openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Completion /HierarchyWords.lean
| import OAI.LinearAlgebra.MatrixMultiplication.Entropy.ComplexConditionalPrefixWords | |
| import OAI.LinearAlgebra.MatrixMultiplication.Entropy.ConditionalLabels | |
| namespace OAI | |
| /-! Readable tensor completion and its finite arithmetic realization. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.CompletionHierarchyWords | |
| open MatrixMultiplication.Foundation ConditionalPrefixWords | |
| open scoped BigOperators Classical | |
| universe u v x y z p q | |
| structure ReadableHierarchy (A : Type u) (X : Type x) (Y : Type y) (Z : Type z) where | |
| Label : ℕ → Type v | |
| fintypeLabel : ∀ n, Fintype (Label n) | |
| labels : ∀ n, A → Label n | |
| depth : ℕ | |
| pair : ℕ → ReaderPair | |
| x : A → X | |
| y : A → Y | |
| z : A → Z | |
| coordinates_injective : Function.Injective (fun a => (x a, y a, z a)) | |
| readXYX : (n : ℕ) → pair n = .xy → LabelRecord Label n → X → Label n | |
| readXYY : (n : ℕ) → pair n = .xy → LabelRecord Label n → Y → Label n | |
| readXZX : (n : ℕ) → pair n = .xz → LabelRecord Label n → X → Label n | |
| readXZZ : (n : ℕ) → pair n = .xz → LabelRecord Label n → Z → Label n | |
| readYZY : (n : ℕ) → pair n = .yz → LabelRecord Label n → Y → Label n | |
| readYZZ : (n : ℕ) → pair n = .yz → LabelRecord Label n → Z → Label n | |
| readXYX_readable : ∀ n, n < depth → ∀ (hp : pair n = .xy) (a : A), | |
| readXYX n hp (labelRecordOf labels n a) (x a) = labels n a | |
| readXYY_readable : ∀ n, n < depth → ∀ (hp : pair n = .xy) (a : A), | |
| readXYY n hp (labelRecordOf labels n a) (y a) = labels n a | |
| readXZX_readable : ∀ n, n < depth → ∀ (hp : pair n = .xz) (a : A), | |
| readXZX n hp (labelRecordOf labels n a) (x a) = labels n a | |
| readXZZ_readable : ∀ n, n < depth → ∀ (hp : pair n = .xz) (a : A), | |
| readXZZ n hp (labelRecordOf labels n a) (z a) = labels n a | |
| readYZY_readable : ∀ n, n < depth → ∀ (hp : pair n = .yz) (a : A), | |
| readYZY n hp (labelRecordOf labels n a) (y a) = labels n a | |
| readYZZ_readable : ∀ n, n < depth → ∀ (hp : pair n = .yz) (a : A), | |
| readYZZ n hp (labelRecordOf labels n a) (z a) = labels n a | |
| attribute [instance] ReadableHierarchy.fintypeLabel | |
| variable {A : Type u} {X : Type x} {Y : Type y} {Z : Type z} | |
| variable (H : ReadableHierarchy A X Y Z) | |
| variable {Position : Type p} [Fintype Position] | |
| def longX (word : Position → A) : Position → X := fun i => H.x (word i) | |
| def longY (word : Position → A) : Position → Y := fun i => H.y (word i) | |
| def longZ (word : Position → A) : Position → Z := fun i => H.z (word i) | |
| omit [Fintype Position] in | |
| theorem longCoordinates_injective : Function.Injective | |
| (fun word : Position → A => (longX H word, longY H word, longZ H word)) := by | |
| intro word other h | |
| funext i | |
| apply H.coordinates_injective | |
| exact Prod.ext (congrFun (congrArg Prod.fst h) i) | |
| (Prod.ext (congrFun (congrArg (fun q => q.2.1) h) i) | |
| (congrFun (congrArg (fun q => q.2.2) h) i)) | |
