openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Completion /GeometricValues.lean
| import OAI.LinearAlgebra.MatrixMultiplication.Completion.HierarchyWords | |
| import OAI.LinearAlgebra.MatrixMultiplication.Entropy.ComplexConditionalHierarchyStage | |
| import OAI.LinearAlgebra.MatrixMultiplication.Separation.ComplexPrefixSeparationOrientations | |
| import OAI.LinearAlgebra.MatrixMultiplication.Tensor.ComplexGeometricStageCoordinates | |
| namespace OAI | |
| /-! Readable tensor completion and its finite arithmetic realization. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.CompletionGeometricValues | |
| open MatrixMultiplication.Foundation | |
| open CompletionHierarchyWords ConditionalPrefixWords | |
| open ConditionalHierarchyStage PrefixSeparationPolynomialStage PrefixSeparationOrientations | |
| open StageHierarchyResources | |
| open scoped Classical | |
| variable {A X Y Z : Type*} [Fintype A] (H : ReadableHierarchy A X Y Z) | |
| variable {Position : Type*} [Fintype Position] {counts : A → ℕ} {n t : ℕ} | |
| abbrev size (counts : A → ℕ) (n t : ℕ) := stageRefinementCount counts H.labels n t | |
| theorem actualPool_decode (old : ExactPrefix counts H.labels n t Position) | |
| (code : Fin (size H counts n t)) : | |
| decodePool (actualPool counts H.labels n t id) (size H counts n t) | |
| (actualPool_card counts H.labels n t id) old code = | |
| ((conditionalPoolEquivFin old).symm code).val := rfl | |
| theorem decode_eq_iff_eligible (old : ExactPrefix counts H.labels n t Position) | |
| (code : Fin (size H counts n t)) | |
| (u : Position → X) (v : Position → Y) (w : Position → Z) | |
| (word : Position → A) (hword : Represents H id old u v w word) : | |
| decodePool (actualPool counts H.labels n t id) (size H counts n t) | |
| (actualPool_card counts H.labels n t id) old code = (fun i => H.labels n (word i)) ↔ | |
| eligible H id (appendCode old code) u v w := by | |
| rw [actualPool_decode] | |
| exact (eligible_appendCode_iff_of_represents H old code u v w word hword).symm | |
| theorem raw_XY_normal | |
| (old : ExactPrefix counts H.labels n t Position) (hn : n < H.depth) | |
| (hp : H.pair n = .xy) | |
| (x : (Position → X) × InformedOutput (size H counts n t)) | |
| (y : (Position → Y) × InformedOutput (size H counts n t)) | |
| (z : (Position → Z) × MissingOutput (size H counts n t)) | |
| (word : Position → A) (hword : Represents H id old x.1 y.1 z.1 word) : | |
| rawPoolLeading (actualPool counts H.labels n t id) (size H counts n t) | |
| (stageRefinementCount_pos counts H.labels n t) (actualPool_card counts H.labels n t id) | |
| (readXYX H id hp) (readXYY H id hp) old x y z = | |
| if x.2.1 = y.2.1 ∧ x.2.1 = z.2.1 ∧ | |
| eligible H id (appendCode old x.2.1) x.1 y.1 z.1 then | |
| (GeometricStageCoordinates.xy (size H counts n t)).tensor x.2.2 y.2.2 z.2.2 | |
| else 0 := by | |
| have hx : readXYX H id hp old x.1 = fun i => H.labels n (word i) := by | |
| rw [hword.1] | |
| exact readXYX_on_original_word H id hp old word hn hword.2.2.2 | |
| have hy : readXYY H id hp old y.1 = fun i => H.labels n (word i) := by | |
| rw [hword.2.1] | |
| exact readXYY_on_original_word H id hp old word hn hword.2.2.2 | |
| rw [rawPoolLeading_eq_decodedLeading (actualPool counts H.labels n t id) (size H counts n t) (stageRefinementCount_pos counts H.labels n t) (actualPool_card counts H.labels n t id) (readXYX H id hp) (readXYY H id hp) (fun _ _ _ _ => fun i => H.labels n (word i)) old x y z hx hy] | |
| unfold decodedLeading | |
