`Compfiles/Imo1962P1.lean`

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15 declarations, 0 contain sorry, 0 declared as axioms

International Mathematical Olympiad 1962, Problem 1

Find the smallest natural number $n$ which has the following properties:

(a) Its decimal representation has 6 as the last digit.

(b) If the last digit 6 is erased and placed in front of the remaining digits, the resulting number is four times as large as the original number $n$.

Imports (2)

Imported by 1 file

Compfiles.lean

def Imo1962P1.ProblemPredicate line 36 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: And, Eq, List.headI, Nat.digits, OfNat.ofNat, Nat.ofDigits, List.concat, List.tail, HMul.hMul
5 instances, coercions and auxiliary constants left out.
abbrev Imo1962P1.ProblemPredicate' line 44 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: And, Eq, HAdd.hAdd, HMul.hMul, OfNat.ofNat, HPow.hPow, List.length, Nat.digits
9 instances, coercions and auxiliary constants left out.
theorem Imo1962P1.without_digits line 47 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Imo1962P1.ProblemPredicate, Exists, Imo1962P1.ProblemPredicate'
Proof uses: Exists.intro, HDiv.hDiv, OfNat.ofNat, Nat.casesAuxOn, Eq, Not, of_eq_true, Eq.trans, True, False, congrArg, And, List.headI, Nat.digits, Nat.ofDigits, List.concat, List.tail, HMul.hMul, congr, congrFun' and 45 more
36 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
theorem Imo1962P1.case_0_digit line 64 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Eq, List.length, Nat.digits, OfNat.ofNat, Not, Imo1962P1.ProblemPredicate'
Proof uses: HAdd.hAdd, HMul.hMul, HPow.hPow, Eq.mpr, id, congrArg, Eq.refl
11 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
theorem Imo1962P1.case_1_digit line 69 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Eq, List.length, Nat.digits, OfNat.ofNat, Not, Imo1962P1.ProblemPredicate'
Proof uses: HAdd.hAdd, HMul.hMul, HPow.hPow, Eq.mpr, id, congrArg, Eq.refl
11 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
theorem Imo1962P1.case_2_digit line 74 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Eq, List.length, Nat.digits, OfNat.ofNat, Not, Imo1962P1.ProblemPredicate'
Proof uses: HAdd.hAdd, HMul.hMul, HPow.hPow, Eq.mpr, id, congrArg, Eq.refl
11 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
theorem Imo1962P1.case_3_digit line 79 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Eq, List.length, Nat.digits, OfNat.ofNat, Not, Imo1962P1.ProblemPredicate'
Proof uses: HAdd.hAdd, HMul.hMul, HPow.hPow, Eq.mpr, id, congrArg, Eq.refl
11 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
theorem Imo1962P1.case_4_digit line 84 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Eq, List.length, Nat.digits, OfNat.ofNat, Not, Imo1962P1.ProblemPredicate'
Proof uses: HAdd.hAdd, HMul.hMul, HPow.hPow, Eq.mpr, id, congrArg, Eq.refl
11 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
theorem Imo1962P1.helper_5_digit Putting this inline causes a deep recursion error, so we separate it out line 90 source
Source documentation
Putting this inline causes a deep recursion error, so we separate it out.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Int, Eq, HAdd.hAdd, HMul.hMul, OfNat.ofNat, HPow.hPow, Nat
Proof uses: Mathlib.Tactic.Linarith.eq_of_not_lt_of_not_gt, Not.intro, LT.lt, Mathlib.Tactic.Linarith.lt_irrefl, Eq.mp, Neg.neg, HSub.hSub, congrArg, Mathlib.Tactic.Ring.of_eq, Mathlib.Tactic.Ring.Common.add_congr, Nat.rawCast, Mathlib.Meta.NormNum.instAddMonoidWithOne, inferInstance, Ring, Int.rawCast, Int.negOfNat, Mathlib.Tactic.Ring.Common.mul_congr, Mathlib.Tactic.Ring.cast_pos, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl and 64 more
80 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem helper_5_digit {c : ℤ} (h : 6 * 10 ^ 5 + c = 4 * (10 * c + 6)) : c = 15384 := by linarith
```
Mentions: h
theorem Imo1962P1.case_5_digit line 92 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Eq, List.length, Nat.digits, OfNat.ofNat, Imo1962P1.ProblemPredicate'
Proof uses: HAdd.hAdd, HMul.hMul, HPow.hPow, Eq.mpr, id, congrArg, Eq.refl
11 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem Imo1962P1.case_more_digits linarith fails on numbers this large, so this lemma spells out some of the arithmetic that normally would be automated line 100 source
Source documentation
linarith fails on numbers this large, so this lemma spells out some of the arithmetic that normally would be automated.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, GE.ge, List.length, Nat.digits, OfNat.ofNat, Imo1962P1.ProblemPredicate'
Proof uses: Ne, Eq, of_eq_true, Eq.trans, True, congrFun', congrArg, List, List.nil, Nat.digits_zero, eq_self, Imo1962P1.case_0_digit, Trans.trans, HPow.hPow, HMul.hMul, Nat.le.intro, Eq.symm, HAdd.hAdd, And.left, Nat.base_pow_length_digits_le and 17 more
32 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
theorem case_more_digits {c n : ℕ} (h1 : (digits 10 c).length ≥ 6) (h2 : ProblemPredicate' c n) :
    n ≥ 153846 := by
…
```
Mentions: Imo1962P1.ProblemPredicate'
theorem Imo1962P1.satisfied_by_153846 line 116 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Imo1962P1.ProblemPredicate, OfNat.ofNat, Nat
Proof uses: Eq.mpr, Eq, Nat.ofDigits, List.cons, List.nil, id, Eq.trans, And, List.headI, Nat.digits, List.concat, List.tail, HMul.hMul, True, congr, congrArg, congrFun', List, HMod.hMod, HDiv.hDiv and 21 more
20 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem Imo1962P1.no_smaller_solutions line 120 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Imo1962P1.ProblemPredicate, GE.ge, OfNat.ofNat
Proof uses: Exists, Imo1962P1.ProblemPredicate', Imo1962P1.without_digits, Or, LT.lt, List.length, Nat.digits, lt_or_ge, Eq, Or.inl, dite, LE.le, Imo1962P1.case_5_digit, HAdd.hAdd, HMul.hMul, Eq.rec, And.left, HPow.hPow, Nat.le_of_eq, Eq.symm and 17 more
20 instances, coercions and auxiliary constants left out.

