56 lines
1.4 KiB
Text
56 lines
1.4 KiB
Text
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open nat
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inductive type : Type :=
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| Nat : type
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| Func : type → type → type
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open type
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section var
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variable {var : type → Type}
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inductive term : type → Type :=
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| Var : ∀ {t}, var t → term t
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| Const : nat → term Nat
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| Plus : term Nat → term Nat → term Nat
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| Abs : ∀ {dom ran}, (var dom → term ran) → term (Func dom ran)
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| App : ∀ {dom ran}, term (Func dom ran) → term dom → term ran
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| Let : ∀ {t1 t2}, term t1 → (var t1 → term t2) → term t2
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end var
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open term
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definition Term t := Π (var : type → Type), @term var t
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open unit
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-- Define count_vars using tactics
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definition count_vars1 {t : type} (T : @term (λ x, unit) t) : nat :=
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begin
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induction T,
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{exact 1},
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{exact 0},
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{exact v_0 + v_1},
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{exact v_0 star},
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{exact v_0 + v_1},
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{exact v_0 + v_1 star},
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end
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-- Define count_vars using recursor
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definition count_vars2 {t : type} (T : @term (λ x, unit) t) : nat :=
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term.rec_on T
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(λ t u, 1)
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(λ n, 0)
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(λ T₁ T₂ n₁ n₂, n₁ + n₂)
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(λ d r f n, n star)
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(λ d r T₁ T₂ n₁ n₂, n₁ + n₂)
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(λ t₁ t₂ T₁ T₂ n₁ n₂, n₁ + n₂ star)
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definition var (t : type) : @term (λ x, unit) t :=
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Var star
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example : count_vars1 (App (App (var (Func Nat (Func Nat Nat))) (var Nat)) (var Nat)) = 3 :=
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rfl
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example : count_vars2 (App (App (var (Func Nat (Func Nat Nat))) (var Nat)) (var Nat)) = 3 :=
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rfl
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