lean2/tests/lean/induction2.lean

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import macros -- loads the λ, λ, obtain macros
using Nat -- using the Nat namespace (it allows us to suppress the Nat:: prefix)
axiom Induction : ∀ P : Nat → Bool, P 0 → (∀ n, P n → P (n + 1)) → ∀ n, P n.
-- induction on n
theorem Comm1 : ∀ n m, n + m = m + n
:= Induction
_ -- I use a placeholder because I do not want to write the P
(λ m, -- Base case
calc 0 + m = m : add::zerol m
... = m + 0 : symm (add::zeror m))
(λ n iH m, -- Inductive case
calc n + 1 + m = (n + m) + 1 : add::succl n m
... = (m + n) + 1 : { iH } -- Error is here
... = m + (n + 1) : symm (add::succr m n))