| import Mathlib |
| import OAI.LinearAlgebra.MatrixFields.Construction.GroupAssignmentRetention |
| import OAI.LinearAlgebra.MatrixFields.Histories.CWShapePhysicalPermutation |
| import OAI.LinearAlgebra.MatrixFields.Parameters.GroupStatisticComplement |
| import OAI.LinearAlgebra.MatrixFields.Tensors.JointPopulationCWNonzero |
|
|
| namespace OAI |
|
|
| namespace MatrixAllFields |
|
|
| open scoped BigOperators Topology Polynomial |
|
|
| section |
| noncomputable section |
|
|
| namespace MatrixMultiplication.AllFieldGroupSourceSupport |
|
|
| open MatrixMultiplication.Foundation AllFieldHistory AllFieldHistorySupport |
| open AllFieldHistoryChildLaws AllFieldHistoryGroupMasks AllFieldHistoryGroupedRecovery |
| open AllFieldGroupOrbitData JointPopulation JointCanonicalization JointCanonicalCW |
| open scoped BigOperators |
| attribute [local instance] Classical.propDecidable |
|
|
| variable {K tick : ℕ} |
|
|
| theorem masks_of_prepared (F : Type*) [CommRing F] (allocation : Allocation) |
| (m : ℕ) (ε : ℝ) (sigma : Placement) |
| (x y z : GroupRaw (K := K) (tick := tick) allocation m sigma) |
| (hT : groupPreparedSource F allocation m ε sigma x y z ≠ 0) : |
| completeKeep allocation m ε 0 sigma x ∧ |
| completeKeep allocation m ε 1 sigma y ∧ |
| completeKeep allocation m ε 2 sigma z := by |
| by_contra hn |
| apply hT |
| exact ite_eq_right hn |
|
|
| theorem raw_nonzero (F : Type*) [CommRing F] (allocation : Allocation) |
| (m : ℕ) (ε : ℝ) (sigma : Placement) |
| (x y z : GroupRaw (K := K) (tick := tick) allocation m sigma) |
| (hT : groupPreparedSource F allocation m ε sigma x y z ≠ 0) : |
| groupRawSource F allocation m sigma x y z ≠ 0 := by |
| have hm := masks_of_prepared F allocation m ε sigma x y z hT |
| simpa only [groupPreparedSource, ExactRecovery.delete, ite_eq_left hm] using hT |
|
|
| theorem marginal_sideMask (allocation : Allocation) (m : ℕ) (side : Fin 3) |
| (sigma : Placement) (w : GroupRaw (K := K) (tick := tick) allocation m sigma) |
| (hm : marginal allocation m side sigma w) : |
| JointCanonicalMixed.sideMask (groupCounts allocation m sigma) |
| (fun h => activeHalfLength h.val) (fun h => activeHalfLength h.val) |
| (fun h => AllFieldHistoryRecovery.halfLength_le_eight h.val) side w := by |
| intro h a |
| exact (hm h a).trans |
| (marginal_population_eq (groupCounts allocation m sigma) |
| (projectTarget allocation m sigma (canonicalTarget (activeCounts allocation m))) |
| (canonicalTarget (groupCounts allocation m sigma)) h side a) |
|
|
| theorem marginal_source_nonzero (F : Type*) [CommRing F] (allocation : Allocation) |
| (m : ℕ) (ε : ℝ) (sigma : Placement) |
| (x y z : GroupRaw (K := K) (tick := tick) allocation m sigma) |
| (hT : groupPreparedSource F allocation m ε sigma x y z ≠ 0) : |
| JointCanonicalMixed.maskedSource (F := F) (groupCounts allocation m sigma) |
| (fun h => activeHalfLength h.val) (fun h => activeHalfLength h.val) |
| (fun h => activeParentShape h.val) (fun _ => true) |
| (fun h => AllFieldHistoryRecovery.halfLength_le_eight h.val) x y z ≠ 0 := by |
| have hm := masks_of_prepared F allocation m ε sigma x y z hT |
