`Compfiles/Usa2001P5.lean`

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11 declarations, 0 contain sorry, 0 declared as axioms

USA Mathematical Olympiad 2001, Problem 5

Let S be a set of integers (not necessarily positive) such that

(a) there exist a, b ∈ S with gcd(a, b) = gcd(a − 2, b − 2) = 1; (b) if x and y are elements of S (possibly equal), then x² − y also belongs to S.

Prove that S is the set of all integers.

Imports (2)

Imported by 1 file

Compfiles.lean

def Usa2001P5.Shifty An integer d is shifty for a set S if shifting by d does not change membership in S line 34 source
Source documentation
An integer d is shifty for a set S if shifting by d does not change membership in S. The shifty integers form a subgroup of ℤ, and the proof below shows that 1 is shifty, which forces S = ℤ.

no axioms what this means
Statement uses: Set, Int
Proof uses: Iff, Membership.mem, HAdd.hAdd
3 instances, coercions and auxiliary constants left out.
```
def Shifty (S : Set ℤ) (d : ℤ) : Prop := ∀ x : ℤ, x ∈ S ↔ x + d ∈ S
```
Mentions: S
lemma Usa2001P5.shifty_zero line 36 source
standard axioms: propext what this means
Statement uses: Set, Int, Usa2001P5.Shifty, OfNat.ofNat
Proof uses: of_eq_true, Iff, Membership.mem, HAdd.hAdd, Eq.trans, True, congrArg, add_zero, iff_self
7 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Usa2001P5.shifty_neg line 38 source
standard axioms: propext what this means
Statement uses: Set, Int, Usa2001P5.Shifty, Neg.neg
Proof uses: Iff, Membership.mem, HAdd.hAdd, Eq, Mathlib.Tactic.Ring.of_eq, HMul.hMul, HPow.hPow, Nat, Nat.rawCast, OfNat.ofNat, Mathlib.Tactic.Ring.Common.add_congr, Int.rawCast, inferInstance, Ring, Int.negOfNat, Mathlib.Tactic.Ring.Common.atom_pf, rfl, Eq.mpr, id, congrArg and 20 more
30 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma Usa2001P5.shifty_add line 45 source
standard axioms: propext what this means
Statement uses: Set, Int, Usa2001P5.Shifty, HAdd.hAdd
Proof uses: Eq, Mathlib.Tactic.Ring.of_eq, HMul.hMul, HPow.hPow, Nat, Nat.rawCast, OfNat.ofNat, 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, Iff, Membership.mem, Iff.trans
22 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma Usa2001P5.shifty_nsmul line 52 source
standard axioms: propext what this means
Statement uses: Set, Int, Usa2001P5.Shifty, Nat, HSMul.hSMul
Proof uses: Nat.recAux, Eq.mpr, OfNat.ofNat, id, Eq, congrArg, zero_nsmul, Usa2001P5.shifty_zero, HAdd.hAdd, succ_nsmul, Usa2001P5.shifty_add
16 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
lemma Usa2001P5.shifty_zsmul line 60 source
standard axioms: propext what this means
Statement uses: Set, Int, Usa2001P5.Shifty, HSMul.hSMul
Proof uses: Nat, Or, Eq, Neg.neg, Exists, Int.eq_nat_or_neg, id, Usa2001P5.shifty_nsmul, Eq.symm, Eq.mpr, HMul.hMul, congrArg, neg_mul, Usa2001P5.shifty_neg
26 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma Usa2001P5.shifty_sq_sub_sq If u and v belong to S, then v^2 - u^2 is shifty for S line 67 source
Source documentation
If u and v belong to S, then v^2 - u^2 is shifty for S.

standard axioms: propext what this means
Statement uses: Set, Int, Membership.mem, HSub.hSub, HPow.hPow, Nat, OfNat.ofNat, Usa2001P5.Shifty
Proof uses: Iff.intro, HAdd.hAdd, Eq, Mathlib.Tactic.Ring.of_eq, HMul.hMul, Nat.rawCast, Int.rawCast, inferInstance, Ring, Int.negOfNat, 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.sub_congr, Mathlib.Tactic.Ring.Common.pow_congr and 36 more
41 instances, coercions and auxiliary constants left out.

