`Compfiles/Usa1997P3.lean`

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17 declarations, 0 contain sorry, 0 declared as axioms

USA Mathematical Olympiad 1997, Problem 3

Prove that for any integer n, there exists a unique polynomial Q with coefficients in {0, 1, ..., 9} such that Q(-2) = Q(-5) = n.

Imports (8)

Imported by 1 file

Compfiles.lean

def Usa1997P3.digit The unique digit in {0, …, 9} congruent to u modulo 2 and to v modulo 5 (which exist and are unique by the Chinese remainder theorem) line 47 source
Source documentation
The unique digit in {0, …, 9} congruent to u modulo 2 and to v modulo 5 (which exist and are unique by the Chinese remainder theorem).

no axioms what this means
Statement uses: Int
Proof uses: HMod.hMod, HSub.hSub, HMul.hMul, OfNat.ofNat
7 instances, coercions and auxiliary constants left out.
```
def digit (u v : ℤ) : ℤ := (6 * v - 5 * u) % 10
```
lemma Usa1997P3.digit_props line 49 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Int, And, LE.le, OfNat.ofNat, Usa1997P3.digit, Eq, HMod.hMod
Proof uses: id, And.intro, HSub.hSub, HMul.hMul
16 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma Usa1997P3.step_props Documented declaration: step_props line 58 source
Source documentation
One step of the extraction: the successor pair ((digit u v - u)/2, (digit u v - v)/5) keeps the invariant u ≡ v (mod 3), and its components are strictly smaller than those of (u, v) as soon as |v| > 2 (for the second component) or |u| > 9 (for the first one).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Int, Eq, HMod.hMod, HSub.hSub, OfNat.ofNat, And, LE.le, Usa1997P3.digit, HMul.hMul, HDiv.hDiv, LT.lt, Nat, Int.natAbs
Proof uses: Usa1997P3.digit_props, Dvd.dvd, Int.mul_ediv_cancel', And.intro
34 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
```
lemma step_props (u v : ℤ) (h3 : (u - v) % 3 = 0) :
    0 ≤ digit u v ∧ digit u v ≤ 9 ∧
    2 * ((digit u v - u) / 2) = digit u v - u ∧
    5 * ((digit u v - v) / 5) = digit u v - v ∧
    ((digit u v - u) / 2 - (digit u v - v) / 5) % 3 = 0 ∧
    (2 < v.natAbs → ((digit u v - v) / 5).natAbs < v.natAbs) ∧
    (v.natAbs ≤ 2 → 9 < u.natAbs → ((digit u v - u) / 2).natAbs < u.natAbs) ∧
    (v.natAbs ≤ 2 → ((digit u v - v) / 5).natAbs ≤ 2) := by
…
```
Mentions: Usa1997P3.digit
def Usa1997P3.polyOfDigits The polynomial whose coefficients are the given list of digits, in little-endian order line 75 source
Source documentation
The polynomial whose coefficients are the given list of digits, in little-endian order.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: List, Int, Polynomial
4 instances, coercions and auxiliary constants left out.
```
noncomputable def polyOfDigits : List ℤ → ℤ[X]
  | [] => 0
  | d :: ds => C d + X * polyOfDigits ds
```
lemma Usa1997P3.eval_polyOfDigits_cons line 79 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Int, List, Eq, Polynomial.eval, Usa1997P3.polyOfDigits, List.cons, HAdd.hAdd, HMul.hMul
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, Polynomial, Polynomial.X, RingHom, Polynomial.C, eq_intCast, Polynomial.eval_add, congr, Polynomial.eval_intCast, Polynomial.eval_mul, Polynomial.eval_X, eq_self
31 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Usa1997P3.coeff_polyOfDigits_cons line 83 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Int, List, Nat, Eq, Finsupp, Polynomial.coeff, Usa1997P3.polyOfDigits, List.cons, ite, OfNat.ofNat, HSub.hSub
Proof uses: Nat.casesAuxOn, Eq.mpr, id, congrArg, ite_eq_left, rfl, of_eq_true, Eq.trans, True, congrFun', HAdd.hAdd, Polynomial, RingHom, Polynomial.C, HMul.hMul, Polynomial.X, Polynomial.coeff_add, congr, Polynomial.coeff_C_zero, Polynomial.mul_coeff_zero and 12 more
34 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
lemma Usa1997P3.coeff_polyOfDigits line 95 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: List, Int, Membership.mem, And, LE.le, OfNat.ofNat, Nat, Finsupp, Polynomial.coeff, Usa1997P3.polyOfDigits
Proof uses: List.rec, List.nil, of_eq_true, Eq.trans, True, congr, congrArg, and_self, List.cons, Or, Eq, congrFun', eq_self, true_or, List.mem_cons_of_mem, Eq.mpr, ite, HSub.hSub, id, Usa1997P3.coeff_polyOfDigits_cons and 9 more
55 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
def Usa1997P3.build Runs the extraction algorithm for at most fuel steps, returning the list of extracted digits if the pair (0, 0) is reached in time line 109 source
Source documentation
Runs the extraction algorithm for at most fuel steps, returning the list of extracted digits if the pair (0, 0) is reached in time.

