132 lines
4.1 KiB
Coq
132 lines
4.1 KiB
Coq
From Coq Require Import Strings.String.
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Require Import terms.
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Include Terms.
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Module Subst.
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(* Type Variable "x" is a free variable in type *)
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Inductive type_var_free (x:string) : type_term -> Prop :=
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| TFree_Var :
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(type_var_free x (type_var x))
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| TFree_Ladder : forall τ1 τ2,
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(type_var_free x τ1) ->
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(type_var_free x τ2) ->
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(type_var_free x (type_ladder τ1 τ2))
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| TFree_Fun : forall τ1 τ2,
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(type_var_free x τ1) ->
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(type_var_free x τ2) ->
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(type_var_free x (type_fun τ1 τ2))
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| TFree_Morph : forall τ1 τ2,
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(type_var_free x τ1) ->
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(type_var_free x τ2) ->
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(type_var_free x (type_morph τ1 τ2))
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| TFree_Spec : forall τ1 τ2,
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(type_var_free x τ1) ->
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(type_var_free x τ2) ->
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(type_var_free x (type_spec τ1 τ2))
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| TFree_Univ : forall y τ,
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~(y = x) ->
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(type_var_free x τ) ->
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(type_var_free x (type_univ y τ))
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.
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Open Scope ladder_type_scope.
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Example ex_type_free_var1 :
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(type_var_free "T" (type_univ "U" (type_var "T")))
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.
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Proof.
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apply TFree_Univ.
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easy.
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apply TFree_Var.
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Qed.
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Open Scope ladder_type_scope.
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Example ex_type_free_var2 :
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~(type_var_free "T" (type_univ "T" (type_var "T")))
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.
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Proof.
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intro H.
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inversion H.
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contradiction.
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Qed.
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(* scoped variable substitution in type terms $\label{coq:subst-type}$ *)
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Fixpoint type_subst (v:string) (n:type_term) (t0:type_term) :=
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match t0 with
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| type_var name => if (eqb v name) then n else t0
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| type_fun t1 t2 => (type_fun (type_subst v n t1) (type_subst v n t2))
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| type_univ x t => if (eqb v x) then t0 else type_univ x (type_subst v n t)
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| type_spec t1 t2 => (type_spec (type_subst v n t1) (type_subst v n t2))
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| type_ladder t1 t2 => (type_ladder (type_subst v n t1) (type_subst v n t2))
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| t => t
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end.
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Inductive type_subst1 (x:string) (σ:type_term) : type_term -> type_term -> Prop :=
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| TSubst_VarReplace :
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(type_subst1 x σ (type_var x) σ)
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| TSubst_VarKeep : forall y,
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~(x = y) ->
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(type_subst1 x σ (type_var y) (type_var y))
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| TSubst_UnivReplace : forall y τ τ',
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~(x = y) ->
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~(type_var_free y σ) ->
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(type_subst1 x σ τ τ') ->
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(type_subst1 x σ (type_univ y τ) (type_univ y τ'))
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| TSubst_Id : forall n,
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(type_subst1 x σ (type_id n) (type_id n))
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| TSubst_Spec : forall τ1 τ2 τ1' τ2',
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(type_subst1 x σ τ1 τ1') ->
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(type_subst1 x σ τ2 τ2') ->
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(type_subst1 x σ (type_spec τ1 τ2) (type_spec τ1' τ2'))
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| TSubst_Fun : forall τ1 τ1' τ2 τ2',
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(type_subst1 x σ τ1 τ1') ->
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(type_subst1 x σ τ2 τ2') ->
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(type_subst1 x σ (type_fun τ1 τ2) (type_fun τ1' τ2'))
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| TSubst_Morph : forall τ1 τ1' τ2 τ2',
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(type_subst1 x σ τ1 τ1') ->
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(type_subst1 x σ τ2 τ2') ->
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(type_subst1 x σ (type_morph τ1 τ2) (type_morph τ1' τ2'))
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| TSubst_Ladder : forall τ1 τ1' τ2 τ2',
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(type_subst1 x σ τ1 τ1') ->
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(type_subst1 x σ τ2 τ2') ->
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(type_subst1 x σ (type_ladder τ1 τ2) (type_ladder τ1' τ2'))
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.
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(* scoped variable substitution, replaces free occurences of v with n in expression e *)
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Fixpoint expr_subst (v:string) (n:expr_term) (e0:expr_term) :=
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match e0 with
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| expr_var name => if (eqb v name) then n else e0
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| expr_ty_abs x e => if (eqb v x) then e0 else expr_ty_abs x (expr_subst v n e)
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| expr_ty_app e t => expr_ty_app (expr_subst v n e) t
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| expr_tm_abs x t e => if (eqb v x) then e0 else expr_tm_abs x t (expr_subst v n e)
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| 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)
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| expr_tm_app e a => expr_tm_app (expr_subst v n e) (expr_subst v n a)
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| expr_let x t a e => expr_let x t (expr_subst v n a) (expr_subst v n e)
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| expr_ascend t e => expr_ascend t (expr_subst v n e)
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| expr_descend t e => expr_descend t (expr_subst v n e)
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end.
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(* replace only type variables in expression *)
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Fixpoint expr_specialize (v:string) (n:type_term) (e0:expr_term) :=
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match e0 with
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| expr_ty_app e t => expr_ty_app (expr_specialize v n e) (type_subst v n t)
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| expr_ascend t e => expr_ascend (type_subst v n t) (expr_specialize v n e)
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| expr_descend t e => expr_descend (type_subst v n t) (expr_specialize v n e)
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| e => e
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end.
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End Subst.
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