| simp only [decode_eq_iff_eligible H 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 H.labels n t Position) (hn : n < H.depth) | |
| (hp : H.pair n = .xz) | |
| (x : (Position → X) × InformedOutput (size H counts n t)) | |
| (y : (Position → Y) × MissingOutput (size H counts n t)) | |
| (z : (Position → Z) × InformedOutput (size H counts n t)) | |
| (word : Position → A) (hword : Represents H id old x.1 y.1 z.1 word) : | |
| rawPoolLeadingXZ (actualPool counts H.labels n t id) (size H counts n t) | |
| (stageRefinementCount_pos counts H.labels n t) (actualPool_card counts H.labels n t id) | |
| (readXZX H id hp) (readXZZ H id hp) old x y z = | |
| if x.2.1 = y.2.1 ∧ x.2.1 = z.2.1 ∧ | |
| eligible H id (appendCode old x.2.1) x.1 y.1 z.1 then | |
| (GeometricStageCoordinates.xz (size H counts n t)).tensor x.2.2 y.2.2 z.2.2 | |
| else 0 := by | |
| have hx : readXZX H id hp old x.1 = fun i => H.labels n (word i) := by | |
| rw [hword.1] | |
| exact readXZX_on_original_word H id hp old word hn hword.2.2.2 | |
| have hz : readXZZ H id hp old z.1 = fun i => H.labels n (word i) := by | |
| rw [hword.2.2.1] | |
| exact readXZZ_on_original_word H id hp old word hn hword.2.2.2 | |
| unfold rawPoolLeadingXZ | |
| rw [rawPoolLeading_eq_decodedLeading (actualPool counts H.labels n t id) (size H counts n t) (stageRefinementCount_pos counts H.labels n t) (actualPool_card counts H.labels n t id) (readXZX H id hp) (readXZZ H id hp) (fun _ _ _ _ => fun i => H.labels n (word i)) old x z y hx hz] | |
| unfold decodedLeading | |
| simp only [decode_eq_iff_eligible H 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 H 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 H 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 H.labels n t Position) (hn : n < H.depth) | |
| (hp : H.pair n = .yz) | |
| (x : (Position → X) × MissingOutput (size H counts n t)) | |
| (y : (Position → Y) × InformedOutput (size H counts n t)) | |
| (z : (Position → Z) × InformedOutput (size H counts n t)) | |
| (word : Position → A) (hword : Represents H id old x.1 y.1 z.1 word) : | |
| rawPoolLeadingYZ (actualPool counts H.labels n t id) (size H counts n t) | |
| (stageRefinementCount_pos counts H.labels n t) (actualPool_card counts H.labels n t id) | |
| (readYZY H id hp) (readYZZ H id hp) old x y z = | |
| if x.2.1 = y.2.1 ∧ x.2.1 = z.2.1 ∧ | |
| eligible H id (appendCode old x.2.1) x.1 y.1 z.1 then | |
| (GeometricStageCoordinates.yz (size H counts n t)).tensor x.2.2 y.2.2 z.2.2 | |
| else 0 := by | |
| have hy : readYZY H id hp old y.1 = fun i => H.labels n (word i) := by | |
| rw [hword.2.1] | |
| exact readYZY_on_original_word H id hp old word hn hword.2.2.2 | |
| have hz : readYZZ H id hp old z.1 = fun i => H.labels n (word i) := by | |
| rw [hword.2.2.1] | |
| exact readYZZ_on_original_word H id hp old word hn hword.2.2.2 | |
| unfold rawPoolLeadingYZ | |
| rw [rawPoolLeading_eq_decodedLeading (actualPool counts H.labels n t id) (size H counts n t) (stageRefinementCount_pos counts H.labels n t) (actualPool_card counts H.labels n t id) (readYZY H id hp) (readYZZ H id hp) (fun _ _ _ _ => fun i => H.labels 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 H.labels n t id) (size H counts n t) | |
| (actualPool_card counts H.labels n t id) old y.2.1 = (fun i => H.labels n (word i)) ∧ | |
| y.2.2 = z.2.2) ↔ | |
| (x.2.1 = y.2.1 ∧ x.2.1 = z.2.1 ∧ | |
| eligible H 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 H 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 H 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.CompletionGeometricValues | |
| end | |
| end OAI | |