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import OAI.Computability.MinUncut.PCP.ContextLoad
import OAI.Computability.MinUncut.Machines.MachineUnaryAdd
namespace OAI
namespace MinUncutGames.Foundations.Hastad.SourceAddressArithmetic
open scoped BigOperators
open Target SourceContexts SourceOccurrences
theorem variable_rank (F : Formula) (u : ℕ) (v : VariableContext F u) :
((variableEncoding F u).code v).val =
∑ t : Fin u, (v t).val * F.«variables» ^ t.val := rfl
theorem clause_rank (F : Formula) (u : ℕ) (c : ClauseContext F u) :
((clauseEncoding F u).code c).val =
∑ t : Fin u, (c t).val * F.clauses.length ^ t.val := rfl
theorem left_address (F : Formula) (u : ℕ) (v : VariableContext F u) (f : Cube (I u)) :
((proofEncoding F u).code (.inl (v, f))).val =
(((iEncoding u).function Encoding.bool).code f).val +
2 ^ (2 ^ u) * ((variableEncoding F u).code v).val := rfl
theorem right_address (F : Formula) (u : ℕ) (c : ClauseContext F u) (g : Cube (J u)) :
((proofEncoding F u).code (.inr (.inl (c, g)))).val =
F.«variables» ^ u * 2 ^ (2 ^ u) +
((((jEncoding u).function Encoding.bool).code g).val +
2 ^ (8 ^ u) * ((clauseEncoding F u).code c).val) := rfl
theorem dummy_address (F : Formula) (u : ℕ) :
(dummyIndex F u).val =
F.«variables» ^ u * 2 ^ (2 ^ u) + F.clauses.length ^ u * 2 ^ (8 ^ u) := by
change F.«variables» ^ u * 2 ^ (2 ^ u) +
(F.clauses.length ^ u * 2 ^ (8 ^ u) + 0) = _
omega
theorem left_address_radix (F : Formula) (u : ℕ)
(v : VariableContext F u) (f : Cube (I u)) :
((proofEncoding F u).code (.inl (v, f))).val =
(((iEncoding u).function Encoding.bool).code f).val +
2 ^ (2 ^ u) * ∑ t : Fin u, (v t).val * F.«variables» ^ t.val := by
rw [left_address, variable_rank]
theorem right_address_radix (F : Formula) (u : ℕ)
(c : ClauseContext F u) (g : Cube (J u)) :
((proofEncoding F u).code (.inr (.inl (c, g)))).val =
F.«variables» ^ u * 2 ^ (2 ^ u) +
((((jEncoding u).function Encoding.bool).code g).val +
2 ^ (8 ^ u) * ∑ t : Fin u, (c t).val * F.clauses.length ^ t.val) := by
rw [right_address, clause_rank]
theorem nBits_eq_dummy_succ (F : Formula) (u : ℕ) :
nBits F u = (dummyIndex F u).val + 1 := by
rw [dummy_address, nBits_eq]
omega
end MinUncutGames.Foundations.Hastad.SourceAddressArithmetic
namespace MinUncutGames.Foundations.Hastad.SourceRankPhase
open Turing Complexity
open scoped BigOperators
open Target SourceContexts SourceOccurrences SourceAddressArithmetic
def rank {width : Nat} (radix : Nat) (digits : Fin width → Nat) : Nat :=
∑ i : Fin width, digits i * radix ^ i.val
def reverseDigits {width : Nat} (digits : Fin width → Nat) (i : Nat) : Nat :=
if h : i < width then digits (Fin.rev ⟨i, h⟩) else 0
@[simp] theorem reverseDigits_apply {width : Nat} (digits : Fin width → Nat) (i : Fin width) :
reverseDigits digits i.val = digits i.rev := by
simp [reverseDigits, i.isLt]
theorem horner_value_sum (radix : Nat) (digits : Nat → Nat) (n : Nat) :
MachineHorner.value radix digits n =
∑ i ∈ Finset.range n, digits i * radix ^ (n - 1 - i) := by
induction n with
| zero => simp [MachineHorner.value]
| succ n ih =>
rw [MachineHorner.value, ih, Finset.sum_range_succ, Finset.mul_sum]
simp only [Nat.add_sub_cancel, Nat.sub_self, pow_zero, Nat.mul_one]
congr 1
apply Finset.sum_congr rfl
intro i hi
have hi' := Finset.mem_range.mp hi
