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refactor(library/data/nat/div): simplify proof of dvd_of_dvd_add_left

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Leonardo de Moura 2015-02-17 18:54:53 -08:00
parent eec57bef33
commit 74b8499fa9
1 changed files with 9 additions and 25 deletions
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library/data/nat/div.lean
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@ -310,31 +310,15 @@ calc
n = n * k div k : mul_div_cancel _ H1
... = m div k : H2
theorem dvd_of_dvd_add_left {m n1 n2 : ℕ} : m | (n1 + n2) → m | n1 → m | n2 :=
by_cases_zero_pos m
(assume (H1 : 0 | n1 + n2) (H2 : 0 | n1),
have H3 : n1 + n2 = 0, from eq_zero_of_zero_dvd H1,
have H4 : n1 = 0, from eq_zero_of_zero_dvd H2,
have H5 : n2 = 0, from calc
n2 = 0 + n2 : zero_add
... = n1 + n2 : H4
... = 0 : H3,
show 0 | n2, from H5 ▸ dvd.refl n2)
(take m,
assume mpos : m > 0,
assume H1 : m | (n1 + n2),
assume H2 : m | n1,
have H3 : n1 + n2 = n1 + n2 div m * m, from calc
n1 + n2 = (n1 + n2) div m * m : div_mul_cancel H1
... = (n1 div m * m + n2) div m * m : div_mul_cancel H2
... = (n2 + n1 div m * m) div m * m : add.comm
... = (n2 div m + n1 div m) * m : add_mul_div_self_right mpos
... = n2 div m * m + n1 div m * m : mul.right_distrib
... = n1 div m * m + n2 div m * m : add.comm
... = n1 + n2 div m * m : div_mul_cancel H2,
have H4 : n2 = n2 div m * m, from add.cancel_left H3,
have H5 : m * (n2 div m) = n2, from !mul.comm ▸ H4⁻¹,
dvd.intro H5)
theorem dvd_of_dvd_add_left {m n₁ n₂ : ℕ} (H₁ : m | n₁ + n₂) (H₂ : m | n₁) : m | n₂ :=
obtain (c₁ : nat) (Hc₁ : n₁ + n₂ = m * c₁), from H₁,
obtain (c₂ : nat) (Hc₂ : n₁ = m * c₂), from H₂,
have aux : m * (c₁ - c₂) = n₂, from calc
m * (c₁ - c₂) = m * c₁ - m * c₂ : mul_sub_left_distrib
... = n₁ + n₂ - m * c₂ : Hc₁
... = n₁ + n₂ - n₁ : Hc₂
... = n₂ : add_sub_cancel_left,
dvd.intro aux
theorem dvd_of_dvd_add_right {m n1 n2 : ℕ} (H : m | (n1 + n2)) : m | n2 → m | n1 :=
dvd_of_dvd_add_left (!add.comm ▸ H)