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720fa25 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 | import OAI.GameTheory.SnakyConditional.Opening.Opening
import OAI.GameTheory.SnakyConditional.Semantics.Policy
namespace OAI
namespace SnakyConditional
open OrdinaryStrategy
theorem snaky_winning_strategy :
∃ σ : Policy Cell,
(∀ r M B, σ r M B ∉ M ∧ σ r M B ∉ B) ∧
∀ β : ℕ → Cell,
LegalRepliesBeforeFinal σ 35 β ∅ ∅ →
HasSnaky (playState σ 35 β ∅ ∅ 35).1 := by
exact winsIn_has_ordinary_policy
(fun M N hMN hw => HasSnaky.mono hw hMN) empty_board_wins
theorem snaky_winning_strategy_with_legal_states :
∃ σ : Policy Cell,
(∀ r M B, σ r M B ∉ M ∧ σ r M B ∉ B) ∧
∀ β : ℕ → Cell, LegalRepliesBeforeFinal σ 35 β ∅ ∅ →
HasSnaky (playState σ 35 β ∅ ∅ 35).1 ∧
(playState σ 35 β ∅ ∅ 35).1.card = 35 ∧
Disjoint (playState σ 35 β ∅ ∅ 35).1 (playState σ 35 β ∅ ∅ 34).2 ∧
∀ k, k < 35 →
Disjoint (playState σ 35 β ∅ ∅ k).1 (playState σ 35 β ∅ ∅ k).2 ∧
(playState σ 35 β ∅ ∅ k).2.card = k := by
refine ⟨policy HasSnaky, policy_fresh HasSnaky, ?_⟩
intro β hβ
refine ⟨policy_wins (fun M N hMN hw => HasSnaky.mono hw hMN)
35 β ∅ ∅ empty_board_wins hβ, ?_, ?_, ?_⟩
· simpa using playState_maker_card (policy HasSnaky) (policy_fresh HasSnaky)
35 β ∅ ∅ 35
· exact final_maker_disjoint (policy HasSnaky) (policy_fresh HasSnaky)
34 β ∅ ∅ (by simp) hβ
· intro k hk
simpa using playState_legal_prefix (policy HasSnaky) (policy_fresh HasSnaky)
35 β ∅ ∅ (by simp) hβ k hk
end SnakyConditional
end OAI
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