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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 | import OAI.LinearAlgebra.MatrixMultiplication.CoppersmithWinograd.CWOrientedZero
import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.ShapeEncoding
namespace OAI
/-! Coppersmith–Winograd tensors, tensor powers and local restrictions. -/
namespace MatrixMultiplication.CWZeroGeometry
open AllFieldParameters AllFieldHistory CWOrientedZero
def zeroAxis (g : Shape) : Fin 3 :=
if g 0 = 0 then 0 else if g 1 = 0 then 1 else 2
def maxAxis (g : Shape) : Fin 3 :=
if g 0 = shapeMax g then 0 else if g 1 = shapeMax g then 1 else 2
theorem zeroAxis_spec (g : Shape) (hz : ∃ i, g i = 0) : g (zeroAxis g) = 0 := by
unfold zeroAxis
split_ifs with h0 h1
· exact h0
· exact h1
· obtain ⟨i, hi⟩ := hz
fin_cases i <;> simp_all
theorem maxAxis_spec (g : Shape) : g (maxAxis g) = shapeMax g := by
have hm : shapeMax g = g 0 ∨ shapeMax g = g 1 ∨ shapeMax g = g 2 := by
unfold shapeMax
rcases le_total (g 0) (max (g 1) (g 2)) with h | h
· rw [max_eq_right h]
rcases le_total (g 1) (g 2) with h | h
· exact Or.inr (Or.inr (max_eq_right h))
· exact Or.inr (Or.inl (max_eq_left h))
· exact Or.inl (max_eq_left h)
unfold maxAxis
split_ifs with h0 h1
· exact h0
· exact h1
· rcases hm with hm | hm | hm
· exact (h0 hm.symm).elim
· exact (h1 hm.symm).elim
· exact hm.symm
theorem shapeMax_le_total (g : Shape) : shapeMax g ≤ shapeTotal g := by
exact max_le (coordinate_le_total g 0)
(max_le (coordinate_le_total g 1) (coordinate_le_total g 2))
theorem shapeMax_pos (g : Shape) (ht : 0 < shapeTotal g) : 0 < shapeMax g := by
have h0 : g 0 ≤ shapeMax g := le_max_left _ _
have h1 : g 1 ≤ shapeMax g := (le_max_left _ _).trans (le_max_right _ _)
have h2 : g 2 ≤ shapeMax g := (le_max_right _ _).trans (le_max_right _ _)
unfold shapeTotal at ht
omega
theorem zero_ne_selected (g : Shape) (zero selected : Fin 3)
(ht : 0 < shapeTotal g) (hz : g zero = 0) (hs : g selected = shapeMax g) :
zero ≠ selected := by
intro he
have hp := shapeMax_pos g ht
rw [← he, hz] at hs
omega
theorem max_add_complement (g : Shape) :
shapeMax g + (shapeTotal g - shapeMax g) = shapeTotal g :=
Nat.add_sub_of_le (shapeMax_le_total g)
theorem eq_placedShape (g : Shape) (zero selected : Fin 3)
(hz : g zero = 0) (hs : g selected = shapeMax g) (hne : zero ≠ selected) :
g = placedShape zero selected (shapeMax g) (shapeTotal g - shapeMax g) := by
funext side
fin_cases zero <;> fin_cases selected
all_goals try contradiction
all_goals fin_cases side <;>
simp_all [placedShape, shapeTotal]
theorem physicalShape_eq_placedShape (g : Shape) (phi : Equiv.Perm (Fin 3))
(zero selected : Fin 3) (hz : g zero = 0)
(hs : g selected = shapeMax g) (hne : zero ≠ selected) :
physicalShape phi g = placedShape (phi zero) (phi selected)
(shapeMax g) (shapeTotal g - shapeMax g) := by
funext side
calc
physicalShape phi g side =
placedShape zero selected (shapeMax g) (shapeTotal g - shapeMax g)
(phi.symm side) := congrFun (eq_placedShape g zero selected hz hs hne) _
_ = _ := by simp only [placedShape, Equiv.symm_apply_eq]
theorem selected_add_remaining (g : Shape) (zero selected side : Fin 3)
(hz : g zero = 0) (hs : g selected = shapeMax g) (hne : zero ≠ selected)
(hzs : side ≠ zero) (hss : side ≠ selected) :
g selected + g side = shapeTotal g := by
have hside : g side = shapeTotal g - shapeMax g := by
rw [congrFun (eq_placedShape g zero selected hz hs hne) side]
simp only [placedShape, ite_eq_right hzs, ite_eq_right hss]
rw [hs, hside]
exact max_add_complement g
end MatrixMultiplication.CWZeroGeometry
end OAI
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