Tactic trace: 16 steps
Open to load the checked before-and-after goal states.
```
lemma shifty_sq_sub_sq {S : Set ℤ} (hS : ∀ x ∈ S, ∀ y ∈ S, x^2 - y ∈ S)
    {u v : ℤ} (hu : u ∈ S) (hv : v ∈ S) : Shifty S (v^2 - u^2) := by
…
```
Mentions: S, Usa2001P5.Shifty
lemma Usa2001P5.not_prime_dvd_three The number-theoretic core: no prime divides all three of a^2 - b^2, a^3 * (a - 2) and b^3 * (b - 2) line 84 source
Source documentation
The number-theoretic core: no prime divides all three of a^2 - b^2, a^3 * (a - 2) and b^3 * (b - 2).

standard axioms: Quot.sound, propext what this means
Statement uses: Int, Eq, Nat, Int.gcd, OfNat.ofNat, HSub.hSub, Prime, Not, And, Dvd.dvd, HPow.hPow, HMul.hMul
Proof uses: False, Or, Prime.dvd_or_dvd, Or.inl, Prime.dvd_of_dvd_pow, Or.inr, HAdd.hAdd, Mathlib.Tactic.Ring.of_eq, Nat.rawCast, Int.rawCast, inferInstance, Ring, Int.negOfNat, Mathlib.Tactic.Ring.Common.mul_congr, Mathlib.Tactic.Ring.Common.add_congr, Mathlib.Tactic.Ring.Common.atom_pf, rfl, Eq.mpr, id, congrArg and 53 more
63 instances, coercions and auxiliary constants left out.

Tactic trace: 49 steps
Open to load the checked before-and-after goal states.
```
lemma not_prime_dvd_three {a b : ℤ} (h1 : Int.gcd a b = 1)
    (h2 : Int.gcd (a - 2) (b - 2) = 1) {p : ℤ} (hp : Prime p) :
    ¬ (p ∣ a^2 - b^2 ∧ p ∣ a^3 * (a - 2) ∧ p ∣ b^3 * (b - 2)) := by
…
```
lemma Usa2001P5.gcd_gcd_eq_one The gcd of the three shifty integers a^2 - b^2, a^3 * (a - 2), b^3 * (b - 2) is 1 line 122 source
Source documentation
The gcd of the three shifty integers a^2 - b^2, a^3 * (a - 2), b^3 * (b - 2) is 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Int, Eq, Nat, Int.gcd, OfNat.ofNat, HSub.hSub, HPow.hPow, HMul.hMul
Proof uses: Decidable.byContradiction, Not, And, Prime, Dvd.dvd, Exists, False, Int.exists_prime_and_dvd, id, Ne, Int.natAbs, dvd_trans, Int.gcd_dvd_left, Int.gcd_dvd_right, Usa2001P5.not_prime_dvd_three, And.intro
31 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
```
lemma gcd_gcd_eq_one {a b : ℤ} (h1 : Int.gcd a b = 1)
    (h2 : Int.gcd (a - 2) (b - 2) = 1) :
    Int.gcd (↑(Int.gcd (a^2 - b^2) (a^3 * (a - 2)))) (b^3 * (b - 2)) = 1 := by
…
```
lemma Usa2001P5.bezout_three Bézout's identity for three integers whose gcd is 1 line 137 source
Source documentation
Bézout's identity for three integers whose gcd is 1.

standard axioms: Quot.sound, propext what this means
Statement uses: Int, Eq, Nat, Int.gcd, OfNat.ofNat, Exists, HAdd.hAdd, HMul.hMul
Proof uses: Int.gcdA, rfl, Int.gcdB, Int.gcd_eq_gcd_ab, Eq.mp, congrFun', congrArg, Nat.cast_one, Eq.symm, Exists.intro, Eq.mpr, id, Mathlib.Tactic.Ring.of_eq, HPow.hPow, Nat.rawCast, Mathlib.Tactic.Ring.Common.add_congr, Mathlib.Tactic.Ring.Common.mul_congr, Mathlib.Tactic.Ring.Common.atom_pf, Eq.refl, Mathlib.Tactic.Ring.Common.add_mul and 11 more
37 instances, coercions and auxiliary constants left out.

Tactic trace: 13 steps
Open to load the checked before-and-after goal states.
```
lemma bezout_three {A B C : ℤ} (h : Int.gcd (↑(Int.gcd A B)) C = 1) :
    ∃ m n r : ℤ, m * A + n * B + r * C = 1 := by
…
```
Mentions: h
problem Usa2001P5.usa2001_p5 line 153 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Int, Membership.mem, HSub.hSub, HPow.hPow, Nat, OfNat.ofNat, Eq, Int.gcd, Set.univ
Proof uses: Usa2001P5.Shifty, Usa2001P5.shifty_sq_sub_sq, HMul.hMul, Mathlib.Tactic.Ring.of_eq, HAdd.hAdd, Nat.rawCast, Int.rawCast, inferInstance, Ring, Int.negOfNat, Mathlib.Tactic.Ring.Common.sub_congr, Mathlib.Tactic.Ring.Common.pow_congr, Mathlib.Tactic.Ring.Common.atom_pf, rfl, Eq.mpr, id, congrArg, Eq.symm, Eq.refl, Mathlib.Tactic.Ring.cast_pos and 56 more
54 instances, coercions and auxiliary constants left out.

Tactic trace: 28 steps
Open to load the checked before-and-after goal states.