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2f887db | 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 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 | import OAI.Computability.PerfectCompleteness.Foundations.WholeCutExteriorTransportLemmas
import OAI.Computability.PerfectCompleteness.Foundations.WholeCutReplayKeys
import OAI.Computability.PerfectCompleteness.Sampling.CutSamplerKeyLocalityLemmas
namespace OAI
section
namespace PerfectCompleteness.CutSamplerReplayGroupingTransport
open PointwiseSpaces RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays
open RecursiveSampler TerminalCalls CutSamplerKeyLocality
open scoped Classical
abbrev F2 := ZMod 2
noncomputable section
universe u v
theorem cast_pi_apply {I : Type u} {A B : I → Type v}
(h : ((i : I) → A i) = ((i : I) → B i)) (hA : ∀ i, A i = B i)
(f : (i : I) → A i) (i : I) :
(cast h f) i = cast (hA i) (f i) := by
have hAB : A = B := funext hA
cases hAB
rfl
theorem cast_prod_fst {A A' : Type u} {B B' : Type v}
(h : (A × B) = (A' × B')) (hA : A = A') (hB : B = B') (x : A × B) :
(cast h x).1 = cast hA x.1 := by
cases hA
cases hB
rfl
theorem cast_prod_snd {A A' : Type u} {B B' : Type v}
(h : (A × B) = (A' × B')) (hA : A = A') (hB : B = B') (x : A × B) :
(cast h x).2 = cast hB x.2 := by
cases hA
cases hB
rfl
variable {branch : Nat → Nat} {n m t : Nat}
theorem outside_kept (p : Path branch n (m + 1)) (clean : Fin (branch m) → Prop) :
∀ s, WholeCutExteriorTransport.Outside p s → keptLeaf p clean s := by
induction n generalizing m with
| zero =>
have h := p.height_le
omega
| succ n ih =>
cases p with
| refl =>
intro s hs
exact False.elim (hs ⟨s, rfl⟩)
| step i p =>
rintro ⟨j, s⟩ hs
by_cases hji : j = i
· subst j
apply Or.inr
apply ih p clean s
rintro ⟨z, hz⟩
exact hs ⟨z, congrArg (fun a => (i, a)) hz⟩
· exact Or.inl hji
theorem outside_eq (p : Path branch n (m + 1))
(slots target : Slots branch n → Fin t → MixedSupport.Slot)
(clean : Fin (branch m) → Prop)
(hs : ∀ s, keptLeaf p clean s → slots s = target s) :
∀ s, WholeCutExteriorTransport.Outside p s → slots s = target s :=
fun s hout => hs s (outside_kept p clean s hout)
theorem retainedChildSlots_eq (p : Path branch n (m + 1))
(slots target : Slots branch n → Fin t → MixedSupport.Slot)
(clean : Fin (branch m) → Prop)
(hs : ∀ s, keptLeaf p clean s → slots s = target s)
(i : {i : Fin (branch m) // ¬ clean i}) :
childSlots (WholeCutGrouping.cutSlots p slots) i.val =
childSlots (WholeCutGrouping.cutSlots p target) i.val := by
funext s
exact hs (p.slotEmbedding (i.val, s)) ((keptLeaf_at_cut p clean i.val s).mpr i.property)
theorem squareEquiv_eq_cast
(slots target : Slots branch n → Fin t → MixedSupport.Slot) (hs : slots = target)
(f : squareSpace (H slots)) :
(RecursiveSpaceEquiv.squareEquiv (𝕜 := F2)
(fun s => Equiv.cast (congrArg MixedSupport.Assignment (congrFun hs s)))).symm f =
cast (congrArg
(fun ss : Slots branch n → Fin t → MixedSupport.Slot => (squareSpace (H ss) : Type))
hs) f := by
cases hs
apply Subtype.ext
funext x
rfl
def castTerminal (repeats : Nat → Nat) (p : Path branch n (m + 1))
(slots target : Slots branch n → Fin t → MixedSupport.Slot)
(clean : Fin (branch m) → Prop)
(hs : ∀ s, keptLeaf p clean s → slots s = target s)
(terminal : CutSamplerReplayGrouping.RetainedTerminal F2 repeats p (LeafDomain slots) clean) :
CutSamplerReplayGrouping.RetainedTerminal F2 repeats p (LeafDomain target) clean :=
fun call child => cast (congrArg
(fun ss : Slots branch m → Fin t → MixedSupport.Slot => (squareSpace (H ss) : Type))
(retainedChildSlots_eq p slots target clean hs child)) (terminal call child)
theorem subExterior_cast (repeats : Nat → Nat) (i : Fin (branch n))
(p : Path branch n (m + 1))
(slots target : Slots branch (n + 1) → Fin t → MixedSupport.Slot)
(clean : Fin (branch m) → Prop)
(hs : ∀ s, keptLeaf (.step i p) clean s → slots s = target s)
(call : Fin (repeats (n + 1)) × Bool)
(exterior : CutTerminalSplit.ExteriorTape F2 repeats (.step i p) (LeafDomain slots)) :
CutSamplerReplayGrouping.subExterior F2 repeats i p (LeafDomain target) call
(cast (WholeCutExteriorTransport.scalarExterior_eq repeats (.step i p) slots target
(outside_eq (.step i p) slots target clean hs)) exterior) =
cast (WholeCutExteriorTransport.scalarExterior_eq repeats p
(childSlots slots i) (childSlots target i)
(outside_eq p (childSlots slots i) (childSlots target i) clean
(fun s h => hs (i, s) (Or.inr h))))
(CutSamplerReplayGrouping.subExterior F2 repeats i p (LeafDomain slots) call exterior) := by
