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f0f05de | 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 | import OAI.Combinatorics.BalancedRyser.Local.AlignedGlobalProduct
import OAI.Combinatorics.BalancedRyser.Local.AlignedProjectedExposure
namespace OAI
namespace Balanced.AlignedListProbability
open TupleAlignment PartialAlignment
noncomputable section
open scoped Classical
variable {F : Type*} [Field F] [Fintype F] [DecidableEq F]
theorem generatedPositions_reconstruct (d : Fin 4 → Direction F)
(hd : ∀ i, d i ≠ d (i+1)) (σ : Fin 4 → Bool)
(g : Direction F) (hg : ∀ i, d i ≠ g) (c : F)
(J : Finset (Fin 4)) (hJ : J.card ≤ 1)
(a o : Fin 4 → Bool → F) (hold : ∀ j ∈ J, o j = a j)
(halign : ∀ j, OnLine (tuplePositionEquiv d hd σ o j) (g,c))
(x : {j : Fin 4 // j ∈ freePositions J} → F)
(hx : ∀ j, linePointEquiv g c
⟨tuplePositionEquiv d hd σ o j.1, halign j.1⟩ = x j) :
generatedPositions d hd σ g hg c J hJ a x = tuplePositionEquiv d hd σ o := by
let v : AlignedPartialOffsets d hd σ g c J a := ⟨⟨o,hold⟩,halign⟩
let e := alignedFreeCoordinatesEquiv d hd σ g hg c J hJ a
have he : e v = x := funext hx
have hi : e.symm x = v := by rw [← he]; exact e.symm_apply_apply v
change tuplePositionEquiv d hd σ (e.symm x).1.1 = _
rw [hi]
end
end Balanced.AlignedListProbability
namespace Balanced.Local.AlignedGlobal
open RootChildPartition AlignedListProbability TupleAlignment PartialAlignment TokenExposure
noncomputable section
open scoped Classical
variable {F : Type*} [Field F] [Fintype F] [DecidableEq F]
variable {b κ : ℕ}
def currentOldIndices (P : EqualityPattern b κ) (R : Finset (Fin b))
(i : Fin b) : Finset (Fin 4) :=
if i ∈ R then ∅ else rootGeneratorIndices P R i
theorem currentOldIndices_card_le_one (P : EqualityPattern b κ)
(R : Finset (Fin b))
(hchild : ∀ i, i ∉ R → (P.classSet i ∩ R.biUnion P.classSet).card = 1)
(i : Fin b) : (currentOldIndices P R i).card ≤ 1 := by
by_cases hi : i ∈ R
· simp [currentOldIndices, hi]
· rw [currentOldIndices, ite_eq_right hi, card_rootGeneratorIndices, hchild i hi]
def currentFreeIndex (P : EqualityPattern b κ) (R : Finset (Fin b)) (i : Fin b)
(j : {j : Fin 4 // j ∈ freePositions (currentOldIndices P R i)}) :
FreeIndex P R := by
refine ⟨(i,j.1), ?_⟩
by_cases hi : i ∈ R
· simp [mem_dependentOccurrences, hi]
· have hj := j.2
simp only [freePositions, Finset.mem_filter, Finset.mem_univ, true_and,
currentOldIndices, ite_eq_right hi, rootGeneratorIndices] at hj
simp only [mem_dependentOccurrences, hi, not_false_eq_true, true_and, not_or]
exact hj
theorem generated_current_tuple (P : EqualityPattern b κ) (hκ : κ ≤ 2)
(R : Finset (Fin b)) (hD : (dependentOccurrences P R).card = 2*κ)
(bits : Bits b) (p : Point F) (e : DirectionToken b κ ↪ Direction F)
(i : Fin b) (hJ : (currentOldIndices P R i).card ≤ 1)
