`Compfiles/Imo1995P6.lean`

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28 declarations, 0 contain sorry, 0 declared as axioms

International Mathematical Olympiad 1995, Problem 6

Let p be an odd prime number. How many p-element subsets A of {1, 2, ..., 2p} are there, the sum of whose elements is divisible by p?

Imports (6)

Imported by 1 file

Compfiles.lean

def Imo1995P6.shift The cyclic shift x ↦ x + 1 on {1, ..., p} (sending p back to 1), extended by the identity on all other natural numbers line 34 source
Source documentation
The cyclic shift x ↦ x + 1 on {1, ..., p} (sending p back to 1), extended by the identity on all other natural numbers.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: ite, Membership.mem, Finset, Finset.Icc, OfNat.ofNat, HAdd.hAdd, HMod.hMod
11 instances, coercions and auxiliary constants left out.
```
def shift (p x : ℕ) : ℕ := if x ∈ Finset.Icc 1 p then x % p + 1 else x
```
def Imo1995P6.shiftSet The action of shift p on sets of naturals: the elements of A lying in {1, ..., p} are cyclically shifted and all other elements stay fixed line 38 source
Source documentation
The action of shift p on sets of naturals: the elements of A lying in {1, ..., p} are cyclically shifted and all other elements stay fixed.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset
Proof uses: Finset.image, Imo1995P6.shift
1 instances, coercions and auxiliary constants left out.
```
def shiftSet (p : ℕ) (A : Finset ℕ) : Finset ℕ := A.image (shift p)
```
Mentions: Imo1995P6.shift
theorem Imo1995P6.shiftSet_eq line 40 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset, Eq, Imo1995P6.shiftSet, Finset.image, Imo1995P6.shift
Proof uses: rfl
1 instances, coercions and auxiliary constants left out.
theorem Imo1995P6.shift_of_mem line 42 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Membership.mem, Finset, Finset.Icc, OfNat.ofNat, Eq, Imo1995P6.shift, HAdd.hAdd, HMod.hMod
Proof uses: ite_eq_left
20 instances, coercions and auxiliary constants left out.
theorem Imo1995P6.shift_of_not_mem line 44 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Not, Membership.mem, Finset, Finset.Icc, OfNat.ofNat, Eq, Imo1995P6.shift
Proof uses: ite_eq_right, HAdd.hAdd, HMod.hMod
16 instances, coercions and auxiliary constants left out.
theorem Imo1995P6.shift_mem_Icc line 46 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Membership.mem, Finset, Finset.Icc, Imo1995P6.shift
Proof uses: Eq, HAdd.hAdd, HMod.hMod, ite_eq_left, Eq.mpr, And, LE.le, id, congrArg, propext, Finset.mem_Icc, Nat.mod_lt
20 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
theorem Imo1995P6.shift_mem_iff line 54 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Iff, Membership.mem, Finset, Finset.Icc, Imo1995P6.shift
Proof uses: Iff.intro, dite, Not, Eq.mpr, id, Eq, congrArg, Imo1995P6.shift_of_not_mem, absurd, Imo1995P6.shift_mem_Icc
14 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
theorem Imo1995P6.shift_injective line 63 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Function.Injective, Imo1995P6.shift
Proof uses: Eq, dite, Membership.mem, Finset, Finset.Icc, HAdd.hAdd, HMod.hMod, ite_eq_left, Eq.mp, congrArg, Eq.mpr, id, Eq.symm, Nat.mod_self, Eq.refl, Dvd.dvd, Nat.dvd_of_mod_eq_zero, LE.le, Nat.le_of_dvd, And and 7 more
24 instances, coercions and auxiliary constants left out.

