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import OAI.Computability.BinPacking.PCP.RawInitialMachineLoopData
namespace OAI
namespace BinPackingGames.Foundations.PCP.RawInitialMachineReadClause
open Turing Complexity Hastad RawInitialMachineModel
def fieldValue {n : ℕ} (c : Target.Clause n) (j : Fin 6) : ℕ :=
(Complexity.clauseWords c)[j.val]'(by
rw [Complexity.clauseWords_length]
exact j.isLt)
def recordTapes {n : ℕ} (base : Tape → List Bool) (c : Target.Clause n)
(suffix : List Bool) : Tape → List Bool
| .input => suffix
| .field j => encodeWord (fieldValue c j)
| k => base k
@[simp] theorem recordTapes_input {n : ℕ} (base : Tape → List Bool)
(c : Target.Clause n) (suffix : List Bool) :
recordTapes base c suffix .input = suffix := rfl
@[simp] theorem recordTapes_field {n : ℕ} (base : Tape → List Bool)
(c : Target.Clause n) (suffix : List Bool) (j : Fin 6) :
recordTapes base c suffix (.field j) = encodeWord (fieldValue c j) := rfl
theorem recordTapes_other {n : ℕ} (base : Tape → List Bool)
(c : Target.Clause n) (suffix : List Bool) (k : Tape)
(hi : k ≠ .input) (hf : ∀ j, k ≠ .field j) :
recordTapes base c suffix k = base k := by
cases k <;> try rfl
· exact (hi rfl).elim
· exact (hf _ rfl).elim
@[simp] theorem recordTapes_name {n : ℕ} (base : Tape → List Bool)
(c : Target.Clause n) (suffix : List Bool) (j : Fin 3) :
recordTapes base c suffix (.field (nameField j)) =
encodeWord ((c)[j].variableIndex.val) := by
fin_cases j <;> rfl
private theorem trace_trans {α : Type*} (f : α → α) {a b : ℕ} {x y z : α}
(first : f^[a] x = y) (second : f^[b] y = z) : f^[a + b] x = z := by
rw [Nat.add_comm, Function.iterate_add_apply, first, second]
theorem fieldsTrace {n : ℕ} (base : Tape → List Bool) (c : Target.Clause n)
(suffix : List Bool)
(hinput : base .input = encodeWords (Complexity.clauseWords c) ++ suffix)
(hempty : ∀ j, base (.field j) = []) (state : State) :
(MachineComposition.advance (TM2.step program))^[
(encodeWords (Complexity.clauseWords c)).length + 6]
(some ⟨some (.fieldStart 0), state, base⟩) =
some ⟨some .loadSigns, (state.1, none), recordTapes base c suffix⟩ := by
let a := fieldValue c 0
let b := fieldValue c 1
let d := fieldValue c 2
let e := fieldValue c 3
let f := fieldValue c 4
let g := fieldValue c 5
have hwords : Complexity.clauseWords c = [a, b, d, e, f, g] := rfl
have hin : base .input = encodeWord a ++ (encodeWords [b, d, e, f, g] ++ suffix) := by
rw [hinput, hwords]
simp only [encodeWords, List.append_assoc]
let t1 := SourceMachine.afterField .input (.field 0) base a
(encodeWords [b, d, e, f, g] ++ suffix)
let t2 := SourceMachine.afterField .input (.field 1) t1 b
(encodeWords [d, e, f, g] ++ suffix)
let t3 := SourceMachine.afterField .input (.field 2) t2 d
(encodeWords [e, f, g] ++ suffix)
let t4 := SourceMachine.afterField .input (.field 3) t3 e
(encodeWords [f, g] ++ suffix)
let t5 := SourceMachine.afterField .input (.field 4) t4 f
(encodeWords [g] ++ suffix)
let t6 := SourceMachine.afterField .input (.field 5) t5 g suffix
have h1 := (SourceMachine.fieldInTime .input (.field 0) (by decide)
(.fieldStart 0) (.fieldLoop 0) (some (.fieldStart 1)) program rfl rfl
base a (encodeWords [b, d, e, f, g] ++ suffix) hin state.1 state.2).evals_in_steps
change (MachineComposition.advance (TM2.step program))^[a + 2]
(some ⟨some (.fieldStart 0), state, base⟩) =
some ⟨some (.fieldStart 1), (state.1, none), t1⟩ at h1
have h2 := (SourceMachine.fieldInTime .input (.field 1) (by decide)
(.fieldStart 1) (.fieldLoop 1) (some (.fieldStart 2)) program rfl rfl
t1 b (encodeWords [d, e, f, g] ++ suffix)
(by simp [t1, SourceMachine.afterField, encodeWords, List.append_assoc])
state.1 none).evals_in_steps
change (MachineComposition.advance (TM2.step program))^[b + 2]
(some ⟨some (.fieldStart 1), (state.1, none), t1⟩) =
some ⟨some (.fieldStart 2), (state.1, none), t2⟩ at h2
have h3 := (SourceMachine.fieldInTime .input (.field 2) (by decide)
(.fieldStart 2) (.fieldLoop 2) (some (.fieldStart 3)) program rfl rfl
t2 d (encodeWords [e, f, g] ++ suffix)
(by simp [t2, SourceMachine.afterField, encodeWords, List.append_assoc])
