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import OAI.Computability.PerfectCompleteness.Decoding.CutProjectionGeometry
import OAI.Computability.PerfectCompleteness.Foundations.OriginalTerminalSplitLemmas
import OAI.Computability.PerfectCompleteness.Sampling.CutSamplerReplayGroupingTransport
namespace OAI
section
namespace PerfectCompleteness.CutGroupedProjection
open PointwiseSpaces RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays
open RecursiveSampler TerminalCalls
open scoped BigOperators Classical
abbrev F2 := ZMod 2
noncomputable section
variable {branch : Nat → Nat} {n m t : Nat}
def cutProjection (p : Path branch n (m + 1))
{left right : Slots branch n → Fin t → MixedSupport.Slot}
(q : ∀ s k, MixedSupport.Projection (left s k) (right s k)) :
∀ s k, MixedSupport.Projection
(WholeCutGrouping.cutSlots p left s k) (WholeCutGrouping.cutSlots p right s k) :=
fun s k => q (p.slotEmbedding s) k
def terminalPullback (repeats : Nat → Nat) (p : Path branch n (m + 1))
{left right : Slots branch n → Fin t → MixedSupport.Slot}
(q : ∀ s k, MixedSupport.Projection (left s k) (right s k))
(terminal : CutTerminalSplit.TerminalTape F2 repeats p (LeafDomain right)) :
CutTerminalSplit.TerminalTape F2 repeats p (LeafDomain left) :=
fun call child => ChildBlockProjection.squarePullback
(ChildAssemblyProjection.childProjection (cutProjection p q) child) (terminal call child)
def factorPullback (repeats : Nat → Nat) :
{n m : Nat} → (p : Path branch n (m + 1)) →
(left right : Slots branch n → Fin t → MixedSupport.Slot) →
(∀ s k, MixedSupport.Projection (left s k) (right s k)) →
(j : DrawIndex repeats p) →
CutSamplerRefinement.RefinedSpace F2 repeats p (LeafDomain right) j →
CutSamplerRefinement.RefinedSpace F2 repeats p (LeafDomain left) j
| _ + 1, _, .refl _, _, _, q, _, x =>
fun child => ChildBlockProjection.squarePullback
(ChildAssemblyProjection.childProjection q child) (x child)
| _ + 1, _, .step i p, left, right, q, j, x =>
match j with
| Sum.inl j =>
ChildBlockProjection.squarePullback (ChildAssemblyProjection.childProjection q j.val) x
| Sum.inr j =>
factorPullback repeats p (childSlots left i) (childSlots right i)
(ChildAssemblyProjection.childProjection q i) j.2 x
def tapePullback (repeats : Nat → Nat) (p : Path branch n (m + 1))
(left right : Slots branch n → Fin t → MixedSupport.Slot)
(q : ∀ s k, MixedSupport.Projection (left s k) (right s k))
(tape : CutSamplerRefinement.Tape F2 repeats p (LeafDomain right)) :
CutSamplerRefinement.Tape F2 repeats p (LeafDomain left) :=
fun j => factorPullback repeats p left right q j (tape j)
theorem evaluate_tapePullback (repeats : Nat → Nat) (p : Path branch n (m + 1)) :
∀ (left right : Slots branch n → Fin t → MixedSupport.Slot)
(q : ∀ s k, MixedSupport.Projection (left s k) (right s k))
(tape : CutSamplerRefinement.Tape F2 repeats p (LeafDomain right)) (x : Domain left),
(CutSamplerRefinement.evaluate F2 repeats p (LeafDomain left)
(tapePullback repeats p left right q tape)).val x =
(CutSamplerRefinement.evaluate F2 repeats p (LeafDomain right) tape).val
(sourceProjection q x) := by
induction n generalizing m with
| zero =>
have h := p.height_le
omega
| succ n ih =>
cases p with
| refl =>
intro left right q tape x
exact congrArg (fun f : H left => f.val x)
(ChildAssemblyProjection.recursiveSum_pullback q (tape ()))
| step i p =>
intro left right q tape x
simp only [CutSamplerLocality.evaluate_step, RecursiveSampler.combine,
