32 lines
650 B
Text
32 lines
650 B
Text
open nat
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definition iter (f : nat → nat) (n : nat) : nat :=
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nat.rec_on n
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(f 1)
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(λ (n₁ : nat) (r : nat), f r)
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definition ack (m : nat) : nat → nat :=
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nat.rec_on m
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nat.succ
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(λ (m₁ : nat) (r : nat → nat), iter r)
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theorem ack_0_n (n : nat) : ack 0 n = n + 1 :=
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rfl
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theorem ack_m_0 (m : nat) : ack (m + 1) 0 = ack m 1 :=
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rfl
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theorem ack_m_n (m n : nat) : ack (m + 1) (n + 1) = ack m (ack (m + 1) n) :=
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rfl
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example : ack 3 2 = 29 :=
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rfl
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example : ack 3 3 = 61 :=
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rfl
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/- Defining Ack using well founded recursion -/
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def Ack : nat → nat → nat
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| 0 y := y+1
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| (x+1) 0 := Ack x 1
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| (x+1) (y+1) := Ack x (Ack (x+1) y)
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