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97c3405 | 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 | import OAI.Combinatorics.ContingencyTables.Counting.CompanionStoredRepresentation
import OAI.Combinatorics.ContingencyTables.Counting.CompanionOuterPhaseSemantics
import OAI.Combinatorics.ContingencyTables.Counting.CompanionCanonicalHistoryLaw
namespace OAI
/-! # Reading a represented physical history as source matrix data -/
namespace ContingencyTables.CompanionHistoryReadback
open MatchingFPRAS CompanionMultiplierMatrixProgram CompanionStoredRepresentation
open CompanionFactorialApplication CompanionComputedInitial CompanionEvaluatedWeightProgram
open GridBoundary
open scoped Classical
noncomputable section
abbrev Stored := CompanionPhaseFinishProgram.Stored
def lookup (initial : Stored) (history : List Stored) : ℕ→Stored
| 0 => initial
| i+1 => history.getD i initial
def matrixAt (p : ℕ) (initial : Stored) (history : List Stored) (i : ℕ) :
Fin p→Fin p→TreeFraction.Fraction := entries p (lookup initial history i).1
def estimateAt (initial : Stored) (history : List Stored) (i : ℕ) : TreeFraction.Fraction :=
(lookup initial history i).2
lemma getD_rep {p : ℕ} (history : List Stored) (initial : Stored)
(hi : StoredRep p initial) (hh : ∀ s∈history,StoredRep p s) (i : ℕ) :
StoredRep p (history.getD i initial) := by
by_cases h : i < history.length
· rw [List.getD_eq_getElem _ _ h]
exact hh _ (List.getElem_mem h)
· rw [List.getD_eq_default _ _ (Nat.le_of_not_gt h)]
exact hi
lemma lookup_rep {p : ℕ} (initial : Stored) (history : List Stored)
(hi : StoredRep p initial) (hh : ∀ s∈history,StoredRep p s) (i : ℕ) :
StoredRep p (lookup initial history i) := by
cases i with
| zero => exact hi
| succ i => exact getD_rep history initial hi hh i
theorem lookup_readback {p : ℕ} (initial : Stored) (history : List Stored)
(hi : StoredRep p initial) (hh : ∀ s∈history,StoredRep p s) (i : ℕ) :
(matrixCode (matrixAt p initial history i),estimateAt initial history i)=
lookup initial history i :=
stored_readback (lookup_rep initial history hi hh i)
theorem initial_readback {p : ℕ} (initial : Stored) (history : List Stored)
(hi : StoredRep p initial) :
(matrixCode (matrixAt p initial history 0),estimateAt initial history 0)=initial :=
stored_readback hi
theorem history_readback {p : ℕ} (initial : Stored) (history : List Stored)
(hi : StoredRep p initial) (hh : ∀ s∈history,StoredRep p s) :
List.ofFn (fun i : Fin history.length =>
(matrixCode (matrixAt p initial history (i.val+1)),estimateAt initial history (i.val+1)))=
history := by
apply List.ext_getElem
· exact List.length_ofFn
· intro i _ hj
simp only [List.getElem_ofFn]
have h := lookup_readback initial history hi hh (i+1)
change (matrixCode (matrixAt p initial history (i+1)),estimateAt initial history (i+1))=
history.getD i initial at h
rw [List.getD_eq_getElem _ _ hj] at h
exact h
def viewLookup {Z : Type} (initial : Z) (history : List Z) : ℕ→Z
| 0 => initial
| i+1 => history.getD i initial
theorem view_lookup (p : ℕ) (initial : Stored) (history : List Stored) (i : ℕ) :
CompanionPhaseFinishProgram.view (p := p) (lookup initial history i)=
viewLookup (CompanionPhaseFinishProgram.view (p := p) initial)
(history.map (CompanionPhaseFinishProgram.view (p := p))) i := by
cases i with
| zero => rfl
| succ i => exact (List.getD_map history initial
(CompanionPhaseFinishProgram.view (p := p))).symm
theorem old_eq_lookup (fixed : CompanionOuterHistoryProgram.Fixed) (history : List Stored)
(bits : List Bool) :
CompanionOuterPhaseProgram.old ((fixed,history),bits)=lookup fixed.2 history history.length := by
cases history with
| nil => rfl
| cons s history =>
change (s::history).getD history.length default=(s::history).getD history.length fixed.2
rw [List.getD_eq_getElem _ _ (by simp : history.length < (s::history).length),
List.getD_eq_getElem _ _ (by simp : history.length < (s::history).length)]
theorem view_old (p : ℕ) (fixed : CompanionOuterHistoryProgram.Fixed) (history : List Stored)
(bits : List Bool) :
CompanionPhaseFinishProgram.view (p := p) (CompanionOuterPhaseProgram.old ((fixed,history),bits))=
viewLookup (CompanionPhaseFinishProgram.view (p := p) fixed.2)
(history.map (CompanionPhaseFinishProgram.view (p := p))) history.length := by
rw [old_eq_lookup,view_lookup]
theorem view_old_canonical {I J : Type} [Fintype I] [Fintype J]
(A : Instance I J) {p : ℕ} (g : Fin p ≃ Labels A)
(fixed : CompanionOuterHistoryProgram.Fixed) (history : List Stored) (bits : List Bool)
(hi : CompanionPhaseFinishProgram.view (p := p) fixed.2=
CompanionCanonicalHistory.project A g (RootedStoredEstimate.initialStored A)) :
CompanionPhaseFinishProgram.view (p := p) (CompanionOuterPhaseProgram.old ((fixed,history),bits))=
CompanionCanonicalHistory.canonicalLookup A g
(history.map (CompanionPhaseFinishProgram.view (p := p))) history.length := by
rw [view_old,hi]
rfl
variable {m n : ℕ}
theorem outer_data_readback (A : Instance (Fin m) (Fin n)) (C K H T N R L : ℕ)
(epsilon : TreeFraction.Fraction) (z : View A→ℕ)
(v : Grid (I := Fin (RootedCatalogGrid.freeCount A)) K)
(initial : Stored) (history : List Stored)
(hi : StoredRep (CanonicalInitialEncoding.copyOrder (capacities A)).length initial)
(hh : ∀ s∈history,StoredRep (CanonicalInitialEncoding.copyOrder (capacities A)).length s)
(bits : List Bool) :
CompanionOuterPhaseProgram.data A C K H T N R L history.length epsilon z v
(matrixAt (CanonicalInitialEncoding.copyOrder (capacities A)).length initial history)
(estimateAt initial history) bits=
(((CompanionRestartInitializer.data A C K H T N R L epsilon z v,initial),history),bits) := by
unfold CompanionOuterPhaseProgram.data
rw [initial_readback initial history hi,history_readback initial history hi hh]
end
end ContingencyTables.CompanionHistoryReadback
end OAI
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