`Compfiles/Imo1964P2.lean`

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2 declarations, 0 contain sorry, 0 declared as axioms

International Mathematical Olympiad 1964, Problem 2

Suppose that a,b,c are the side lengths of a triangle. Prove that

a²(b + c - a) + b²(c + a - b) + c²(a + b - c) ≤ 3abc.

Imports (3)

Imported by 1 file

Compfiles.lean

lemma Imo1964P2.schur TODO: get this into mathlib in some form line 31 source
Source comment
TODO: get this into mathlib in some form

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, LE.le, OfNat.ofNat, HAdd.hAdd, HMul.hMul, HSub.hSub
Proof uses: Not, Or, Classical.em, Eq, Mathlib.Tactic.Ring.of_eq, HPow.hPow, Nat, Nat.rawCast, Mathlib.Meta.NormNum.instAddMonoidWithOne, inferInstance, Ring, Int.rawCast, Int.negOfNat, Mathlib.Tactic.Ring.Common.add_congr, Mathlib.Tactic.Ring.Common.mul_congr, Mathlib.Tactic.Ring.Common.atom_pf, rfl, Eq.mpr, id, congrArg and 55 more
82 instances, coercions and auxiliary constants left out.

Tactic trace: 24 steps
Open to load the checked before-and-after goal states.
```
lemma schur {a b c : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (hc : 0 ≤ c) :
    0 ≤ a * (a - b) * (a - c) + b * (b - a) * (b - c) + c * (c - a) * (c - b) := by
…
```
problem Imo1964P2.imo1964_p2 line 56 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Affine.Triangle, Real, EuclideanSpace, Fin, OfNat.ofNat, Nat, ENNReal, Eq, Dist.dist, Affine.Simplex.points, HAdd.hAdd, LE.le, HMul.hMul, HPow.hPow, HSub.hSub
Proof uses: dist_nonneg, Mathlib.Tactic.Ring.of_eq, Nat.rawCast, Int.rawCast, inferInstance, Ring, Int.negOfNat, Mathlib.Meta.NormNum.instAddMonoidWithOne, Mathlib.Tactic.Ring.Common.sub_congr, Mathlib.Tactic.Ring.Common.mul_congr, Mathlib.Tactic.Ring.cast_pos, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.instAtLeastTwo, Mathlib.Tactic.Ring.Common.atom_pf, rfl, Eq.mpr, id, congrArg, Eq.symm and 42 more
126 instances, coercions and auxiliary constants left out.

Tactic trace: 15 steps
Open to load the checked before-and-after goal states.