41 lines
1.1 KiB
Text
41 lines
1.1 KiB
Text
import data.nat.basic data.prod
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open prod
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namespace nat
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namespace manual
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definition brec_on {C : nat → Type} (n : nat) (F : Π (n : nat), @nat.below C n → C n) : C n :=
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have general : C n × @nat.below C n, from
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nat.rec_on n
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(pair (F zero poly_unit.star) poly_unit.star)
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(λ (n₁ : nat) (r₁ : C n₁ × @nat.below C n₁),
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have b : @nat.below C (succ n₁), from
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r₁,
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have c : C (succ n₁), from
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F (succ n₁) b,
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pair c b),
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pr₁ general
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end manual
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definition fib (n : nat) :=
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nat.brec_on n (λ (n : nat),
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nat.cases_on n
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(λ (b₀ : nat.below 0), succ 0)
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(λ (n₁ : nat), nat.cases_on n₁
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(λ b₁ : nat.below (succ 0), succ zero)
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(λ (n₂ : nat) (b₂ : nat.below (succ (succ n₂))), pr₁ b₂ + pr₁ (pr₂ b₂))))
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theorem fib_0 : fib 0 = 1 :=
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rfl
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theorem fib_1 : fib 1 = 1 :=
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rfl
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theorem fib_s_s (n : nat) : fib (succ (succ n)) = fib (succ n) + fib n :=
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rfl
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example : fib 5 = 8 :=
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rfl
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example : fib 9 = 55 :=
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rfl
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end nat
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