openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /CoppersmithWinograd /CWZeroGeometry.lean
| 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 | |