (e:expr_term) (σ:type_term) (τ:type_term),
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Γ |- e \is (type_univ α τ) ->
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Γ |- (expr_ty_app e σ) \is (type_subst α σ τ)
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| T_Abs : forall (Γ:context) (x:string) (σ:type_term) (t:expr_term) (τ:type_term),
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(context_contains Γ x σ) ->
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Γ |- t \is τ ->
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Γ |- (expr_tm_abs x σ t) \is (type_fun σ τ)
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| T_App : forall (Γ:context) (f:expr_term) (a:expr_term) (σ:type_term) (τ:type_term),
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Γ |- f \is (type_fun σ τ) ->
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Γ |- a \is σ ->
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Γ |- (expr_tm_app f a) \is τ
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where "Γ '|-' x '\is' τ" := (expr_type Γ x τ).
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Inductive expr_type_compatible : context -> expr_term -> type_term -> Prop :=
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| T_Compatible : forall Γ x τ,
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(Γ |- x \is τ) ->
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(Γ |- x \compatible τ)
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where "Γ '|-' x '\compatible' τ" := (expr_type_compatible Γ x τ).
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Example typing1 :
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forall Γ,
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(context_contains Γ "x" (type_var "T")) ->
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Γ |- (expr_ty_abs "T" (expr_tm_abs "x" (type_var "T") (expr_var "x"))) \is
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(type_univ "T" (type_fun (type_var "T") (type_var "T"))).
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Proof.
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intros.
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apply T_TypeAbs.
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apply T_Abs.
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apply H.
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apply T_Var.
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apply H.
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Admitted.
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Example typing2 :
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ctx_empty |- (expr_ty_abs "T" (expr_tm_abs "x" (type_var "T") (expr_var "x"))) \is
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(type_univ "T" (type_fun (type_var "T") (type_var "T"))).
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Proof.
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apply T_TypeAbs.
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apply T_Abs.
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Admitted.
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End Typing.
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