| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.Geometry |
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| namespace OAI |
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| /-! Finite extraction histories, inherited masks and recovery bounds. -/ |
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| noncomputable section |
| namespace MatrixMultiplication.AllFieldHistory |
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|
| open AllFieldParameters |
| attribute [local instance] Classical.propDecidable Classical.decEq |
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|
| abbrev ProducedPositions {K tick : ℕ} (allocation : Allocation) (dilation : ℕ) := |
| Σ h : Produced K tick, Fin (population allocation dilation h.val) |
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| abbrev SplitPositions {K tick : ℕ} (allocation : Allocation) (dilation : ℕ) := |
| Σ w : Active K tick, Σ b : w.val.1.Branch, |
| Bool × Fin (branchPopulation allocation dilation w.val b) |
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| def halfPosition {K : ℕ} (allocation : Allocation) (dilation : ℕ) |
| (w : PlacedWork K) (b : w.1.Branch) (right : Bool) |
| (i : Fin (branchPopulation allocation dilation w b)) : |
| Fin (population allocation dilation (w.1.child b right, w.2)) := |
| ⟨i.val, by rw [branchPopulation_half]; exact i.isLt⟩ |
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|
| def unhalfPosition {K : ℕ} (allocation : Allocation) (dilation : ℕ) |
| (w : PlacedWork K) (b : w.1.Branch) (right : Bool) |
| (i : Fin (population allocation dilation (w.1.child b right, w.2))) : |
| Fin (branchPopulation allocation dilation w b) := |
| ⟨i.val, by rw [← branchPopulation_half allocation dilation w b right]; exact i.isLt⟩ |
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| @[simp] theorem unhalf_halfPosition {K : ℕ} (allocation : Allocation) (dilation : ℕ) |
| (w : PlacedWork K) (b : w.1.Branch) (right : Bool) |
| (i : Fin (branchPopulation allocation dilation w b)) : |
| unhalfPosition allocation dilation w b right (halfPosition allocation dilation w b right i) = i := rfl |
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| @[simp] theorem half_unhalfPosition {K : ℕ} (allocation : Allocation) (dilation : ℕ) |
| (w : PlacedWork K) (b : w.1.Branch) (right : Bool) |
| (i : Fin (population allocation dilation (w.1.child b right, w.2))) : |
| halfPosition allocation dilation w b right (unhalfPosition allocation dilation w b right i) = i := rfl |
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| def splitToProduced {K tick : ℕ} (allocation : Allocation) (dilation : ℕ) : |
| SplitPositions (K := K) (tick := tick) allocation dilation → |
| ProducedPositions (K := K) (tick := tick) allocation dilation := by |
| rintro ⟨⟨⟨w, phi⟩, hw⟩, b, right, i⟩ |
| cases w with |
| | stageA h => |
| exact ⟨⟨(.afterA ⟨h, b, right⟩, phi), hw.symm⟩, |
| halfPosition allocation dilation (.stageA h, phi) b right i⟩ |
| | stageB h => |
| let child : AfterB K := ⟨h, b, right⟩ |
| let j := halfPosition allocation dilation (.stageB h, phi) b right i |
| by_cases hp : positive (bShape child) = true |
| · let part := bSubdivisionPositionEquiv allocation dilation ⟨child, hp⟩ phi j |
| exact ⟨⟨(.partC (⟨child, hp⟩, part.1), phi), hw.symm⟩, part.2⟩ |
| · exact ⟨⟨(.afterB child, phi), hw.symm, hp⟩, j⟩ |
| | stageC h => |
| exact ⟨⟨(.afterC (h, b, right), phi), hw.symm⟩, |
| halfPosition allocation dilation (.stageC h, phi) b right i⟩ |
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|
| def producedToSplit {K tick : ℕ} (allocation : Allocation) (dilation : ℕ) : |
| ProducedPositions (K := K) (tick := tick) allocation dilation → |
| SplitPositions (K := K) (tick := tick) allocation dilation := by |
| rintro ⟨⟨⟨h, phi⟩, hp⟩, i⟩ |
| cases h with |
| | initial h => exact False.elim hp |
| | afterA h => |
| exact ⟨⟨(.stageA h.1, phi), hp.symm⟩, h.2.1, h.2.2, |
| unhalfPosition allocation dilation (.stageA h.1, phi) h.2.1 h.2.2 i⟩ |
| | afterB h => |
| exact ⟨⟨(.stageB h.1, phi), hp.1.symm⟩, h.2.1, h.2.2, |
| unhalfPosition allocation dilation (.stageB h.1, phi) h.2.1 h.2.2 i⟩ |
| | partC h => |
| let j := (bSubdivisionPositionEquiv allocation dilation h.1 phi).symm ⟨h.2, i⟩ |
| exact ⟨⟨(.stageB h.1.val.1, phi), hp.symm⟩, h.1.val.2.1, h.1.val.2.2, |
| unhalfPosition allocation dilation (.stageB h.1.val.1, phi) |
| h.1.val.2.1 h.1.val.2.2 j⟩ |
| | afterC h => |
| exact ⟨⟨(.stageC h.1, phi), hp.symm⟩, h.2.1, h.2.2, |
| unhalfPosition allocation dilation (.stageC h.1, phi) h.2.1 h.2.2 i⟩ |
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| theorem producedToSplit_splitToProduced {K tick : ℕ} (allocation : Allocation) (dilation : ℕ) |