Tactic trace: 58 steps
Open to load the checked before-and-after goal states.
theorem Imo1995P6.shift_iterate_mem_Icc line 107 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Membership.mem, Finset, Finset.Icc, Eq, Nat.iterate, Imo1995P6.shift, HAdd.hAdd, HMod.hMod, HSub.hSub
Proof uses: Nat.recAux, id, Eq.mpr, congrArg, Nat.mod_eq_of_lt, Eq.mp, And, LE.le, propext, Finset.mem_Icc, Nat.sub_add_cancel, And.left, Eq.refl, Nat.mod_lt, Nat.succ, Function.iterate_succ_apply', ite_eq_left, Nat.mod_add_mod, Iff.mp, HAdd and 1 more
30 instances, coercions and auxiliary constants left out.

Tactic trace: 15 steps
Open to load the checked before-and-after goal states.
theorem Imo1995P6.shift_iterate_not_mem line 127 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Not, Membership.mem, Finset, Finset.Icc, OfNat.ofNat, Eq, Nat.iterate, Imo1995P6.shift
Proof uses: Nat.recAux, Eq.refl, Eq.mpr, HAdd.hAdd, id, congrArg, Nat.succ, Function.iterate_succ_apply', ite_eq_right, HMod.hMod
16 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
theorem Imo1995P6.shift_iterate_mem_Icc' line 133 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Membership.mem, Finset, Finset.Icc, Nat.iterate, Imo1995P6.shift
Proof uses: Eq.mpr, HAdd.hAdd, HMod.hMod, HSub.hSub, id, Eq, congrArg, Imo1995P6.shift_iterate_mem_Icc, And, LE.le, propext, Finset.mem_Icc, Nat.mod_lt
20 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
theorem Imo1995P6.shift_iterate_p line 140 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Eq, Nat.iterate, Imo1995P6.shift
Proof uses: dite, Membership.mem, Finset, Finset.Icc, Eq.mpr, HAdd.hAdd, HMod.hMod, HSub.hSub, id, congrArg, Imo1995P6.shift_iterate_mem_Icc, Nat.add_mod_right, Nat.mod_eq_of_lt, Eq.mp, And, LE.le, propext, Finset.mem_Icc, Nat.sub_add_cancel, And.left and 3 more
18 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
theorem Imo1995P6.shiftSet_iterate line 147 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset, Eq, Nat.iterate, Imo1995P6.shiftSet, Finset.image, Imo1995P6.shift
Proof uses: Nat.recAux, Eq.mpr, OfNat.ofNat, id, congrArg, Function.iterate_zero, Finset.image_id, id_eq, Eq.refl, HAdd.hAdd, Nat.succ, Function.iterate_succ_apply', Function.comp, Function.iterate_succ', Finset.image_image
5 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
theorem Imo1995P6.shiftSet_iterate_p line 156 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Finset, Eq, Nat.iterate, Imo1995P6.shiftSet
Proof uses: Eq.mpr, Finset.image, Imo1995P6.shift, id, congrArg, Imo1995P6.shiftSet_iterate, funext, Imo1995P6.shift_iterate_p, Finset.image_id, Eq.refl
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Imo1995P6.shiftSet_injective line 159 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Function.Injective, Finset, Imo1995P6.shiftSet
Proof uses: Finset.image_injective, Imo1995P6.shift, Imo1995P6.shift_injective
5 instances, coercions and auxiliary constants left out.
theorem Imo1995P6.sum_shiftSet The key congruence: shifting a set once increases its sum by the number of its elements lying in {1, ..., p}, modulo p line 164 source
Source documentation
The key congruence: shifting a set once increases its sum by the number of its elements lying in {1, ..., p}, modulo p.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Finset, Nat.ModEq, Finset.sum, Imo1995P6.shiftSet, HAdd.hAdd, Finset.card, Finset.filter, Membership.mem, Finset.Icc
Proof uses: Eq.mpr, Finset.image, Imo1995P6.shift, id, Eq, congrArg, Imo1995P6.shiftSet_eq, Finset.sum_image, Set, Imo1995P6.shift_injective, Not, Eq.symm, Finset.sum_filter_add_sum_filter_not, HMod.hMod, ite_eq_left, And.right, Iff.mp, And, Finset.mem_filter, ite_eq_right and 16 more
47 instances, coercions and auxiliary constants left out.