no axioms what this means
Statement uses: Nat, Int, Option, List
2 instances, coercions and auxiliary constants left out.
```
def build : ℕ → ℤ → ℤ → Option (List ℤ)
  | 0, u, v => if u = 0 ∧ v = 0 then some [] else none
  | fuel + 1, u, v =>
    if u = 0 ∧ v = 0 then some []
    else
      match build fuel ((digit u v - u) / 2) ((digit u v - v) / 5) with
      | some ds => some (digit u v :: ds)
      | none => none

/-- Any digit list produced by `build` from a pair `(u, v)` with
`u ≡ v (mod 3)` indeed represents `(u, v)`. -/
```
Mentions: Usa1997P3.digit
lemma Usa1997P3.build_correct Any digit list produced by build from a pair (u, v) with u ≡ v (mod 3) indeed represents (u, v) line 120 source
Source documentation
Any digit list produced by build from a pair (u, v) with u ≡ v (mod 3) indeed represents (u, v).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Int, List, Eq, Option, Usa1997P3.build, Option.some, HMod.hMod, HSub.hSub, OfNat.ofNat, And, Polynomial.eval, Neg.neg, Usa1997P3.polyOfDigits, Membership.mem, LE.le
Proof uses: Nat.recAux, dite, List.nil, And.intro, of_eq_true, Eq.trans, True, congrFun', congrArg, Polynomial.eval_zero, eq_self, False.elim, Eq.mp, False, Option.some.inj, ite, Option.none, ite_eq_left, Eq.symm, Not and 21 more
40 instances, coercions and auxiliary constants left out.