| (p : SplitPositions (K := K) (tick := tick) allocation dilation) : |
| producedToSplit allocation dilation (splitToProduced allocation dilation p) = p := by |
| rcases p with ⟨⟨⟨w, phi⟩, hw⟩, b, right, i⟩ |
| cases w with |
| | stageA h => rfl |
| | stageB h => |
| by_cases hp : positive (bShape (⟨h, b, right⟩ : AfterB K)) = true |
| · dsimp only [splitToProduced] |
| rw [dite_eq_left hp] |
| dsimp only [producedToSplit] |
| simp only [Sigma.eta] |
| let lift : Fin (population allocation dilation |
| (.afterB ⟨h, b, right⟩, phi)) → |
| SplitPositions (K := K) (tick := tick) allocation dilation := |
| fun j => ⟨⟨(.stageB h, phi), hw⟩, b, right, |
| unhalfPosition allocation dilation (.stageB h, phi) b right j⟩ |
| exact congrArg lift ((bSubdivisionPositionEquiv allocation dilation |
| ⟨⟨h, b, right⟩, hp⟩ phi).symm_apply_apply |
| (halfPosition allocation dilation (.stageB h, phi) b right i)) |
| · dsimp only [splitToProduced] |
| rw [dite_eq_right hp] |
| rfl |
| | stageC h => rfl |
|
|
| theorem splitToProduced_producedToSplit {K tick : ℕ} (allocation : Allocation) (dilation : ℕ) |
| (p : ProducedPositions (K := K) (tick := tick) allocation dilation) : |
| splitToProduced allocation dilation (producedToSplit allocation dilation p) = p := by |
| rcases p with ⟨⟨⟨h, phi⟩, hp⟩, i⟩ |
| cases h with |
| | initial h => exact False.elim hp |
| | afterA h => rfl |
| | afterB h => |
| rcases h with ⟨a, b, right⟩ |
| dsimp only [producedToSplit, splitToProduced] |
| rw [dite_eq_right hp.2] |
| rfl |
| | partC h => |
| rcases h with ⟨⟨⟨a, b, right⟩, hb⟩, part⟩ |
| dsimp only [producedToSplit, splitToProduced] |
| rw [dite_eq_left hb] |
| let lift : (Σ j : Fin 3, |
| Fin (population allocation dilation |
| (.partC (⟨⟨a, b, right⟩, hb⟩, j), phi))) → |
| ProducedPositions (K := K) (tick := tick) allocation dilation := |
| fun q => ⟨⟨(.partC (⟨⟨a, b, right⟩, hb⟩, q.1), phi), hp⟩, q.2⟩ |
| exact congrArg lift ((bSubdivisionPositionEquiv allocation dilation |
| ⟨⟨a, b, right⟩, hb⟩ phi).apply_symm_apply ⟨part, i⟩) |
| | afterC h => rfl |
|
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| def producedPositionEquiv {K tick : ℕ} (allocation : Allocation) (dilation : ℕ) : |
| ProducedPositions (K := K) (tick := tick) allocation dilation ≃ |
| SplitPositions (K := K) (tick := tick) allocation dilation where |
| toFun := producedToSplit allocation dilation |
| invFun := splitToProduced allocation dilation |
| left_inv := splitToProduced_producedToSplit allocation dilation |
| right_inv := producedToSplit_splitToProduced allocation dilation |
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|
| theorem producedPositionEquiv_parent {K tick : ℕ} (allocation : Allocation) (dilation : ℕ) |
| (p : ProducedPositions (K := K) (tick := tick) allocation dilation) : |
| (producedPositionEquiv allocation dilation p).1 = productionParent p.1 := by |
| rcases p with ⟨⟨⟨h, phi⟩, hp⟩, i⟩ |
| cases h with |
| | initial h => exact False.elim hp |
| | afterA h => rfl |
| | afterB h => rfl |
| | partC h => rfl |
| | afterC h => rfl |
|
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| theorem producedPositionEquiv_shape {K tick : ℕ} (allocation : Allocation) (dilation : ℕ) |
| (p : ProducedPositions (K := K) (tick := tick) allocation dilation) : |
| currentPhysicalShape p.1.val = |
| physicalShape (producedPositionEquiv allocation dilation p).1.val.2 |
| (halfShape (producedPositionEquiv allocation dilation p).1.val.1.parentShape |
| ((producedPositionEquiv allocation dilation p).1.val.1.splitShape |
| (producedPositionEquiv allocation dilation p).2.1) |
| (producedPositionEquiv allocation dilation p).2.2.1) := by |
| rcases p with ⟨⟨⟨h, phi⟩, hp⟩, i⟩ |
| cases h with |
| | initial h => exact False.elim hp |
| | afterA h => rfl |
| | afterB h => rfl |
| | partC h => rfl |
| | afterC h => rfl |
|
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| theorem producedPositionEquiv_length {K tick : ℕ} (allocation : Allocation) (dilation : ℕ) |
| (p : ProducedPositions (K := K) (tick := tick) allocation dilation) : |
| currentLength p.1.val.1 = (producedPositionEquiv allocation dilation p).1.val.1.halfLength := by |
| rcases p with ⟨⟨⟨h, phi⟩, hp⟩, i⟩ |
| cases h with |
| | initial h => exact False.elim hp |
| | afterA h => rfl |
| | afterB h => rfl |
| | partC h => rfl |
| | afterC h => rfl |
|
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| end MatrixMultiplication.AllFieldHistory |
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| end |
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| end OAI |
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