rw [show n - i = (n - 1 - i) + 1 by omega, pow_succ]
ac_rfl
theorem horner_reverse_eq_rank {width : Nat} (radix : Nat) (digits : Fin width → Nat) :
MachineHorner.value radix (reverseDigits digits) width = rank radix digits := by
rw [horner_value_sum, Finset.sum_range]
calc
(∑ i : Fin width, reverseDigits digits i.val * radix ^ (width - 1 - i.val)) =
∑ i : Fin width, digits i.rev * radix ^ i.rev.val := by
apply Finset.sum_congr rfl
intro i _
rw [reverseDigits_apply]
have he : width - 1 - i.val = i.rev.val := by
simp only [Fin.rev]
omega
rw [he]
_ = rank radix digits := by
simpa only [rank, Fin.revPerm_apply] using
Equiv.sum_comp (Fin.revPerm (n := width)) (fun i => digits i * radix ^ i.val)
variable {K Λ σ : Type} {width : Nat}
def reverseSlots (slots : MachineHorner.Layout width ↪ K) : MachineHorner.Layout width ↪ K :=
(Equiv.sumCongr (Equiv.refl (Fin 6)) (Fin.revPerm (n := width))).toEmbedding.trans slots
@[simp] theorem reverseSlots_control (slots : MachineHorner.Layout width ↪ K) (i : Fin 6) :
reverseSlots slots (.inl i) = slots (.inl i) := rfl
@[simp] theorem reverseSlots_digit (slots : MachineHorner.Layout width ↪ K) (i : Fin width) :
reverseSlots slots (.inr i) = slots (.inr i.rev) := rfl
variable [DecidableEq K]
omit [DecidableEq K] in
theorem clean_reverseSlots (slots : MachineHorner.Layout width ↪ K) (base : K → List Bool)
(clean : MachineHorner.Clean slots base) : MachineHorner.Clean (reverseSlots slots) base where
accA := by simpa only [reverseSlots_control] using clean.accA
accB := by simpa only [reverseSlots_control] using clean.accB
counter := by simpa only [reverseSlots_control] using clean.counter
scratch := by simpa only [reverseSlots_control] using clean.scratch
def statement (slots : MachineHorner.Layout width ↪ K)
(labels : MachineHorner.Label width → Λ) (exit : Option Λ) :
MachineHorner.Label width → TM2.Stmt (fun _ : K => Bool) Λ (MachineHorner.State σ) :=
MachineHorner.statement (reverseSlots slots) labels exit
def program (slots : MachineHorner.Layout width ↪ K) :
MachineHorner.Label width → TM2.Stmt (fun _ : K => Bool)
(MachineHorner.Label width) (MachineHorner.State σ) :=
statement slots id none
theorem rankTrace (slots : MachineHorner.Layout width ↪ K)
(labels : MachineHorner.Label width → Λ) (exit : Option Λ)
(p : Λ → TM2.Stmt (fun _ : K => Bool) Λ (MachineHorner.State σ))
(atLabels : ∀ label, p (labels label) = statement slots labels exit label)
(base : K → List Bool) (radix : Nat) (digits : Fin width → Nat)
(operandRadix : base (slots (.inl 0)) = encodeWord radix)
(operandDigits : ∀ i, base (slots (.inr i)) = encodeWord (digits i))
(clean : MachineHorner.Clean slots base) (ambient : σ) (register : Option Bool) :
(MachineComposition.advance (TM2.step p))^[
MachineHorner.steps radix (reverseDigits digits) width]
(some ⟨some (labels .start), ((ambient, ()), register), base⟩) =
some ⟨exit, ((ambient, ()), none), MachineHorner.resultTapes slots base (rank radix digits)⟩ := by
have h := MachineHorner.hornerTrace (reverseSlots slots) labels exit p atLabels base radix
(reverseDigits digits) operandRadix
(fun i => by simpa only [reverseSlots_digit, reverseDigits_apply] using operandDigits i.rev)
(clean_reverseSlots slots base clean) ambient register
rw [horner_reverse_eq_rank] at h