| variable [Fintype A] {Prefix : Type q} {counts : A → ℕ} {n t : ℕ} | |
| def Represents (priorWord : Prefix → ExactPrefix counts H.labels n t Position) | |
| (prior : Prefix) (u : Position → X) (v : Position → Y) (w : Position → Z) | |
| (word : Position → A) : Prop := | |
| u = longX H word ∧ v = longY H word ∧ w = longZ H word ∧ | |
| ∀ i, (priorWord prior).val i = labelRecordOf H.labels n (word i) | |
| def eligible (priorWord : Prefix → ExactPrefix counts H.labels n t Position) | |
| (prior : Prefix) (u : Position → X) (v : Position → Y) (w : Position → Z) : Prop := | |
| ∃ word : Position → A, Represents H priorWord prior u v w word | |
| def readXYX (priorWord : Prefix → ExactPrefix counts H.labels n t Position) | |
| (hp : H.pair n = .xy) (prior : Prefix) (own : Position → X) : Position → H.Label n := | |
| fun i => H.readXYX n hp ((priorWord prior).val i) (own i) | |
| theorem readXYX_on_original_word | |
| (priorWord : Prefix → ExactPrefix counts H.labels n t Position) | |
| (hp : H.pair n = .xy) (prior : Prefix) (word : Position → A) | |
| (hn : n < H.depth) | |
| (hprior : ∀ i, (priorWord prior).val i = labelRecordOf H.labels n (word i)) : | |
| readXYX H priorWord hp prior (longX H word) = fun i => H.labels n (word i) := by | |
| funext i | |
| change H.readXYX n hp ((priorWord prior).val i) (H.x (word i)) = H.labels n (word i) | |
| rw [hprior i] | |
| exact H.readXYX_readable n hn hp (word i) | |
| def readXYY (priorWord : Prefix → ExactPrefix counts H.labels n t Position) | |
| (hp : H.pair n = .xy) (prior : Prefix) (own : Position → Y) : Position → H.Label n := | |
| fun i => H.readXYY n hp ((priorWord prior).val i) (own i) | |
| theorem readXYY_on_original_word | |
| (priorWord : Prefix → ExactPrefix counts H.labels n t Position) | |
| (hp : H.pair n = .xy) (prior : Prefix) (word : Position → A) | |
| (hn : n < H.depth) | |
| (hprior : ∀ i, (priorWord prior).val i = labelRecordOf H.labels n (word i)) : | |
| readXYY H priorWord hp prior (longY H word) = fun i => H.labels n (word i) := by | |
| funext i | |
| change H.readXYY n hp ((priorWord prior).val i) (H.y (word i)) = H.labels n (word i) | |
| rw [hprior i] | |
| exact H.readXYY_readable n hn hp (word i) | |
| def readXZX (priorWord : Prefix → ExactPrefix counts H.labels n t Position) | |
| (hp : H.pair n = .xz) (prior : Prefix) (own : Position → X) : Position → H.Label n := | |
| fun i => H.readXZX n hp ((priorWord prior).val i) (own i) | |
| theorem readXZX_on_original_word | |
| (priorWord : Prefix → ExactPrefix counts H.labels n t Position) | |
| (hp : H.pair n = .xz) (prior : Prefix) (word : Position → A) | |
| (hn : n < H.depth) | |
| (hprior : ∀ i, (priorWord prior).val i = labelRecordOf H.labels n (word i)) : | |
| readXZX H priorWord hp prior (longX H word) = fun i => H.labels n (word i) := by | |
| funext i | |
| change H.readXZX n hp ((priorWord prior).val i) (H.x (word i)) = H.labels n (word i) | |
| rw [hprior i] | |
| exact H.readXZX_readable n hn hp (word i) | |
| def readXZZ (priorWord : Prefix → ExactPrefix counts H.labels n t Position) | |
