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putnam_2023_a6: fix vacuous strategy-validity condition (Lean) and free-variable typo (Isabelle) #369

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putnam_2023_a6: fix vacuous strategy-validity condition (Lean) and free-variable typo (Isabelle) #369

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161711d
putnam_2023_a6 (Lean): a valid strategy must pick an unused number in…
AaronY77 Oct 13, 2026
5c48885
putnam_2023_a6 (Isabelle): fix free variable G in IsValidGame_def
AaronY77 Oct 13, 2026
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4 changes: 2 additions & 2 deletions isabelle/putnam_2023_a6.thy
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Original file line number Diff line number Diff line change
Expand Up @@ -9,7 +9,7 @@ theorem putnam_2023_a6:
and parityOf :: "nat list \<Rightarrow> nat"
and ConformsToStrategy :: "nat list \<Rightarrow> (nat list \<Rightarrow> nat) \<Rightarrow> bool"
and IsWinningFor :: "nat \<Rightarrow> (nat list \<Rightarrow> nat) \<Rightarrow> bool"
assumes IsValidGame_def : "\<forall> g. (IsValidGame G \<longleftrightarrow>
assumes IsValidGame_def : "\<forall> g. (IsValidGame g \<longleftrightarrow>
distinct g \<and> (\<forall> i \<in> set g. i \<in> {1..length g}))"
and parityOf_def : "\<forall> g. parityOf g = card {n :: nat. n ∈ {0..<length g} \<and> nth g n = n+1} mod 2"
and ConformsToStrategy_def : "\<forall> g s. (ConformsToStrategy g s \<longleftrightarrow>
Expand All @@ -20,4 +20,4 @@ theorem putnam_2023_a6:
shows "{n :: nat. 0 < n \<and> (\<exists> s. IsWinningFor n s)} = putnam_2023_a6_solution"
sorry

end
end
9 changes: 5 additions & 4 deletions lean4/src/putnam_2023_a6.lean
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Expand Up @@ -15,11 +15,12 @@ theorem putnam_2023_a6
(ConformsToStrategy : List ℕ → (List ℕ → ℕ) → Prop)
(ConformsToStrategy_def : ∀ g s, ConformsToStrategy g s ↔
∀ (i) (h : i < g.length), Odd i → g[i] = s (g.take i))
(IsValidStrategy : (List ℕ → ℕ) → Prop)
(IsValidStrategy_def : ∀ s, IsValidStrategy s ↔
∀ (g : List ℕ), g.length > 0 → s g ∈ g)
(IsValidStrategy : ℕ → (List ℕ → ℕ) → Prop)
(IsValidStrategy_def : ∀ n s, IsValidStrategy n s ↔
∀ g : List ℕ, g.Nodup → (∀ i ∈ g, i ∈ Icc 1 n) → g.length < n →
s g ∈ Icc 1 n ∧ s g ∉ g)
(IsWinningFor : ℕ → (List ℕ → ℕ) → Prop)
(IsWinningFor_def : ∀ n s, IsWinningFor n s ↔
IsValidStrategy s ∧ ∃ p, ∀ g, g.length = n → IsValidGame g → ConformsToStrategy g s → parityOf g = p) :
IsValidStrategy n s ∧ ∃ p, ∀ g, g.length = n → IsValidGame g → ConformsToStrategy g s → parityOf g = p) :
{n : ℕ | 0 < n ∧ ∃ s, IsWinningFor n s} = putnam_2023_a6_solution :=
sorry