`Compfiles/Imo2024P6.lean`

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30 declarations, 0 contain sorry, 0 declared as axioms

International Mathematical Olympiad 2024, Problem 6

A function f: ℚ → ℚ is called aquaesulian if the following property holds: for every x, y ∈ ℚ, f(x + f(y)) = f(x) + y or f(f(x) + y) = x + f(y).

Show that there exists an integer c such that for any aquaesulian function f there are at most c different rational numbers of the form f(r)+f(-r) for some rational number r, and find the smallest possible value of c.

Imports (3)

Imported by 1 file

Compfiles.lean

def Imo2024P6.Aquaesulian line 36 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Rat
Proof uses: Or, Eq, HAdd.hAdd
2 instances, coercions and auxiliary constants left out.
def Imo2024P6.General.Aquaesulian The condition on functions in the problem (for a general AddCommGroup and in symmetric form) line 58 source
Source documentation
The condition on functions in the problem (for a general AddCommGroup and in symmetric form).

no axioms what this means
Statement uses: AddCommGroup
Proof uses: Or, Eq, HAdd.hAdd
5 instances, coercions and auxiliary constants left out.
```
def Aquaesulian (f : G → G) : Prop := ∀ x y, f (f y + x) = f x + y ∨ f (f x + y) = f y + x
…
```
Mentions: f
lemma Imo2024P6.General.Aquaesulian.apply_apply_add line 63 source
no axioms what this means
Statement uses: AddCommGroup, Imo2024P6.General.Aquaesulian, Eq, HAdd.hAdd
Proof uses: Or
11 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Imo2024P6.General.Aquaesulian.eq_of_apply_eq_inl line 66 source
standard axioms: propext what this means
Statement uses: AddCommGroup, Imo2024P6.General.Aquaesulian, Eq, HAdd.hAdd
Proof uses: Eq.symm, Eq.mp, congrArg, propext, add_right_inj, Imo2024P6.General.Aquaesulian.apply_apply_add
14 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Imo2024P6.General.Aquaesulian.injective line 71 source
standard axioms: propext what this means
Statement uses: AddCommGroup, Imo2024P6.General.Aquaesulian, Function.Injective
Proof uses: Eq, HAdd.hAdd, Or, Eq.symm, Imo2024P6.General.Aquaesulian.eq_of_apply_eq_inl
6 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma Imo2024P6.General.Aquaesulian.apply_zero line 77 source
standard axioms: propext what this means
Statement uses: AddCommGroup, Imo2024P6.General.Aquaesulian, Eq, OfNat.ofNat
Proof uses: Imo2024P6.General.Aquaesulian.injective, Eq.mpr, HAdd.hAdd, eq_of_heq, HEq, Eq.refl, HEq.refl, Eq.symm, of_eq_true, Eq.trans, True, congrArg, add_zero, eq_self, Imo2024P6.General.Aquaesulian.apply_apply_add
23 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma Imo2024P6.General.Aquaesulian.apply_neg_apply_add line 81 source
standard axioms: propext what this means
Statement uses: AddCommGroup, Imo2024P6.General.Aquaesulian, Eq, HAdd.hAdd, Neg.neg, OfNat.ofNat
Proof uses: Or, Imo2024P6.General.Aquaesulian.injective, Eq.mp, congrArg, Eq.symm, Imo2024P6.General.Aquaesulian.apply_zero, add_neg_cancel
33 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
lemma Imo2024P6.General.Aquaesulian.apply_neg_apply line 88 source
standard axioms: propext what this means
Statement uses: AddCommGroup, Imo2024P6.General.Aquaesulian, Eq, Neg.neg
Proof uses: Eq.mpr, HAdd.hAdd, OfNat.ofNat, id, congrArg, Eq.symm, propext, add_eq_zero_iff_eq_neg, Imo2024P6.General.Aquaesulian.apply_neg_apply_add
20 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Imo2024P6.General.Aquaesulian.apply_neg_apply_neg line 92 source
standard axioms: propext what this means
Statement uses: AddCommGroup, Imo2024P6.General.Aquaesulian, Eq, Neg.neg
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, Imo2024P6.General.Aquaesulian.apply_neg_apply, neg_neg, eq_self
11 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo2024P6.General.Aquaesulian.apply_neg_of_apply_eq line 95 source
standard axioms: propext what this means
Statement uses: AddCommGroup, Imo2024P6.General.Aquaesulian, Eq, Neg.neg
Proof uses: Eq.mpr, id, congrArg, Eq.symm, Imo2024P6.General.Aquaesulian.apply_neg_apply
10 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Imo2024P6.General.Aquaesulian.apply_neg_eq_neg_iff line 99 source
standard axioms: propext what this means
Statement uses: AddCommGroup, Imo2024P6.General.Aquaesulian, Iff, Eq, Neg.neg
Proof uses: Iff.intro, Eq.mpr, eq_of_heq, HEq, Eq.refl, HEq.refl, Eq.symm, id, congrArg, neg_neg, Imo2024P6.General.Aquaesulian.apply_neg_of_apply_eq
14 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma Imo2024P6.General.Aquaesulian.pair_lemma line 103 source
standard axioms: propext what this means
Statement uses: AddCommGroup, Imo2024P6.General.Aquaesulian, Ne, Or, Eq
Proof uses: False.elim, Eq.trans, Eq.symm, Or.inr, Or.inl, Function.Injective.ne, Imo2024P6.General.Aquaesulian.injective
1 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
lemma Imo2024P6.General.Aquaesulian.g_two line 111 source
standard axioms: propext what this means
Statement uses: AddCommGroup, Imo2024P6.General.Aquaesulian, Ne, Eq, HAdd.hAdd, Neg.neg, Or
Proof uses: Imo2024P6.General.Aquaesulian.pair_lemma, of_eq_true, Not, Eq.trans, False, True, congrArg, eq_false, not_false_eq_true, Eq.mpr, eq_of_heq, HEq, Eq.refl, HEq.refl, Eq.symm, id, Imo2024P6.General.Aquaesulian.apply_neg_apply_neg, add_comm, HSMul.hSMul, Int and 28 more
48 instances, coercions and auxiliary constants left out.

