openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /ForcedComputation /Programs /ConfigurationTapes.lean
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.Recorder | |
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.TapeCoordinates | |
| namespace OAI | |
| /-! Exact two-stream coordinates of a one-head tape update. -/ | |
| namespace ForcedComputation.Recorder | |
| variable {Q A : Type*} | |
| def tapeAt (C : Configuration Q A) : Radix.Tape (Symbol Q A) := | |
| (fun n => C.tape (C.head - ((n : ℤ) + 1)), fun n => C.tape (C.head + (n : ℤ))) | |
| theorem tapeAt_right (C : Configuration Q A) (q' : Control Q A) (b : Symbol Q A) : | |
| tapeAt ⟨q', C.head + 1, Function.update C.tape C.head b⟩ = | |
| Radix.writeRight b (tapeAt C) := by | |
| apply Prod.ext | |
| · funext n | |
| cases n with | |
| | zero => simp [tapeAt, Radix.writeRight, Radix.prepend] | |
| | succ n => | |
| dsimp [tapeAt, Radix.writeRight, Radix.prepend] | |
| rw [Function.update_of_ne (by omega)] | |
| congr 1 | |
| omega | |
| · funext n | |
| dsimp [tapeAt, Radix.writeRight] | |
| rw [Function.update_of_ne (by omega)] | |
| congr 1 | |
| omega | |
| theorem tapeAt_stay (C : Configuration Q A) (q' : Control Q A) (b : Symbol Q A) : | |
| tapeAt ⟨q', C.head, Function.update C.tape C.head b⟩ = | |
| Radix.writeStay b (tapeAt C) := by | |
| apply Prod.ext | |
| · funext n | |
| dsimp [tapeAt, Radix.writeStay] | |
| rw [Function.update_of_ne (by omega)] | |
| · funext n | |
| cases n with | |
| | zero => simp [tapeAt, Radix.writeStay, Radix.prepend] | |
| | succ n => | |
| dsimp [tapeAt, Radix.writeStay, Radix.prepend] | |
| rw [Function.update_of_ne (by omega)] | |
| theorem tapeAt_left (C : Configuration Q A) (q' : Control Q A) (b : Symbol Q A) : | |
| tapeAt ⟨q', C.head - 1, Function.update C.tape C.head b⟩ = | |
| Radix.writeLeft b (tapeAt C) := by | |
| apply Prod.ext | |
| · funext n | |
| dsimp [tapeAt, Radix.writeLeft] | |
| rw [Function.update_of_ne (by omega)] | |
| congr 1 | |
| omega | |
| · funext n | |
| cases n with | |
| | zero => | |
| dsimp [tapeAt, Radix.writeLeft, Radix.prepend] | |
| rw [Function.update_of_ne (by omega)] | |
| congr 1 | |
| omega | |
| | succ n => | |
| cases n with | |
| | zero => simp [tapeAt, Radix.writeLeft, Radix.prepend] | |
| | succ n => | |
| dsimp [tapeAt, Radix.writeLeft, Radix.prepend] | |
| rw [Function.update_of_ne (by omega)] | |
| congr 1 | |
| omega | |
| end ForcedComputation.Recorder | |
| end OAI | |