28 lines
818 B
Text
28 lines
818 B
Text
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import data.nat
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open nat
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section
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variables (a b c d e : nat)
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theorem T (H1 : a = b) (H2 : b = c + 1) (H3 : c = d) (H4 : e = 1 + d) : a = e :=
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by rewrite ⟨H1, H2, H3, add.comm, -H4⟩
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end
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example (x y : ℕ) : (x + y) * (x + y) = x * x + y * x + x * y + y * y :=
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by rewrite ⟨*mul.left_distrib, *mul.right_distrib, -add.assoc⟩
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definition even (a : nat) := ∃b, a = 2*b
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theorem even_plus_even {a b : nat} (H1 : even a) (H2 : even b) : even (a + b) :=
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exists.elim H1 (fun (w1 : nat) (Hw1 : a = 2*w1),
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exists.elim H2 (fun (w2 : nat) (Hw2 : b = 2*w2),
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exists.intro (w1 + w2)
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begin
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rewrite ⟨Hw1, Hw2, mul.left_distrib⟩
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end))
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theorem T2 (a b c : nat) (H1 : a = b) (H2 : b = c + 1) : a ≠ 0 :=
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calc
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a = succ c : by rewrite ⟨H1, H2, add_one⟩
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... ≠ 0 : succ_ne_zero c
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