131 lines
5 KiB
Coq
131 lines
5 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_id : string -> type_term
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| type_var : string -> 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_abs : string -> type_term -> expr_term -> expr_term
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| expr_morph : string -> type_term -> expr_term -> expr_term
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| expr_app : expr_term -> expr_term -> expr_term
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| expr_let : string -> 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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(* TODO
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Inductive type_DeBruijn : Type :=
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| id : nat -> type_DeBruijn
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| var : nat -> type_DeBruijn
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| fun : type_DeBruijn -> type_DeBruijn -> type_DeBruijn
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| univ : type_DeBruijn -> type_DeBruijn
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| spec : type_DeBruijn -> type_DeBruijn -> type_DeBruijn
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| morph : type_DeBruijn -> type_DeBruijn -> type_DeBruijn
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| ladder : type_DeBruijn -> type_DeBruijn -> type_DeBruijn
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Inductive expr_DeBruijn : Type :=
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| var : nat -> expr_DeBruijn
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| ty_abs : expr_DeBruijn -> expr_DeBruijn
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| ty_app : expr_DeBruijn -> type_DeBruijn -> expr_Debruijn
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| abs : type_DeBruijn -> expr_DeBruijn -> expr_DeBruijn
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| morph : type_DeBruijn -> expr_DeBruijn -> expr_Debruijn
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| app : expr_DeBruijn -> expr_DeBruijn -> expr_Debruijn
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| let : type_DeBruijn -> expr_DeBruijn -> expr_Debruijn -> expr_Debruijn
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| ascend : type_DeBruijn -> expr_DeBruijn -> expr_DeBruijn
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| descend : type_DeBruijn -> expr_DeBruijn -> expr_DeBruijn
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.
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*)
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(* values *)
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Inductive is_value : expr_term -> Prop :=
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| V_Abs : forall x τ e,
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(is_value (expr_abs x τ e))
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| V_TypAbs : forall τ e,
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(is_value (expr_ty_abs τ e))
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| V_Ascend : forall τ e,
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(is_value e) ->
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(is_value (expr_ascend τ e))
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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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Declare Custom Entry ladder_expr.
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Notation "[< t >]" := t
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(t custom ladder_type at level 80) : ladder_type_scope.
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Notation "'∀' x ',' t" := (type_univ x t)
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(t custom ladder_type at level 80, in custom ladder_type at level 80, x constr).
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Notation "'<' σ τ '>'" := (type_spec σ τ)
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(in custom ladder_type at level 80, left associativity) : ladder_type_scope.
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Notation "'(' τ ')'" := τ
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(in custom ladder_type at level 70) : ladder_type_scope.
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Notation "σ '->' τ" := (type_fun σ τ)
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(in custom ladder_type at level 75, right associativity) : ladder_type_scope.
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Notation "σ '->morph' τ" := (type_morph σ τ)
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(in custom ladder_type at level 75, right associativity, τ at level 80) : ladder_type_scope.
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Notation "σ '~' τ" := (type_ladder σ τ)
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(in custom ladder_type at level 70, right associativity) : ladder_type_scope.
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Notation "'$' x '$'" := (type_id x%string)
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(in custom ladder_type at level 0, x constr) : ladder_type_scope.
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Notation "'%' x '%'" := (type_var x%string)
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(in custom ladder_type at level 0, x constr) : ladder_type_scope.
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Notation "[{ e }]" := e
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(e custom ladder_expr at level 80) : ladder_expr_scope.
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Notation "'%' x '%'" := (expr_var x%string)
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(in custom ladder_expr at level 0, x constr) : ladder_expr_scope.
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Notation "'Λ' t '↦' e" := (expr_ty_abs t e)
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(in custom ladder_expr at level 10, t constr, e custom ladder_expr at level 80) : ladder_expr_scope.
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Notation "'λ' x τ '↦' e" := (expr_abs x τ e)
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(in custom ladder_expr at level 10, x constr, τ custom ladder_type at level 99, e custom ladder_expr at level 99) :ladder_expr_scope.
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Notation "'λ' x τ '↦morph' e" := (expr_morph x τ e)
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(in custom ladder_expr at level 10, x constr, τ custom ladder_type at level 99, e custom ladder_expr at level 99) :ladder_expr_scope.
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Notation "'let' x ':=' e 'in' t" := (expr_let x e t)
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(in custom ladder_expr at level 20, x constr, e custom ladder_expr at level 99, t custom ladder_expr at level 99) : ladder_expr_scope.
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Notation "e 'as' τ" := (expr_ascend τ e)
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(in custom ladder_expr at level 30, e custom ladder_expr, τ custom ladder_type at level 99) : ladder_expr_scope.
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Notation "e1 e2" := (expr_app e1 e2)
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(in custom ladder_expr at level 50) : ladder_expr_scope.
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Notation "'(' e ')'" := e
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(in custom ladder_expr at level 0) : ladder_expr_scope.
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(* EXAMPLES *)
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Open Scope ladder_type_scope.
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Open Scope ladder_expr_scope.
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Check [< ∀"α", (< $"Seq"$ %"α"% > ~ < $"List"$ %"α"% >) >].
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Definition polymorphic_identity1 : expr_term := [{ Λ"T" ↦ λ"x"%"T"% ↦ (%"x"%) }].
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Definition polymorphic_identity2 : expr_term := [{ Λ"T" ↦ λ"y"%"T"% ↦ %"y"% }].
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Compute polymorphic_identity1.
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Close Scope ladder_type_scope.
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Close Scope ladder_expr_scope.
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End Terms.
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