RecursiveSampler.combineFunction, Finset.sum_apply, Pi.add_apply, Pi.mul_apply,
PointwiseSpaces.pullback_apply]
change
(∑ j : OffPath i,
(ChildBlockProjection.squarePullback (ChildAssemblyProjection.childProjection q j.val)
(tape (.inl j))).val (restrictChild left j.val x)) +
(∑ h : Fin (repeats (n + 1)),
(CutSamplerRefinement.evaluate F2 repeats p (LeafDomain (childSlots left i))
(tapePullback repeats p (childSlots left i) (childSlots right i)
(ChildAssemblyProjection.childProjection q i)
(CutSamplerLocality.subTape F2 repeats i p (LeafDomain right) tape h false))).val
(restrictChild left i x) *
(CutSamplerRefinement.evaluate F2 repeats p (LeafDomain (childSlots left i))
(tapePullback repeats p (childSlots left i) (childSlots right i)
(ChildAssemblyProjection.childProjection q i)
(CutSamplerLocality.subTape F2 repeats i p (LeafDomain right) tape h true))).val
(restrictChild left i x)) =
(∑ j : OffPath i, (tape (.inl j)).val (restrictChild right j.val (sourceProjection q x))) +
(∑ h : Fin (repeats (n + 1)),
(CutSamplerRefinement.evaluate F2 repeats p (LeafDomain (childSlots right i))
(CutSamplerLocality.subTape F2 repeats i p (LeafDomain right) tape h false)).val
(restrictChild right i (sourceProjection q x)) *
(CutSamplerRefinement.evaluate F2 repeats p (LeafDomain (childSlots right i))
(CutSamplerLocality.subTape F2 repeats i p (LeafDomain right) tape h true)).val
(restrictChild right i (sourceProjection q x)))
apply congrArg₂ (fun a b : F2 => a + b)
· exact Finset.sum_congr rfl (fun _ _ => rfl)
· apply Finset.sum_congr rfl
intro h _
apply congrArg₂ (fun a b : F2 => a * b)
· exact ih p (childSlots left i) (childSlots right i)
(ChildAssemblyProjection.childProjection q i)
(CutSamplerLocality.subTape F2 repeats i p (LeafDomain right) tape h false)
(restrictChild left i x)
· exact ih p (childSlots left i) (childSlots right i)
(ChildAssemblyProjection.childProjection q i)
(CutSamplerLocality.subTape F2 repeats i p (LeafDomain right) tape h true)
(restrictChild left i x)
theorem terminalFactor_pullback (repeats : Nat → Nat) (p : Path branch n (m + 1)) :
∀ (left right : Slots branch n → Fin t → MixedSupport.Slot)
(q : ∀ s k, MixedSupport.Projection (left s k) (right s k))
(call : TerminalIndex repeats p)
(x : CutSamplerRefinement.RefinedSpace F2 repeats p (LeafDomain right)
(drawIndex repeats p call)),
CutTerminalSplit.terminalFactorEquiv F2 repeats p (LeafDomain left) call
(factorPullback repeats p left right q (drawIndex repeats p call) x) =
fun child => ChildBlockProjection.squarePullback
(ChildAssemblyProjection.childProjection (cutProjection p q) child)
(CutTerminalSplit.terminalFactorEquiv F2 repeats p (LeafDomain right) call x child) := by
induction n generalizing m with
| zero =>
have h := p.height_le
omega
| succ n ih =>
cases p with
| refl =>
intro left right q call x
cases call
rfl
| step i p =>
intro left right q call x
exact ih p (childSlots left i) (childSlots right i)
(ChildAssemblyProjection.childProjection q i) call.2 x
theorem exteriorFactor_pullback (repeats : Nat → Nat) (p : Path branch n (m + 1)) :
∀ (left right : Slots branch n → Fin t → MixedSupport.Slot)
(q : ∀ s k, MixedSupport.Projection (left s k) (right s k))
(hs : ∀ s, WholeCutExteriorTransport.Outside p s → left s = right s)
(_hk : ∀ s (_hout : WholeCutExteriorTransport.Outside p s) k,
HEq (q s k) (MixedSupport.Projection.keep (left s k)))
