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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 76 | import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.FiniteStackArithmetic
import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.ConfigurationTapes
namespace OAI
/-! Exact integer stack formulas for each recorder head move. A single
additional digit accommodates a step; a popped infinite blank tail stays zero. -/
namespace ForcedComputation.Lattice
def StackTailZero (d : ℕ) (a : ℕ → ℕ) : Prop := ∀ i, d ≤ i → a i = 0
theorem stackValue_pop_padding {b k : ℕ} (hb : 0 < b) {a : ℕ → ℕ}
(ha : a 0 < b) (hz : StackTailZero (k + 1) a) (e : ℕ) :
stackValue b (k + e) (fun i => a (i + 1)) = stackValue b (k + 1) a / b := by
rw [stackValue_tail hb ha]
apply stackValue_extend
intro i hi
exact hz (i + 1) (by omega)
def integerTapeCode (b d : ℕ) (T : Radix.Tape ℕ) : ℕ × ℕ :=
(stackValue b d T.1, stackValue b d T.2)
theorem integerTapeCode_right {b k : ℕ} (hb : 0 < b) (s : ℕ) (T : Radix.Tape ℕ)
(hR : T.2 0 < b) (hR0 : StackTailZero (k + 1) T.2) :
integerTapeCode b (k + 2) (Radix.writeRight s T) =
(s + b * (integerTapeCode b (k + 1) T).1,
(integerTapeCode b (k + 1) T).2 / b) := by
apply Prod.ext
· rfl
· exact stackValue_pop_padding hb hR hR0 2
theorem integerTapeCode_stay {b k : ℕ} (hb : 0 < b) (s : ℕ) (T : Radix.Tape ℕ)
(hL0 : StackTailZero (k + 1) T.1)
(hR : T.2 0 < b) (hR0 : StackTailZero (k + 1) T.2) :
integerTapeCode b (k + 2) (Radix.writeStay s T) =
((integerTapeCode b (k + 1) T).1,
s + b * ((integerTapeCode b (k + 1) T).2 / b)) := by
apply Prod.ext
· exact stackValue_extend b (k + 1) 1 T.1 hL0
· change s + b * stackValue b (k + 1) (fun i => T.2 (i + 1)) = _
rw [stackValue_pop_padding hb hR hR0 1]
rfl
theorem integerTapeCode_left {b k : ℕ} (hb : 0 < b) (s : ℕ) (T : Radix.Tape ℕ)
(hL : T.1 0 < b) (hL0 : StackTailZero (k + 1) T.1) (hR : T.2 0 < b) :
integerTapeCode b (k + 2) (Radix.writeLeft s T) =
((integerTapeCode b (k + 1) T).1 / b,
(integerTapeCode b (k + 1) T).1 % b +
b * (s + b * ((integerTapeCode b (k + 1) T).2 / b))) := by
apply Prod.ext
· exact stackValue_pop_padding hb hL hL0 2
· change T.1 0 + b * (s + b * stackValue b k (fun i => T.2 (i + 1))) = _
simp only [integerTapeCode, stackValue_head hL, stackValue_tail hb hR]
theorem integerTapeCode_move {b d : ℕ} (hb : 0 < b) (hd : 0 < d)
(s : ℕ) (T : Radix.Tape ℕ) (hL : T.1 0 < b) (hR : T.2 0 < b)
(hL0 : StackTailZero d T.1) (hR0 : StackTailZero d T.2) (m : Fin 3) :
integerTapeCode b (d + 1)
(if m = 0 then Radix.writeLeft s T else
if m = 1 then Radix.writeStay s T else Radix.writeRight s T) =
(nextLeft b (integerTapeCode b d T).1 s m,
nextRight b (integerTapeCode b d T).1 (integerTapeCode b d T).2 s m) := by
obtain ⟨k, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (Nat.ne_of_gt hd)
fin_cases m
· simpa [nextLeft, nextRight, Nat.succ_eq_add_one] using
integerTapeCode_left hb s T hL hL0 hR
· simpa [nextLeft, nextRight, Nat.succ_eq_add_one] using
integerTapeCode_stay hb s T hL0 hR hR0
· simpa [nextLeft, nextRight, Nat.succ_eq_add_one] using
integerTapeCode_right hb s T hR hR0
end ForcedComputation.Lattice
end OAI
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