Tactic trace: 14 steps
Open to load the checked before-and-after goal states.
```
theorem sum_shiftSet {p : ℕ} (hp : 0 < p) (A : Finset ℕ) :
    ∑ x ∈ shiftSet p A, x ≡ ∑ x ∈ A, x + (A.filter (· ∈ Finset.Icc 1 p)).card [MOD p] := by
…
```
Mentions: Imo1995P6.shiftSet
theorem Imo1995P6.low_card_shiftSet line 189 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Finset, Eq, Finset.card, Finset.filter, Membership.mem, Finset.Icc, Imo1995P6.shiftSet
Proof uses: Finset.image, Imo1995P6.shift, Finset.ext, Eq.mpr, Iff, And, Exists, id, congr, congrArg, Eq.trans, congrFun', funext, Iff.intro, Exists.intro, And.intro, Iff.mp, Imo1995P6.shift_mem_iff, rfl, Iff.mpr and 3 more
21 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
theorem Imo1995P6.low_card_shiftSet_iterate line 202 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Finset, Eq, Finset.card, Finset.filter, Membership.mem, Finset.Icc, Nat.iterate, Imo1995P6.shiftSet
Proof uses: Nat.recAux, Eq.mpr, id, congrArg, Function.iterate_zero, id_eq, Eq.refl, HAdd.hAdd, Nat.succ, Function.iterate_succ_apply', Imo1995P6.low_card_shiftSet
18 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem Imo1995P6.sum_shiftSet_iterate line 208 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Finset, Nat.ModEq, Finset.sum, Nat.iterate, Imo1995P6.shiftSet, HAdd.hAdd, HMul.hMul, Finset.card, Finset.filter, Membership.mem, Finset.Icc
Proof uses: Nat.recAux, Eq.mpr, id, Eq, congrArg, Nat.zero_mul, Nat.ModEq.refl, Imo1995P6.sum_shiftSet, Nat.succ, Function.iterate_succ_apply', Nat.ModEq.trans, Eq.mp, Imo1995P6.low_card_shiftSet_iterate, Nat.ModEq.add, Mathlib.Tactic.Ring.of_eq, HPow.hPow, Nat.rawCast, Mathlib.Tactic.Ring.Common.add_congr, Mathlib.Tactic.Ring.Common.atom_pf, rfl and 18 more
42 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
theorem Imo1995P6.shiftSet_iterate_mem line 229 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Finset, Membership.mem, Finset.powersetCard, Finset.Icc, HMul.hMul, Nat.iterate, Imo1995P6.shiftSet
Proof uses: Eq.mpr, And, LE.le, Eq, Finset.card, id, congrArg, propext, Finset.mem_powersetCard, Eq.mp, Finset.image, Imo1995P6.shift, Imo1995P6.shiftSet_iterate, And.intro, Exists, Iff.mp, Finset.mem_image, dite, Imo1995P6.shift_iterate_mem_Icc', Finset.mem_Icc and 6 more
25 instances, coercions and auxiliary constants left out.

Tactic trace: 17 steps
Open to load the checked before-and-after goal states.
theorem Imo1995P6.shiftSet_iterate_Icc_left line 246 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Eq, Finset, Nat.iterate, Imo1995P6.shiftSet, Finset.Icc
Proof uses: Eq.mpr, Finset.image, Imo1995P6.shift, id, congrArg, Imo1995P6.shiftSet_iterate, Finset.card, Finset.card_image_of_injective, Function.Injective.iterate, Imo1995P6.shift_injective, LE.le, Membership.mem, And, Exists, Iff.mp, Finset.mem_image, Imo1995P6.shift_iterate_mem_Icc', Finset.eq_of_subset_of_card_le, le_of_eq, Eq.symm
17 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
theorem Imo1995P6.shiftSet_iterate_Icc_right line 257 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Eq, Finset, Nat.iterate, Imo1995P6.shiftSet, Finset.Icc, HAdd.hAdd, OfNat.ofNat, HMul.hMul
Proof uses: Eq.mpr, Finset.image, Imo1995P6.shift, id, congrArg, Imo1995P6.shiftSet_iterate, Finset.ext, Iff, Membership.mem, Exists, And, congrFun', Iff.intro, Imo1995P6.shift_iterate_not_mem, Not, LE.le, propext, Finset.mem_Icc, Eq.mp, Iff.mpr and 2 more
23 instances, coercions and auxiliary constants left out.

