`Compfiles/Usa1978P3.lean`

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7 declarations, 0 contain sorry, 0 declared as axioms

USA Mathematical Olympiad 1978, Problem 3

You are told that all integers from 33 to 73 inclusive can be expressed as a sum of positive integers whose reciprocals sum to 1. Show that the same is true for all integers greater than 73.

Imports (6)

Imported by 1 file

Compfiles.lean

def Usa1978P3.Expressible A natural number n is expressible if it can be written as a sum of positive integers whose reciprocals sum to 1 line 32 source
Source documentation
A natural number n is expressible if it can be written as a sum of positive integers whose reciprocals sum to 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Exists, List, And, Membership.mem, LT.lt, OfNat.ofNat, Eq, List.sum, Rat, List.map, Inv.inv
13 instances, coercions and auxiliary constants left out.
```
def Expressible (n : ℕ) : Prop :=
…
```
lemma Usa1978P3.sum_map_two_mul line 37 source
standard axioms: propext what this means
Statement uses: List, Nat, Eq, List.sum, List.map, HMul.hMul, OfNat.ofNat
Proof uses: List.rec, of_eq_true, List.nil, Eq.trans, True, congr, congrArg, List.map_nil, MulZeroClass.mul_zero, eq_self, Eq.mpr, List.cons, HAdd.hAdd, id, congrFun', List.map_cons, Mathlib.Tactic.Ring.of_eq, HPow.hPow, Nat.rawCast, Mathlib.Tactic.Ring.Common.add_congr and 18 more
28 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
lemma Usa1978P3.recip_sum_map_two_mul line 42 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: List, Nat, Eq, Rat, List.sum, List.map, Inv.inv, HMul.hMul, OfNat.ofNat
Proof uses: List.rec, of_eq_true, List.nil, Eq.trans, True, congr, congrArg, List.map_nil, MulZeroClass.mul_zero, eq_self, Eq.mpr, List.cons, HAdd.hAdd, id, List.map_cons, congrFun', Nat.cast_mul, mul_inv_rev, Mathlib.Tactic.Ring.of_eq, HPow.hPow and 35 more
70 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
lemma Usa1978P3.expressible_two_mul_add Documented declaration: expressible_two_mul_add line 55 source
Source documentation
The key construction: given a representation of n, doubling every summand halves the sum of the reciprocals, so adjoining two more positive summands c, d whose reciprocals sum to 1 / 2 yields a representation of 2 * n + (c + d).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Usa1978P3.Expressible, LT.lt, OfNat.ofNat, Eq, Rat, HAdd.hAdd, Inv.inv, HDiv.hDiv, HMul.hMul
Proof uses: List, And, Membership.mem, List.sum, List.map, Exists.intro, HAppend.hAppend, List.cons, List.nil, And.intro, Or, Eq.mp, congrArg, propext, List.mem_append, Exists, List.mem_map, mul_pos, Mathlib.Meta.Positivity.pos_of_isNat, Mathlib.Meta.NormNum.isNat_ofNat and 28 more
76 instances, coercions and auxiliary constants left out.

Tactic trace: 27 steps
Open to load the checked before-and-after goal states.
```
lemma expressible_two_mul_add {n : ℕ} (h : Expressible n) {c d : ℕ}
    (hc : 0 < c) (hd : 0 < d) (hcd : (c : ℚ)⁻¹ + (d : ℚ)⁻¹ = 1 / 2) :
    Expressible (2 * n + (c + d)) := by
…
```
Mentions: h, Usa1978P3.Expressible
lemma Usa1978P3.expressible_two_mul_add_eight Adjoining two 4s: if n is expressible, so is 2 * n + 8 line 80 source
Source documentation
Adjoining two 4s: if n is expressible, so is 2 * n + 8.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Usa1978P3.Expressible, HAdd.hAdd, HMul.hMul, OfNat.ofNat
Proof uses: Usa1978P3.expressible_two_mul_add, Mathlib.Meta.NormNum.isNat_lt_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.false, Mathlib.Meta.NormNum.isNNRat_eq_true, Rat, Inv.inv, HDiv.hDiv, Mathlib.Meta.NormNum.isNNRat_add, Mathlib.Meta.NormNum.isNNRat_inv_pos, Mathlib.Meta.NormNum.IsNat.to_isNNRat, Mathlib.Meta.NormNum.isNat_natCast, Mathlib.Meta.NormNum.isNNRat_div, Mathlib.Meta.NormNum.isNNRat_mul, Nat.mul
41 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
lemma expressible_two_mul_add_eight {n : ℕ} (h : Expressible n) :
    Expressible (2 * n + 8) :=
…
```
Mentions: h, Usa1978P3.Expressible
lemma Usa1978P3.expressible_two_mul_add_nine Adjoining 3 and 6: if n is expressible, so is 2 * n + 9 line 85 source
Source documentation
Adjoining 3 and 6: if n is expressible, so is 2 * n + 9.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Usa1978P3.Expressible, HAdd.hAdd, HMul.hMul, OfNat.ofNat
Proof uses: Usa1978P3.expressible_two_mul_add, Mathlib.Meta.NormNum.isNat_lt_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.false, Mathlib.Meta.NormNum.isNNRat_eq_true, Rat, Inv.inv, HDiv.hDiv, Mathlib.Meta.NormNum.isNNRat_add, Mathlib.Meta.NormNum.isNNRat_inv_pos, Mathlib.Meta.NormNum.IsNat.to_isNNRat, Mathlib.Meta.NormNum.isNat_natCast, Mathlib.Meta.NormNum.isNNRat_div, Mathlib.Meta.NormNum.isNNRat_mul, Nat.mul
41 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
lemma expressible_two_mul_add_nine {n : ℕ} (h : Expressible n) :
    Expressible (2 * n + 9) :=
…
```
Mentions: h, Usa1978P3.Expressible
problem Usa1978P3.usa1978_p3 line 91 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Usa1978P3.Expressible, LT.lt
Proof uses: Nat.strong_induction_on, Even, Odd, Or, Nat.even_or_odd, Eq, HAdd.hAdd, HMul.hMul, HSub.hSub, dite, Not, Eq.mpr, id, congrArg, Usa1978P3.expressible_two_mul_add_eight, Usa1978P3.expressible_two_mul_add_nine
35 instances, coercions and auxiliary constants left out.

Tactic trace: 32 steps
Open to load the checked before-and-after goal states.