(z : (FreeIndex P R → F) × AlignedUnused.UnusedArray P (tokenInjectionEquiv F b κ e).1)
(a : Fin 4 → Bool → F)
(ha : ∀ j ∈ currentOldIndices P R i,
((coordinateEquiv P hκ R hD (tokenInjectionEquiv F b κ e).1
(allTokensDistinct_eligible P _ (tokenInjectionEquiv F b κ e).2) bits p).symm z).1
(e (Sum.inr (P.classOf i j))) = a j) :
generatedPositions (tupleDirections P e i) (tupleDirections_adjacent P e i)
(bits i) (e (Sum.inl i)) (tupleDirections_ne_target P e i)
(lineEval (e (Sum.inl i)) p) (currentOldIndices P R i) hJ a
(fun j => z.1 (currentFreeIndex P R i j)) =
fun j => prescribedPoint P (tokenInjectionEquiv F b κ e).1 bits
((coordinateEquiv P hκ R hD (tokenInjectionEquiv F b κ e).1
(allTokensDistinct_eligible P _ (tokenInjectionEquiv F b κ e).2) bits p).symm z).1
(i,j) := by
let A := (tokenInjectionEquiv F b κ e).1
let hA := allTokensDistinct_eligible P A (tokenInjectionEquiv F b κ e).2
let E := coordinateEquiv P hκ R hD A hA bits p
let M := E.symm z
have hm : ∀ j, OnLine (tuplePositionEquiv (tupleDirections P e i)
(tupleDirections_adjacent P e i) (bits i)
(fun k => M.1 (e (Sum.inr (P.classOf i k)))) j)
(e (Sum.inl i), lineEval (e (Sum.inl i)) p) := fun j => M.2 (i,j)
apply generatedPositions_reconstruct _ _ _ _ _ _ _ _ a
(fun k => M.1 (e (Sum.inr (P.classOf i k)))) ha hm
intro j
have h := congrFun (congrArg Prod.fst (E.apply_symm_apply z)) (currentFreeIndex P R i j)
exact h
theorem currentFreeIndex_injective (P : EqualityPattern b κ)
(R : Finset (Fin b)) (i : Fin b) :
Function.Injective (currentFreeIndex P R i) := by
intro j k h
apply Subtype.ext
exact congrArg (fun t : FreeIndex P R => t.1.2) h
theorem pointOutput_eq_global {Z : Type*}
(P : EqualityPattern b κ) (hκ : κ ≤ 2)
(R : Finset (Fin b)) (hD : (dependentOccurrences P R).card = 2*κ)
(bits : Bits b) (p : Point F) (e : DirectionToken b κ ↪ Direction F)
(i : Fin b) (hJ : (currentOldIndices P R i).card ≤ 1)
(z : (FreeIndex P R → F) × AlignedUnused.UnusedArray P (tokenInjectionEquiv F b κ e).1)
(a : Z → Fin 4 → Bool → F)
(x : Z → {j : Fin 4 // j ∈ freePositions (currentOldIndices P R i)} → F)
(w : Z)
(ha : ∀ j ∈ currentOldIndices P R i,
((coordinateEquiv P hκ R hD (tokenInjectionEquiv F b κ e).1
(allTokensDistinct_eligible P _ (tokenInjectionEquiv F b κ e).2) bits p).symm z).1
(e (Sum.inr (P.classOf i j))) = a w j)
(hx : x w = fun j => z.1 (currentFreeIndex P R i j)) (j : Fin 4) :
pointOutput P bits p i (currentOldIndices P R i) hJ a x e w j =
prescribedPoint P (tokenInjectionEquiv F b κ e).1 bits
((coordinateEquiv P hκ R hD (tokenInjectionEquiv F b κ e).1
(allTokensDistinct_eligible P _ (tokenInjectionEquiv F b κ e).2) bits p).symm z).1
(i,j) := by
unfold pointOutput
rw [hx]
exact congrFun (generated_current_tuple P hκ R hD bits p e i hJ z (a w) ha) j
end
end Balanced.Local.AlignedGlobal
end OAI
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