`Compfiles/Usa1994P1.lean`

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4 declarations, 0 contain sorry, 0 declared as axioms

USA Mathematical Olympiad 1994, Problem 1

a₁, a₂, a₃, ... are positive integers such that aₙ > aₙ₋₁ + 1. Put bₙ = a₁ + a₂ + ... + aₙ. Show that there is always a square in the range bₙ, bₙ+1, bₙ+2, ... , bₙ₊₁-1.

Imports (10)

Imported by 1 file

Compfiles.lean

lemma Usa1994P1.gap_lemma If consecutive terms of a differ by more than one, then terms that are k indices apart differ by at least 2 * k line 38 source
Source documentation
If consecutive terms of a differ by more than one, then terms that are k indices apart differ by at least 2 * k.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, HAdd.hAdd, OfNat.ofNat, LE.le, HMul.hMul
Proof uses: Nat.recAux, of_eq_true, Eq.trans, True, congr, congrArg, MulZeroClass.mul_zero, add_zero, Trans.trans, Eq, Mathlib.Tactic.Ring.of_eq, Nat.rawCast, HPow.hPow, Mathlib.Tactic.Ring.Common.add_congr, Mathlib.Tactic.Ring.Common.atom_pf, rfl, Eq.mpr, id, Eq.symm, Eq.refl and 17 more
40 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
```
lemma gap_lemma {a : ℕ → ℕ} (h : ∀ i, a i + 1 < a (i + 1)) (i k : ℕ) :
    a i + 2 * k ≤ a (i + k) := by
…
```
Mentions: h
lemma Usa1994P1.sum_bound The sum of the first n + 1 terms is at most (n + 1) * (a n + 1) - (n + 1)^2; we state it in an addition-only form line 51 source
Source documentation
The sum of the first n + 1 terms is at most (n + 1) * (a n + 1) - (n + 1)^2; we state it in an addition-only form.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, HAdd.hAdd, OfNat.ofNat, LE.le, Finset.sum, Finset.range, HMul.hMul
Proof uses: Nat.recAux, of_eq_true, Eq.trans, True, congr, congrArg, Singleton.singleton, Finset, congrFun', zero_add, Finset.sum_singleton, mul_one, one_mul, mul_le_mul_right, Usa1994P1.gap_lemma, Eq.mpr, id, Eq, Finset.sum_range_succ, le_of_not_gt and 64 more
93 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
```
lemma sum_bound {a : ℕ → ℕ} (h : ∀ i, a i + 1 < a (i + 1)) (n : ℕ) :
    (∑ i ∈ Finset.range (n + 1), a i) + (n + 1) * (n + 1) ≤
      (n + 1) * (a n + 1) := by
…
```
Mentions: h
lemma Usa1994P1.key The key estimate: 4 * bₙ ≤ (aₙ + 1)² line 63 source
Source documentation
The key estimate: 4 * bₙ ≤ (aₙ + 1)².

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, HAdd.hAdd, OfNat.ofNat, LE.le, HMul.hMul, Finset.sum, Finset.range, HPow.hPow
Proof uses: Usa1994P1.sum_bound, Eq.mpr, Int, id, Eq, Eq.trans, congr, congrArg, Nat.cast_mul, Nat.cast_sum, Nat.cast_pow, congrFun', Nat.cast_add, Nat.cast_one, le_of_not_gt, Mathlib.Tactic.Linarith.lt_irrefl, Eq.mp, Neg.neg, HSub.hSub, Mathlib.Tactic.Ring.of_eq and 68 more
97 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
lemma key {a : ℕ → ℕ} (h : ∀ i, a i + 1 < a (i + 1)) (n : ℕ) :
    4 * (∑ i ∈ Finset.range (n + 1), a i) ≤ (a n + 1) ^ 2 := by
…
```
Mentions: h
problem Usa1994P1.usa1994_p1 line 71 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, HAdd.hAdd, LE.le, Exists, And, Finset.sum, Finset.range, HPow.hPow
Proof uses: Eq, Exists.intro, HSub.hSub, Nat.sqrt, rfl, Nat.sqrt_le', Or, eq_or_lt_of_le, And.intro, le_of_eq, Eq.symm, Eq.mpr, id, congrArg, Finset.sum_range_succ, Nat.lt_add_of_pos_right, Nat.lt_succ_sqrt', le_of_lt, HMul.hMul, le_trans and 96 more
104 instances, coercions and auxiliary constants left out.

Tactic trace: 21 steps
Open to load the checked before-and-after goal states.