| have hx := marginal_sideMask allocation m 0 sigma x hm.1.1 |
| have hy := marginal_sideMask allocation m 1 sigma y hm.2.1.1 |
| have hz := marginal_sideMask allocation m 2 sigma z hm.2.2.1 |
| rw [JointCanonicalMixed.maskedSource, ExactRecovery.delete, ite_eq_left (And.intro hx (And.intro hy hz))] |
| exact raw_nonzero F allocation m ε sigma x y z hT |
|
|
| theorem prepared_source_support (F : Type*) [CommRing F] (allocation : Allocation) |
| (m : ℕ) (ε : ℝ) (sigma : Placement) |
| (x y z : GroupRaw (K := K) (tick := tick) allocation m sigma) |
| (hT : groupPreparedSource F allocation m ε sigma x y z ≠ 0) : |
| reorder sigma |
| (ownWord allocation m sigma x, ownWord allocation m sigma y, |
| ownWord allocation m sigma z) ∈ ambient allocation m sigma := by |
| apply Finset.mem_image.mpr |
| refine ⟨(ownWord allocation m sigma x, ownWord allocation m sigma y, |
| ownWord allocation m sigma z), ?_, rfl⟩ |
| exact JointCanonicalMixed.maskedSource_support (groupCounts allocation m sigma) |
| (fun h => activeHalfLength h.val) (fun h => activeHalfLength h.val) |
| (fun h => activeParentShape h.val) (fun _ => true) |
| (fun h => AllFieldHistoryRecovery.halfLength_le_eight h.val) x y z |
| (marginal_source_nonzero F allocation m ε sigma x y z hT) |
|
|
| end MatrixMultiplication.AllFieldGroupSourceSupport |
|
|
| end |
| end |
|
|
| end MatrixAllFields |
|
|
| namespace MatrixAllFields |
|
|
| open scoped BigOperators Topology Polynomial |
|
|
| section |
| noncomputable section |
|
|
| namespace MatrixMultiplication.AllFieldGroupCompatibilityTransfer |
|
|
| open MatrixMultiplication.Foundation AllFieldHistory AllFieldHistoryChildLaws |
| open AllFieldHistoryGroupMasks AllFieldHistoryGroupedRecovery AllFieldGroupOrbitData |
| open JointPopulation JointCanonicalization JointCanonicalCW CWStrands InheritedMasks |
| open scoped BigOperators |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| variable {K tick : ℕ} |
|
|
| theorem physical_coarseMask (allocation : Allocation) (m : ℕ) (sigma : Placement) |
| (e : Targets (K := K) (tick := tick) allocation m sigma) |
| (a : Fin 3 → Raw (K := K) (tick := tick) allocation m sigma) |
| (hcoarse : reorder sigma (ownWord allocation m sigma (a 0), |
| ownWord allocation m sigma (a 1), ownWord allocation m sigma (a 2)) = |
| AllFieldGroupOrbitData.coarse allocation m sigma e) |
| (s : Fin 3) : |
| coarseMask (Counts allocation m sigma) |
| (fun h => Fin (activeHalfLength h.val) → Fin 7) |
| (fun h => Fin (activeHalfLength h.val) → Fin 7) |
| groupCoarse s e (a s) := by |
| have ht := reorder_injective sigma hcoarse |
| intro h i |
| have he := congrArg (fun t => tripleSide s t ⟨h, i⟩) ht |
| fin_cases s <;> exact he |
|
|
| def halfShape {sigma : Placement} (right : Bool) (h : ActiveOrder K tick sigma) |
| (u : JointPopulation.Shape) : Fin 3 → ℕ := |
| if right then activeParentShape h.val - shapeNat u else shapeNat u |
|
|
| def halfLaw {sigma : Placement} (right : Bool) (side : Fin 3) (h : ActiveOrder K tick sigma) |
| (u : JointPopulation.Shape) : Statistic h.val → ℝ := |
| if right then rightLaw h.val u side else leftLaw h.val u side |
|
|
