`Compfiles/Imo1960P1.lean`

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13 declarations, 0 contain sorry, 0 declared as axioms

International Mathematical Olympiad 1960, Problem 1

Determine all three-digit numbers N having the property that N is divisible by 11, and N/11 is equal to the sum of the squares of the digits of N.

Imports (2)

Imported by 1 file

Compfiles.lean

def Imo1960P1.sumOfSquares line 31 source
standard axioms: propext what this means
Statement uses: List, Nat
Proof uses: List.sum, List.map, HMul.hMul
5 instances, coercions and auxiliary constants left out.
def Imo1960P1.ProblemPredicate line 34 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: And, Eq, List.length, Nat.digits, OfNat.ofNat, Dvd.dvd, HDiv.hDiv, Imo1960P1.sumOfSquares
4 instances, coercions and auxiliary constants left out.
determine Imo1960P1.SolutionPredicate line 37 source
no axioms what this means
Statement uses: Nat
Proof uses: Or, Eq, OfNat.ofNat
1 instances, coercions and auxiliary constants left out.
theorem Imo1960P1.not_zero Proving that three digit numbers are the ones in [100, 1000) line 45 source
Source comment
Proving that three digit numbers are the ones in [100, 1000).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Imo1960P1.ProblemPredicate, Ne, OfNat.ofNat
Proof uses: List, Nat.digits, List.nil, List.ne_nil_of_length_eq_add_one, And.left, Eq, List.length, And, Dvd.dvd, HDiv.hDiv, Imo1960P1.sumOfSquares, Iff.mp, Nat.digits_ne_nil_iff_ne_zero
5 instances, coercions and auxiliary constants left out.
```
theorem not_zero {n : ℕ} (h1 : ProblemPredicate n) : n ≠ 0 :=
…
```
Mentions: Imo1960P1.ProblemPredicate
theorem Imo1960P1.ge_100 line 49 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Imo1960P1.ProblemPredicate, LE.le, OfNat.ofNat
Proof uses: HPow.hPow, HMul.hMul, Eq.mpr, List.length, Nat.digits, id, Eq, congrArg, Eq.symm, And.left, And, Dvd.dvd, HDiv.hDiv, Imo1960P1.sumOfSquares, Nat.base_pow_length_digits_le, of_eq_true, LT.lt, Imo1960P1.not_zero, le_of_not_gt, Mathlib.Tactic.Linarith.lt_irrefl and 73 more
86 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
theorem Imo1960P1.lt_1000 line 56 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Imo1960P1.ProblemPredicate, LT.lt, OfNat.ofNat
Proof uses: HPow.hPow, Eq.mpr, List.length, Nat.digits, id, Eq, congrArg, Eq.symm, And.left, And, Dvd.dvd, HDiv.hDiv, Imo1960P1.sumOfSquares, Nat.lt_base_pow_length_digits', lt_of_not_ge, LE.le, Mathlib.Tactic.Linarith.lt_irrefl, Int, Eq.mp, HAdd.hAdd and 60 more
73 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
def Imo1960P1.SearchUpTo We do an exhaustive search to show that all results are covered by SolutionPredicate line 65 source
Source comment
We do an exhaustive search to show that all results are covered by `SolutionPredicate`.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: And, Eq, HMul.hMul, OfNat.ofNat, LT.lt, Imo1960P1.ProblemPredicate, Imo1960P1.SolutionPredicate
4 instances, coercions and auxiliary constants left out.
```
def SearchUpTo (c n : ℕ) : Prop :=
…
```
theorem Imo1960P1.searchUpTo_start line 68 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Imo1960P1.SearchUpTo, OfNat.ofNat, Nat
Proof uses: And.intro, Eq, HMul.hMul, LT.lt, Imo1960P1.ProblemPredicate, Imo1960P1.SolutionPredicate, rfl, False.elim, Mathlib.Tactic.Linarith.lt_irrefl, Int, Eq.mp, HAdd.hAdd, Neg.neg, HSub.hSub, congrArg, Mathlib.Tactic.Ring.of_eq, Mathlib.Tactic.Ring.Common.add_congr, Int.rawCast, inferInstance, Ring and 50 more
68 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Imo1960P1.searchUpTo_step line 71 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Imo1960P1.SearchUpTo, Eq, HAdd.hAdd, OfNat.ofNat, List, Nat.digits, Imo1960P1.sumOfSquares, Or
Proof uses: HMul.hMul, LT.lt, Imo1960P1.ProblemPredicate, Imo1960P1.SolutionPredicate, And.intro, Mathlib.Tactic.Ring.of_eq, Nat.rawCast, HPow.hPow, Mathlib.Tactic.Ring.Common.add_congr, Mathlib.Tactic.Ring.Common.mul_congr, Mathlib.Tactic.Ring.Common.atom_pf, rfl, Eq.mpr, id, congrArg, Eq.symm, Eq.refl, Mathlib.Tactic.Ring.cast_pos, Mathlib.Meta.NormNum.isNat_ofNat, Mathlib.Tactic.Ring.Common.add_mul and 32 more
44 instances, coercions and auxiliary constants left out.

Tactic trace: 17 steps
Open to load the checked before-and-after goal states.
theorem Imo1960P1.searchUpTo_end line 83 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Imo1960P1.SearchUpTo, OfNat.ofNat, Imo1960P1.ProblemPredicate, Imo1960P1.SolutionPredicate
Proof uses: And.right, Eq, HMul.hMul, LT.lt, lt_of_not_ge, LE.le, Mathlib.Tactic.Linarith.lt_irrefl, Int, Eq.mp, HAdd.hAdd, Neg.neg, HSub.hSub, congrArg, Mathlib.Tactic.Ring.of_eq, Mathlib.Tactic.Ring.Common.add_congr, Nat.rawCast, Mathlib.Meta.NormNum.instAddMonoidWithOne, inferInstance, Ring, HPow.hPow and 50 more
71 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Imo1960P1.right_direction line 87 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Imo1960P1.ProblemPredicate, Imo1960P1.SolutionPredicate
Proof uses: Imo1960P1.SearchUpTo, OfNat.ofNat, Imo1960P1.searchUpTo_start, Imo1960P1.searchUpTo_step, Eq.mpr, Eq, HAdd.hAdd, id, congrFun', congrArg, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_add, Eq.refl, Mathlib.Meta.NormNum.isNat_ofNat, Nat.digits, rfl, List, Imo1960P1.sumOfSquares, Or, Not and 32 more
19 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
theorem Imo1960P1.left_direction Now we just need to prove the equivalence, for the precise problem statement line 98 source
Source comment
Now we just need to prove the equivalence, for the precise problem statement.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Imo1960P1.SolutionPredicate, Imo1960P1.ProblemPredicate
Proof uses: Eq, OfNat.ofNat, And.intro, List.length, Nat.digits, And, Dvd.dvd, HDiv.hDiv, Imo1960P1.sumOfSquares, of_eq_true, Eq.trans, True, congrFun', congrArg, List.cons, List.nil, HAdd.hAdd, List, HMod.hMod, Nat.digits_of_two_le_of_pos and 21 more
19 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
```
theorem left_direction (n : ℕ) (spn : SolutionPredicate n) : ProblemPredicate n := by
…
```
Mentions: Imo1960P1.SolutionPredicate, Imo1960P1.ProblemPredicate
problem Imo1960P1.imo1960_p1 line 103 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Iff, Imo1960P1.ProblemPredicate, Imo1960P1.SolutionPredicate
Proof uses: Iff.intro, Imo1960P1.right_direction, Imo1960P1.left_direction