`Compfiles/Imo1994P6.lean`

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10 declarations, 0 contain sorry, 0 declared as axioms

International Mathematical Olympiad 1994, Problem 6

Show that there exists a set A of positive integers with the following property: For any infinite set S of primes there exist two positive integers m ∈ A and n ∉ A each of which is a product of k distinct elements of S for some k ≥ 2.

Imports (4)

Imported by 1 file

Compfiles.lean

def Imo1994P6.IsProductOfkDistinctMembers line 31 source
standard axioms: Quot.sound, propext what this means
Statement uses: Nat, Set
Proof uses: Exists, Finset, And, Eq, Finset.card, Finset.prod, Membership.mem
4 instances, coercions and auxiliary constants left out.
def Imo1994P6.Primes line 36 source
standard axioms: propext what this means
Statement uses: Set, Nat
Proof uses: Set.ofPred, Nat.Prime
instance Imo1994P6.(instance at line 37) line 37 source
def Imo1994P6.primes_iso line 38 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OrderIso, Nat, Imo1994P6.Primes, Membership.mem, Set
Proof uses: Nat.Subtype.orderIsoOfNat
5 instances, coercions and auxiliary constants left out.
def Imo1994P6.ω Number of distinct prime factors of x line 41 source
Source comment
Number of distinct prime factors of x

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.card, Nat.primeFactors
```
def ω : ℕ → ℕ := fun x ↦ (Nat.primeFactors x).card
```
lemma Imo1994P6.prod_of_primes line 43 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat, Finset, Membership.mem, Nat.Prime, Eq, Nat.primeFactors, Finset.prod, Finset.map, Subtype, Function.Embedding.mk, Subtype.val_injective
Proof uses: Eq.symm, Finset.prod_map, Eq.mpr, id, congrArg, Nat.primeFactors_prod, Subtype.mk, Iff.mpr, Iff.of_eq, Eq.refl, Exists, Eq.mp, Eq.trans, And, Function.Embedding, funext, propext, exists_prop_congr, of_eq_true, eq_true
17 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma Imo1994P6.minFac_prod_primes line 50 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat, OrderIso, Membership.mem, Nat.Prime, LT.lt, OfNat.ofNat, Eq, Nat.minFac, Finset.prod, Finset.image, HAdd.hAdd, Finset.range
Proof uses: Finset, Eq.mpr, id, congrArg, rfl, Iff.mpr, Exists, And, Finset.mem_image, Exists.intro, And.intro, Finset.mem_range, of_eq_true, Eq.trans, True, congrFun', zero_add, eq_self, Subtype.prop, Ne and 24 more
54 instances, coercions and auxiliary constants left out.

Tactic trace: 20 steps
Open to load the checked before-and-after goal states.
lemma Imo1994P6.prod_of_distinct_members line 77 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Set, OrderIso, Membership.mem, Imo1994P6.IsProductOfkDistinctMembers, Finset.prod, Finset.image, HAdd.hAdd, Finset.range
Proof uses: id, Finset, Exists.intro, And, Eq, Finset.card, Function.Injective, Eq.mpr, congrArg, Finset.card_image_of_injective, And.intro, rfl
26 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
lemma Imo1994P6.primeFactors_card line 89 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat, OrderIso, Membership.mem, Nat.Prime, Eq, Imo1994P6.ω, Finset.prod, Finset.image, HAdd.hAdd, Finset.range
Proof uses: Finset, id, Eq.mpr, Finset.card, Nat.primeFactors, Finset.map, Subtype, Function.Embedding.mk, Subtype.val_injective, congrArg, Imo1994P6.prod_of_primes, congrFun', Finset.card_map, Finset.card_image_of_injective, Nat.add_right_cancel, OrderIso.injective, Finset.card_range
24 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
problem Imo1994P6.imo1994_p6 line 103 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Exists, Set, Nat, And, Membership.mem, LT.lt, OfNat.ofNat, Infinite, Nat.Prime, LE.le, Imo1994P6.IsProductOfkDistinctMembers, Not
Proof uses: Set.ofPred, Imo1994P6.Primes, OrderIso, Imo1994P6.primes_iso, HSub.hSub, Imo1994P6.ω, Nat.minFac, Exists.intro, And.intro, Nat.Subtype.orderIsoOfNat, HAdd.hAdd, OrderIso.symm, Subtype.mk, Subtype.val_prop, id, Subtype, Subtype.prop, Iff.mpr, Subtype.mk_lt_mk, OrderIso.strictMono and 27 more
60 instances, coercions and auxiliary constants left out.

Tactic trace: 35 steps
Open to load the checked before-and-after goal states.