| theorem physical_half_product_nonzero (F : Type*) [Field F] |
| (allocation : Allocation) (m : ℕ) (sigma : Placement) |
| (e : Targets (K := K) (tick := tick) allocation m sigma) |
| (a : Fin 3 → Raw (K := K) (tick := tick) allocation m sigma) |
| (hraw : groupRawSource F allocation m sigma (a 0) (a 1) (a 2) ≠ 0) |
| (hcoarse : reorder sigma (ownWord allocation m sigma (a 0), |
| ownWord allocation m sigma (a 1), ownWord allocation m sigma (a 2)) = |
| AllFieldGroupOrbitData.coarse allocation m sigma e) |
| (right : Bool) (h : ActiveOrder K tick sigma) (u : JointPopulation.Shape) |
| (hu : 0 < Counts allocation m sigma h u) : |
| (∏ i : Class (Counts allocation m sigma) e h u, |
| shapeTensor (F := F) (Fin (activeHalfLength h.val)) (halfShape right h u) |
| (if right then (a 0 h i.val).2 else (a 0 h i.val).1) |
| (if right then (a 1 h i.val).2 else (a 1 h i.val).1) |
| (if right then (a 2 h i.val).2 else (a 2 h i.val).1)) ≠ 0 := by |
| have hsupp : ∀ (h : ActiveOrder K tick sigma) (u : JointPopulation.Shape), |
| 0 < Counts allocation m sigma h u → |
| ∀ s, shapeNat u s ≤ activeParentShape h.val s := by |
| intro h u hu s |
| exact jointCounts_support_le allocation m h.val.val u hu s |
| have hsource : sourceTensor (F := F) (Counts (K := K) (tick := tick) allocation m sigma) |
| (Left (fun h => activeHalfLength h.val)) (Right (fun h => activeHalfLength h.val)) |
| (parentTensor (F := F) (fun h : ActiveOrder K tick sigma => activeHalfLength h.val) |
| (fun h : ActiveOrder K tick sigma => activeHalfLength h.val) |
| (fun h : ActiveOrder K tick sigma => activeParentShape h.val)) |
| (a 0) (a 1) (a 2) ≠ 0 := by |
| exact hraw |
| have hp := JointPopulationCWNonzero.class_half_products_nonzero |
| (Counts allocation m sigma) (fun h => activeHalfLength h.val) |
| (fun h => activeHalfLength h.val) (fun h => activeParentShape h.val) |
| (fun h => AllFieldHistoryRecovery.halfLength_le_eight h.val) hsupp e |
| (a 0) (a 1) (a 2) hsource |
| (physical_coarseMask allocation m sigma e a hcoarse 0) |
| (physical_coarseMask allocation m sigma e a hcoarse 1) |
| (physical_coarseMask allocation m sigma e a hcoarse 2) h u hu |
| cases right |
| · exact hp.1 |
| · exact hp.2 |
|
|
| theorem priority_half_product_nonzero (F : Type*) [Field F] |
| (allocation : Allocation) (m : ℕ) (sigma : Placement) |
| (e : Targets (K := K) (tick := tick) allocation m sigma) |
| (a : Fin 3 → Raw (K := K) (tick := tick) allocation m sigma) |
| (hraw : groupRawSource F allocation m sigma (a 0) (a 1) (a 2) ≠ 0) |
| (hcoarse : reorder sigma (ownWord allocation m sigma (a 0), |
| ownWord allocation m sigma (a 1), ownWord allocation m sigma (a 2)) = |
| AllFieldGroupOrbitData.coarse allocation m sigma e) |
| (right : Bool) (h : ActiveOrder K tick sigma) (u : JointPopulation.Shape) |
| (hu : 0 < Counts allocation m sigma h u) : |
| (∏ i : Class (Counts allocation m sigma) e h u, |
| shapeTensor (F := F) (Fin (activeHalfLength h.val)) (halfShape right h u ∘ sigma) |
| (if right then (a (sigma 0) h i.val).2 else (a (sigma 0) h i.val).1) |
| (if right then (a (sigma 1) h i.val).2 else (a (sigma 1) h i.val).1) |