Tactic trace: 27 steps
Open to load the checked before-and-after goal states.
lemma Imo2024P6.General.Aquaesulian.u_eq_zero line 131 source
standard axioms: propext what this means
Statement uses: AddCommGroup, Imo2024P6.General.Aquaesulian, Ne, Eq, HAdd.hAdd, Neg.neg, OfNat.ofNat
Proof uses: Or, HSub.hSub, Eq.mp, congrArg, Eq.symm, propext, eq_sub_iff_add_eq, add_comm, neg_add_cancel_left, neg_sub, Imo2024P6.General.Aquaesulian.apply_neg_eq_neg_iff, HSMul.hSMul, Int, Eq.trans, neg_zsmul, one_zsmul, Mathlib.Tactic.Abel.termg, Mathlib.Tactic.Abel.subst_into_addg, Mathlib.Tactic.Abel.subst_into_negg, Mathlib.Tactic.Abel.term_atom_pfg and 28 more
60 instances, coercions and auxiliary constants left out.

Tactic trace: 13 steps
Open to load the checked before-and-after goal states.
lemma Imo2024P6.General.Aquaesulian.u_eq_zero_or_v_eq_zero line 145 source
standard axioms: propext what this means
Statement uses: AddCommGroup, Imo2024P6.General.Aquaesulian, Ne, Eq, HAdd.hAdd, Neg.neg, Or, OfNat.ofNat
Proof uses: Imo2024P6.General.Aquaesulian.g_two, Or.inl, Imo2024P6.General.Aquaesulian.u_eq_zero, Or.inr, Ne.symm, Eq.mp, congrArg, add_comm
25 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma Imo2024P6.General.Aquaesulian.card_le_two line 152 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: AddCommGroup, Imo2024P6.General.Aquaesulian, LE.le, Cardinal, Cardinal.mk, Set.range, HAdd.hAdd, Neg.neg, OfNat.ofNat, Nat
Proof uses: dite, Eq, Classical.propDecidable, of_eq_true, Eq.trans, True, congrFun', congrArg, Fintype.card, Singleton.singleton, Set, funext, Set.range_const, Cardinal.mk_fintype, Fintype.card_unique, Nat.cast_one, Not, Exists, Eq.mp, propext and 34 more
74 instances, coercions and auxiliary constants left out.