| (hp : H.pair n = .xz) (prior : Prefix) (own : Position → Z) : Position → H.Label n := | |
| fun i => H.readXZZ n hp ((priorWord prior).val i) (own i) | |
| theorem readXZZ_on_original_word | |
| (priorWord : Prefix → ExactPrefix counts H.labels n t Position) | |
| (hp : H.pair n = .xz) (prior : Prefix) (word : Position → A) | |
| (hn : n < H.depth) | |
| (hprior : ∀ i, (priorWord prior).val i = labelRecordOf H.labels n (word i)) : | |
| readXZZ H priorWord hp prior (longZ H word) = fun i => H.labels n (word i) := by | |
| funext i | |
| change H.readXZZ n hp ((priorWord prior).val i) (H.z (word i)) = H.labels n (word i) | |
| rw [hprior i] | |
| exact H.readXZZ_readable n hn hp (word i) | |
| def readYZY (priorWord : Prefix → ExactPrefix counts H.labels n t Position) | |
| (hp : H.pair n = .yz) (prior : Prefix) (own : Position → Y) : Position → H.Label n := | |
| fun i => H.readYZY n hp ((priorWord prior).val i) (own i) | |
| theorem readYZY_on_original_word | |
| (priorWord : Prefix → ExactPrefix counts H.labels n t Position) | |
| (hp : H.pair n = .yz) (prior : Prefix) (word : Position → A) | |
| (hn : n < H.depth) | |
| (hprior : ∀ i, (priorWord prior).val i = labelRecordOf H.labels n (word i)) : | |
| readYZY H priorWord hp prior (longY H word) = fun i => H.labels n (word i) := by | |
| funext i | |
| change H.readYZY n hp ((priorWord prior).val i) (H.y (word i)) = H.labels n (word i) | |
| rw [hprior i] | |
| exact H.readYZY_readable n hn hp (word i) | |
| def readYZZ (priorWord : Prefix → ExactPrefix counts H.labels n t Position) | |
| (hp : H.pair n = .yz) (prior : Prefix) (own : Position → Z) : Position → H.Label n := | |
| fun i => H.readYZZ n hp ((priorWord prior).val i) (own i) | |
| theorem readYZZ_on_original_word | |
| (priorWord : Prefix → ExactPrefix counts H.labels n t Position) | |
| (hp : H.pair n = .yz) (prior : Prefix) (word : Position → A) | |
| (hn : n < H.depth) | |
| (hprior : ∀ i, (priorWord prior).val i = labelRecordOf H.labels n (word i)) : | |
| readYZZ H priorWord hp prior (longZ H word) = fun i => H.labels n (word i) := by | |
| funext i | |
| change H.readYZZ n hp ((priorWord prior).val i) (H.z (word i)) = H.labels n (word i) | |
| rw [hprior i] | |
| exact H.readYZZ_readable n hn hp (word i) | |
| theorem readable_XY | |
| (priorWord : Prefix → ExactPrefix counts H.labels n t Position) | |
| (hp : H.pair n = .xy) (prior : Prefix) | |
| (u : Position → X) (v : Position → Y) (w : Position → Z) | |
| (hn : n < H.depth) (h : eligible H priorWord prior u v w) : | |
| readXYX H priorWord hp prior u = | |
| readXYY H priorWord hp prior v := by | |
| obtain ⟨word, hu, hv, hw, hprior⟩ := h | |
| rw [hu, hv] | |
| rw [readXYX_on_original_word H priorWord hp prior word hn hprior, | |
| readXYY_on_original_word H priorWord hp prior word hn hprior] | |
| theorem readable_XZ | |
| (priorWord : Prefix → ExactPrefix counts H.labels n t Position) | |
| (hp : H.pair n = .xz) (prior : Prefix) | |
| (u : Position → X) (v : Position → Y) (w : Position → Z) | |
| (hn : n < H.depth) (h : eligible H priorWord prior u v w) : | |
| readXZX H priorWord hp prior u = | |
