23 lines
1.1 KiB
Text
23 lines
1.1 KiB
Text
Set: pp::colors
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Set: pp::unicode
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Imported 'tactic'
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Using: Nat
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Proved: add_assoc
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Nat::add_zerol : ∀ a : ℕ, 0 + a = a
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Nat::add_succl : ∀ a b : ℕ, a + 1 + b = a + b + 1
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@eq_id : ∀ (A : TypeU) (a : A), a = a ↔ ⊤
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theorem add_assoc (a b c : ℕ) : a + (b + c) = a + b + c :=
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Nat::induction_on
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a
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(let have_expr : 0 + (b + c) = 0 + b + c :=
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eqt_elim (trans (congr (congr2 eq (Nat::add_zerol (b + c)))
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(congr1 c (congr2 Nat::add (Nat::add_zerol b))))
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(eq_id (b + c)))
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in have_expr)
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(λ (n : ℕ) (iH : n + (b + c) = n + b + c),
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let have_expr : n + 1 + (b + c) = n + 1 + b + c :=
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eqt_elim (trans (congr (congr2 eq (trans (Nat::add_succl n (b + c)) (congr1 1 (congr2 Nat::add iH))))
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(trans (congr1 c (congr2 Nat::add (Nat::add_succl n b)))
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(Nat::add_succl (n + b) c)))
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(eq_id (n + b + c + 1)))
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in have_expr)
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