openAI-math-dataset / lean /OAI /Computability /UniqueGames /Machines /MachineExpanderFamilyTapes.lean
| import OAI.Computability.UniqueGames.Machines.MachineExpanderTableProgramLemmas | |
| namespace OAI | |
| section | |
| /-! Static, pointwise Boolean presentation of the actual family tape alphabet. | |
| Every conversion below is an identity on symbols and has no machine runtime. -/ | |
| namespace UniqueGamesTheorem.Foundations.Complexity.MachineExpanderFamily | |
| open Turing | |
| def boolWord : (tape : Tape) → List Bool → List (Alphabet tape) | |
| | .inl (.inl _), word => word | |
| | .inl (.inr _), word => word | |
| | .inr _, word => word | |
| def toBoolWord : (tape : Tape) → List (Alphabet tape) → List Bool | |
| | .inl (.inl _), word => word | |
| | .inl (.inr _), word => word | |
| | .inr _, word => word | |
| def fromBoolTapes (base : Tape → List Bool) : (tape : Tape) → List (Alphabet tape) := | |
| fun tape => boolWord tape (base tape) | |
| def toBoolTapes (base : (tape : Tape) → List (Alphabet tape)) : Tape → List Bool := | |
| fun tape => toBoolWord tape (base tape) | |
| @[simp] theorem toBoolWord_boolWord (tape : Tape) (word : List Bool) : | |
| toBoolWord tape (boolWord tape word) = word := by | |
| rcases tape with tape | tape | |
| · cases tape <;> rfl | |
| · rfl | |
| @[simp] theorem boolWord_toBoolWord (tape : Tape) (word : List (Alphabet tape)) : | |
| boolWord tape (toBoolWord tape word) = word := by | |
| rcases tape with tape | tape | |
| · cases tape <;> rfl | |
| · rfl | |
| @[simp] theorem toBool_fromBool (base : Tape → List Bool) : | |
| toBoolTapes (fromBoolTapes base) = base := by | |
| funext tape | |
| exact toBoolWord_boolWord tape (base tape) | |
| @[simp] theorem fromBool_toBool (base : (tape : Tape) → List (Alphabet tape)) : | |
| fromBoolTapes (toBoolTapes base) = base := by | |
| funext tape | |
| exact boolWord_toBoolWord tape (base tape) | |
| private theorem transport_tapes_apply {K : Type} {Γ Δ : K → Type} | |
| (h : Γ = Δ) (base : (k : K) → List (Γ k)) (k : K) : | |
| MachineAlphabetTransport.tapes h base k = | |
| Eq.mp (congrArg (fun alphabet => List (alphabet k)) h) (base k) := by | |
| cases h | |
| rfl | |
| theorem toBoolTapes_eq_transport (base : (tape : Tape) → List (Alphabet tape)) : | |
| toBoolTapes base = MachineAlphabetTransport.tapes alphabet_eq base := by | |
| funext tape | |
| rw [transport_tapes_apply] | |
| rcases tape with tape | tape | |
| · cases tape <;> rfl | |
| · rfl | |
| theorem fromBoolTapes_eq_transport (base : Tape → List Bool) : | |
| fromBoolTapes base = MachineAlphabetTransport.tapes alphabet_eq.symm base := by | |
| funext tape | |
| rw [transport_tapes_apply] | |
| rcases tape with tape | tape | |
| · cases tape <;> rfl | |
| · rfl | |
| @[simp] theorem fromBoolTapes_update (base : Tape → List Bool) (tape : Tape) (word : List Bool) : | |
| fromBoolTapes (Function.update base tape word) = | |
| Function.update (fromBoolTapes base) tape (boolWord tape word) := by | |
| funext k | |
| by_cases h : k = tape | |
| · subst k | |
| simp [fromBoolTapes] | |
| · simp [fromBoolTapes, h] | |
| @[simp] theorem toBoolTapes_update (base : (tape : Tape) → List (Alphabet tape)) | |
| (tape : Tape) (word : List (Alphabet tape)) : | |
| toBoolTapes (Function.update base tape word) = | |
| Function.update (toBoolTapes base) tape (toBoolWord tape word) := by | |
| funext k | |
| by_cases h : k = tape | |
| · subst k | |
| simp [toBoolTapes] | |
| · simp [toBoolTapes, h] | |
| theorem configuration_fromBool {ρ : Type} {d : Nat} (label : Option (Label d)) | |
| (state : State ρ d) (base : Tape → List Bool) : | |
| MachineAlphabetTransport.configuration alphabet_eq.symm ⟨label, state, base⟩ = | |
| ⟨label, state, fromBoolTapes base⟩ := by | |
| rw [MachineAlphabetTransport.configuration_mk, fromBoolTapes_eq_transport] | |
| theorem configuration_toBool {ρ : Type} {d : Nat} (label : Option (Label d)) | |
| (state : State ρ d) (base : (tape : Tape) → List (Alphabet tape)) : | |
| MachineAlphabetTransport.configuration alphabet_eq ⟨label, state, base⟩ = | |
| ⟨label, state, toBoolTapes base⟩ := by | |
| rw [MachineAlphabetTransport.configuration_mk, toBoolTapes_eq_transport] | |
| end UniqueGamesTheorem.Foundations.Complexity.MachineExpanderFamily | |
| end | |
| end OAI | |