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refactor(library/data/int/order): cleanup

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Leonardo de Moura 2015-03-28 12:50:28 -07:00
parent ea3407d06d
commit 5b9d52c0dd
1 changed files with 16 additions and 15 deletions
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library/data/int/order.lean
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@ -112,24 +112,25 @@ have H3 : a + of_nat (n + m) = c, from
... = c : Hm, ... = c : Hm,
le.intro H3 le.intro H3
theorem le.antisymm {a b : ℤ} (H1 : a ≤ b) (H2 : b ≤ a) : a = b := theorem le.antisymm : ∀ {a b : ℤ}, a ≤ b → b ≤ a → a = b :=
obtain (n : ℕ) (Hn : a + n = b), from le.elim H1, take a b : ℤ, assume (H₁ : a ≤ b) (H₂ : b ≤ a),
obtain (m : ℕ) (Hm : b + m = a), from le.elim H2, obtain (n : ℕ) (Hn : a + n = b), from le.elim H₁,
have H3 : a + of_nat (n + m) = a + 0, from obtain (m : ℕ) (Hm : b + m = a), from le.elim H₂,
have H₃ : a + of_nat (n + m) = a + 0, from
calc calc
a + of_nat (n + m) = a + (of_nat n + m) : {(of_nat_add_of_nat n m)⁻¹} a + of_nat (n + m) = a + (of_nat n + m) : of_nat_add_of_nat
... = a + n + m : (add.assoc a n m)⁻¹ ... = a + n + m : add.assoc
... = b + m : {Hn} ... = b + m : Hn
... = a : Hm ... = a : Hm
... = a + 0 : (add_zero a)⁻¹, ... = a + 0 : add_zero,
have H4 : of_nat (n + m) = of_nat 0, from add.left_cancel H3, have H₄ : of_nat (n + m) = of_nat 0, from add.left_cancel H₃,
have H5 : n + m = 0, from of_nat.inj H4, have H₅ : n + m = 0, from of_nat.inj H₄,
have H6 : n = 0, from nat.eq_zero_of_add_eq_zero_right H5, have H₆ : n = 0, from nat.eq_zero_of_add_eq_zero_right H₅,
show a = b, from show a = b, from
calc calc
a = a + of_nat 0 : (add_zero a)⁻¹ a = a + 0 : add_zero
... = a + n : {H6⁻¹} ... = a + n : H₆
... = b : Hn ... = b : Hn
theorem lt.irrefl (a : ℤ) : ¬ a < a := theorem lt.irrefl (a : ℤ) : ¬ a < a :=
(assume H : a < a, (assume H : a < a,