`Compfiles/Usa1993P3.lean`

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6 declarations, 0 contain sorry, 0 declared as axioms

USA Mathematical Olympiad 1993, Problem 3

Consider functions f : [0,1] → ℝ which satisfy

(i) f(x) ≥ 0 for all x ∈ [0,1] (ii) f(1) = 1 (iii) f(x) + f(y) ≤ f (x + y) whenever x, y and x + y are all in [0,1].

Determine the smallest constant c such that

f(x) ≤ cx

for every function satisfying (i) - (iii) and every x ∈ [0,1].

Imports (2)

Imported by 1 file

Compfiles.lean

theorem Usa1993P3.lemma1 This lemma was proved automatically by Kimina-Prover 72B line 38 source
Source comment
This lemma was proved automatically by Kimina-Prover 72B

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Set.Icc, OfNat.ofNat, ite, LE.le, Membership.mem, Set, HDiv.hDiv, Nat, And, Subtype.mk, HMul.hMul, Eq, HAdd.hAdd
Proof uses: Decidable.byContradiction, Not, GE.ge, of_eq_true, Eq.trans, True, congr, congrArg, eq_true, Mathlib.Meta.NormNum.isRat_le_true, Int.ofNat, Mathlib.Meta.NormNum.IsNNRat.to_isRat, Mathlib.Meta.NormNum.IsNat.to_isNNRat, Mathlib.Meta.NormNum.isNat_ofNat, Nat.cast_zero, Mathlib.Meta.NormNum.isNNRat_div, Mathlib.Meta.NormNum.isNNRat_mul, Inv.inv, Eq.refl, Nat.cast_one and 123 more
141 instances, coercions and auxiliary constants left out.

Tactic trace: 25 steps
Open to load the checked before-and-after goal states.
```
theorem lemma1 (c1 : ℝ) :
  let f := fun x : (Set.Icc (0:ℝ) 1) ↦ if (x:ℝ) ≤ 1 / 2 then 0 else 1;
…
```
Mentions: f
lemma Usa1993P3.dyadicBracket line 67 source
theorem Usa1993P3.lemma2 This lemma was proved by GPT5, with some help from dwrensha to fix up some errors line 85 source
Source comment
This lemma was proved by GPT5, with some help from dwrensha to fix up some errors.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Set.Icc, OfNat.ofNat, And, LE.le, Eq, Membership.mem, Set, HAdd.hAdd, Subtype.mk, HSub.hSub, Nat, HDiv.hDiv, HPow.hPow, HMul.hMul
Proof uses: of_eq_true, Eq.trans, True, congr, congrArg, and_self, Eq.mpr, Subtype, id, Eq.mp, congrFun', Subtype.mk.congr_simp, sub_self, le_of_not_gt, LT.lt, Mathlib.Tactic.Linarith.lt_irrefl, Mathlib.Tactic.Ring.of_eq, Mathlib.Tactic.Ring.Common.add_congr, Nat.rawCast, Int.rawCast and 74 more
182 instances, coercions and auxiliary constants left out.

Tactic trace: 73 steps
Open to load the checked before-and-after goal states.
```
theorem lemma2 (f : ↑(Set.Icc 0 1) → ℝ) (x : ℝ) (hx : 0 ≤ x ∧ x ≤ 1)
    (h1 : ∀ (x : ↑(Set.Icc 0 1)), 0 ≤ f x) (h2 : f 1 = 1)
    (h3 : ∀ (x y : ↑(Set.Icc 0 1)) (h : x.val + y.val ∈ Set.Icc 0 1),
               f x + f y ≤ f ⟨x.val + y.val, h⟩)
    (h4 : ∀ (x : ↑(Set.Icc (0 : ℝ) 1)), 1 - x.val ∈ Set.Icc 0 1)
    (h5 : ∀ (x : ↑(Set.Icc (0 : ℝ) 1)), f x + f ⟨1 - x.val, h4 x⟩ ≤ 1)
    (h6 : ∀ (x : ↑(Set.Icc (0 : ℝ) 1)), f x ≤ 1)
    (h8 : ∀ n : ℕ, ∀ x : Set.Icc (0 : ℝ) (1/2^n), 2^n * (x : ℝ) ∈ Set.Icc (0:ℝ) 1)
    : f ⟨x, hx⟩ ≤ 2 * x := by
…
```
Mentions: f, h
def Usa1993P3.valid line 186 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Set.Icc, OfNat.ofNat
Proof uses: And, LE.le, Eq, Membership.mem, Set, HAdd.hAdd, Subtype.mk
21 instances, coercions and auxiliary constants left out.
determine Usa1993P3.min_c line 191 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
Proof uses: OfNat.ofNat
3 instances, coercions and auxiliary constants left out.
problem Usa1993P3.usa1993_p3 line 193 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: IsLeast, Real, Set.ofPred, Set.Icc, OfNat.ofNat, Usa1993P3.valid, LE.le, HMul.hMul, Membership.mem, Set, Usa1993P3.min_c
Proof uses: Eq.mpr, And, Subtype.mk, Eq.substr, id, Eq, congrFun', congrArg, funext, Subtype, forall_congr, implies_congr, Eq.refl, Eq.trans, forall_prop_domain_congr, And.intro, lowerBounds, HAdd.hAdd, HSub.hSub, Iff.mp and 125 more
148 instances, coercions and auxiliary constants left out.

Tactic trace: 49 steps
Open to load the checked before-and-after goal states.