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feat(library/algebra/ordered_group): add variant of triangle inequality

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This commit is contained in:
Jeremy Avigad 2015-09-12 21:21:34 -04:00
parent b48b33c412
commit 948cdee366
1 changed files with 11 additions and 1 deletions
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10
library/algebra/ordered_group.lean
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@ -698,6 +698,10 @@ section
theorem abs_eq_zero_iff_eq_zero (a : A) : abs a = 0 ↔ a = 0 :=
iff.intro eq_zero_of_abs_eq_zero (assume H, congr_arg abs H ⬝ !abs_zero)
theorem eq_of_abs_sub_eq_zero {a b : A} (H : abs (a - b) = 0) : a = b :=
have a - b = 0, from eq_zero_of_abs_eq_zero H,
show a = b, from eq_of_sub_eq_zero this
theorem abs_pos_of_ne_zero (H : a ≠ 0) : abs a > 0 :=
or.elim (lt_or_gt_of_ne H) abs_pos_of_neg abs_pos_of_pos
@ -770,6 +774,12 @@ section
... ≤ abs (a - b) + abs b : abs_add_le_abs_add_abs,
algebra.le_of_add_le_add_right H1
theorem abs_sub_le (a b c : A) : abs (a - c) ≤ abs (a - b) + abs (b - c) :=
calc
abs (a - c) = abs (a - b + (b - c)) :
by rewrite [sub_eq_add_neg, add.assoc, neg_add_cancel_left]
... ≤ abs (a - b) + abs (b - c) : abs_add_le_abs_add_abs
theorem abs_add_three (a b c : A) : abs (a + b + c) ≤ abs a + abs b + abs c :=
begin
apply le.trans,