`Compfiles/Imo2008P4.lean`

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6 declarations, 0 contain sorry, 0 declared as axioms

International Mathematical Olympiad 2008, Problem 4

Determine all functions f from the positive reals to the positive reals such that

(f(w)² + f(x)²) / (f(y)² + f(z)²) = (w² + x²) / (y² + z²)

for all positive real numbers w,x,y,z satisfying xw = yz.

Imports (3)

Imported by 1 file

Compfiles.lean

abbrev Imo2008P4.PosReal line 35 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Proof uses: Subtype, Real, LT.lt, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.
lemma Imo2008P4.positive_pow_eq_pow line 39 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Imo2008P4.PosReal, Iff, Eq, HPow.hPow, Real
Proof uses: Eq.mpr, id, congrArg, Subtype, Eq.symm, propext, Subtype.coe_inj, Positive.val_pow, pow_left_inj₀, le_of_lt, Subtype.property, Nat.ne_of_gt
33 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Imo2008P4.PosReal.mul_two In mathlib3, mul_two automatically worked on PosReal line 46 source
Source comment
In mathlib3, mul_two automatically worked on PosReal. In mathlib4 it does not. See https://github.com/leanprover-community/mathlib4/blob/7cc262f36953b78637c096c4bc6634c2af0b2a0a/Mathlib/Algebra/Ring/Defs.lean#L176-L179.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Imo2008P4.PosReal, Eq, HMul.hMul, Real, Subtype.mk, LT.lt, OfNat.ofNat, Nat, two_pos, HAdd.hAdd
Proof uses: Eq.mpr, id, congrArg, Subtype, propext, Subtype.ext_iff, of_eq_true, Eq.trans, True, congrFun', mul_two, eq_self
82 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
lemma PosReal.mul_two (x : PosReal) : x * ⟨2, two_pos⟩ = x + x := by
…
```
Mentions: Imo2008P4.PosReal
lemma Imo2008P4.PosReal.two_mul line 51 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Imo2008P4.PosReal, Eq, HMul.hMul, Real, Subtype.mk, LT.lt, OfNat.ofNat, Nat, two_pos, HAdd.hAdd
Proof uses: Eq.mpr, id, congrArg, Subtype, propext, Subtype.ext_iff, of_eq_true, Eq.trans, True, congrFun', two_mul, eq_self
82 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
determine Imo2008P4.solution_set line 58 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Imo2008P4.PosReal
Proof uses: Set.ofPred, Or, Eq, id, HDiv.hDiv, Real, OfNat.ofNat
12 instances, coercions and auxiliary constants left out.
problem Imo2008P4.imo2008_p4 line 60 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Imo2008P4.PosReal, Iff, Membership.mem, Set, Imo2008P4.solution_set, Eq, HMul.hMul, Real, HAdd.hAdd, HPow.hPow, Nat, OfNat.ofNat
Proof uses: Iff.intro, id, HDiv.hDiv, Eq.refl, Eq.symm, of_eq_true, Eq.trans, True, congrFun', congrArg, Inv.inv, funext, one_div, mul_inv_cancel, eq_self, Eq.mpr, propext, mul_right_inj, LT.lt, mul_assoc and 63 more
197 instances, coercions and auxiliary constants left out.

Tactic trace: 58 steps
Open to load the checked before-and-after goal states.