`Compfiles/Imo1959P1.lean`

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2 declarations, 0 contain sorry, 0 declared as axioms

International Mathematical Olympiad 1959, Problem 1.

Prove that the fraction (21n+4)/(14n+3) is irreducible for every natural number n.

Imports (4)

Imported by 1 file

Compfiles.lean

lemma Imo1959P1.calculation line 33 source
standard axioms: propext what this means
Statement uses: Nat, Dvd.dvd, HAdd.hAdd, HMul.hMul, OfNat.ofNat
Proof uses: Dvd.dvd.mul_left, Eq, Mathlib.Tactic.Ring.of_eq, Nat.rawCast, HPow.hPow, Mathlib.Tactic.Ring.Common.mul_congr, Mathlib.Tactic.Ring.cast_pos, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Tactic.Ring.Common.add_congr, Mathlib.Tactic.Ring.Common.atom_pf, rfl, Eq.mpr, id, congrArg, Eq.symm, Mathlib.Tactic.Ring.Common.add_mul, Mathlib.Tactic.Ring.Common.mul_add, Mathlib.Tactic.Ring.Common.mul_pf_right, Mathlib.Meta.NormNum.IsNat.to_raw_eq and 13 more
29 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
problem Imo1959P1.imo1959_p1 IMO 1959 Problem 1: an irreducible fraction line 50 source
Source comment
Since Lean doesn't have a concept of "irreducible fractions" per se, we just formalize this as saying the numerator and denominator are relatively prime.
L4DEL explanation
The fraction (21n+4)/(14n+3) is in lowest terms for every natural number n.
1. Problem: prove that (21n+4)/(14n+3) is irreducible for all natural n. The formalization states this as Nat.Coprime (21 * n + 4) (14 * n + 3), since Lean has no notion of an irreducible fraction.
2. Solution (complete, no sorry): if k divides both numbers then it divides 3 * (14n+3) - 2 * (21n+4), and that difference equals 1, so k divides 1.
3. The file says the solution is imported from the Mathlib Archive (solutionImportedFrom).
Sources and attribution: Pinned Lean source (47b19bbfe5f3)
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Nat.Coprime, HAdd.hAdd, HMul.hMul, OfNat.ofNat
Proof uses: Nat.coprime_of_dvd', Nat.Prime, Dvd.dvd, Imo1959P1.calculation
11 instances, coercions and auxiliary constants left out.

Nat.Coprime: two natural numbers have no common divisor other than 1, equivalently their gcd is 1.
```
problem imo1959_p1 : ∀ n : ℕ, Nat.Coprime (21 * n + 4) (14 * n + 3) :=
…
```