simpa only [MachineHorner.resultTapes, reverseSlots_control] using h
def rankInTime (slots : MachineHorner.Layout width ↪ K)
(labels : MachineHorner.Label width → Λ) (exit : Option Λ)
(p : Λ → TM2.Stmt (fun _ : K => Bool) Λ (MachineHorner.State σ))
(atLabels : ∀ label, p (labels label) = statement slots labels exit label)
(base : K → List Bool) (radix : Nat) (digits : Fin width → Nat)
(operandRadix : base (slots (.inl 0)) = encodeWord radix)
(operandDigits : ∀ i, base (slots (.inr i)) = encodeWord (digits i))
(clean : MachineHorner.Clean slots base) (ambient : σ) (register : Option Bool) :
StateTransition.EvalsToInTime (TM2.step p)
⟨some (labels .start), ((ambient, ()), register), base⟩
(some ⟨exit, ((ambient, ()), none), MachineHorner.resultTapes slots base (rank radix digits)⟩)
(MachineHorner.steps radix (reverseDigits digits) width) where
steps := MachineHorner.steps radix (reverseDigits digits) width
evals_in_steps := rankTrace slots labels exit p atLabels base radix digits operandRadix
operandDigits clean ambient register
steps_le_m := Nat.le_refl _
def rankInPolynomialTime (slots : MachineHorner.Layout width ↪ K)
(labels : MachineHorner.Label width → Λ) (exit : Option Λ)
(p : Λ → TM2.Stmt (fun _ : K => Bool) Λ (MachineHorner.State σ))
(atLabels : ∀ label, p (labels label) = statement slots labels exit label)
(base : K → List Bool) (radix : Nat) (digits : Fin width → Nat)
(operandRadix : base (slots (.inl 0)) = encodeWord radix)
(operandDigits : ∀ i, base (slots (.inr i)) = encodeWord (digits i))
(clean : MachineHorner.Clean slots base) (ambient : σ) (register : Option Bool)
(magnitude : Nat) (radixBound : radix ≤ magnitude) (digitBound : ∀ i, digits i ≤ magnitude) :
StateTransition.EvalsToInTime (TM2.step p)
⟨some (labels .start), ((ambient, ()), register), base⟩
(some ⟨exit, ((ambient, ()), none), MachineHorner.resultTapes slots base (rank radix digits)⟩)
((MachineHorner.timePolynomial width).eval magnitude) where
steps := MachineHorner.steps radix (reverseDigits digits) width
evals_in_steps := rankTrace slots labels exit p atLabels base radix digits operandRadix
operandDigits clean ambient register
steps_le_m := MachineHorner.steps_le_timePolynomial radix (reverseDigits digits) width magnitude
radixBound (fun i hi => by simpa only [reverseDigits, dite_eq_left hi] using digitBound (Fin.rev ⟨i, hi⟩))
def variableRankInTime (F : Formula) (u : Nat) (v : VariableContext F u)
(slots : MachineHorner.Layout u ↪ K) (labels : MachineHorner.Label u → Λ) (exit : Option Λ)
(p : Λ → TM2.Stmt (fun _ : K => Bool) Λ (MachineHorner.State σ))
(atLabels : ∀ label, p (labels label) = statement slots labels exit label)
(base : K → List Bool) (operandRadix : base (slots (.inl 0)) = encodeWord F.«variables»)
(operandDigits : ∀ i, base (slots (.inr i)) = encodeWord (v i).val)
(clean : MachineHorner.Clean slots base) (ambient : σ) (register : Option Bool) :
StateTransition.EvalsToInTime (TM2.step p)
⟨some (labels .start), ((ambient, ()), register), base⟩
(some ⟨exit, ((ambient, ()), none),
MachineHorner.resultTapes slots base (((variableEncoding F u).code v).val)⟩)
((MachineHorner.timePolynomial u).eval F.«variables») := by
have h := rankInPolynomialTime slots labels exit p atLabels base F.«variables» (fun i => (v i).val)
operandRadix operandDigits clean ambient register F.«variables» (Nat.le_refl _) (fun i => (v i).isLt.le)