| (if right then (a (sigma 2) h i.val).2 else (a (sigma 2) h i.val).1)) ≠ 0 := by |
| have he := fun i : Class (Counts allocation m sigma) e h u => |
| CWShapePhysicalPermutation.shapeTensor_permute (F := F) sigma (halfShape right h u) |
| (fun s => if right then (a s h i.val).2 else (a s h i.val).1) |
| simp_rw [he] |
| exact physical_half_product_nonzero F allocation m sigma e a hraw hcoarse right h u hu |
|
|
| theorem useful_half_window (allocation : Allocation) (m : ℕ) (ε : ℝ) |
| (sigma : Placement) (s : Fin 3) |
| (e : Targets (K := K) (tick := tick) allocation m sigma) |
| (w : Raw (K := K) (tick := tick) allocation m sigma) |
| (hw : groupIdealSide allocation m ε s sigma e w) |
| (right : Bool) (h : ActiveOrder K tick sigma) (u : JointPopulation.Shape) |
| (hu : 0 < Counts allocation m sigma h u) : |
| typeWindow (halfLaw right s h u) (AllFieldHistoryMasks.childWidth ε h.val) |
| (fun i : Class (Counts allocation m sigma) e h u => |
| statistic h.val (if right then (w h i.val).2 else (w h i.val).1)) := by |
| cases right |
| · exact (hw.2 h u hu).1 |
| · exact (hw.2 h u hu).2 |
|
|
| theorem own_coarse_guard (allocation : Allocation) (m : ℕ) (sigma : Placement) |
| (e : Targets (K := K) (tick := tick) allocation m sigma) |
| (a : Fin 3 → Raw (K := K) (tick := tick) allocation m sigma) |
| (hcoarse : reorder sigma (ownWord allocation m sigma (a 0), |
| ownWord allocation m sigma (a 1), ownWord allocation m sigma (a 2)) = |
| AllFieldGroupOrbitData.coarse allocation m sigma e) |
| (side : Fin 3) : |
| tripleSide side (AllFieldGroupOrbitData.coarse allocation m sigma e) = |
| ownWord allocation m sigma (a (sigma side)) := by |
| rw [← hcoarse, tripleSide_reorder] |
| generalize sigma side = s |
| fin_cases s <;> rfl |
|
|
| theorem transferY (F : Type*) [Field F] (allocation : Allocation) (m : ℕ) (ε : ℝ) |
| (sigma : Placement) (e : Targets (K := K) (tick := tick) allocation m sigma) |
| (a : Fin 3 → Raw (K := K) (tick := tick) allocation m sigma) |
| (hraw : groupRawSource F allocation m sigma (a 0) (a 1) (a 2) ≠ 0) |
| (hcoarse : reorder sigma (ownWord allocation m sigma (a 0), |
| ownWord allocation m sigma (a 1), ownWord allocation m sigma (a 2)) = |
| AllFieldGroupOrbitData.coarse allocation m sigma e) |
| (huseX : groupIdealSide allocation m ε (sigma 0) sigma e (a (sigma 0))) : |
| Compatible allocation m ε sigma 1 e (a (sigma 1)) := by |
| refine ⟨own_coarse_guard allocation m sigma e a hcoarse 1, ?_⟩ |
| intro right h u hd |
| apply JointPopulationCompatibility.shape_count_window_of_typeWindow |
| (Counts allocation m sigma) e h u _ _ _ |
| intro hu |
| have hz : halfShape right h u (sigma 2) = 0 := by |
| cases right <;> simpa [designated, halfShape, shapeNat] using hd.2 |
| have hw := CWStrandCompatibility.zeroZ_transfer |
| (fun _ : Class (Counts allocation m sigma) e h u => halfShape right h u ∘ sigma) |
| (fun i => if right then (a (sigma 0) h i.val).2 else (a (sigma 0) h i.val).1) |
| (fun i => if right then (a (sigma 1) h i.val).2 else (a (sigma 1) h i.val).1) |
| (fun i => if right then (a (sigma 2) h i.val).2 else (a (sigma 2) h i.val).1) |
