| import OAI.GameTheory.SnakyTwentyOne.Cards.Height21Part01 |
| import OAI.GameTheory.SnakyTwentyOne.Certificate.Summary |
| import OAI.GameTheory.SnakyTwentyOne.Model |
| import OAI.GameTheory.SnakyTwentyOne.Semantics.Game |
| import OAI.GameTheory.SnakyTwentyOne.Semantics.Geometry |
| import OAI.GameTheory.SnakyTwentyOne.Semantics.Policy |
| import OAI.GameTheory.SnakyTwentyOne.Semantics.Replies |
|
|
| namespace OAI |
|
|
| namespace SnakyPrototype |
| theorem empty_board_wins_21 : WinsIn HasSnaky 21 ∅ ∅ := by |
| exact Certificate21.row_works_727 ∅ ∅ (by simp) |
| (by simp [Certificate21.required_727]) (by simp) |
| end SnakyPrototype |
| namespace SnakyPrototype |
| open OrdinaryStrategy |
| theorem snaky_winning_strategy_21_with_legal_states : |
| ∃ σ : Policy Cell, |
| (∀ r M B, σ r M B ∉ M ∧ σ r M B ∉ B) ∧ |
| ∀ β : ℕ → Cell, LegalRepliesBeforeFinal σ 21 β ∅ ∅ → |
| HasSnaky (playState σ 21 β ∅ ∅ 21).1 ∧ |
| (playState σ 21 β ∅ ∅ 21).1.card = 21 ∧ |
| Disjoint (playState σ 21 β ∅ ∅ 21).1 (playState σ 21 β ∅ ∅ 20).2 ∧ |
| ∀ k, k < 21 → |
| Disjoint (playState σ 21 β ∅ ∅ k).1 (playState σ 21 β ∅ ∅ k).2 ∧ |
| (playState σ 21 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) |
| 21 β ∅ ∅ empty_board_wins_21 hβ, ?_, ?_, ?_⟩ |
| · simpa using playState_maker_card (policy HasSnaky) (policy_fresh HasSnaky) |
| 21 β ∅ ∅ 21 |
| · exact final_maker_disjoint (policy HasSnaky) (policy_fresh HasSnaky) |
| 20 β ∅ ∅ (by simp) hβ |
| · intro k hk |
| simpa using playState_legal_prefix (policy HasSnaky) (policy_fresh HasSnaky) |
| 21 β ∅ ∅ (by simp) hβ k hk |
| theorem snaky_winning_strategy_21 : |
| ∃ σ : Policy Cell, |
| (∀ r M B, σ r M B ∉ M ∧ σ r M B ∉ B) ∧ |
| ∀ β : ℕ → Cell, |
| LegalRepliesBeforeFinal σ 21 β ∅ ∅ → |
| HasSnaky (playState σ 21 β ∅ ∅ 21).1 := by |
| obtain ⟨σ, hσ, hw⟩ := snaky_winning_strategy_21_with_legal_states |
| exact ⟨σ, hσ, fun β hβ => (hw β hβ).1⟩ |
| end SnakyPrototype |
|
|
| end OAI |
|
|