(j : DrawIndex repeats p) (hj : ¬ IsTerminal repeats p j)
(x : CutSamplerRefinement.RefinedSpace F2 repeats p (LeafDomain right) j),
factorPullback repeats p left right q j x =
cast (WholeCutExteriorTransport.factor_eq repeats p right left
(fun s h => (hs s h).symm) j hj) x := by
induction n generalizing m with
| zero =>
have h := p.height_le
omega
| succ n ih =>
cases p with
| refl =>
intro left right q hs hk j hj x
exact False.elim (hj True.intro)
| step i p =>
intro left right q hs hk j hj x
cases j with
| inl j =>
exact CutProjectionGeometry.squarePullback_eq_cast
(childSlots left j.val) (childSlots right j.val)
(ChildAssemblyProjection.childProjection q j.val)
(WholeCutExteriorTransport.ordinarySlots_eq i p left right hs j)
(fun s k => hk (j.val, s) (WholeCutExteriorTransport.outside_ordinary i p j s) k) x
| inr j =>
exact ih p (childSlots left i) (childSlots right i)
(ChildAssemblyProjection.childProjection q i)
(fun s h => hs (i, s) (WholeCutExteriorTransport.outside_selected i p s h))
(fun s h k => hk (i, s) (WholeCutExteriorTransport.outside_selected i p s h) k)
j.2 hj x
theorem splitTape_tapePullback (repeats : Nat → Nat) (p : Path branch n (m + 1))
(left right : Slots branch n → Fin t → MixedSupport.Slot)
(q : ∀ s k, MixedSupport.Projection (left s k) (right s k))
(hs : ∀ s, WholeCutExteriorTransport.Outside p s → left s = right s)
(hk : ∀ s (_hout : WholeCutExteriorTransport.Outside p s) k,
HEq (q s k) (MixedSupport.Projection.keep (left s k)))
(tape : CutSamplerRefinement.Tape F2 repeats p (LeafDomain right)) :
CutTerminalSplit.splitTape F2 repeats p (LeafDomain left)
(tapePullback repeats p left right q tape) =
(terminalPullback repeats p q (CutTerminalSplit.splitTape F2 repeats p (LeafDomain right) tape).1,
cast (WholeCutExteriorTransport.scalarExterior_eq repeats p right left
(fun s h => (hs s h).symm))
(CutTerminalSplit.splitTape F2 repeats p (LeafDomain right) tape).2) := by
apply Prod.ext
· funext call
exact terminalFactor_pullback repeats p left right q call (tape (drawIndex repeats p call))
· funext j
exact (exteriorFactor_pullback repeats p left right q hs hk j.val j.property (tape j.val)).trans
(CutSamplerReplayGroupingTransport.cast_pi_apply
(WholeCutExteriorTransport.scalarExterior_eq repeats p right left
(fun s h => (hs s h).symm))
(fun j => WholeCutExteriorTransport.factor_eq repeats p right left
(fun s h => (hs s h).symm) j.val j.property)
(CutTerminalSplit.splitTape F2 repeats p (LeafDomain right) tape).2 j).symm
theorem evaluate_rebuild (repeats : Nat → Nat) (p : Path branch n (m + 1))
(left right : Slots branch n → Fin t → MixedSupport.Slot)
(q : ∀ s k, MixedSupport.Projection (left s k) (right s k))
(hs : ∀ s, WholeCutExteriorTransport.Outside p s → left s = right s)
(hk : ∀ s (_hout : WholeCutExteriorTransport.Outside p s) k,
HEq (q s k) (MixedSupport.Projection.keep (left s k)))
(terminal : CutTerminalSplit.TerminalTape F2 repeats p (LeafDomain right))
(exterior : CutTerminalSplit.ExteriorTape F2 repeats p (LeafDomain right)) (x : Domain left) :
(CutSamplerRefinement.evaluate F2 repeats p (LeafDomain left)
((CutTerminalSplit.splitTape F2 repeats p (LeafDomain left)).symm
(terminalPullback repeats p q terminal,
cast (WholeCutExteriorTransport.scalarExterior_eq repeats p right left
(fun s h => (hs s h).symm)) exterior))).val x =
(CutSamplerRefinement.evaluate F2 repeats p (LeafDomain right)
