`Compfiles/Usa2002P4.lean`

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11 declarations, 0 contain sorry, 0 declared as axioms

USA Mathematical Olympiad 2002, Problem 4

Find all functions f : ℝ → ℝ such that for all x, y ∈ ℝ: f(x² - y²) = x f(x) - y f(y).

Imports (2)

Imported by 1 file

Compfiles.lean

def Usa2002P4.FE line 28 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
Proof uses: Eq, HSub.hSub, HPow.hPow, Nat, OfNat.ofNat, HMul.hMul
9 instances, coercions and auxiliary constants left out.
lemma Usa2002P4.f_zero f(0)=0 line 36 source
Source comment
f(0)=0

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Usa2002P4.FE, Eq, OfNat.ofNat
Proof uses: Eq.mp, HSub.hSub, HPow.hPow, Nat, HMul.hMul, congr, congrArg, Eq.trans, zero_pow, of_eq_true, Not, False, True, not_false_eq_true, sub_self, MulZeroClass.zero_mul
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
lemma f_zero {f : ℝ → ℝ} (hf : FE f) : f 0 = 0 := by
…
```
Mentions: f, Usa2002P4.FE
lemma Usa2002P4.squarish f(x^2) = xf(x) line 40 source
Source comment
f(x^2) = xf(x)

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Usa2002P4.FE, Eq, HPow.hPow, Nat, OfNat.ofNat, HMul.hMul
Proof uses: Eq.mp, HSub.hSub, congr, congrArg, Eq.trans, zero_pow, of_eq_true, Not, False, True, not_false_eq_true, sub_zero, Usa2002P4.f_zero, MulZeroClass.mul_zero
33 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
lemma squarish {f : ℝ → ℝ} (hf : FE f) (x : ℝ) : f (x^2) = x * f x := by
…
```
Mentions: f, Usa2002P4.FE
lemma Usa2002P4.square_additive f(x^2) = f(x^2)-f(y^2) line 44 source
Source comment
f(x^2) = f(x^2)-f(y^2)

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Usa2002P4.FE, Eq, HSub.hSub, HPow.hPow, Nat, OfNat.ofNat
Proof uses: Eq.mpr, HMul.hMul, id, congrArg, congr, Usa2002P4.squarish
16 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
lemma square_additive {f : ℝ → ℝ} (hf : FE f) (x : ℝ) (y : ℝ)  :
    f (x^2 - y^2) = f (x^2) - f (y^2) := by
…
```
Mentions: f, Usa2002P4.FE
lemma Usa2002P4.f_odd f is odd line 50 source
Source comment
f is odd

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Usa2002P4.FE, Eq, Neg.neg
Proof uses: dite, GE.ge, OfNat.ofNat, HSub.hSub, HPow.hPow, Nat, Real.sqrt, Usa2002P4.square_additive, of_eq_true, Eq.trans, True, congrArg, id, Eq.mp, congr, zero_pow, Not, False, not_false_eq_true, Real.sq_sqrt and 10 more
56 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
lemma f_odd {f : ℝ → ℝ} (hf : FE f) (x : ℝ)  :
    -f x = f (-x) := by
…
```
Mentions: f, Usa2002P4.FE
lemma Usa2002P4.additive_pos f(x+y) = f(x)+f(y) with x,y ≥ 0 first line 60 source
Source comment
f(x+y) = f(x)+f(y) with x,y ≥ 0 first

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Usa2002P4.FE, LE.le, OfNat.ofNat, Eq, HAdd.hAdd
Proof uses: HSub.hSub, HPow.hPow, Nat, Real.sqrt, Usa2002P4.square_additive
18 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
lemma additive_pos {f : ℝ → ℝ} (hf : FE f) {x : ℝ} {y : ℝ} (hx : 0 ≤ x) (hy : 0 ≤ y):
    f (x + y) = f x + f y := by
…
```
Mentions: f, Usa2002P4.FE
lemma Usa2002P4.additive Now show Cauchy FE holds with x,y not necessarily positive (lots of cases) line 66 source
Source comment
Now show Cauchy FE holds with x,y not necessarily positive (lots of cases)

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Usa2002P4.FE, Eq, HAdd.hAdd
Proof uses: LE.le, Not, Or, Classical.em, dite, OfNat.ofNat, Usa2002P4.additive_pos, Neg.neg, of_eq_true, Eq.trans, True, congrArg, id, Eq.mp, congrFun', add_neg_cancel_comm, Eq.symm, Usa2002P4.f_odd, add_neg_cancel_comm_assoc, eq_self and 59 more
77 instances, coercions and auxiliary constants left out.

Tactic trace: 39 steps
Open to load the checked before-and-after goal states.
```
lemma additive {f : ℝ → ℝ} (hf : FE f) (x : ℝ) (y : ℝ)  :
    f (x + y) = f x + f y := by
…
```
Mentions: f, Usa2002P4.FE
lemma Usa2002P4.four_additive Apply Cauchy four times to f line 93 source
Source comment
Apply Cauchy four times to f

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Usa2002P4.FE, Eq, HAdd.hAdd
Proof uses: Usa2002P4.additive, Mathlib.Tactic.Linarith.eq_of_not_lt_of_not_gt, Not.intro, LT.lt, Mathlib.Tactic.Linarith.lt_irrefl, OfNat.ofNat, Eq.mp, Neg.neg, HSub.hSub, congrArg, Mathlib.Tactic.Ring.of_eq, Mathlib.Tactic.Ring.Common.add_congr, HMul.hMul, HPow.hPow, Nat, Nat.rawCast, Int.rawCast, inferInstance, Ring, Int.negOfNat and 38 more
54 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
lemma four_additive {f : ℝ → ℝ} (hf : FE f) (a : ℝ) (b : ℝ) (c : ℝ) (d : ℝ) :
    f (a + b + c + d) = f a + f b + f c + f d := by
…
```
Mentions: f, Usa2002P4.FE
lemma Usa2002P4.main_proof Show the FE implies f(x) = f(1)*x line 101 source
Source comment
Show the FE implies f(x) = f(1)*x

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Usa2002P4.FE, Eq, HMul.hMul, OfNat.ofNat
Proof uses: HPow.hPow, Nat, HAdd.hAdd, Usa2002P4.squarish, Eq.mp, congrArg, Usa2002P4.four_additive, 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, Eq.symm, Eq.refl, Mathlib.Tactic.Ring.cast_pos, Mathlib.Meta.NormNum.isNat_ofNat, Nat.cast_one and 29 more
33 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
```
lemma main_proof {f : ℝ → ℝ} (hf : FE f) (x : ℝ)  :
    f x = f 1 * x := by
…
```
Mentions: f, Usa2002P4.FE
determine Usa2002P4.solution_set line 112 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Real
Proof uses: Set.ofPred, Exists, Eq, HMul.hMul
2 instances, coercions and auxiliary constants left out.
problem Usa2002P4.usa2002_p4 line 114 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Iff, Membership.mem, Set, Usa2002P4.solution_set, Usa2002P4.FE
Proof uses: Iff.intro, id, Exists.intro, Eq, HMul.hMul, OfNat.ofNat, Usa2002P4.main_proof
7 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.