simpa only [rank, ← variable_rank F u v] using h
def clauseRankInTime (F : Formula) (u : Nat) (c : ClauseContext F u)
(slots : MachineHorner.Layout u ↪ K) (labels : MachineHorner.Label u → Λ) (exit : Option Λ)
(p : Λ → TM2.Stmt (fun _ : K => Bool) Λ (MachineHorner.State σ))
(atLabels : ∀ label, p (labels label) = statement slots labels exit label)
(base : K → List Bool) (operandRadix : base (slots (.inl 0)) = encodeWord F.clauses.length)
(operandDigits : ∀ i, base (slots (.inr i)) = encodeWord (c i).val)
(clean : MachineHorner.Clean slots base) (ambient : σ) (register : Option Bool) :
StateTransition.EvalsToInTime (TM2.step p)
⟨some (labels .start), ((ambient, ()), register), base⟩
(some ⟨exit, ((ambient, ()), none),
MachineHorner.resultTapes slots base (((clauseEncoding F u).code c).val)⟩)
((MachineHorner.timePolynomial u).eval F.clauses.length) := by
have h := rankInPolynomialTime slots labels exit p atLabels base F.clauses.length (fun i => (c i).val)
operandRadix operandDigits clean ambient register F.clauses.length (Nat.le_refl _) (fun i => (c i).isLt.le)
simpa only [rank, ← clause_rank F u c] using h
def programInTime (slots : MachineHorner.Layout width ↪ K)
(base : K → List Bool) (radix : Nat) (digits : Fin width → Nat)
(operandRadix : base (slots (.inl 0)) = encodeWord radix)
(operandDigits : ∀ i, base (slots (.inr i)) = encodeWord (digits i))
(clean : MachineHorner.Clean slots base) (ambient : σ) (register : Option Bool) :
StateTransition.EvalsToInTime (TM2.step (program (σ := σ) slots))
⟨some .start, ((ambient, ()), register), base⟩
(some ⟨none, ((ambient, ()), none), MachineHorner.resultTapes slots base (rank radix digits)⟩)
(MachineHorner.steps radix (reverseDigits digits) width) :=
rankInTime slots id none (program slots) (fun _ => rfl) base radix digits operandRadix
operandDigits clean ambient register
def machine (width : Nat) : FinTM2 where
K := MachineHorner.Layout width
k₀ := .inl 0
k₁ := .inl 3
Γ _ := Bool
Λ := MachineHorner.Label width
main := .start
σ := MachineHorner.State Unit
initialState := (((), ()), none)
m := program (Function.Embedding.refl (MachineHorner.Layout width))
def machineInTime (width radix : Nat) (digits : Fin width → Nat)
(base : MachineHorner.Layout width → List Bool)
(operandRadix : base (.inl 0) = encodeWord radix)
(operandDigits : ∀ i, base (.inr i) = encodeWord (digits i))
(clean : MachineHorner.Clean (Function.Embedding.refl _) base) (register : Option Bool) :
StateTransition.EvalsToInTime (machine width).step
⟨some (MachineHorner.Label.start), (((), ()), register), base⟩
(some ⟨none, (((), ()), none),
MachineHorner.resultTapes (Function.Embedding.refl _) base (rank radix digits)⟩)
(MachineHorner.steps radix (reverseDigits digits) width) :=
programInTime (Function.Embedding.refl _) base radix digits operandRadix operandDigits clean () register
variable {u : Nat} {Extra : Type}
def clauseSlots (controls : Fin 6 ↪ Extra) :
MachineHorner.Layout u ↪ SourceContextLoad.Tape u Extra where
toFun
| .inl i => .extra (controls i)
| .inr i => .current i
inj' := by
intro a b h
cases a with
| inl a =>
cases b with
| inl b => exact congrArg Sum.inl (controls.injective (SourceContextLoad.Tape.extra.inj h))
| inr b => cases h
| inr a =>
cases b with
