openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /ComplexBounds /CW75ReorderedGeometricValues.lean
| import OAI.LinearAlgebra.MatrixMultiplication.ComplexBounds.CW75ReorderedWords | |
| import OAI.LinearAlgebra.MatrixMultiplication.Entropy.ComplexConditionalHierarchyStage | |
| import OAI.LinearAlgebra.MatrixMultiplication.Separation.ComplexPrefixSeparationOrientations | |
| import OAI.LinearAlgebra.MatrixMultiplication.Tensor.ComplexGeometricStageCoordinates | |
| namespace OAI | |
| /-! Explicit complex square and rectangular matrix multiplication bounds. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.CW75ReorderedGeometricValues | |
| open MatrixMultiplication.Foundation | |
| open ComplexWitness CW75LabelHierarchy CW75ReorderedWords ConditionalPrefixWords | |
| open ConditionalHierarchyStage PrefixSeparationPolynomialStage PrefixSeparationOrientations | |
| open StageHierarchyResources | |
| open scoped Classical | |
| variable {Position : Type*} [Fintype Position] {counts : Scalar → ℕ} {n t : ℕ} | |
| abbrev size (counts : Scalar → ℕ) (n t : ℕ) := stageRefinementCount counts curveLabels n t | |
| theorem actualPool_decode (old : ExactPrefix counts curveLabels n t Position) | |
| (code : Fin (size counts n t)) : | |
| decodePool (actualPool counts curveLabels n t id) (size counts n t) | |
| (actualPool_card counts curveLabels n t id) old code = | |
| ((conditionalPoolEquivFin old).symm code).val := rfl | |
| theorem decode_eq_iff_eligible (old : ExactPrefix counts curveLabels n t Position) | |
| (code : Fin (size counts n t)) (u v w : LongWord Position) | |
| (word : Position → Scalar) (hword : Represents id old u v w word) : | |
| decodePool (actualPool counts curveLabels n t id) (size counts n t) | |
| (actualPool_card counts curveLabels n t id) old code = (fun i => curveLabels n (word i)) ↔ | |
| eligible id (appendCode old code) u v w := by | |
| rw [actualPool_decode] | |
| exact (eligible_appendCode_iff_of_represents old code u v w word hword).symm | |
| private theorem first_read (old : ExactPrefix counts curveLabels n t Position) | |
| (u v w : LongWord Position) (word : Position → Scalar) (hn : n < 5) | |
| (hword : Represents id old u v w word) : | |
| firstRead id old (coordinateOf (curveFirstSide n) u v w) = fun i => curveLabels n (word i) := by | |
| rw [hword.1, hword.2.1, hword.2.2.1, coordinateOf_original] | |
| exact firstRead_on_original_word id old word hn hword.2.2.2 | |
| private theorem second_read (old : ExactPrefix counts curveLabels n t Position) | |
| (u v w : LongWord Position) (word : Position → Scalar) (hn : n < 5) | |
| (hword : Represents id old u v w word) : | |
| secondRead id old (coordinateOf (curveSecondSide n) u v w) = fun i => curveLabels n (word i) := by | |
| rw [hword.1, hword.2.1, hword.2.2.1, coordinateOf_original] | |
| exact secondRead_on_original_word id old word hn hword.2.2.2 | |
| theorem raw_XY_normal | |
| (old : ExactPrefix counts curveLabels n t Position) (hn : n < 5) | |
| (hfirst : curveFirstSide n = .x) (hsecond : curveSecondSide n = .y) | |
| (x : LongWord Position × InformedOutput (size counts n t)) | |
| (y : LongWord Position × InformedOutput (size counts n t)) | |
| (z : LongWord Position × MissingOutput (size counts n t)) | |
| (word : Position → Scalar) (hword : Represents id old x.1 y.1 z.1 word) : | |
| rawPoolLeading (actualPool counts curveLabels n t id) (size counts n t) | |
| (stageRefinementCount_pos counts curveLabels n t) (actualPool_card counts curveLabels n t id) | |
| (firstRead id) (secondRead id) old x y z = | |
| if x.2.1 = y.2.1 ∧ x.2.1 = z.2.1 ∧ | |
| eligible id (appendCode old x.2.1) x.1 y.1 z.1 then | |
| (GeometricStageCoordinates.xy (size counts n t)).tensor x.2.2 y.2.2 z.2.2 | |
| else 0 := by | |
| have hx := first_read old x.1 y.1 z.1 word hn hword | |
| have hy := second_read old x.1 y.1 z.1 word hn hword | |
| simp only [hfirst, coordinateOf] at hx | |
| simp only [hsecond, coordinateOf] at hy | |
| rw [rawPoolLeading_eq_decodedLeading (actualPool counts curveLabels n t id) (size counts n t) (stageRefinementCount_pos counts curveLabels n t) (actualPool_card counts curveLabels n t id) (firstRead id) (secondRead id) (fun _ _ _ _ => fun i => curveLabels n (word i)) old x y z hx hy] | |
| unfold decodedLeading | |
