File size: 2,292 Bytes
37bcf52 | 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 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 | 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
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