| (priority_half_product_nonzero F allocation m sigma e a hraw hcoarse right h u hu) |
| (statistic h.val) (complement h) |
| (AllFieldGroupStatisticComplement.statistic_wordComplement h) |
| (halfLaw right (sigma 0) h u) (AllFieldHistoryMasks.childWidth ε h.val) |
| (fun _ => hz) (useful_half_window allocation m ε sigma (sigma 0) e _ huseX right h u hu) |
| simpa [compatibilityLaw, halfLaw] using hw |
|
|
| theorem transferZ (F : Type*) [Field F] (allocation : Allocation) (m : ℕ) (ε : ℝ) |
| (sigma : Placement) (e : Targets (K := K) (tick := tick) allocation m sigma) |
| (a : Fin 3 → Raw (K := K) (tick := tick) allocation m sigma) |
| (hraw : groupRawSource F allocation m sigma (a 0) (a 1) (a 2) ≠ 0) |
| (hcoarse : reorder sigma (ownWord allocation m sigma (a 0), |
| ownWord allocation m sigma (a 1), ownWord allocation m sigma (a 2)) = |
| AllFieldGroupOrbitData.coarse allocation m sigma e) |
| (huseX : groupIdealSide allocation m ε (sigma 0) sigma e (a (sigma 0))) |
| (huseY : groupIdealSide allocation m ε (sigma 1) sigma e (a (sigma 1))) : |
| Compatible allocation m ε sigma 2 e (a (sigma 2)) := by |
| refine ⟨own_coarse_guard allocation m sigma e a hcoarse 2, ?_⟩ |
| intro right h u hd |
| apply JointPopulationCompatibility.shape_count_window_of_typeWindow |
| (Counts allocation m sigma) e h u _ _ _ |
| intro hu |
| have hz : halfShape right h u (sigma 0) * halfShape right h u (sigma 1) = 0 := by |
| cases right <;> simpa [designated, halfShape, shapeNat] using hd.2 |
| have hp := priority_half_product_nonzero F allocation m sigma e a hraw hcoarse right h u hu |
| by_cases hy : halfShape right h u (sigma 1) = 0 |
| · have hw := CWStrandCompatibility.zeroY_transfer |
| (fun _ : Class (Counts allocation m sigma) e h u => halfShape right h u ∘ sigma) |
| (fun i => if right then (a (sigma 0) h i.val).2 else (a (sigma 0) h i.val).1) |
| (fun i => if right then (a (sigma 1) h i.val).2 else (a (sigma 1) h i.val).1) |
| (fun i => if right then (a (sigma 2) h i.val).2 else (a (sigma 2) h i.val).1) |
| hp (statistic h.val) (complement h) |
| (AllFieldGroupStatisticComplement.statistic_wordComplement h) |
| (halfLaw right (sigma 0) h u) (AllFieldHistoryMasks.childWidth ε h.val) |
| (fun _ => hy) (useful_half_window allocation m ε sigma (sigma 0) e _ huseX right h u hu) |
| have hy' : (if right then activeParentShape h.val (sigma 1) - |
| (shapeSide (sigma 1) u).val else (shapeSide (sigma 1) u).val) = 0 := by |
| cases right <;> simpa [halfShape, shapeNat] using hy |
| simpa [compatibilityLaw, halfLaw, hy'] using hw |
| · have hx : halfShape right h u (sigma 0) = 0 := (Nat.mul_eq_zero.mp hz).resolve_right hy |
| have hw := CWStrandCompatibility.zeroX_transfer |
| (fun _ : Class (Counts allocation m sigma) e h u => halfShape right h u ∘ sigma) |
| (fun i => if right then (a (sigma 0) h i.val).2 else (a (sigma 0) h i.val).1) |
| (fun i => if right then (a (sigma 1) h i.val).2 else (a (sigma 1) h i.val).1) |
| (fun i => if right then (a (sigma 2) h i.val).2 else (a (sigma 2) h i.val).1) |
| hp (statistic h.val) (complement h) |
| (AllFieldGroupStatisticComplement.statistic_wordComplement h) |