| readXZZ H priorWord hp prior w := by | |
| obtain ⟨word, hu, hv, hw, hprior⟩ := h | |
| rw [hu, hw] | |
| rw [readXZX_on_original_word H priorWord hp prior word hn hprior, | |
| readXZZ_on_original_word H priorWord hp prior word hn hprior] | |
| theorem readable_YZ | |
| (priorWord : Prefix → ExactPrefix counts H.labels n t Position) | |
| (hp : H.pair n = .yz) (prior : Prefix) | |
| (u : Position → X) (v : Position → Y) (w : Position → Z) | |
| (hn : n < H.depth) (h : eligible H priorWord prior u v w) : | |
| readYZY H priorWord hp prior v = | |
| readYZZ H priorWord hp prior w := by | |
| obtain ⟨word, hu, hv, hw, hprior⟩ := h | |
| rw [hv, hw] | |
| rw [readYZY_on_original_word H priorWord hp prior word hn hprior, | |
| readYZZ_on_original_word H priorWord hp prior word hn hprior] | |
| theorem represented_word_unique | |
| (priorWord : Prefix → ExactPrefix counts H.labels n t Position) (prior : Prefix) | |
| (u : Position → X) (v : Position → Y) (w : Position → Z) {word other : Position → A} | |
| (hword : Represents H priorWord prior u v w word) | |
| (hother : Represents H priorWord prior u v w other) : word = other := by | |
| apply longCoordinates_injective H | |
| exact Prod.ext (hword.1.symm.trans hother.1) | |
| (Prod.ext (hword.2.1.symm.trans hother.2.1) | |
| (hword.2.2.1.symm.trans hother.2.2.1)) | |
| theorem eligible_appendCode_iff | |
| (old : ExactPrefix counts H.labels n t Position) | |
| (code : Fin (StageHierarchyResources.stageRefinementCount counts H.labels n t)) | |
| (u : Position → X) (v : Position → Y) (w : Position → Z) : | |
| eligible H id (appendCode old code) u v w ↔ | |
| ∃ word : Position → A, Represents H id old u v w word ∧ | |
| ((conditionalPoolEquivFin old).symm code).val = | |
| (fun i => H.labels n (word i)) := by | |
| constructor | |
| · rintro ⟨word, hu, hv, hw, hprior⟩ | |
| refine ⟨word, ⟨hu, hv, hw, ?_⟩, ?_⟩ | |
| · intro i | |
| have hi := hprior i | |
| change (old.val i, ((conditionalPoolEquivFin old).symm code).val i) = | |
| (labelRecordOf H.labels n (word i), H.labels n (word i)) at hi | |
| exact congrArg Prod.fst hi | |
| · funext i | |
| have hi := hprior i | |
| change (old.val i, ((conditionalPoolEquivFin old).symm code).val i) = | |
| (labelRecordOf H.labels n (word i), H.labels n (word i)) at hi | |
| exact congrArg Prod.snd hi | |
| · rintro ⟨word, ⟨hu, hv, hw, hprior⟩, hfine⟩ | |
| refine ⟨word, hu, hv, hw, ?_⟩ | |
| intro i | |
| change (old.val i, ((conditionalPoolEquivFin old).symm code).val i) = | |
| (labelRecordOf H.labels n (word i), H.labels n (word i)) | |
| exact Prod.ext (hprior i) (congrFun hfine i) | |
| theorem represented_fine_eq | |
| (priorWord : Prefix → ExactPrefix counts H.labels n t Position) (prior : Prefix) | |
| (u : Position → X) (v : Position → Y) (w : Position → Z) {word other : Position → A} | |
| (hword : Represents H priorWord prior u v w word) | |
| (hother : Represents H priorWord prior u v w other) : | |
| (fun i => H.labels n (word i)) = (fun i => H.labels n (other i)) := by | |
| rw [represented_word_unique H priorWord prior u v w hword hother] | |