Tactic trace: 25 steps
Open to load the checked before-and-after goal states.
determine Imo1962P1.solution line 138 source
no axioms what this means
Statement uses: Nat
Proof uses: OfNat.ofNat
1 instances, coercions and auxiliary constants left out.
problem Imo1962P1.imo1962_p1 IMO 1962 Problem 1: moving the last digit 6 to the front line 140 source
L4DEL explanation
The smallest natural number ending in 6 whose value becomes four times larger when that 6 is moved to the front is 153846.
1. Problem: find the smallest natural n whose last digit is 6 and such that moving that 6 to the front gives 4n. The statement is IsLeast {n | ProblemPredicate n} solution, where the answer solution is declared as 153846.
2. Solution (complete, no sorry): write n = 10c + 6 so the condition becomes 6 * 10^(number of digits of c) + c = 4n.
3. Cases on the digit count of c: zero to four digits are impossible, five digits forces c = 15384, and six or more digits give nothing smaller. A separate lemma checks that 153846 satisfies the condition.
See also: Imo1960P1.imo1960_p1
Sources and attribution: Pinned Lean source (47b19bbfe5f3)
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: IsLeast, Nat, Set.ofPred, Imo1962P1.ProblemPredicate, Imo1962P1.solution
Proof uses: And.intro, Membership.mem, Set, lowerBounds, Imo1962P1.satisfied_by_153846, Imo1962P1.no_smaller_solutions
3 instances, coercions and auxiliary constants left out.
```
problem imo1962_p1 : IsLeast {n | ProblemPredicate n} solution :=
…
```
Mentions: Imo1962P1.ProblemPredicate, Imo1962P1.solution