`Compfiles/UK2024R1P1.lean`

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21 declarations, 0 contain sorry, 0 declared as axioms

British Mathematical Olympiad 2024, Round 1, Problem 1

An unreliable typist can guarantee that when they try to type a word with different letters, every letter of the word will appear exactly once in what they type, and each letter will occur at most one letter late (though it may occur more than one letter early). Thus, when trying to type MATHS, the typist may type MATHS, MTAHS or TMASH, but not ATMSH.

Determine, with proof, the number of possible spellings of OLYMPIADS that might be typed.

Imports (4)

Imported by 1 file

Compfiles.lean

def UK2024R1P1.S line 53 source
standard axioms: Quot.sound what this means
Statement uses: Nat, Set, Equiv.Perm, Fin
Proof uses: Set.ofPred, LE.le, HAdd.hAdd, OfNat.ofNat, Fin.castSucc, Fin.succ
7 instances, coercions and auxiliary constants left out.
def UK2024R1P1.S' line 54 source
standard axioms: Quot.sound, propext what this means
Statement uses: Nat, Fin, HAdd.hAdd, OfNat.ofNat, Set, Equiv.Perm
Proof uses: Set.ofPred, And, Membership.mem, UK2024R1P1.S, Eq
12 instances, coercions and auxiliary constants left out.
lemma UK2024R1P1.g_nonzero_ne_m line 56 source
standard axioms: Quot.sound, propext what this means
Statement uses: Nat, Fin, HAdd.hAdd, OfNat.ofNat, Equiv.Perm, Membership.mem, Set, UK2024R1P1.S', Ne
Proof uses: Mathlib.Tactic.Contrapose.contrapose₄, Eq, Eq.mpr, Equiv, id, congrArg, Eq.symm, propext, Equiv.apply_eq_iff_eq, And.right, UK2024R1P1.S
21 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
def UK2024R1P1.f_toFun line 61 source
standard axioms: Quot.sound, propext what this means
Statement uses: Nat, Fin, HAdd.hAdd, OfNat.ofNat, Equiv.Perm, Membership.mem, Set, UK2024R1P1.S'
Proof uses: dite, LE.le, Fin.succ, Fin.castPred, Not, Fin.pred
15 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
def UK2024R1P1.f_invFun line 73 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, HAdd.hAdd, OfNat.ofNat, Equiv.Perm, Membership.mem, Set, UK2024R1P1.S'
Proof uses: dite, LT.lt, Fin.castSucc, Fin.pred, Equiv, Equiv.symm, Not, Fin.succ
15 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
lemma UK2024R1P1.f_left_inv line 88 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, HAdd.hAdd, OfNat.ofNat, Equiv.Perm, Membership.mem, Set, UK2024R1P1.S', Function.LeftInverse, UK2024R1P1.f_invFun, UK2024R1P1.f_toFun
Proof uses: id, Eq, LE.le, Fin.succ, Not, Or, dite, LT.lt, Fin.castSucc, Fin.castPred, Fin.pred, Equiv, Equiv.symm, Ne, Eq.refl, em, Eq.mpr, True, Eq.mpr_prop, eq_true and 32 more
33 instances, coercions and auxiliary constants left out.

Tactic trace: 14 steps
Open to load the checked before-and-after goal states.
lemma UK2024R1P1.f_right_inv line 103 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, HAdd.hAdd, OfNat.ofNat, Equiv.Perm, Membership.mem, Set, UK2024R1P1.S', Function.RightInverse, UK2024R1P1.f_invFun, UK2024R1P1.f_toFun
Proof uses: id, Eq, LT.lt, Fin.castSucc, Not, Or, dite, LE.le, Fin.succ, Fin.pred, Equiv, Equiv.symm, Fin.castPred, Ne, Fin.last, Eq.refl, em, Eq.mpr, True, Eq.mpr_prop and 22 more
29 instances, coercions and auxiliary constants left out.

