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cdec422 | 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 | import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.Transitions
namespace OAI
/-! Finite extraction histories, inherited masks and recovery bounds. -/
noncomputable section
namespace MatrixMultiplication.AllFieldHistory
open AllFieldParameters
attribute [local instance] Classical.propDecidable Classical.decEq
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
def currentPhysicalShape {K : ℕ} (h : History K) : Shape :=
physicalShape h.2 (currentShape h.1)
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
theorem source_length {K : ℕ} (w : Work K) : currentLength w.source = 2 * w.halfLength := by
cases w <;> rfl
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
theorem child_length {K : ℕ} (w : Work K) (b : w.Branch) (right : Bool) :
currentLength (w.child b right) = w.halfLength := by
cases w <;> rfl
end MatrixMultiplication.AllFieldHistory
end
end OAI
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