| (halfLaw right (sigma 1) h u) (AllFieldHistoryMasks.childWidth ε h.val) |
| (fun _ => hx) (useful_half_window allocation m ε sigma (sigma 1) e _ huseY right h u hu) |
| have hy' : (if right then activeParentShape h.val (sigma 1) - |
| (shapeSide (sigma 1) u).val else (shapeSide (sigma 1) u).val) ≠ 0 := by |
| cases right <;> simpa [halfShape, shapeNat] using hy |
| simpa [compatibilityLaw, halfLaw, hy'] using hw |
|
|
| end MatrixMultiplication.AllFieldGroupCompatibilityTransfer |
|
|
| end |
| end |
|
|
| end MatrixAllFields |
|
|
| namespace MatrixAllFields |
|
|
| open scoped BigOperators Topology Polynomial |
|
|
| section |
| noncomputable section |
|
|
| namespace MatrixMultiplication.AllFieldGroupAssignments |
|
|
| open AllFieldHistory AllFieldHistoryGroupMasks AllFieldHistoryGroupedRecovery |
| open AllFieldGroupOrbitData AllFieldGroupAssignmentRetention JointCoarseHashing |
| attribute [local instance] Classical.propDecidable |
|
|
| variable {K tick : ℕ} |
|
|
| theorem actualCompatible_own (allocation : Allocation) (m : ℕ) (ε : ℝ) |
| (sigma : Placement) (t : Triple (Pos (K := K) (tick := tick) allocation m sigma)) |
| (v : Variable (K := K) (tick := tick) allocation m sigma) |
| (hc : actualCompatible allocation m ε sigma t v) : |
| tripleSide v.1 t = ownWord allocation m sigma v.2 := by |
| obtain ⟨e, he, hc⟩ := hc |
| rw [← he] |
| exact hc.1 |
|
|
| theorem physicalAssignment_coherent (F : Type*) [Field F] |
| (allocation : Allocation) (m : ℕ) (ε : ℝ) (sigma : Placement) |
| (k : ℕ) (U : Finset (ZMod (37 ^ k))) |
| (sample : Sample (Pos (K := K) (tick := tick) allocation m sigma) k) : |
| JointExtraction.Coherent (groupPreparedSource F allocation m ε sigma) |
| (physicalAssignment allocation m ε sigma k U sample) := by |
| intro x y z tx ty tz hT htx hty htz |
| let a : Fin 3 → Raw (K := K) (tick := tick) allocation m sigma := (Matrix.vecCons (x) (Matrix.vecCons (y) (Matrix.vecCons (z) Matrix.vecEmpty))) |
| let physicalTag : Fin 3 → Triple (Pos (K := K) (tick := tick) allocation m sigma) := |
| (Matrix.vecCons (tx) (Matrix.vecCons (ty) (Matrix.vecCons (tz) Matrix.vecEmpty))) |
| let b : Fin 3 → Raw (K := K) (tick := tick) allocation m sigma := fun r => a (sigma r) |
| let tag : Fin 3 → Triple (Pos (K := K) (tick := tick) allocation m sigma) := |
| fun r => physicalTag (sigma r) |
| have hphysical (r : Fin 3) : |
| assigned allocation m ε sigma k U sample (sigma.symm r, a r) = |
| some (physicalTag r) := by |
| fin_cases r |
| · exact htx |
| · exact hty |
| · exact htz |
| have hassign (r : Fin 3) : |
| assigned allocation m ε sigma k U sample (r, b r) = some (tag r) := by |
| simpa only [b, tag, sigma.symm_apply_apply] using hphysical (sigma r) |
| have hspec (r : Fin 3) := |
| assigned_spec allocation m ε sigma k U sample (r, b r) (tag r) (hassign r) |
| let d := data (K := K) (tick := tick) allocation m ε sigma |
| let hx := JointMaskedSelection.OrbitData.xFlag k U sample |
| let hy := JointMaskedSelection.OrbitData.yFlag k U sample |
| let hz := d.zFlag k U sample |
| have heligible (r : Fin 3) := |
| (JointCoarseAssignment.mem_eligible_iff d.ambient d.actualTargets hx hy hz (tag r)).mp |
