71 lines
2.9 KiB
Coq
71 lines
2.9 KiB
Coq
(* Define the abstract syntax of the calculus
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* by inductive definition of type-terms
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* and expression-terms.
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*)
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From Coq Require Import Strings.String.
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Module Terms.
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(* types *)
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Inductive type_term : Type :=
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| type_unit : type_term
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| type_id : string -> type_term
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| type_var : string -> type_term
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| type_num : nat -> type_term
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| type_fun : type_term -> type_term -> type_term
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| type_univ : string -> type_term -> type_term
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| type_spec : type_term -> type_term -> type_term
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| type_morph : type_term -> type_term -> type_term
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| type_ladder : type_term -> type_term -> type_term
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.
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(* expressions *)
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Inductive expr_term : Type :=
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| expr_var : string -> expr_term
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| expr_ty_abs : string -> expr_term -> expr_term
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| expr_ty_app : expr_term -> type_term -> expr_term
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| expr_tm_abs : string -> type_term -> expr_term -> expr_term
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| expr_tm_abs_morph : string -> type_term -> expr_term -> expr_term
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| expr_tm_app : expr_term -> expr_term -> expr_term
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| expr_let : string -> type_term -> expr_term -> expr_term -> expr_term
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| expr_ascend : type_term -> expr_term -> expr_term
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| expr_descend : type_term -> expr_term -> expr_term
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.
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Coercion type_var : string >-> type_term.
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Coercion expr_var : string >-> expr_term.
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(*
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Coercion type_var : string >-> type_term.
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Coercion expr_var : string >-> expr_term.
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*)
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Declare Scope ladder_type_scope.
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Declare Scope ladder_expr_scope.
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Declare Custom Entry ladder_type.
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Notation "[ e ]" := e (e custom ladder_type at level 80) : ladder_type_scope.
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(* TODO: allow any variable names in notation, not just α,β,γ *)
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Notation "'∀α.' τ" := (type_univ "α" τ) (in custom ladder_type at level 80) : ladder_type_scope.
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Notation "'∀β.' τ" := (type_univ "β" τ) (in custom ladder_type at level 80) : ladder_type_scope.
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Notation "'∀γ.' τ" := (type_univ "γ" τ) (in custom ladder_type at level 80) : ladder_type_scope.
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Notation "'<' σ τ '>'" := (type_spec σ τ) (in custom ladder_type at level 80, left associativity) : ladder_type_scope.
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Notation "'(' τ ')'" := τ (in custom ladder_type at level 70) : ladder_type_scope.
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Notation "σ '->' τ" := (type_fun σ τ) (in custom ladder_type at level 75, right associativity) : ladder_type_scope.
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Notation "σ '->morph' τ" := (type_morph σ τ) (in custom ladder_type at level 75, right associativity) : ladder_type_scope.
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Notation "σ '~' τ" := (type_ladder σ τ) (in custom ladder_type at level 70, right associativity) : ladder_type_scope.
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Notation "'α'" := (type_var "α") (in custom ladder_type at level 60, right associativity) : ladder_type_scope.
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Notation "'β'" := (type_var "β") (in custom ladder_type at level 60, right associativity) : ladder_type_scope.
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Notation "'γ'" := (type_var "γ") (in custom ladder_type at level 60, right associativity) : ladder_type_scope.
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Open Scope ladder_type_scope.
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Definition t1 : type_term := [ ∀α.∀β.(α~β~γ)->β->(α->α)->β ].
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Compute t1.
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Close Scope ladder_type_scope.
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End Terms.
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