Theorems · Theorem · order theory
two_mul_le_add_of_sq_le_mul
∀ {R : Type u} [inst : CommSemiring R] [inst_1 : LinearOrder R] [ExistsAddOfLE R] [MulPosStrictMono R]
[PosMulStrictMono R] [AddLeftReflectLE R] [AddLeftMono R] {a b r : R}, 0 ≤ a → 0 ≤ b → r ^ 2 ≤ a * b → 2 * r ≤ a + bBinary and division-free arithmetic mean-geometric mean inequality (aka AM-GM inequality) for linearly ordered commutative semirings.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 24 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites20
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommSemiringstatement and proof · cited by 10,911
- LinearOrderstatement and proof · cited by 8,572
- le_reflproof · cited by 2,061
- mul_assocproof · cited by 1,667
- AddLeftMonostatement and proof · cited by 687
- le_imp_le_of_le_of_leproof · cited by 576
- mul_le_mul_of_nonneg_leftproof · cited by 361
- ExistsAddOfLEstatement and proof · cited by 330
- sqproof · cited by 280
- two_mulproof · cited by 232
- PosMulStrictMonostatement and proof · cited by 151
- pow_twoproof · cited by 150
Cited by2
Results whose statement or proof uses this declaration.
- Finset.sum_sq_le_sum_mul_sum_of_sq_le_mulproof · cited by 3
- two_mul_le_add_of_sq_eq_mulproof · cited by 0