| (hspec r).1 |
| let observed : Triple (Pos (K := K) (tick := tick) allocation m sigma) := |
| reorder sigma (ownWord allocation m sigma x, |
| ownWord allocation m sigma y, ownWord allocation m sigma z) |
| have hword (r : Fin 3) : |
| ownWord allocation m sigma (a r) = |
| tripleSide r (ownWord allocation m sigma x, |
| ownWord allocation m sigma y, ownWord allocation m sigma z) := by |
| fin_cases r <;> rfl |
| have hobserved (r : Fin 3) : tripleSide r observed = tripleSide r (tag r) := by |
| calc |
| tripleSide r observed = ownWord allocation m sigma (b r) := by |
| rw [show observed = reorder sigma (ownWord allocation m sigma x, ownWord allocation m sigma y, ownWord allocation m sigma z) from rfl, tripleSide_reorder] |
| exact (hword (sigma r)).symm |
| _ = tripleSide r (tag r) := |
| (actualCompatible_own allocation m ε sigma (tag r) (r, b r) (hspec r).2.1).symm |
| have hobsX : observed.1 = (tag 0).1 := hobserved 0 |
| have hobsY : observed.2.1 = (tag 1).2.1 := hobserved 1 |
| have hobsZ : observed.2.2 = (tag 2).2.2 := hobserved 2 |
| have hsurvives : JointCoarseAssignment.survives hx hy hz observed := by |
| refine ⟨?_, ?_, ?_⟩ |
| · rw [hobsX] |
| exact (heligible 0).2.2.1.1 |
| · rw [hobsY] |
| exact (heligible 1).2.2.1.2.1 |
| · rw [hobsZ] |
| exact (heligible 2).2.2.1.2.2 |
| have hsupport : observed ∈ d.ambient := |
| AllFieldGroupSourceSupport.prepared_source_support F allocation m ε sigma x y z hT |
| have hobs : observed = tag 0 := |
| (heligible 0).2.2.2 observed hsupport hsurvives hobsX |
| obtain ⟨e, he, huseX⟩ := (hspec 0).2.2.1 |
| have hcoarse : reorder sigma (ownWord allocation m sigma (a 0), |
| ownWord allocation m sigma (a 1), ownWord allocation m sigma (a 2)) = |
| coarse allocation m sigma e := hobs.trans he.symm |
| have hraw : groupRawSource F allocation m sigma (a 0) (a 1) (a 2) ≠ 0 := |
| AllFieldGroupSourceSupport.raw_nonzero F allocation m ε sigma x y z hT |
| have hcompY : actualCompatible allocation m ε sigma (tag 0) (1, b 1) := |
| ⟨e, he, AllFieldGroupCompatibilityTransfer.transferY F allocation m ε sigma e a |
| hraw hcoarse huseX⟩ |
| have h01 : tag 0 = tag 1 := (hspec 1).2.2.2 (tag 0) (hspec 0).1 hcompY |
| have huseY : groupIdealSide allocation m ε (sigma 1) sigma e (b 1) := by |
| apply (actualUseful_coarse allocation m ε sigma e (1, b 1)).mp |
| rw [he, h01] |
| exact (hspec 1).2.2.1 |
| have hcompZ : actualCompatible allocation m ε sigma (tag 0) (2, b 2) := |
| ⟨e, he, AllFieldGroupCompatibilityTransfer.transferZ F allocation m ε sigma e a |
| hraw hcoarse huseX huseY⟩ |
| have h02 : tag 0 = tag 2 := (hspec 2).2.2.2 (tag 0) (hspec 0).1 hcompZ |
| have htag (r : Fin 3) : tag r = tag 0 := by |
| fin_cases r |
| · rfl |
| · exact h01.symm |
| · exact h02.symm |
| have hphysicalTag (r : Fin 3) : physicalTag r = tag 0 := by |
| simpa only [tag, sigma.apply_symm_apply] using htag (sigma.symm r) |
| exact ⟨(hphysicalTag 0).trans (hphysicalTag 1).symm, |
| (hphysicalTag 0).trans (hphysicalTag 2).symm⟩ |
|
|
| end MatrixMultiplication.AllFieldGroupAssignments |
|
|
| end |
| end |
|
|
| end MatrixAllFields |
|
|
| end OAI |
|
|
