`Compfiles/Imo2009P5.lean`

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8 declarations, 0 contain sorry, 0 declared as axioms

International Mathematical Olympiad 2009, Problem 5

Determine all functions f: ℤ>0 → ℤ>0 such that for all positive integers a and b, the numbers

a, f(b), and f(b + f(a) - 1)

form the sides of a nondegenerate triangle.

Imports (4)

Imported by 1 file

Compfiles.lean

lemma Imo2009P5.exists_sup_fn_fin line 40 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Exists, LT.lt, LE.le
Proof uses: Exists.intro, Finset.sup, Finset.range, Finset.le_sup, Iff.mpr, Membership.mem, Finset, Finset.mem_range
9 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma Imo2009P5.pnat_to_nat_prop line 45 source
lemma Imo2009P5.pnat_to_nat_prop2 line 49 source
lemma Imo2009P5.succ_pnat_add_succ_pnat line 53 source
theorem Imo2009P5.final_solution_nat Final solution, nat version line 62 source
Source documentation
Final solution, nat version

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Iff, And, LE.le, HAdd.hAdd, Eq
Proof uses: Iff.intro, OfNat.ofNat, Mathlib.Tactic.Contrapose.contrapose₁, Eq.mpr, Not, Ne, Exists, LT.lt, id, implies_congr, Eq.refl, Eq.trans, Mathlib.Tactic.Push.not_forall_eq, congrArg, funext, Imo2009P5.exists_sup_fn_fin, Exists.intro, Nat.lt_succ_of_le, add_le_add, Eq.mp and 31 more
50 instances, coercions and auxiliary constants left out.

Tactic trace: 50 steps
Open to load the checked before-and-after goal states.
```
theorem final_solution_nat (f : ℕ → ℕ) :
  ((∀ x y : ℕ, f (y + f x) ≤ f y + x)
    ∧ (∀ x y : ℕ, x ≤ f y + f (y + f x))
    ∧ (∀ x y : ℕ, f y ≤ f (y + f x) + x))
      ↔ f = λ x ↦ x := by
…
```
Mentions: f
theorem Imo2009P5.final_solution_pnat Final solution, pnat version line 122 source
Source documentation
Final solution, pnat version

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: PNat, Iff, And, LT.lt, HSub.hSub, HAdd.hAdd, OfNat.ofNat, Nat, Eq
Proof uses: Nat.succPNat, PNat.natPred, Exists, Exists.intro, funext, Eq.mpr, id, congrArg, PNat.succPNat_natPred, Eq.refl, congrFun', congr, forall_congr, Eq.trans, PNat.add_sub, LE.le, Imo2009P5.final_solution_nat, Eq.symm
41 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
theorem final_solution_pnat (f : ℕ+ → ℕ+) :
    ((∀ x y : ℕ+, f (y + f x - 1) < f y + x)
      ∧ (∀ x y : ℕ+, x < f y + f (y + f x - 1))
      ∧ (∀ x y : ℕ+, f y < f (y + f x - 1) + x))
        ↔ f = λ x ↦ x := by
…
```
Mentions: f
determine Imo2009P5.solution_set line 137 source
no axioms what this means
Statement uses: Set, PNat
Proof uses: Singleton.singleton, id
1 instances, coercions and auxiliary constants left out.
problem Imo2009P5.imo2009_p5 line 139 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: PNat, Iff, Membership.mem, Set, Imo2009P5.solution_set, And, LT.lt, HSub.hSub, HAdd.hAdd, OfNat.ofNat, Nat
Proof uses: Eq, Imo2009P5.final_solution_pnat, Iff.intro, id, Iff.mpr, rfl, And.intro, Eq.symm, Eq.mpr, congrArg, Singleton.singleton, propext, Set.mem_singleton_iff, Iff.mp, And.left, And.right
33 instances, coercions and auxiliary constants left out.

Tactic trace: 20 steps
Open to load the checked before-and-after goal states.