state.1 none).evals_in_steps
change (MachineComposition.advance (TM2.step program))^[d + 2]
(some ⟨some (.fieldStart 2), (state.1, none), t2⟩) =
some ⟨some (.fieldStart 3), (state.1, none), t3⟩ at h3
have h4 := (SourceMachine.fieldInTime .input (.field 3) (by decide)
(.fieldStart 3) (.fieldLoop 3) (some (.fieldStart 4)) program rfl rfl
t3 e (encodeWords [f, g] ++ suffix)
(by simp [t3, SourceMachine.afterField, encodeWords, List.append_assoc])
state.1 none).evals_in_steps
change (MachineComposition.advance (TM2.step program))^[e + 2]
(some ⟨some (.fieldStart 3), (state.1, none), t3⟩) =
some ⟨some (.fieldStart 4), (state.1, none), t4⟩ at h4
have h5 := (SourceMachine.fieldInTime .input (.field 4) (by decide)
(.fieldStart 4) (.fieldLoop 4) (some (.fieldStart 5)) program rfl rfl
t4 f (encodeWords [g] ++ suffix)
(by simp [t4, SourceMachine.afterField, encodeWords, List.append_assoc])
state.1 none).evals_in_steps
change (MachineComposition.advance (TM2.step program))^[f + 2]
(some ⟨some (.fieldStart 4), (state.1, none), t4⟩) =
some ⟨some (.fieldStart 5), (state.1, none), t5⟩ at h5
have h6 := (SourceMachine.fieldInTime .input (.field 5) (by decide)
(.fieldStart 5) (.fieldLoop 5) (some .loadSigns) program rfl rfl
t5 g suffix
(by simp [t5, SourceMachine.afterField, encodeWords])
state.1 none).evals_in_steps
change (MachineComposition.advance (TM2.step program))^[g + 2]
(some ⟨some (.fieldStart 5), (state.1, none), t5⟩) =
some ⟨some .loadSigns, (state.1, none), t6⟩ at h6
have total := trace_trans _ (trace_trans _ (trace_trans _
(trace_trans _ (trace_trans _ h1 h2) h3) h4) h5) h6
have htime : (((((a + 2) + (b + 2)) + (d + 2)) + (e + 2)) + (f + 2)) + (g + 2) =
(encodeWords (Complexity.clauseWords c)).length + 6 := by
rw [hwords, encodeWords_length]
simp
omega
rw [htime] at total
have frame : t6 = recordTapes base c suffix := by
funext k
cases k with
| field j =>
fin_cases j <;>
simp [t6, t5, t4, t3, t2, t1, SourceMachine.afterField, SourceMachine.fieldTapes,
recordTapes, hempty, a, b, d, e, f, g]
| _ =>
simp [t6, t5, t4, t3, t2, t1, SourceMachine.afterField, SourceMachine.fieldTapes,
recordTapes]
rw [frame] at total
exact total
theorem readClauseTrace {n : ℕ} (base : Tape → List Bool) (c : Target.Clause n)
(suffix : List Bool)
(hinput : base .input = encodeWords (Complexity.clauseWords c) ++ suffix)
(hempty : ∀ j, base (.field j) = []) (state : State) :
(MachineComposition.advance (TM2.step program))^[
(encodeWords (Complexity.clauseWords c)).length + 7]
(some ⟨some (.fieldStart 0), state, base⟩) =
some ⟨some (.scan (.tailVariables 0)), (RawInitialRows.clauseSigns c, none),
recordTapes base c suffix⟩ := by
have first := fieldsTrace base c suffix hinput hempty state
have last := RawInitialMachinePhases.loadSigns_step (recordTapes base c suffix)
(c)[0].positive (c)[1].positive (c)[2].positive rfl rfl rfl (state.1, none)
have lastTrace : (MachineComposition.advance (TM2.step program))^[1]
(some ⟨some .loadSigns, (state.1, none), recordTapes base c suffix⟩) =
some ⟨some (.scan (.tailVariables 0)), (RawInitialRows.clauseSigns c, none),
recordTapes base c suffix⟩ := last
have total := trace_trans _ first lastTrace
simpa only [show (encodeWords (Complexity.clauseWords c)).length + 6 + 1 =
(encodeWords (Complexity.clauseWords c)).length + 7 by omega] using total
def readClauseInTime {n : ℕ} (base : Tape → List Bool) (c : Target.Clause n)
(suffix : List Bool)
(hinput : base .input = encodeWords (Complexity.clauseWords c) ++ suffix)
(hempty : ∀ j, base (.field j) = []) (state : State) :
StateTransition.EvalsToInTime (TM2.step program)
⟨some (.fieldStart 0), state, base⟩
(some ⟨some (.scan (.tailVariables 0)), (RawInitialRows.clauseSigns c, none),
recordTapes base c suffix⟩)
((encodeWords (Complexity.clauseWords c)).length + 7) where
steps := (encodeWords (Complexity.clauseWords c)).length + 7
evals_in_steps := readClauseTrace base c suffix hinput hempty state
steps_le_m := Nat.le_refl _
end BinPackingGames.Foundations.PCP.RawInitialMachineReadClause
end OAI
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