((CutTerminalSplit.splitTape F2 repeats p (LeafDomain right)).symm (terminal, exterior))).val
(sourceProjection q x) := by
let tape := (CutTerminalSplit.splitTape F2 repeats p (LeafDomain right)).symm (terminal, exterior)
have hsplit :
CutTerminalSplit.splitTape F2 repeats p (LeafDomain left)
(tapePullback repeats p left right q tape) =
(terminalPullback repeats p q terminal,
cast (WholeCutExteriorTransport.scalarExterior_eq repeats p right left
(fun s h => (hs s h).symm)) exterior) := by
simpa only [tape, Equiv.apply_symm_apply] using
splitTape_tapePullback repeats p left right q hs hk tape
have hleft := congrArg (CutTerminalSplit.splitTape F2 repeats p (LeafDomain left)).symm hsplit
simp only [Equiv.symm_apply_apply] at hleft
rw [← hleft]
exact evaluate_tapePullback repeats p left right q tape x
end
end PerfectCompleteness.CutGroupedProjection
end
section
namespace PerfectCompleteness.OriginalScalarProjection
open PointwiseSpaces RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays
open RecursiveSampler TerminalCalls
open scoped BigOperators Classical
abbrev F2 := ZMod 2
noncomputable section
variable {branch : Nat → Nat} {n m t : Nat}
theorem factor_eq (repeats : Nat → Nat) (p : Path branch n m) :
∀ (left right : Slots branch n → Fin t → MixedSupport.Slot),
(∀ s, WholeCutExteriorTransport.Outside p s → left s = right s) →
∀ (j : DrawIndex repeats p), ¬ IsTerminal repeats p j →
DrawSpace F2 repeats p (LeafDomain left) j =
DrawSpace F2 repeats p (LeafDomain right) j := by
induction p with
| refl n =>
intro left right hs j hj
exact False.elim (hj True.intro)
| @step n m i p ih =>
intro left right hs j hj
cases j with
| inl j =>
exact congrArg
(fun ss : Slots branch n → Fin t → MixedSupport.Slot =>
(squareSpace (space F2 branch n (LeafDomain ss)) : Type))
(WholeCutExteriorTransport.ordinarySlots_eq i p left right hs j)
| inr j =>
exact ih (childSlots left i) (childSlots right i)
(fun s h => hs (i, s) (WholeCutExteriorTransport.outside_selected i p s h)) j.2 hj
theorem scalarExterior_eq (repeats : Nat → Nat) (p : Path branch n (m + 1))
(left right : Slots branch n → Fin t → MixedSupport.Slot)
(hs : ∀ s, WholeCutExteriorTransport.Outside p s → left s = right s) :
OriginalTerminalSplit.ExteriorTape F2 repeats p (LeafDomain left) =
OriginalTerminalSplit.ExteriorTape F2 repeats p (LeafDomain right) :=
congrArg
(fun A : OriginalTerminalSplit.ExteriorIndex repeats p → Type =>
(j : OriginalTerminalSplit.ExteriorIndex repeats p) → A j)
(funext (fun j => factor_eq repeats p left right hs j.val j.property))
def terminalPullback (repeats : Nat → Nat) (p : Path branch n (m + 1))
{left right : Slots branch n → Fin t → MixedSupport.Slot}
(q : ∀ s k, MixedSupport.Projection (left s k) (right s k))
(values : OriginalScalarReconstruction.TerminalValues F2 repeats p (LeafDomain right)) :
OriginalScalarReconstruction.TerminalValues F2 repeats p (LeafDomain left) :=
fun call => HPullback (CutGroupedProjection.cutProjection p q) (values call)
def factorPullback (repeats : Nat → Nat) :
{n m : Nat} → (p : Path branch n (m + 1)) →
(left right : Slots branch n → Fin t → MixedSupport.Slot) →
(∀ s k, MixedSupport.Projection (left s k) (right s k)) →
(j : DrawIndex repeats p) →
RecursiveSampler.DrawSpace F2 repeats p (LeafDomain right) j →
RecursiveSampler.DrawSpace F2 repeats p (LeafDomain left) j
| _ + 1, _, .refl _, _, _, q, _, x => HPullback q x
| _ + 1, _, .step i p, left, right, q, j, x =>