Tactic trace: 18 steps
Open to load the checked before-and-after goal states.
theorem Imo1995P6.shiftSet_iterate_mem_U line 272 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Finset, Membership.mem, SDiff.sdiff, Finset.powersetCard, Finset.Icc, HMul.hMul, Insert.insert, Singleton.singleton, HAdd.hAdd, Nat.iterate, Imo1995P6.shiftSet
Proof uses: Eq.mpr, And, Not, id, Eq, congrArg, propext, Finset.mem_sdiff, Eq.mp, Or, Finset.mem_insert, Finset.mem_singleton, Ne, And.intro, Imo1995P6.shiftSet_iterate_mem, And.left, Function.Injective.iterate, Imo1995P6.shiftSet_injective, Imo1995P6.shiftSet_iterate_Icc_left, Eq.refl and 2 more
32 instances, coercions and auxiliary constants left out.

Tactic trace: 13 steps
Open to load the checked before-and-after goal states.
theorem Imo1995P6.low_card_bounds If A is a p-element subset of {1, ..., 2p} different from the two "interval" subsets, then A ∩ {1, ..., p} has between 1 and p - 1 elements line 291 source
Source documentation
If A is a p-element subset of {1, ..., 2p} different from the two "interval" subsets, then A ∩ {1, ..., p} has between 1 and p - 1 elements.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Finset, Membership.mem, SDiff.sdiff, Finset.powersetCard, Finset.Icc, HMul.hMul, Insert.insert, Singleton.singleton, HAdd.hAdd, And, LE.le, Finset.card, Finset.filter, HSub.hSub
Proof uses: Eq, Ne, Eq.mp, Not, Or, congrArg, propext, Finset.mem_powersetCard, Finset.mem_singleton, Finset.mem_insert, Finset.mem_sdiff, And.right, Iff.mp, Finset.mem_filter, Trans.trans, Finset.card_le_card, Eq.mpr, id, Nat.card_Icc, Nat.add_sub_cancel and 11 more
52 instances, coercions and auxiliary constants left out.

Tactic trace: 38 steps
Open to load the checked before-and-after goal states.
```
theorem low_card_bounds {p : ℕ} (hp : 0 < p) {A : Finset ℕ}
    (hA : A ∈ (Finset.Icc 1 (2 * p)).powersetCard p \
      {Finset.Icc 1 p, Finset.Icc (p + 1) (2 * p)}) :
    1 ≤ (A.filter (· ∈ Finset.Icc 1 p)).card ∧
    (A.filter (· ∈ Finset.Icc 1 p)).card ≤ p - 1 := by
…
```
theorem Imo1995P6.card_range_filter_eq_one For A as above, among the p sets A, σ A, ..., σ^{p-1} A exactly one has sum divisible by p (here σ is shiftSet p) line 331 source
Source documentation
For A as above, among the p sets A, σ A, ..., σ^{p-1} A exactly one has sum divisible by p (here σ is shiftSet p).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Nat.Prime, Finset, Membership.mem, SDiff.sdiff, Finset.powersetCard, Finset.Icc, OfNat.ofNat, HMul.hMul, Insert.insert, Singleton.singleton, HAdd.hAdd, Eq, Finset.card, Finset.filter, Dvd.dvd, Finset.sum, Nat.iterate, Imo1995P6.shiftSet, Finset.range
Proof uses: LT.lt, Nat.Prime.pos, NeZero, NeZero.mk, LT.lt.ne', Fact, Fact.mk, LE.le, HSub.hSub, And, Imo1995P6.low_card_bounds, Iff, ZMod, Nat.ModEq, Imo1995P6.sum_shiftSet_iterate, Eq.mpr, id, congrArg, Eq.symm, Nat.cast_sum and 30 more
86 instances, coercions and auxiliary constants left out.