| simp only [decode_eq_iff_eligible old x.2.1 x.1 y.1 z.1 word hword, GeometricStageCoordinates.xy_tensor_apply] | |
| simp only [ite_and] | |
| theorem raw_XZ_normal | |
| (old : ExactPrefix counts curveLabels n t Position) (hn : n < 5) | |
| (hfirst : curveFirstSide n = .x) (hsecond : curveSecondSide n = .z) | |
| (x : LongWord Position × InformedOutput (size counts n t)) | |
| (y : LongWord Position × MissingOutput (size counts n t)) | |
| (z : LongWord Position × InformedOutput (size counts n t)) | |
| (word : Position → Scalar) (hword : Represents id old x.1 y.1 z.1 word) : | |
| rawPoolLeadingXZ (actualPool counts curveLabels n t id) (size counts n t) | |
| (stageRefinementCount_pos counts curveLabels n t) (actualPool_card counts curveLabels n t id) | |
| (firstRead id) (secondRead id) old x y z = | |
| if x.2.1 = y.2.1 ∧ x.2.1 = z.2.1 ∧ | |
| eligible id (appendCode old x.2.1) x.1 y.1 z.1 then | |
| (GeometricStageCoordinates.xz (size counts n t)).tensor x.2.2 y.2.2 z.2.2 | |
| else 0 := by | |
| have hx := first_read old x.1 y.1 z.1 word hn hword | |
| have hz := second_read old x.1 y.1 z.1 word hn hword | |
| simp only [hfirst, coordinateOf] at hx | |
| simp only [hsecond, coordinateOf] at hz | |
| unfold rawPoolLeadingXZ | |
| rw [rawPoolLeading_eq_decodedLeading (actualPool counts curveLabels n t id) (size counts n t) (stageRefinementCount_pos counts curveLabels n t) (actualPool_card counts curveLabels n t id) (firstRead id) (secondRead id) (fun _ _ _ _ => fun i => curveLabels n (word i)) old x z y hx hz] | |
| unfold decodedLeading | |
| simp only [decode_eq_iff_eligible old x.2.1 x.1 y.1 z.1 word hword, GeometricStageCoordinates.xz_tensor_apply] | |
| have hswap : | |
| (x.2.1 = z.2.1 ∧ x.2.1 = y.2.1 ∧ | |
| eligible id (appendCode old x.2.1) x.1 y.1 z.1 ∧ x.2.2 = z.2.2) ↔ | |
| (x.2.1 = y.2.1 ∧ x.2.1 = z.2.1 ∧ | |
| eligible id (appendCode old x.2.1) x.1 y.1 z.1 ∧ x.2.2 = z.2.2) := | |
| and_left_comm | |
| simp only [hswap, ite_and] | |
| theorem raw_YZ_normal | |
| (old : ExactPrefix counts curveLabels n t Position) (hn : n < 5) | |
| (hfirst : curveFirstSide n = .y) (hsecond : curveSecondSide n = .z) | |
| (x : LongWord Position × MissingOutput (size counts n t)) | |
| (y : LongWord Position × InformedOutput (size counts n t)) | |
| (z : LongWord Position × InformedOutput (size counts n t)) | |
| (word : Position → Scalar) (hword : Represents id old x.1 y.1 z.1 word) : | |
| rawPoolLeadingYZ (actualPool counts curveLabels n t id) (size counts n t) | |
| (stageRefinementCount_pos counts curveLabels n t) (actualPool_card counts curveLabels n t id) | |
| (firstRead id) (secondRead id) old x y z = | |
| if x.2.1 = y.2.1 ∧ x.2.1 = z.2.1 ∧ | |
| eligible id (appendCode old x.2.1) x.1 y.1 z.1 then | |
| (GeometricStageCoordinates.yz (size counts n t)).tensor x.2.2 y.2.2 z.2.2 | |
| else 0 := by | |
| have hy := first_read old x.1 y.1 z.1 word hn hword | |
| have hz := second_read old x.1 y.1 z.1 word hn hword | |
| simp only [hfirst, coordinateOf] at hy | |
| simp only [hsecond, coordinateOf] at hz | |
| unfold rawPoolLeadingYZ | |
| rw [rawPoolLeading_eq_decodedLeading (actualPool counts curveLabels n t id) (size counts n t) (stageRefinementCount_pos counts curveLabels n t) (actualPool_card counts curveLabels n t id) (firstRead id) (secondRead id) (fun _ _ _ _ => fun i => curveLabels n (word i)) old y z x hy hz] | |
| have hguard : | |
| (y.2.1 = z.2.1 ∧ y.2.1 = x.2.1 ∧ | |
| decodePool (actualPool counts curveLabels n t id) (size counts n t) | |
| (actualPool_card counts curveLabels n t id) old y.2.1 = (fun i => curveLabels n (word i)) ∧ | |
| y.2.2 = z.2.2) ↔ | |
| (x.2.1 = y.2.1 ∧ x.2.1 = z.2.1 ∧ | |
| eligible id (appendCode old x.2.1) x.1 y.1 z.1 ∧ y.2.2 = z.2.2) := by | |
| constructor | |
| · rintro ⟨hyz, hyx, hdecode, hpair⟩ | |
| refine ⟨hyx.symm, hyx.symm.trans hyz, ?_, hpair⟩ | |
| apply (decode_eq_iff_eligible old x.2.1 x.1 y.1 z.1 word hword).mp | |
| simpa only [hyx] using hdecode | |
| · rintro ⟨hxy, hxz, he, hpair⟩ | |
| refine ⟨hxy.symm.trans hxz, hxy.symm, ?_, hpair⟩ | |
| have hdecode := (decode_eq_iff_eligible old x.2.1 x.1 y.1 z.1 word hword).mpr he | |
| simpa only [hxy] using hdecode | |
| unfold decodedLeading | |
| simp only [hguard, GeometricStageCoordinates.yz_tensor_apply] | |
| simp only [ite_and] | |
| end MatrixMultiplication.CW75ReorderedGeometricValues | |
| end | |
| end OAI | |