ladder-calculus/coq/subst.v

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Coq
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From Coq Require Import Strings.String.
Require Import terms.
Include Terms.
Module Subst.
(* scoped variable substitution in type terms *)
Fixpoint type_subst (v:string) (n:type_term) (t0:type_term) :=
match t0 with
| type_var name => if (eqb v name) then n else t0
| type_fun t1 t2 => (type_fun (type_subst v n t1) (type_subst v n t2))
| type_univ x t => if (eqb v x) then t0 else type_univ x (type_subst v n t)
| type_spec t1 t2 => (type_spec (type_subst v n t1) (type_subst v n t2))
| type_ladder t1 t2 => (type_ladder (type_subst v n t1) (type_subst v n t2))
| t => t
end.
(* scoped variable substitution, replaces free occurences of v with n in expression e *)
Fixpoint expr_subst (v:string) (n:expr_term) (e0:expr_term) :=
match e0 with
| expr_var name => if (eqb v name) then n else e0
| 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)
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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)
| 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)
| expr_descend t e => expr_descend t (expr_subst v n e)
end.
(* replace only type variables in expression *)
Fixpoint expr_specialize (v:string) (n:type_term) (e0:expr_term) :=
match e0 with
| expr_ty_app e t => expr_ty_app (expr_specialize v n e) (type_subst v n t)
| expr_ascend t e => expr_ascend (type_subst v n t) (expr_specialize v n e)
| expr_descend t e => expr_descend (type_subst v n t) (expr_specialize v n e)
| e => e
end.
End Subst.