match j with
| Sum.inl j =>
ChildBlockProjection.squarePullback (ChildAssemblyProjection.childProjection q j.val) x
| Sum.inr j =>
factorPullback repeats p (childSlots left i) (childSlots right i)
(ChildAssemblyProjection.childProjection q i) j.2 x
def tapePullback (repeats : Nat → Nat) (p : Path branch n (m + 1))
(left right : Slots branch n → Fin t → MixedSupport.Slot)
(q : ∀ s k, MixedSupport.Projection (left s k) (right s k))
(tape : RecursiveSampler.Tape F2 repeats p (LeafDomain right)) :
RecursiveSampler.Tape F2 repeats p (LeafDomain left) :=
fun j => factorPullback repeats p left right q j (tape j)
theorem evaluate_tapePullback (repeats : Nat → Nat) (p : Path branch n (m + 1)) :
∀ (left right : Slots branch n → Fin t → MixedSupport.Slot)
(q : ∀ s k, MixedSupport.Projection (left s k) (right s k))
(tape : RecursiveSampler.Tape F2 repeats p (LeafDomain right)) (x : Domain left),
(RecursiveSampler.evaluate F2 repeats p (LeafDomain left)
(tapePullback repeats p left right q tape)).val x =
(RecursiveSampler.evaluate F2 repeats p (LeafDomain right) tape).val
(sourceProjection q x) := by
induction n generalizing m with
| zero =>
have h := p.height_le
omega
| succ n ih =>
cases p with
| refl =>
intro left right q tape x
rfl
| step i p =>
intro left right q tape x
simp only [RecursiveSampler.evaluate_step, RecursiveSampler.combine,
RecursiveSampler.combineFunction, Finset.sum_apply, Pi.add_apply, Pi.mul_apply,
PointwiseSpaces.pullback_apply]
change
(∑ j : OffPath i,
(ChildBlockProjection.squarePullback (ChildAssemblyProjection.childProjection q j.val)
(tape (.inl j))).val (restrictChild left j.val x)) +
(∑ h : Fin (repeats (n + 1)),
(RecursiveSampler.evaluate F2 repeats p (LeafDomain (childSlots left i))
(tapePullback repeats p (childSlots left i) (childSlots right i)
(ChildAssemblyProjection.childProjection q i)
(RecursiveSampler.subTape F2 repeats i p (LeafDomain right) tape h false))).val
(restrictChild left i x) *
(RecursiveSampler.evaluate F2 repeats p (LeafDomain (childSlots left i))
(tapePullback repeats p (childSlots left i) (childSlots right i)
(ChildAssemblyProjection.childProjection q i)
(RecursiveSampler.subTape F2 repeats i p (LeafDomain right) tape h true))).val
(restrictChild left i x)) =
(∑ j : OffPath i, (tape (.inl j)).val (restrictChild right j.val (sourceProjection q x))) +
(∑ h : Fin (repeats (n + 1)),
(RecursiveSampler.evaluate F2 repeats p (LeafDomain (childSlots right i))
(RecursiveSampler.subTape F2 repeats i p (LeafDomain right) tape h false)).val
(restrictChild right i (sourceProjection q x)) *
(RecursiveSampler.evaluate F2 repeats p (LeafDomain (childSlots right i))
(RecursiveSampler.subTape F2 repeats i p (LeafDomain right) tape h true)).val
(restrictChild right i (sourceProjection q x)))
apply congrArg₂ (fun a b : F2 => a + b)
· exact Finset.sum_congr rfl (fun _ _ => rfl)
· apply Finset.sum_congr rfl
intro h _
apply congrArg₂ (fun a b : F2 => a * b)
· exact ih p (childSlots left i) (childSlots right i)
(ChildAssemblyProjection.childProjection q i)
(RecursiveSampler.subTape F2 repeats i p (LeafDomain right) tape h false)
(restrictChild left i x)
· exact ih p (childSlots left i) (childSlots right i)
(ChildAssemblyProjection.childProjection q i)
(RecursiveSampler.subTape F2 repeats i p (LeafDomain right) tape h true)
(restrictChild left i x)
theorem terminalFactor_pullback (repeats : Nat → Nat) (p : Path branch n (m + 1)) :
