| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.Transitions |
|
|
| 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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|
| def currentShape {K : ℕ} : CanonicalHistory K → Shape |
| | .initial h => initialShape h |
| | .afterA h => aShape h |
| | .afterB h => bShape h |
| | .partC h => cShapeParent h |
| | .afterC h => cShape h |
|
|
| def currentLength {K : ℕ} : CanonicalHistory K → ℕ |
| | .initial _ => 8 |
| | .afterA _ => 4 |
| | .afterB _ | .partC _ => 2 |
| | .afterC _ => 1 |
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|
| def currentPhysicalShape {K : ℕ} (h : History K) : Shape := |
| physicalShape h.2 (currentShape h.1) |
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|
| theorem currentShape_spec {K : ℕ} (h : CanonicalHistory K) : |
| ShapeBounded (currentShape h) ∧ shapeTotal (currentShape h) = 2 * currentLength h := by |
| cases h with |
| | initial h => exact root_shape_spec _ (initialShape_mem h) |
| | afterA h => exact ⟨mem_shapes_bounded (aShape_size h) (by omega), mem_shapes_total (aShape_size h)⟩ |
| | afterB h => exact ⟨mem_shapes_bounded (bShape_size h) (by omega), mem_shapes_total (bShape_size h)⟩ |
| | partC h => exact ⟨mem_shapes_bounded (bShape_size h.1.val) (by omega), |
| mem_shapes_total (bShape_size h.1.val)⟩ |
| | afterC h => |
| cases hh : h.2.2 with |
| | false => |
| simpa only [currentShape, currentLength, cShape, cSplit, cShapeParent, |
| halfShape, hh, Bool.false_eq_true, ite_false, mul_one] using |
| stageC_shape_spec _ (bShape_size h.1.1.val) h.1.1.property h.2.1 |
| | true => |
| simpa only [currentShape, currentLength, cShape, cSplit, cShapeParent, |
| halfShape, hh, ite_true, mul_one] using |
| stageC_complement_spec _ (bShape_size h.1.1.val) h.1.1.property h.2.1 |
|
|
| theorem currentPhysicalShape_total {K : ℕ} (h : History K) : |
| shapeTotal (currentPhysicalShape h) = 2 * currentLength h.1 := |
| (physicalShape_total h.2 _).trans (currentShape_spec h.1).2 |
|
|
| theorem positive_iff_no_zero (u : Shape) : positive u = true ↔ ∀ i, u i ≠ 0 := by |
| simp only [positive, Bool.and_eq_true, decide_eq_true_eq] |
| constructor |
| · intro h i |
| fin_cases i |
| · change u 0 ≠ 0 |
| omega |
| · change u 1 ≠ 0 |
| omega |
| · change u 2 ≠ 0 |
| omega |
| · intro h |
| have h0 := h 0 |
| have h1 := h 1 |
| have h2 := h 2 |
| omega |
|
|
| theorem not_positive_has_zero (u : Shape) (hu : positive u ≠ true) : ∃ i, u i = 0 := by |
| by_contra hn |
| apply hu |
| rw [positive_iff_no_zero] |
| intro i hi |
| exact hn ⟨i, hi⟩ |
|
|
| theorem terminal_has_zero {K : ℕ} (h : CanonicalHistory K) (ht : terminal h) : |
| ∃ i, currentShape h i = 0 := by |
| cases h with |
| | initial h => |
| apply not_positive_has_zero |
| intro hp |
| exact ht (List.mem_filter.mpr ⟨initialShape_mem h, hp⟩) |
| | afterA h => |
| apply not_positive_has_zero |
| intro hp |
| exact ht (List.mem_filter.mpr ⟨aShape_size h, hp⟩) |
| | afterB h => exact not_positive_has_zero _ ht |
| | partC h => exact False.elim ht |
| | afterC h => exact cShape_zero h |
|
|
| theorem terminal_physical_zero {K : ℕ} (h : History K) (ht : terminal h.1) : |
| ∃ s, currentPhysicalShape h s = 0 := by |
| obtain ⟨i, hi⟩ := terminal_has_zero h.1 ht |
| exact ⟨h.2 i, by simpa only [currentPhysicalShape, physicalShape_source] using hi⟩ |
|
|
| theorem final_physical_zero {K : ℕ} (h : State K (K + 2)) : |
| ∃ s, currentPhysicalShape h.val s = 0 := |
| terminal_physical_zero h.val (final_state_terminal h) |
|
|
| theorem source_shape {K : ℕ} (w : Work K) : currentShape w.source = w.parentShape := by |
| cases w <;> rfl |
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| theorem source_length {K : ℕ} (w : Work K) : currentLength w.source = 2 * w.halfLength := by |
| cases w <;> rfl |
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| theorem child_shape {K : ℕ} (w : Work K) (b : w.Branch) (right : Bool) : |
| currentShape (w.child b right) = halfShape w.parentShape (w.splitShape b) right := by |
| cases w <;> rfl |
|
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| theorem child_length {K : ℕ} (w : Work K) (b : w.Branch) (right : Bool) : |
| currentLength (w.child b right) = w.halfLength := by |
| cases w <;> rfl |
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| end MatrixMultiplication.AllFieldHistory |
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| end |
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| end OAI |
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