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feat(algebra/ordered_field): ad missing theorem

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Rob Lewis 2016-02-01 15:39:08 -05:00 committed by Leonardo de Moura
parent 40a1371cd0
commit dcfc496992
1 changed files with 17 additions and 2 deletions
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19
library/algebra/ordered_field.lean
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@ -317,11 +317,16 @@ section linear_ordered_field
symm (iff.mpr (!eq_div_iff_mul_eq (ne_of_gt (add_pos zero_lt_one zero_lt_one)))
(by krewrite [left_distrib, *mul_one]))
theorem two_ge_one : (2:A) ≥ 1 :=
theorem two_gt_one : (2:A) > 1 :=
calc (2:A) = 1+1 : one_add_one_eq_two
... ≥ 1+0 : add_le_add_left (le_of_lt zero_lt_one)
... > 1+0 : add_lt_add_left zero_lt_one
... = 1 : add_zero
theorem two_ge_one : (2:A) ≥ 1 :=
le_of_lt two_gt_one
theorem four_pos : (4 : A) > 0 := add_pos two_pos two_pos
theorem mul_le_mul_of_mul_div_le (H : a * (b / c) ≤ d) (Hc : c > 0) : b * a ≤ d * c :=
begin
rewrite [-mul_div_assoc at H, mul.comm b],
@ -534,4 +539,14 @@ section discrete_linear_ordered_field
have abs a ≠ 0, from assume H, this (eq_zero_of_abs_eq_zero H),
!eq_div_of_mul_eq this !eq_sign_mul_abs⁻¹)
theorem add_quarters (a : A) : a / 4 + a / 4 = a / 2 :=
have H4 [visible] : (4 : A) = 2 * 2, by norm_num,
calc
a / 4 + a / 4 = (a + a) / (2 * 2) : by rewrite [-H4, div_add_div_same]
... = (a * 1 + a * 1) / (2 * 2) : by rewrite mul_one
... = (a * (1 + 1)) / (2 * 2) : by rewrite left_distrib
... = (a * 2) / (2 * 2) : rfl
... = ((a * 2) / 2) / 2 : by rewrite -div_div_eq_div_mul
... = a / 2 : by rewrite (mul_div_cancel a two_ne_zero)
end discrete_linear_ordered_field