Tactic trace: 13 steps
Open to load the checked before-and-after goal states.
def UK2024R1P1.f line 117 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, HAdd.hAdd, OfNat.ofNat, Equiv.Perm, Membership.mem, Set, UK2024R1P1.S', Equiv
Proof uses: Equiv.mk, UK2024R1P1.f_toFun, UK2024R1P1.f_invFun, UK2024R1P1.f_left_inv, UK2024R1P1.f_right_inv
8 instances, coercions and auxiliary constants left out.
def UK2024R1P1.f'_toFun line 120 source
standard axioms: Quot.sound, propext what this means
Statement uses: Nat, Equiv.Perm, Fin, HAdd.hAdd, OfNat.ofNat
Proof uses: dite, Eq, Not, ite, LT.lt, Fin.castSucc, Fin.pred, Fin.succ
14 instances, coercions and auxiliary constants left out.
def UK2024R1P1.f'_invFun line 129 source
standard axioms: Quot.sound, propext what this means
Statement uses: Nat, Equiv.Perm, Fin, HAdd.hAdd, OfNat.ofNat
Proof uses: dite, Eq, Not, LT.lt, Fin.succ, Equiv, Equiv.symm, Fin.castPred, Fin.pred
16 instances, coercions and auxiliary constants left out.
lemma UK2024R1P1.f'_left_inv line 138 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, HAdd.hAdd, OfNat.ofNat, Equiv.Perm, Function.LeftInverse, UK2024R1P1.f'_invFun, UK2024R1P1.f'_toFun
Proof uses: id, Eq, Not, Or, dite, ite, LT.lt, Fin.castSucc, Fin.pred, Fin.succ, Equiv, Equiv.symm, Fin.castPred, Ne, Fin.last, Eq.refl, em, of_eq_true, Eq.trans, True and 28 more
32 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
lemma UK2024R1P1.f'_right_inv line 150 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, HAdd.hAdd, OfNat.ofNat, Equiv.Perm, Function.RightInverse, UK2024R1P1.f'_invFun, UK2024R1P1.f'_toFun
Proof uses: id, Eq, Not, Or, dite, LT.lt, Fin.succ, Equiv, Equiv.symm, Fin.castPred, Fin.pred, ite, Fin.castSucc, Ne, Eq.refl, em, of_eq_true, Eq.trans, True, congr and 33 more
31 instances, coercions and auxiliary constants left out.

Tactic trace: 13 steps
Open to load the checked before-and-after goal states.
def UK2024R1P1.f' line 165 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, HAdd.hAdd, OfNat.ofNat, Equiv.Perm, Equiv
Proof uses: Equiv.mk, UK2024R1P1.f'_toFun, UK2024R1P1.f'_invFun, UK2024R1P1.f'_left_inv, UK2024R1P1.f'_right_inv
6 instances, coercions and auxiliary constants left out.
lemma UK2024R1P1.f'_in_S'_k line 168 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, HAdd.hAdd, OfNat.ofNat, LE.le, Equiv.Perm, Membership.mem, Set, UK2024R1P1.S, Equiv, UK2024R1P1.S', UK2024R1P1.f'
Proof uses: And.intro, Eq, id, Fin.castSucc, Fin.succ, Not, Or, dite, ite, LT.lt, Fin.pred, em, Eq.mpr, congr, congrArg, dite_eq_left_of_eq_true, Eq.trans, True, congrFun', eq_self and 17 more
27 instances, coercions and auxiliary constants left out.

Tactic trace: 21 steps
Open to load the checked before-and-after goal states.
lemma UK2024R1P1.f_f'_left_inv line 185 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, HAdd.hAdd, OfNat.ofNat, Equiv.Perm, Membership.mem, Set, UK2024R1P1.S', Eq, Equiv, UK2024R1P1.f', UK2024R1P1.f
Proof uses: Equiv.ext, Eq.mpr, dite, Not, ite, LT.lt, Fin.castSucc, LE.le, Fin.castPred, Fin.succ, Fin.pred, Eq.mpr_not, Eq.refl, Ne, Fin.last, Eq.mpr_prop, congrFun', congrArg, Fin.succ_pred, id and 35 more
34 instances, coercions and auxiliary constants left out.