Tactic trace: 46 steps
Open to load the checked before-and-after goal states.
```
lemma build_correct (fuel : ℕ) (u v : ℤ) (ds : List ℤ) (h : build fuel u v = some ds)
    (h3 : (u - v) % 3 = 0) :
    (polyOfDigits ds).eval (-2) = u ∧ (polyOfDigits ds).eval (-5) = v ∧
      ∀ d ∈ ds, 0 ≤ d ∧ d ≤ 9 := by
…
```
Mentions: h, Usa1997P3.build, Usa1997P3.polyOfDigits
lemma Usa1997P3.build_region_all On the finite region |u| ≤ 9, |v| ≤ 2 the algorithm always terminates in at most 8 steps (checked here by exhaustive computation over the 95 pairs with u ≡ v (mod 3)) line 159 source
Source documentation
On the finite region |u| ≤ 9, |v| ≤ 2 the algorithm always terminates in at most 8 steps (checked here by exhaustive computation over the 95 pairs with u ≡ v (mod 3)).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Membership.mem, Finset, Finset.range, OfNat.ofNat, Eq, Int, HMod.hMod, HSub.hSub, Bool, Option.isSome, List, Usa1997P3.build, Bool.true
Proof uses: of_decide_eq_true, id, Decidable.decide, Eq.refl
24 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
lemma build_region_all :
    ∀ a ∈ Finset.range 19, ∀ b ∈ Finset.range 5,
      ((a : ℤ) - 9 - ((b : ℤ) - 2)) % 3 = 0 →
      (build 8 ((a : ℤ) - 9) ((b : ℤ) - 2)).isSome := by
…
```
Mentions: Usa1997P3.build
lemma Usa1997P3.build_region_some line 165 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Int, And, LE.le, Neg.neg, OfNat.ofNat, Eq, HMod.hMod, HSub.hSub, Exists, List, Option, Usa1997P3.build, Nat, Option.some
Proof uses: Membership.mem, Finset, Finset.range, Exists.intro, Int.toNat, HAdd.hAdd, And.intro, Iff.mpr, LT.lt, Finset.mem_range, Eq.mpr, id, congrArg, Eq.symm, Iff.mp, Bool, Option.isSome, Bool.true, Option.isSome_iff_exists, Usa1997P3.build_region_all and 1 more
29 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
def Usa1997P3.Good The specification of the problem: Q has coefficients in {0, …, 9} and takes the values u at -2 and v at -5 line 180 source
Source documentation
The specification of the problem: Q has coefficients in {0, …, 9} and takes the values u at -2 and v at -5.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Int, Polynomial
Proof uses: And, Nat, LE.le, OfNat.ofNat, Finsupp, Polynomial.coeff, Eq, Polynomial.eval, Neg.neg
9 instances, coercions and auxiliary constants left out.
```
def Good (u v : ℤ) (Q : ℤ[X]) : Prop :=
…
```
lemma Usa1997P3.good_of_good_step If the successor pair of (u, v) is representable, so is (u, v): prepend the extracted digit as the new constant coefficient line 185 source
Source documentation
If the successor pair of (u, v) is representable, so is (u, v): prepend the extracted digit as the new constant coefficient.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Int, Eq, HMod.hMod, HSub.hSub, OfNat.ofNat, Exists, Polynomial, Usa1997P3.Good, HDiv.hDiv, Usa1997P3.digit
Proof uses: LE.le, And, HMul.hMul, LT.lt, Nat, Int.natAbs, Usa1997P3.step_props, Finsupp, Polynomial.coeff, Polynomial.eval, Neg.neg, Exists.intro, HAdd.hAdd, RingHom, Polynomial.C, Polynomial.X, And.intro, Nat.casesAuxOn, of_eq_true, Eq.trans and 24 more
65 instances, coercions and auxiliary constants left out.

Tactic trace: 19 steps
Open to load the checked before-and-after goal states.
```
lemma good_of_good_step (u v : ℤ) (h3 : (u - v) % 3 = 0)
    (h : ∃ Q, Good ((digit u v - u) / 2) ((digit u v - v) / 5) Q) :
    ∃ Q, Good u v Q := by
…
```
Mentions: h, Usa1997P3.Good, Usa1997P3.digit
theorem Usa1997P3.exists_good_region Documented declaration: exists_good_region line 209 source
Source documentation
Existence for all pairs in the region |v| ≤ 2, by strong descent on |u|: either |u| ≤ 9 and we are in the finite region checked by build_region_all, or one extraction step strictly decreases |u|.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Int, LE.le, Nat, Int.natAbs, OfNat.ofNat, Eq, HMod.hMod, HSub.hSub, Exists, Polynomial, Usa1997P3.Good
Proof uses: Nat.strong_induction_on, LT.lt, dite, List, Option, Usa1997P3.build, Option.some, Usa1997P3.build_region_some, Polynomial.eval, Neg.neg, Usa1997P3.polyOfDigits, And, Membership.mem, Usa1997P3.build_correct, Exists.intro, And.intro, Finsupp, Polynomial.coeff, Usa1997P3.coeff_polyOfDigits, Not and 7 more
35 instances, coercions and auxiliary constants left out.

Tactic trace: 15 steps
Open to load the checked before-and-after goal states.
```
theorem exists_good_region (u v : ℤ) (hv2 : v.natAbs ≤ 2) (h3 : (u - v) % 3 = 0) :
    ∃ Q, Good u v Q := by
…
```
Mentions: Usa1997P3.Good
theorem Usa1997P3.exists_good Documented declaration: exists_good line 224 source
Source documentation
Existence for all pairs (u, v) with u ≡ v (mod 3), by strong descent on |v|: either |v| ≤ 2 and exists_good_region applies, or one extraction step strictly decreases |v|.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Int, Eq, HMod.hMod, HSub.hSub, OfNat.ofNat, Exists, Polynomial, Usa1997P3.Good
Proof uses: Nat.strong_induction_on, Nat, Int.natAbs, LT.lt, dite, LE.le, Usa1997P3.exists_good_region, Not, Usa1997P3.digit, And, HMul.hMul, HDiv.hDiv, Usa1997P3.step_props, rfl, Usa1997P3.good_of_good_step, Exists.intro, Eq.refl
25 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
```
theorem exists_good (u v : ℤ) (h3 : (u - v) % 3 = 0) : ∃ Q, Good u v Q := by
…
```
Mentions: Usa1997P3.Good
theorem Usa1997P3.good_unique Uniqueness: the difference D of two good polynomials has coefficients in {-9, …, 9} and vanishes at -2 and -5 line 237 source
Source documentation
Uniqueness: the difference D of two good polynomials has coefficients in {-9, …, 9} and vanishes at -2 and -5. Its trailing coefficient is then divisible by both 2 and 5, hence by 10, hence it is zero — a contradiction unless D = 0.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Polynomial, Int, Usa1997P3.Good, Eq
Proof uses: Decidable.byContradiction, Not, Nat, And, LE.le, OfNat.ofNat, Finsupp, Polynomial.coeff, Polynomial.eval, Neg.neg, False, HSub.hSub, rfl, Ne, Iff.mpr, sub_ne_zero, of_eq_true, Eq.trans, True, congrFun' and 45 more
90 instances, coercions and auxiliary constants left out.

Tactic trace: 56 steps
Open to load the checked before-and-after goal states.
```
theorem good_unique (Q₁ Q₂ : ℤ[X]) (n : ℤ) (h₁ : Good n n Q₁) (h₂ : Good n n Q₂) :
    Q₁ = Q₂ := by
…
```
Mentions: Usa1997P3.Good
problem Usa1997P3.usa1997_p3 line 289 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Int, ExistsUnique, Polynomial, And, Nat, LE.le, OfNat.ofNat, Finsupp, Polynomial.coeff, Eq, Polynomial.eval, Neg.neg
Proof uses: Usa1997P3.Good, Exists, Usa1997P3.exists_good, of_eq_true, HMod.hMod, HSub.hSub, Eq.trans, True, congrFun', congrArg, sub_self, EuclideanDomain.zero_mod, eq_self, Exists.intro, And.intro, Usa1997P3.good_unique
32 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.