| inl b => cases h
| inr b => exact congrArg Sum.inr (SourceContextLoad.Tape.current.inj h)
def variableSlots (controls : Fin 6 ↪ Extra) (selected : SlotContext u) :
MachineHorner.Layout u ↪ SourceContextLoad.Tape u Extra where
toFun
| .inl i => .extra (controls i)
| .inr i => SourceContextLoad.variableField i (slotEncoding.code (selected i))
inj' := by
intro a b h
cases a with
| inl a =>
cases b with
| inl b => exact congrArg Sum.inl (controls.injective (SourceContextLoad.Tape.extra.inj h))
| inr b => cases h
| inr a =>
cases b with
| inl b => cases h
| inr b => exact congrArg Sum.inr (SourceContextLoad.Tape.field.inj h).1
@[simp] theorem clauseSlots_control (controls : Fin 6 ↪ Extra) (i : Fin 6) :
clauseSlots (u := u) controls (.inl i) = .extra (controls i) := rfl
@[simp] theorem clauseSlots_digit (controls : Fin 6 ↪ Extra) (i : Fin u) :
clauseSlots controls (.inr i) = .current i := rfl
@[simp] theorem variableSlots_control (controls : Fin 6 ↪ Extra) (selected : SlotContext u) (i : Fin 6) :
variableSlots controls selected (.inl i) = .extra (controls i) := rfl
@[simp] theorem variableSlots_digit (controls : Fin 6 ↪ Extra) (selected : SlotContext u) (i : Fin u) :
variableSlots controls selected (.inr i) =
SourceContextLoad.variableField i (slotEncoding.code (selected i)) := rfl
theorem selectedDigits_loaded (F : Formula) (c : ClauseContext F u) (selected : SlotContext u)
(controls : Fin 6 ↪ Extra) (base : SourceContextLoad.Tape u Extra → List Bool)
(empty : ∀ i, base (SourceContextLoad.variableField i (slotEncoding.code (selected i))) = [])
(i : Fin u) :
SourceContextLoad.stageTapes F c base u (variableSlots controls selected (.inr i)) =
encodeWord (sampledVariables F c selected i).val := by
rw [variableSlots_digit]
erw [SourceContextLoad.output_variable, empty i, List.append_nil]
congr 1
cases hs : selected i <;> simp [sampledVariables, PCP.nameAt, PCP.clauseAt, slotEncoding, hs] <;> rfl
end MinUncutGames.Foundations.Hastad.SourceRankPhase
namespace MinUncutGames.Foundations.Hastad.SourceAddressDescriptors
open Target SourceContexts SourceOccurrences SourceLocalEquation SourceLocalSignature
open SourceAddressArithmetic
open MinUncutGames.Reduction.CloneGap
abbrev Descriptor := Fin 3 × ℕ
def descriptor (u : ℕ) : LocalKey u → Descriptor
| .inl f => (0, (((iEncoding u).function Encoding.bool).code f).val)
| .inr (.inl g) => (1, (((jEncoding u).function Encoding.bool).code g).val)
| .inr (.inr _) => (2, 0)
def baseValue (F : Formula) (u : ℕ) (c : ClauseContext F u)
(v : VariableContext F u) (side : Fin 3) : ℕ :=
if side = 0 then 2 ^ (2 ^ u) * ((variableEncoding F u).code v).val
else if side = 1 then F.«variables» ^ u * 2 ^ (2 ^ u) +
2 ^ (8 ^ u) * ((clauseEncoding F u).code c).val
else F.«variables» ^ u * 2 ^ (2 ^ u) + F.clauses.length ^ u * 2 ^ (8 ^ u)
def realize (base : Fin 3 → ℕ) (d : Descriptor) : ℕ := d.2 + base d.1
def words (base : Fin 3 → ℕ) (e : Equation Descriptor) : List ℕ :=
[realize base e.first, realize base e.second, realize base e.third,
if e.rhs then 1 else 0]
theorem realize_descriptor (F : Formula) (u : ℕ) (c : ClauseContext F u)
(v : VariableContext F u) (key : LocalKey u) :
realize (baseValue F u c v) (descriptor u key) = (addressMap F u c v key).val := by
cases key with
| inl f =>
simpa [realize, descriptor, baseValue, addressMap] using
