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

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@AaronY77

@AaronY77 AaronY77 commented Oct 13, 2026 •

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Fixes #365.

  • Lean: IsValidStrategy s ↔ ∀ g, g.length > 0 → s g ∈ g forces Bob to repeat a chosen number, so no legal game of length ≥ 2 conforms and every valid strategy wins vacuously. The new condition requires, on every legal unfinished history, an unused number in 1..n:
    (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)
    and IsWinningFor_def uses IsValidStrategy n s. Range and freshness are both needed (missing range was the Fix misformalizations #285 exploit). The statement elaborates on Lean v4.35.0 + current Mathlib; we have not proved the fixed theorem.
  • Isabelle: IsValidGame G → IsValidGame g in IsValidGame_def (the free G made the assumptions contradictory). The Isabelle version still has no strategy-validity condition (see putnam_2023_a6 (Lean, Isabelle): statement is vacuous — "valid strategy" requires Bob to pick an already-chosen number #365); not changed here because we could not machine-check an Isabelle edit.

… 1..n

The current condition `s g ∈ g` forces Bob to repeat a chosen number, so no legal game conforms and the statement is vacuous (trishullab#365).
`IsValidGame G` (free G) made the assumption contradictory (trishullab#365).

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putnam_2023_a6 (Lean, Isabelle): statement is vacuous — "valid strategy" requires Bob to pick an already-chosen number

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