| 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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