`Compfiles/Usa2013P4.lean`

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9 declarations, 0 contain sorry, 0 declared as axioms

USA Mathematical Olympiad 2013, Problem 4

Find all real numbers x, y, z ≥ 1 satisfying

min(√(x + xyz), √(y + xyz), √(z + xyz)) = √(x - 1) + √(y - 1) + √(z - 1).

Imports (2)

Imported by 1 file

Compfiles.lean

def Usa2013P4.Distinguished Documented declaration: Distinguished line 32 source
Source documentation
The condition that x is the "distinguished" coordinate of a solution (the coordinate at which the minimum of the left-hand side is attained): the other two shifted coordinates y - 1, z - 1 multiply to 1, and x - 1 is the reciprocal of (√(y - 1) + √(z - 1))².

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
Proof uses: And, Eq, HMul.hMul, HSub.hSub, OfNat.ofNat, HPow.hPow, Nat, HAdd.hAdd, Real.sqrt
13 instances, coercions and auxiliary constants left out.
```
def Distinguished (x y z : ℝ) : Prop :=
…
```
lemma Usa2013P4.key_identity The key algebraic identity: the difference between (1+u)(1+(1+v)(1+w)) and (√u + √v + √w)² is a sum of two squares line 56 source
Source documentation
The key algebraic identity: the difference between (1+u)(1+(1+v)(1+w)) and (√u + √v + √w)² is a sum of two squares.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, LE.le, OfNat.ofNat, Eq, HSub.hSub, HMul.hMul, HAdd.hAdd, HPow.hPow, Nat, Real.sqrt
Proof uses: Mathlib.Tactic.Ring.of_eq, Nat.rawCast, Mathlib.Tactic.Ring.Common.pow_congr, Mathlib.Tactic.Ring.Common.add_congr, Mathlib.Tactic.Ring.Common.atom_pf, rfl, Eq.mpr, id, congrArg, Eq.symm, Eq.refl, Mathlib.Tactic.Ring.Common.add_pf_add_lt, Mathlib.Tactic.Ring.Common.add_pf_zero_add, Mathlib.Tactic.Ring.cast_pos, Mathlib.Meta.NormNum.isNat_ofNat, Mathlib.Tactic.Ring.Common.pow_add, Mathlib.Tactic.Ring.Common.pow_nat, Mathlib.Tactic.Ring.Common.coeff_one, Mathlib.Tactic.Ring.Common.pow_one_cast_of_isNat, Mathlib.Meta.NormNum.IsNat.mk and 48 more
67 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
```
lemma key_identity (u v w : ℝ) (hu : 0 ≤ u) (hv : 0 ≤ v) (hw : 0 ≤ w) :
    (1 + u) * (1 + (1 + v) * (1 + w)) - (√u + √v + √w) ^ 2 =
      (√u * (√v + √w) - 1) ^ 2 + (1 + u) * (√v * √w - 1) ^ 2 := by
…
```
lemma Usa2013P4.eq_conds Equality conditions: if (1+u)(1+(1+v)(1+w)) = (√u + √v + √w)² for nonnegative u, v, w, then v * w = 1 and u * (√v + √w)² = 1 line 71 source
Source documentation
Equality conditions: if (1+u)(1+(1+v)(1+w)) = (√u + √v + √w)² for nonnegative u, v, w, then v * w = 1 and u * (√v + √w)² = 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, LE.le, OfNat.ofNat, Eq, HMul.hMul, HAdd.hAdd, HPow.hPow, Nat, Real.sqrt, And
Proof uses: HSub.hSub, Usa2013P4.key_identity, sq_nonneg, mul_nonneg, le_of_not_gt, LT.lt, Mathlib.Tactic.Linarith.lt_irrefl, Eq.mp, Neg.neg, congrArg, Mathlib.Tactic.Ring.of_eq, Mathlib.Tactic.Ring.Common.add_congr, Int.rawCast, inferInstance, Ring, Int.negOfNat, Nat.rawCast, Mathlib.Tactic.Ring.Common.neg_congr, Mathlib.Tactic.Ring.cast_pos, Mathlib.Meta.NormNum.isNat_ofNat and 77 more
107 instances, coercions and auxiliary constants left out.