(left_address F u v f).symm
| inr key =>
cases key with
| inl g =>
simp [realize, descriptor, baseValue, addressMap, right_address]
omega
| inr _dummy =>
simp only [realize, descriptor, baseValue, show (2 : Fin 3) ≠ 0 by decide,
show (2 : Fin 3) ≠ 1 by decide, ite_false, Nat.zero_add, addressMap,
dummy_address]
theorem descriptor_offset_bound (u : ℕ) (key : LocalKey u) :
(descriptor u key).2 < 2 ^ (2 ^ u) + 2 ^ (8 ^ u) + 1 := by
cases key with
| inl f =>
have h := (((iEncoding u).function Encoding.bool).code f).isLt
change _ < 2 ^ (2 ^ u) at h
change (((iEncoding u).function Encoding.bool).code f).val < _
calc
_ < (2 ^ (2 ^ u) : ℕ) := h
_ ≤ 2 ^ (2 ^ u) + 2 ^ (8 ^ u) := Nat.le_add_right _ _
_ ≤ _ := Nat.le_add_right _ _
| inr key =>
cases key with
| inl g =>
have h := (((jEncoding u).function Encoding.bool).code g).isLt
change _ < 2 ^ (8 ^ u) at h
change (((jEncoding u).function Encoding.bool).code g).val < _
calc
_ < (2 ^ (8 ^ u) : ℕ) := h
_ ≤ 2 ^ (2 ^ u) + 2 ^ (8 ^ u) := Nat.le_add_left _ _
_ ≤ _ := Nat.le_add_right _ _
| inr _dummy => simp [descriptor]
theorem sourceEquation_numeric (F : Formula) (u D : ℕ) (p : SourceIndex F u D) :
mapEquation Fin.val (sourceEquation F u D p) =
mapEquation (realize (baseValue F u p.1.1 (sampledVariables F p.1.1 p.1.2)))
(mapEquation (descriptor u) (emit u D (ofContext F p.1.1 p.1.2, p.2))) := by
rw [sourceEquation_eq_map_emit, mapEquation_comp, mapEquation_comp]
congr 1
funext key
exact (realize_descriptor F u p.1.1 (sampledVariables F p.1.1 p.1.2) key).symm
theorem sourceEquation_words (F : Formula) (u D : ℕ) (p : SourceIndex F u D) :
MinUncutGames.Reduction.SourceEncoding.equationWords (sourceEquation F u D p) =
words (baseValue F u p.1.1 (sampledVariables F p.1.1 p.1.2))
(mapEquation (descriptor u) (emit u D (ofContext F p.1.1 p.1.2, p.2))) := by
rw [sourceEquation_eq_map_emit]
simp only [MinUncutGames.Reduction.SourceEncoding.equationWords, words, mapEquation]
rw [realize_descriptor, realize_descriptor, realize_descriptor]
rfl
theorem baseValue_lt (F : Formula) (u : ℕ) (c : ClauseContext F u)
(v : VariableContext F u) (side : Fin 3) :
baseValue F u c v side < nBits F u := by
have hside : side = 0 ∨ side = 1 ∨ side = 2 := by omega
rcases hside with rfl | rfl | rfl
· have h := (addressMap F u c v (.inl (fun _ => false))).isLt
rw [← realize_descriptor] at h
change _ + baseValue F u c v 0 < _ at h
exact (Nat.le_add_left _ _).trans_lt h
· have h := (addressMap F u c v (.inr (.inl (fun _ => false)))).isLt
rw [← realize_descriptor] at h
change _ + baseValue F u c v 1 < _ at h
exact (Nat.le_add_left _ _).trans_lt h
· have h := (addressMap F u c v (.inr (.inr ()))).isLt
rw [← realize_descriptor] at h
simpa only [realize, descriptor, Nat.zero_add] using h
theorem equation_emit_steps_le (F : Formula) (u : ℕ) (c : ClauseContext F u)
(v : VariableContext F u) (e : Equation Descriptor) :
2 * (baseValue F u c v e.first.1 + baseValue F u c v e.second.1 +
baseValue F u c v e.third.1) + 16 ≤ 6 * nBits F u + 16 := by
have h₁ := (baseValue_lt F u c v e.first.1).le
have h₂ := (baseValue_lt F u c v e.second.1).le
have h₃ := (baseValue_lt F u c v e.third.1).le
omega
end MinUncutGames.Foundations.Hastad.SourceAddressDescriptors
end OAI
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