Tactic trace: 23 steps
Open to load the checked before-and-after goal states.
lemma Imo2024P6.aquaesulian_iff_general line 178 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Rat, Iff, Imo2024P6.Aquaesulian, Imo2024P6.General.Aquaesulian
Proof uses: Eq.mpr, Or, Eq, HAdd.hAdd, id, congrArg, Imo2024P6.Aquaesulian.eq_1, Imo2024P6.General.Aquaesulian.eq_1, forall_congr', add_comm, Iff.rfl
9 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma Imo2024P6.Aquaesulian.card_le_two line 183 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Rat, Imo2024P6.Aquaesulian, LE.le, Cardinal, Cardinal.mk, Set.range, HAdd.hAdd, Neg.neg, OfNat.ofNat, Nat
Proof uses: Imo2024P6.General.Aquaesulian.card_le_two, Eq.mp, Imo2024P6.General.Aquaesulian, congrArg, propext, Imo2024P6.aquaesulian_iff_general
11 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
def Imo2024P6.fExample An example of a function achieving the maximum number of values of f(r)+f(-r) line 189 source
Source documentation
An example of a function achieving the maximum number of values of f(r)+f(-r).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Rat
Proof uses: HSub.hSub, Int.floor, Int.fract
8 instances, coercions and auxiliary constants left out.
```
def fExample (x : ℚ) : ℚ := ⌊x⌋ - Int.fract x
```
lemma Imo2024P6.fExample_add line 191 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Rat, Eq, HAdd.hAdd, Imo2024P6.fExample, Int.floor, HSub.hSub, Int.fract
Proof uses: Eq.mpr, id, congrArg, Imo2024P6.fExample.eq_1, Eq.symm, Int.floor_add_fract
26 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma Imo2024P6.add_fExample line 196 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Rat, Eq, HAdd.hAdd, Imo2024P6.fExample, Int.floor, HSub.hSub, Int.fract
Proof uses: Eq.mpr, id, congrArg, add_comm, Imo2024P6.fExample_add
24 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Imo2024P6.fExample_int_add line 200 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Int, Rat, Eq, Imo2024P6.fExample, HAdd.hAdd
Proof uses: id, Eq.mpr, HSub.hSub, Int.floor, Int.fract, congrFun', congrArg, Int.floor_intCast_add, Int.fract_intCast_add, cast, Eq.symm, rfl
33 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Imo2024P6.fExample_of_mem_Ico line 204 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Rat, Membership.mem, Set, Set.Ico, OfNat.ofNat, Eq, Imo2024P6.fExample, Neg.neg
Proof uses: Eq.mpr, HSub.hSub, Int.floor, Int.fract, id, congrArg, Imo2024P6.fExample.eq_1, Iff.mpr, And, LE.le, LT.lt, Int.fract_eq_self, Int, Int.floor_eq_zero_iff, of_eq_true, Eq.trans, True, congrFun', Int.cast_zero, zero_sub and 1 more
42 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Imo2024P6.apply_fExample_add_apply_of_fract_le line 208 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Rat, LE.le, Int.fract, Eq, Imo2024P6.fExample, HAdd.hAdd
Proof uses: LT.lt, HSub.hSub, OfNat.ofNat, LE.le.trans_lt, sub_le_self, Int.fract_nonneg, Int.fract_lt_one, Eq.mpr, Int.floor, id, congrArg, Imo2024P6.fExample_add, Imo2024P6.add_fExample, add_assoc, Imo2024P6.fExample_int_add, Neg.neg, Imo2024P6.fExample_of_mem_Ico, And.intro, Eq.mp, Eq.symm and 2 more
47 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
lemma Imo2024P6.aquaesulian_fExample line 217 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Imo2024P6.Aquaesulian, Imo2024P6.fExample
Proof uses: Rat, LT.lt, Int.fract, LE.le, Or, Eq, HAdd.hAdd, lt_or_ge, Eq.mpr, id, congrArg, add_comm, Or.inr, Imo2024P6.apply_fExample_add_apply_of_fract_le, LT.lt.le, Or.inl
15 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
lemma Imo2024P6.fract_fExample line 224 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Rat, Eq, Int.fract, Imo2024P6.fExample, ite, OfNat.ofNat, HSub.hSub
Proof uses: dite, of_eq_true, Eq.trans, True, congr, congrArg, Int.floor, sub_zero, Int.fract_intCast, ite_eq_left_of_eq_true, congrFun', eq_self, Not, HAdd.hAdd, Neg.neg, sub_eq_add_neg, Int.fract_intCast_add, Int.fract_neg, Ne, Int.fract_fract and 4 more
49 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma Imo2024P6.floor_fExample line 230 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Rat, Eq, Int.floor, Imo2024P6.fExample, ite, Int.fract, OfNat.ofNat, HSub.hSub
Proof uses: dite, Eq.mpr, id, congr, congrArg, Int, Eq.trans, sub_zero, Int.floor_intCast, True, ite_congr, congrFun', eq_self, Eq.refl, Not, ite_true, Eq.symm, Eq.mp, propext, sub_eq_zero and 27 more
89 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
lemma Imo2024P6.card_range_fExample line 242 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Cardinal, Cardinal.mk, Rat, Set.range, HAdd.hAdd, Imo2024P6.fExample, Neg.neg, OfNat.ofNat, Nat
Proof uses: Set, Insert.insert, Singleton.singleton, Set.ext, Eq.mpr, Iff, Membership.mem, Exists, Or, id, congr, congrArg, Eq.trans, Iff.intro, Int.floor, Int.fract, Eq.symm, Int.floor_add_fract, dite, of_eq_true and 98 more
138 instances, coercions and auxiliary constants left out.

Tactic trace: 30 steps
Open to load the checked before-and-after goal states.
determine Imo2024P6.solution line 268 source
no axioms what this means
Statement uses: Nat
Proof uses: OfNat.ofNat
1 instances, coercions and auxiliary constants left out.
problem Imo2024P6.imo2024_p6 line 270 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: And, Rat, Imo2024P6.Aquaesulian, LE.le, Cardinal, Cardinal.mk, Set.range, HAdd.hAdd, Neg.neg, Imo2024P6.solution, Nat
Proof uses: And.intro, Imo2024P6.Aquaesulian.card_le_two, Imo2024P6.fExample, Imo2024P6.aquaesulian_fExample, cast, OfNat.ofNat, Eq.mp, congrArg, Imo2024P6.card_range_fExample
34 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.