Theorems · Theorem · commutative algebra
IsSumSq.nonneg
∀ {R : Type u_2} [inst : Semiring R] [inst_1 : LinearOrder R] [IsStrictOrderedRing R] [ExistsAddOfLE R] {s : R},
IsSumSq s → 0 ≤ sIn a linearly ordered semiring with the property a ≤ b → ∃ c, a + c = b (e.g. ℕ),
sums of squares are non-negative.
- Defined in
- Mathlib.Algebra.Ring.SumsOfSquares
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 20 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
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
- IsStrictOrderedRingstatement and proof · cited by 2,490
- ExistsAddOfLEstatement and proof · cited by 330
- IsSquareproof · cited by 132
- add_nonnegproof · cited by 104
- IsSumSqstatement and proof · cited by 38
- IsSquare.nonnegproof · cited by 4
- IsSumSq.rec'proof · cited by 2
Cited by2
Results whose statement or proof uses this declaration.
- RootPairing.zero_le_rootFormproof · cited by 3
- RootPairing.exists_ge_zero_eq_rootFormproof · cited by 1