Tactic trace: 47 steps
Open to load the checked before-and-after goal states.
```
theorem card_range_filter_eq_one {p : ℕ} (hp : p.Prime) {A : Finset ℕ}
    (hA : A ∈ (Finset.Icc 1 (2 * p)).powersetCard p \
      {Finset.Icc 1 p, Finset.Icc (p + 1) (2 * p)}) :
    ((Finset.range p).filter (fun i ↦ p ∣ ∑ x ∈ (shiftSet p)^[i] A, x)).card = 1 := by
…
```
Mentions: Imo1995P6.shiftSet
theorem Imo1995P6.card_U_filter_eq Shifting i times is a bijection on the good subsets, so for each fixed i the number of good A with p ∣ ∑ σⁱ A equals the number of good B with p ∣ ∑ B line 392 source
Source documentation
Shifting i times is a bijection on the good subsets, so for each fixed i the number of good A with p ∣ ∑ σⁱ A equals the number of good B with p ∣ ∑ B.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Nat.Prime, Membership.mem, Finset, Finset.range, Eq, Finset.card, Finset.filter, Dvd.dvd, Finset.sum, Nat.iterate, Imo1995P6.shiftSet, SDiff.sdiff, Finset.powersetCard, Finset.Icc, OfNat.ofNat, HMul.hMul, Insert.insert, Singleton.singleton, HAdd.hAdd
Proof uses: LT.lt, Nat.Prime.pos, Finset.card_bij, Eq.mpr, And, id, congrArg, propext, Finset.mem_filter, And.intro, Imo1995P6.shiftSet_iterate_mem_U, And.left, Eq.mp, And.right, Function.Injective.iterate, Imo1995P6.shiftSet_injective, LE.le, Nat.le_of_lt, Iff.mp, Finset.mem_range and 8 more
38 instances, coercions and auxiliary constants left out.

Tactic trace: 22 steps
Open to load the checked before-and-after goal states.
```
theorem card_U_filter_eq {p : ℕ} (hp : p.Prime) (i : ℕ) (hi : i ∈ Finset.range p) :
    (((Finset.Icc 1 (2 * p)).powersetCard p \
        {Finset.Icc 1 p, Finset.Icc (p + 1) (2 * p)}).filter
      (fun A ↦ p ∣ ∑ x ∈ (shiftSet p)^[i] A, x)).card =
    (((Finset.Icc 1 (2 * p)).powersetCard p \
        {Finset.Icc 1 p, Finset.Icc (p + 1) (2 * p)}).filter
      (fun A ↦ p ∣ ∑ x ∈ A, x)).card := by
…
```
Mentions: Imo1995P6.shiftSet
determine Imo1995P6.solution line 419 source
no axioms what this means
Statement uses: Nat
Proof uses: HAdd.hAdd, OfNat.ofNat, HDiv.hDiv, HSub.hSub, Nat.choose, HMul.hMul
9 instances, coercions and auxiliary constants left out.
problem Imo1995P6.imo1995_p6 line 421 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Nat.Prime, Odd, Eq, Finset.card, Finset, Finset.filter, Dvd.dvd, Finset.sum, Finset.powersetCard, Finset.Icc, OfNat.ofNat, HMul.hMul, Imo1995P6.solution
Proof uses: LT.lt, Nat.Prime.pos, Membership.mem, Eq.mpr, And, LE.le, id, congrArg, propext, Finset.mem_powersetCard, And.intro, Finset.Icc_subset_Icc, le_refl, HSub.hSub, HAdd.hAdd, Nat.card_Icc, Nat.add_sub_cancel, Eq.refl, Ne, Eq.rec and 95 more
104 instances, coercions and auxiliary constants left out.

Tactic trace: 105 steps
Open to load the checked before-and-after goal states.