80 lines
2.4 KiB
Coq
80 lines
2.4 KiB
Coq
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 smallstep.
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Include Terms.
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Include Subst.
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Include Smallstep.
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Open Scope ladder_type_scope.
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Open Scope ladder_expr_scope.
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(* let bb_zero = Λα ↦ λs: α->α ↦ λz: α ↦ z
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* ∀α.(α->α)->α->α
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*)
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Definition bb_zero : expr_term :=
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(expr_ty_abs "α"
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(expr_abs "s" (type_fun (type_var "α") (type_var "α"))
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(expr_abs "z" (type_var "α")
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(expr_var "z")))).
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(* let bb_one = Λα ↦ λs: α->α ↦ λz: α ↦ s z
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*)
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Definition bb_one : expr_term :=
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(expr_ty_abs "α"
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(expr_abs "s" (type_fun (type_var "α") (type_var "α"))
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(expr_abs "z" (type_var "α")
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(expr_app (expr_var "s") (expr_var "z"))))).
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(* let bb_two = Λα ↦ λs: α->α ↦ λz: α ↦ s (s z)
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*)
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Definition bb_two : expr_term :=
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(expr_ty_abs "α"
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(expr_abs "s" (type_fun (type_var "α") (type_var "α"))
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(expr_abs "z" (type_var "α")
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(expr_app (expr_var "s") (expr_app (expr_var "s") (expr_var "z")))))).
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Definition bb_succ : expr_term :=
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(expr_abs "n" (type_ladder (type_id "ℕ")
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(type_ladder (type_id "BBNat")
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(type_univ "α"
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(type_fun (type_fun (type_var "α") (type_var "α"))
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(type_fun (type_var "α") (type_var "α"))))))
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(expr_ascend (type_ladder (type_id "ℕ") (type_id "BBNat"))
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(expr_ty_abs "α"
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(expr_abs "s" (type_fun (type_var "α") (type_var "α"))
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(expr_abs "z" (type_var "α")
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(expr_app (expr_var "s")
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(expr_app (expr_app
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(expr_ty_app (expr_var "n") (type_var "α"))
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(expr_var "s"))
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(expr_var "z")))))))).
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Definition e1 : expr_term :=
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(expr_let "bb-zero" bb_zero
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(expr_app (expr_app (expr_var "+") (expr_var "bb-zero")) (expr_var "bb-zero"))
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).
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Definition t1 : expr_term := (expr_app (expr_var "x") (expr_var "x")).
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Compute (expr_subst "x"
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(expr_ty_abs "α" (expr_abs "a" (type_var "α") (expr_var "a")))
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bb_one
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).
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Example example_let_reduction :
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e1 -->β (expr_app (expr_app (expr_var "+") bb_zero) bb_zero).
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Proof.
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apply E_Let.
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Qed.
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Compute (expr_app bb_succ bb_zero) -->β bb_one.
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Example example_succ :
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(expr_app bb_succ bb_zero) -->β bb_one.
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Proof.
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Admitted.
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