Tactic trace: 31 steps
Open to load the checked before-and-after goal states.
```
lemma eq_conds {u v w : ℝ} (hu : 0 ≤ u) (hv : 0 ≤ v) (hw : 0 ≤ w)
    (h : (1 + u) * (1 + (1 + v) * (1 + w)) = (√u + √v + √w) ^ 2) :
    v * w = 1 ∧ u * (√v + √w) ^ 2 = 1 := by
…
```
Mentions: h
lemma Usa2013P4.min_eq_of_conds The conditions v * w = 1 and u * (√v + √w)² = 1 imply the equation, with the minimum attained at the term corresponding to u line 104 source
Source documentation
The conditions v * w = 1 and u * (√v + √w)² = 1 imply the equation, with the minimum attained at the term corresponding to u.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, LE.le, OfNat.ofNat, Eq, HMul.hMul, HPow.hPow, Nat, HAdd.hAdd, Real.sqrt, Min.min
Proof uses: Eq.mpr, id, congrArg, Eq.symm, Real.sqrt_mul, Real.sqrt_one, Eq.refl, add_nonneg, Real.sqrt_nonneg, LT.lt, Decidable.byContradiction, Not, le_antisymm, Eq.mp, sq_nonneg, zero_ne_one, MulZeroClass.mul_zero, mul_pow, Real.sq_sqrt, mul_nonneg and 97 more
134 instances, coercions and auxiliary constants left out.

Tactic trace: 61 steps
Open to load the checked before-and-after goal states.
```
lemma min_eq_of_conds {u v w : ℝ} (hu : 0 ≤ u) (hv : 0 ≤ v) (hw : 0 ≤ w)
    (h1 : v * w = 1) (h2 : u * (√v + √w) ^ 2 = 1) :
    min (min (√((1 + u) * (1 + (1 + v) * (1 + w))))
            (√((1 + v) * (1 + (1 + u) * (1 + w)))))
        (√((1 + w) * (1 + (1 + u) * (1 + v)))) = √u + √v + √w := by
…
```
lemma Usa2013P4.solution_of_conds Backward direction for one distinguished coordinate: the conditions v * w = 1 and u * (√v + √w)² = 1 imply u, v, w > 0 and the equation line 174 source
Source documentation
Backward direction for one distinguished coordinate: the conditions v * w = 1 and u * (√v + √w)² = 1 imply u, v, w > 0 and the equation.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Eq, HMul.hMul, OfNat.ofNat, HPow.hPow, Nat, HAdd.hAdd, Real.sqrt, And, LT.lt, Min.min
Proof uses: Ne, False.elim, False, Eq.mp, Eq.trans, congrFun', congrArg, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_pow, Eq.refl, Mathlib.Meta.NormNum.isNat_ofNat, Nat.cast_zero, Mathlib.Meta.NormNum.zero_natPow, MulZeroClass.mul_zero, eq_false, not_false, Decidable.byContradiction, Not, Real.sqrt_eq_zero_of_nonpos, LE.le and 82 more
99 instances, coercions and auxiliary constants left out.

Tactic trace: 39 steps
Open to load the checked before-and-after goal states.
```
lemma solution_of_conds {u v w : ℝ} (h1 : v * w = 1)
    (h2 : u * (√v + √w) ^ 2 = 1) :
    1 < 1 + u ∧ 1 < 1 + v ∧ 1 < 1 + w ∧
      min (min (√((1 + u) * (1 + (1 + v) * (1 + w))))
              (√((1 + v) * (1 + (1 + u) * (1 + w)))))
          (√((1 + w) * (1 + (1 + u) * (1 + v)))) = √u + √v + √w := by
…
```
lemma Usa2013P4.min_swap12 line 215 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Eq, Min.min
Proof uses: Eq.mpr, id, congrArg, min_comm, Eq.refl
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Usa2013P4.min_rotate line 218 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Eq, Min.min
Proof uses: Eq.mpr, id, congrArg, min_comm, Eq.symm, min_assoc, Eq.refl
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
determine Usa2013P4.solution_set line 223 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Prod, Real
Proof uses: Set.ofPred, Or, Usa2013P4.Distinguished, Prod.fst, Prod.snd
problem Usa2013P4.usa2013_p4 line 228 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Iff, And, LE.le, OfNat.ofNat, Eq, Min.min, Real.sqrt, HAdd.hAdd, HMul.hMul, HSub.hSub, Membership.mem, Prod, Set, Usa2013P4.solution_set, Prod.mk
Proof uses: Iff.intro, le_of_not_gt, LT.lt, Mathlib.Tactic.Linarith.lt_irrefl, Eq.mp, congrArg, Mathlib.Tactic.Ring.of_eq, Mathlib.Tactic.Ring.Common.add_congr, Nat.rawCast, HPow.hPow, Nat, Int.rawCast, inferInstance, Ring, Int.negOfNat, Mathlib.Tactic.Ring.Common.sub_congr, Mathlib.Tactic.Ring.cast_pos, Mathlib.Meta.NormNum.isNat_ofNat, Nat.cast_one, Mathlib.Tactic.Ring.Common.atom_pf and 77 more
85 instances, coercions and auxiliary constants left out.

Tactic trace: 98 steps
Open to load the checked before-and-after goal states.