37 lines
No EOL
1.2 KiB
Text
37 lines
No EOL
1.2 KiB
Text
import macros -- loads the λ, λ, obtain macros
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using Nat -- using the Nat namespace (it allows us to suppress the Nat:: prefix)
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axiom Induction : ∀ P : Nat → Bool, P 0 → (∀ n, P n → P (n + 1)) → ∀ n, P n.
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-- induction on n
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theorem Comm1 : ∀ n m, n + m = m + n
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:= Induction
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_ -- I use a placeholder because I do not want to write the P
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(λ m, -- Base case
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calc 0 + m = m : add_zerol m
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... = m + 0 : symm (add_zeror m))
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(λ n, -- Inductive case
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λ (iH : ∀ m, n + m = m + n),
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λ m,
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calc n + 1 + m = (n + m) + 1 : add_succl n m
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... = (m + n) + 1 : { iH m }
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... = m + (n + 1) : symm (add_succr m n))
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-- indunction on m
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theorem Comm2 : ∀ n m, n + m = m + n
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:= λ n,
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Induction
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_
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(calc n + 0 = n : add_zeror n
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... = 0 + n : symm (add_zerol n))
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(λ m,
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λ (iH : n + m = m + n),
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calc n + (m + 1) = (n + m) + 1 : add_succr n m
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... = (m + n) + 1 : { iH }
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... = (m + 1) + n : symm (add_succl m n))
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print environment 1 |