∀ (left right : Slots branch n → Fin t → MixedSupport.Slot)
(q : ∀ s k, MixedSupport.Projection (left s k) (right s k))
(call : TerminalIndex repeats p)
(x : RecursiveSampler.DrawSpace F2 repeats p (LeafDomain right)
(drawIndex repeats p call)),
TerminalCalls.terminalSpaceEquiv F2 repeats p (LeafDomain left) call
(factorPullback repeats p left right q (drawIndex repeats p call) x) =
HPullback (CutGroupedProjection.cutProjection p q)
(TerminalCalls.terminalSpaceEquiv F2 repeats p (LeafDomain right) call x) := by
induction n generalizing m with
| zero =>
have h := p.height_le
omega
| succ n ih =>
cases p with
| refl =>
intro left right q call x
cases call
rfl
| step i p =>
intro left right q call x
exact ih p (childSlots left i) (childSlots right i)
(ChildAssemblyProjection.childProjection q i) call.2 x
theorem exteriorFactor_pullback (repeats : Nat → Nat) (p : Path branch n (m + 1)) :
∀ (left right : Slots branch n → Fin t → MixedSupport.Slot)
(q : ∀ s k, MixedSupport.Projection (left s k) (right s k))
(hs : ∀ s, WholeCutExteriorTransport.Outside p s → left s = right s)
(_hk : ∀ s (_hout : WholeCutExteriorTransport.Outside p s) k,
HEq (q s k) (MixedSupport.Projection.keep (left s k)))
(j : DrawIndex repeats p) (hj : ¬ IsTerminal repeats p j)
(x : RecursiveSampler.DrawSpace F2 repeats p (LeafDomain right) j),
factorPullback repeats p left right q j x =
cast (factor_eq repeats p right left
(fun s h => (hs s h).symm) j hj) x := by
induction n generalizing m with
| zero =>
have h := p.height_le
omega
| succ n ih =>
cases p with
| refl =>
intro left right q hs hk j hj x
exact False.elim (hj True.intro)
| step i p =>
intro left right q hs hk j hj x
cases j with
| inl j =>
exact CutProjectionGeometry.squarePullback_eq_cast
(childSlots left j.val) (childSlots right j.val)
(ChildAssemblyProjection.childProjection q j.val)
(WholeCutExteriorTransport.ordinarySlots_eq i p left right hs j)
(fun s k => hk (j.val, s) (WholeCutExteriorTransport.outside_ordinary i p j s) k) x
| inr j =>
exact ih p (childSlots left i) (childSlots right i)
(ChildAssemblyProjection.childProjection q i)
(fun s h => hs (i, s) (WholeCutExteriorTransport.outside_selected i p s h))
(fun s h k => hk (i, s) (WholeCutExteriorTransport.outside_selected i p s h) k)
j.2 hj x
theorem prefixSplit_tapePullback (repeats : Nat → Nat) (p : Path branch n (m + 1))
(left right : Slots branch n → Fin t → MixedSupport.Slot)
(q : ∀ s k, MixedSupport.Projection (left s k) (right s k))
(hs : ∀ s, WholeCutExteriorTransport.Outside p s → left s = right s)
(hk : ∀ s (_hout : WholeCutExteriorTransport.Outside p s) k,
HEq (q s k) (MixedSupport.Projection.keep (left s k)))
(tape : RecursiveSampler.Tape F2 repeats p (LeafDomain right)) :
OriginalScalarReconstruction.prefixSplit F2 repeats p (LeafDomain left)
(tapePullback repeats p left right q tape) =
(terminalPullback repeats p q (OriginalScalarReconstruction.prefixSplit F2 repeats p (LeafDomain right) tape).1,
cast (scalarExterior_eq repeats p right left
(fun s h => (hs s h).symm))
(OriginalScalarReconstruction.prefixSplit F2 repeats p (LeafDomain right) tape).2) := by
apply Prod.ext
· funext call
exact terminalFactor_pullback repeats p left right q call (tape (drawIndex repeats p call))
· funext j
exact (exteriorFactor_pullback repeats p left right q hs hk j.val j.property (tape j.val)).trans
(CutSamplerReplayGroupingTransport.cast_pi_apply
