Theorems · Theorem · order theory
sq_nonneg
∀ {R : Type u} [inst : Semiring R] [inst_1 : LinearOrder R] [ExistsAddOfLE R] [PosMulMono R] [AddLeftMono R] (a : R),
0 ≤ a ^ 2- Cited by
- 106 results in Mathlib
- Foundations
- Depth 17 from the axioms, rests on 218 definitions · uses propext
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.
- Semiringstatement and proof · cited by 13,802
- LinearOrderstatement and proof · cited by 8,572
- add_zeroproof · cited by 2,707
- LT.lt.leproof · cited by 2,189
- MulZeroClass.mul_zeroproof · cited by 2,091
- MulZeroClass.zero_mulproof · cited by 1,625
- add_commproof · cited by 1,535
- LT.lt.ne'proof · cited by 1,417
- add_assocproof · cited by 746
- AddLeftMonostatement and proof · cited by 687
- mul_addproof · cited by 413
- add_mulproof · cited by 363
Cited by106
Results whose statement or proof uses this declaration.
- mul_self_nonnegproof · cited by 35
- Real.le_sqrt_of_sq_leproof · cited by 10
- Complex.cos_argproof · cited by 7
- Complex.abs_im_le_normproof · cited by 6
- norm_jacobiTheta₂_term_leproof · cited by 5
- ProbabilityTheory.variance_const_mulproof · cited by 5
- Finset.sum_mul_sq_le_sq_mul_sqproof · cited by 5
- ProbabilityTheory.gaussianReal_map_const_mulstatement and proof · cited by 4
- InnerProductGeometry.angle_add_eq_arctan_of_inner_eq_zeroproof · cited by 4
- Real.lt_sqrtproof · cited by 4
- Real.cos_arcsinproof · cited by 4
- LinearMap.BilinForm.apply_apply_same_eq_zero_iffproof · cited by 3