funext j
let hext := WholeCutExteriorTransport.scalarExterior_eq repeats (.step i p) slots target
(outside_eq (.step i p) slots target clean hs)
let hsub := WholeCutExteriorTransport.scalarExterior_eq repeats p
(childSlots slots i) (childSlots target i)
(outside_eq p (childSlots slots i) (childSlots target i) clean
(fun s h => hs (i, s) (Or.inr h)))
change (cast hext exterior) (CutSamplerReplayGrouping.exteriorStepIndex repeats i p call j) =
(cast hsub
(CutSamplerReplayGrouping.subExterior F2 repeats i p (LeafDomain slots) call exterior)) j
rw [cast_pi_apply hext
(fun j => WholeCutExteriorTransport.factor_eq repeats (.step i p) slots target
(outside_eq (.step i p) slots target clean hs) j.val j.property)]
rw [cast_pi_apply hsub
(fun j => WholeCutExteriorTransport.factor_eq repeats p (childSlots slots i)
(childSlots target i)
(outside_eq p (childSlots slots i) (childSlots target i) clean
(fun s h => hs (i, s) (Or.inr h))) j.val j.property)]
rfl
theorem transport_assemble (repeats : Nat → Nat) (p : Path branch n (m + 1)) :
∀ (slots target : Slots branch n → Fin t → MixedSupport.Slot)
(clean : Fin (branch m) → Prop)
(hs : ∀ s, keptLeaf p clean s → slots s = target s)
(exterior : CutTerminalSplit.ExteriorTape F2 repeats p (LeafDomain slots))
(terminal : CutSamplerReplayGrouping.RetainedTerminal F2 repeats p (LeafDomain slots) clean),
CutSamplerReplayTransport.transport F2 repeats p (LeafDomain slots) (LeafDomain target)
clean (WholeCutReplayKeys.leafEquiv p slots target clean hs)
(CutSamplerReplayGrouping.assemble F2 repeats p (LeafDomain slots) clean exterior terminal) =
CutSamplerReplayGrouping.assemble F2 repeats p (LeafDomain target) clean
(cast (WholeCutExteriorTransport.scalarExterior_eq repeats p slots target
(outside_eq p slots target clean hs)) exterior)
(castTerminal repeats p slots target clean hs terminal) := by
induction n generalizing m with
| zero =>
have h := p.height_le
omega
| succ n ih =>
cases p with
| refl =>
intro slots target clean hs exterior terminal
funext child
exact squareEquiv_eq_cast (childSlots slots child.val) (childSlots target child.val)
(retainedChildSlots_eq (.refl (n + 1)) slots target clean hs child) (terminal () child)
| step i p =>
intro slots target clean hs exterior terminal
apply Prod.ext
· funext j
let hext := WholeCutExteriorTransport.scalarExterior_eq repeats (.step i p)
slots target (outside_eq (.step i p) slots target clean hs)
let hslots := WholeCutExteriorTransport.ordinarySlots_eq i p slots target
(outside_eq (.step i p) slots target clean hs) j
change (RecursiveSpaceEquiv.squareEquiv (𝕜 := F2)
(fun s => WholeCutReplayKeys.leafEquiv (.step i p) slots target clean hs
(j.val, s) (Or.inl j.property))).symm
(exterior ⟨.inl j, not_isTerminal_ordinary repeats i p j⟩) =
(cast hext exterior) ⟨.inl j, not_isTerminal_ordinary repeats i p j⟩
rw [cast_pi_apply hext
(fun j => WholeCutExteriorTransport.factor_eq repeats (.step i p) slots target
(outside_eq (.step i p) slots target clean hs) j.val j.property)]
exact squareEquiv_eq_cast (childSlots slots j.val) (childSlots target j.val) hslots
(exterior ⟨.inl j, not_isTerminal_ordinary repeats i p j⟩)
· funext call
change CutSamplerReplayTransport.transport F2 repeats p
(LeafDomain (childSlots slots i)) (LeafDomain (childSlots target i)) clean
(WholeCutReplayKeys.leafEquiv p (childSlots slots i) (childSlots target i) clean
(fun s h => hs (i, s) (Or.inr h)))
(CutSamplerReplayGrouping.assemble F2 repeats p (LeafDomain (childSlots slots i))
clean (CutSamplerReplayGrouping.subExterior F2 repeats i p (LeafDomain slots)
call exterior) (fun location child => terminal (call, location) child)) =
CutSamplerReplayGrouping.assemble F2 repeats p (LeafDomain (childSlots target i))
clean (CutSamplerReplayGrouping.subExterior F2 repeats i p (LeafDomain target) call
(cast (WholeCutExteriorTransport.scalarExterior_eq repeats (.step i p) slots target
(outside_eq (.step i p) slots target clean hs)) exterior))
(fun location child => castTerminal repeats (.step i p) slots target clean hs
terminal (call, location) child)
rw [subExterior_cast repeats i p slots target clean hs call exterior]
exact ih p (childSlots slots i) (childSlots target i) clean
(fun s h => hs (i, s) (Or.inr h))
(CutSamplerReplayGrouping.subExterior F2 repeats i p (LeafDomain slots) call exterior)
(fun location child => terminal (call, location) child)
end
end PerfectCompleteness.CutSamplerReplayGroupingTransport
end
end OAI
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