Theorems · Theorem · order theory
four_mul_le_sq_add
∀ {R : Type u} [inst : CommSemiring R] [inst_1 : LinearOrder R] [ExistsAddOfLE R] [MulPosStrictMono R]
[AddLeftReflectLE R] [AddLeftMono R] (a b : R), 4 * a * b ≤ (a + b) ^ 2Binary, squared, and division-free arithmetic mean-geometric mean inequality (aka AM-GM inequality) for linearly ordered commutative semirings.
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 23 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
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
- add_le_addproof · cited by 666
- add_mulproof · cited by 363
- ExistsAddOfLEstatement and proof · cited by 330
- AddLeftReflectLEstatement and proof · cited by 119
- MulPosStrictMonostatement and proof · cited by 94
- add_right_commproof · cited by 85
- add_sqproof · cited by 8
Cited by3
Results whose statement or proof uses this declaration.
- NNReal.sqrt_mul_le_half_addproof · cited by 3
- two_mul_le_add_of_sq_le_mulproof · cited by 2
- four_mul_le_pow_two_addproof · cited by 0