`Compfiles/Usa1973P2.lean`

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7 declarations, 0 contain sorry, 0 declared as axioms

USA Mathematical Olympiad 1973, Problem 2

The sequence an is defined by a1 = a2 = 1, an+2 = an+1 + 2an. The sequence bn is defined by b1 = 1, b2 = 7, bn+2 = 2bn+1 + 3bn.

Show that the only integer in both sequences is 1.

Imports (3)

Imported by 1 file

Compfiles.lean

def Usa1973P2.a reindexed to start at 0 line 30 source
Source comment
reindexed to start at 0

no axioms what this means
Statement uses: Nat
2 instances, coercions and auxiliary constants left out.
```
def a : ℕ → ℕ
| 0 => 1
| 1 => 1
| n + 2 => a (n + 1) + 2 * a n
```
def Usa1973P2.b line 35 source
no axioms what this means
Statement uses: Nat
2 instances, coercions and auxiliary constants left out.
lemma Usa1973P2.b_mod_8_eq line 42 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, And, Even, Eq, HMod.hMod, Usa1973P2.b, OfNat.ofNat, Odd
Proof uses: Nat.twoStepInduction, of_eq_true, Eq.trans, True, congr, congrArg, implies_congr, congrFun', Nat.one_mod, eq_self, False, and_self, Nat.mod_succ, HAdd.hAdd, And.intro, Eq.mpr, HMul.hMul, id, Nat.succ, Usa1973P2.b.eq_3
37 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
lemma Usa1973P2.b_mod_8_mem line 48 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Or, Eq, HMod.hMod, Usa1973P2.b, OfNat.ofNat
Proof uses: Even, Odd, Nat.even_or_odd, Or.inl, And.left, Usa1973P2.b_mod_8_eq, Or.inr, And.right, Eq.refl
9 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma Usa1973P2.a_add_2_mod_8_eq line 53 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, And, Even, Eq, HMod.hMod, Usa1973P2.a, HAdd.hAdd, OfNat.ofNat, Odd
Proof uses: Nat.twoStepInduction, of_eq_true, Eq.trans, True, congr, congrArg, implies_congr, congrFun', HMul.hMul, mul_one, eq_self, False, eq_false_of_decide, Eq.refl, Bool, Bool.false, and_self, And.intro, Even.add_one, Eq.mp and 10 more
35 instances, coercions and auxiliary constants left out.

Tactic trace: 15 steps
Open to load the checked before-and-after goal states.
lemma Usa1973P2.a_add_2_mod_8_mem line 70 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Or, Eq, HMod.hMod, Usa1973P2.a, HAdd.hAdd, OfNat.ofNat
Proof uses: Even, Odd, Nat.even_or_odd, Or.inl, And.left, Usa1973P2.a_add_2_mod_8_eq, Or.inr, And.right, Eq.refl
12 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
problem Usa1973P2.usa1973_p2 line 77 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Eq, Usa1973P2.a, Usa1973P2.b, OfNat.ofNat
Proof uses: dite, LT.lt, LE.le, Eq.refl, Eq.symm, le_antisymm, Nat.ge_of_not_lt, Mathlib.Tactic.IntervalCases.of_lt_right, Nat.rawCast, Mathlib.Meta.NormNum.IsNat.to_raw_eq, Mathlib.Meta.NormNum.isNat_ofNat, Not, Nat.zero_le, Exists, HAdd.hAdd, Nat.exists_eq_add_of_le, Eq.mp, Or, HMod.hMod, Usa1973P2.a_add_2_mod_8_mem and 3 more
20 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.