Tactic trace: 14 steps
Open to load the checked before-and-after goal states.
lemma UK2024R1P1.f_f'_right_inv line 198 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, HAdd.hAdd, OfNat.ofNat, LE.le, Equiv.Perm, Membership.mem, Set, UK2024R1P1.S, Eq, Equiv, UK2024R1P1.f, UK2024R1P1.f', UK2024R1P1.f'_in_S'_k
Proof uses: Equiv.ext, Eq.mpr, dite, ite, LT.lt, Fin.castSucc, Fin.succ, Fin.castPred, Equiv.mk, UK2024R1P1.f'_toFun, UK2024R1P1.f'_invFun, UK2024R1P1.f'_left_inv, UK2024R1P1.f'_right_inv, Ne, Fin.last, Eq.mpr_prop, congrFun', congrArg, Eq.trans, Not and 27 more
34 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
lemma UK2024R1P1.S'_k_card_eq_S_succ_k_card line 211 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, HAdd.hAdd, OfNat.ofNat, LE.le, Eq, Set.ncard, Equiv.Perm, UK2024R1P1.S', UK2024R1P1.S
Proof uses: Set.ncard_congr, Equiv, UK2024R1P1.f, Membership.mem, Set, id, Fin.succ, Not, Or, Fin.castSucc, dite, Fin.castPred, Fin.pred, em, Eq.mpr, congrFun', congrArg, Eq.trans, Eq.mpr_prop, True and 29 more
27 instances, coercions and auxiliary constants left out.

Tactic trace: 22 steps
Open to load the checked before-and-after goal states.
lemma UK2024R1P1.S_succ_succ_k_card_eq_two_mul_S_succ_k_card line 230 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Eq, Set.ncard, Equiv.Perm, Fin, HAdd.hAdd, OfNat.ofNat, UK2024R1P1.S, HMul.hMul
Proof uses: Set, UK2024R1P1.S', UK2024R1P1.S'_k_card_eq_S_succ_k_card, of_eq_true, LE.le, Eq.trans, True, eq_true, True.intro, Disjoint, Eq.mpr, Membership.mem, Not, id, congrArg, propext, Set.disjoint_left, And, congrFun', Set.ofPred and 22 more
77 instances, coercions and auxiliary constants left out.

Tactic trace: 34 steps
Open to load the checked before-and-after goal states.
lemma UK2024R1P1.S_succ_k_card_eq_two_pow_k line 259 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Eq, Set.ncard, Equiv.Perm, Fin, HAdd.hAdd, OfNat.ofNat, UK2024R1P1.S, HPow.hPow
Proof uses: Nat.recAux, Eq.mpr, Exists, Set, Singleton.singleton, id, Eq.trans, congrArg, pow_zero, Equiv.mk, of_decide_eq_true, Function.LeftInverse, Bool, Decidable.decide, Bool.true, Eq.refl, Function.RightInverse, Exists.intro, Set.ext, Iff.intro and 19 more
47 instances, coercions and auxiliary constants left out.

Tactic trace: 17 steps
Open to load the checked before-and-after goal states.
determine UK2024R1P1.solution_value line 277 source
no axioms what this means
Statement uses: Nat
Proof uses: OfNat.ofNat
1 instances, coercions and auxiliary constants left out.
problem UK2024R1P1.uk2024_r1_p1 Since OLYMPIADS has no duplicate letters, then the set of spellings is just a subset of the permutations of 9 elements line 283 source
Source comment
Since OLYMPIADS has no duplicate letters, then the set of spellings is just a subset of the permutations of 9 elements.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Set.ncard, Equiv.Perm, Fin, OfNat.ofNat, Set.ofPred, LE.le, HAdd.hAdd, UK2024R1P1.solution_value
Proof uses: id, Eq.mpr, UK2024R1P1.S, HPow.hPow, congrArg, UK2024R1P1.S_succ_k_card_eq_two_pow_k, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_pow, Eq.refl, Mathlib.Meta.NormNum.isNat_ofNat, Mathlib.Meta.NormNum.IsNatPowT.run, Mathlib.Meta.NormNum.IsNatPowT.trans, Nat.pow, Mathlib.Meta.NormNum.IsNatPowT.bit0
22 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
problem uk2024_r1_p1 :
  {f : Equiv.Perm (Fin 9) | ∀ k, (f k : ℕ) ≤ k + 1}.ncard = solution_value := by
…
```
Mentions: UK2024R1P1.f, UK2024R1P1.solution_value