(scalarExterior_eq repeats p right left
(fun s h => (hs s h).symm))
(fun j => factor_eq repeats p right left
(fun s h => (hs s h).symm) j.val j.property)
(OriginalScalarReconstruction.prefixSplit F2 repeats p (LeafDomain right) tape).2 j).symm
theorem reconstruct_pullback_apply (repeats : Nat → Nat) (p : Path branch n (m + 1))
(left right : Slots branch n → Fin t → MixedSupport.Slot)
(q : ∀ s k, MixedSupport.Projection (left s k) (right s k))
(hs : ∀ s, WholeCutExteriorTransport.Outside p s → left s = right s)
(hk : ∀ s (_hout : WholeCutExteriorTransport.Outside p s) k,
HEq (q s k) (MixedSupport.Projection.keep (left s k)))
(values : OriginalScalarReconstruction.TerminalValues F2 repeats p (LeafDomain right))
(exterior : OriginalTerminalSplit.ExteriorTape F2 repeats p (LeafDomain right))
(x : Domain left) :
(OriginalScalarReconstruction.reconstruct F2 repeats p (LeafDomain left)
(fun call => HPullback (CutGroupedProjection.cutProjection p q) (values call))
(cast (scalarExterior_eq repeats p right left
(fun s h => (hs s h).symm)) exterior)).val x =
(OriginalScalarReconstruction.reconstruct F2 repeats p (LeafDomain right)
values exterior).val (sourceProjection q x) := by
let tape := (OriginalScalarReconstruction.prefixSplit F2 repeats p (LeafDomain right)).symm
(values, exterior)
have hsplit :
OriginalScalarReconstruction.prefixSplit F2 repeats p (LeafDomain left)
(tapePullback repeats p left right q tape) =
(terminalPullback repeats p q values,
cast (scalarExterior_eq repeats p right left
(fun s h => (hs s h).symm)) exterior) := by
simpa only [tape, Equiv.apply_symm_apply] using
prefixSplit_tapePullback repeats p left right q hs hk tape
have hleft := congrArg
(OriginalScalarReconstruction.prefixSplit F2 repeats p (LeafDomain left)).symm hsplit
simp only [Equiv.symm_apply_apply] at hleft
unfold OriginalScalarReconstruction.reconstruct
change (RecursiveSampler.evaluate F2 repeats p (LeafDomain left)
((OriginalScalarReconstruction.prefixSplit F2 repeats p (LeafDomain left)).symm
(terminalPullback repeats p q values,
cast (scalarExterior_eq repeats p right left
(fun s h => (hs s h).symm)) exterior))).val x =
(RecursiveSampler.evaluate F2 repeats p (LeafDomain right) tape).val (sourceProjection q x)
rw [← hleft]
exact evaluate_tapePullback repeats p left right q tape x
theorem reconstruct_pullback (repeats : Nat → Nat) (p : Path branch n (m + 1))
(left right : Slots branch n → Fin t → MixedSupport.Slot)
(q : ∀ s k, MixedSupport.Projection (left s k) (right s k))
(hs : ∀ s, WholeCutExteriorTransport.Outside p s → left s = right s)
(hk : ∀ s (_hout : WholeCutExteriorTransport.Outside p s) k,
HEq (q s k) (MixedSupport.Projection.keep (left s k)))
(values : OriginalScalarReconstruction.TerminalValues F2 repeats p (LeafDomain right))
(exterior : OriginalTerminalSplit.ExteriorTape F2 repeats p (LeafDomain right)) :
OriginalScalarReconstruction.reconstruct F2 repeats p (LeafDomain left)
(fun call => HPullback (CutGroupedProjection.cutProjection p q) (values call))
(cast (scalarExterior_eq repeats p right left
(fun s h => (hs s h).symm)) exterior) =
HPullback q (OriginalScalarReconstruction.reconstruct F2 repeats p (LeafDomain right)
values exterior) := by
apply Subtype.ext
funext x
exact reconstruct_pullback_apply repeats p left right q hs hk values exterior x
end
end PerfectCompleteness.OriginalScalarProjection
end
end OAI
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