| theorem eligible_appendCode_iff_of_represents | |
| (old : ExactPrefix counts H.labels n t Position) | |
| (code : Fin (StageHierarchyResources.stageRefinementCount counts H.labels n t)) | |
| (u : Position → X) (v : Position → Y) (w : Position → Z) (word : Position → A) | |
| (hword : Represents H id old u v w word) : | |
| eligible H id (appendCode old code) u v w ↔ | |
| ((conditionalPoolEquivFin old).symm code).val = (fun i => H.labels n (word i)) := by | |
| rw [eligible_appendCode_iff] | |
| constructor | |
| · rintro ⟨other, hother, hfine⟩ | |
| exact hfine.trans (represented_fine_eq H id old u v w hother hword) | |
| · intro hfine | |
| exact ⟨word, hword, hfine⟩ | |
| theorem root_eligible_iff | |
| (old : ExactPrefix counts H.labels 0 t Position) | |
| (u : Position → X) (v : Position → Y) (w : Position → Z) : | |
| eligible H id old u v w ↔ ∀ i, ∃ a : A, | |
| H.x a = u i ∧ H.y a = v i ∧ H.z a = w i := by | |
| constructor | |
| · rintro ⟨word, hu, hv, hw, _⟩ i | |
| refine ⟨word i, ?_, ?_, ?_⟩ | |
| · exact (congrFun hu i).symm | |
| · exact (congrFun hv i).symm | |
| · exact (congrFun hw i).symm | |
| · intro h | |
| choose word hx hy hz using h | |
| refine ⟨word, ?_, ?_, ?_, ?_⟩ | |
| · funext i | |
| exact (hx i).symm | |
| · funext i | |
| exact (hy i).symm | |
| · funext i | |
| exact (hz i).symm | |
| · intro i | |
| exact Subsingleton.elim _ _ | |
| def terminalWord (complete : Function.Injective (labelRecordOf H.labels H.depth)) | |
| (last : ExactPrefix counts H.labels H.depth t Position) : | |
| PopulationWords Position (fun a => t * counts a) := by | |
| let source := sourcePrefixEquiv counts H.labels H.depth t Position complete last | |
| refine ⟨source.val, ?_⟩ | |
| intro a | |
| refine Eq.trans ?_ (source.property a) | |
| unfold wordPopulation | |
| exact congrArg | |
| (fun inst : Fintype {i : Position // source.val i = a} => @Fintype.card _ inst) | |
| (Subsingleton.elim _ _) | |
| theorem terminalWord_record | |
| (complete : Function.Injective (labelRecordOf H.labels H.depth)) | |
| (last : ExactPrefix counts H.labels H.depth t Position) (i : Position) : | |
| labelRecordOf H.labels H.depth ((terminalWord H complete last).val i) = last.val i := | |
| sourceSymbol_record last i | |
| theorem terminalWord_of_represents | |
| (complete : Function.Injective (labelRecordOf H.labels H.depth)) | |
| (last : ExactPrefix counts H.labels H.depth t Position) | |
| (u : Position → X) (v : Position → Y) (w : Position → Z) | |
| (word : Position → A) (hword : Represents H id last u v w word) : | |
| (terminalWord H complete last).val = word := by | |
| funext i | |
| exact (sourceSymbol_eq_iff complete last i (word i)).2 (hword.2.2.2 i) | |
| theorem terminal_eligible_iff | |
| (complete : Function.Injective (labelRecordOf H.labels H.depth)) | |
| (last : ExactPrefix counts H.labels H.depth t Position) | |
| (u : Position → X) (v : Position → Y) (w : Position → Z) : | |
| eligible H id last u v w ↔ | |
| u = longX H (terminalWord H complete last).val ∧ | |
| v = longY H (terminalWord H complete last).val ∧ | |
| w = longZ H (terminalWord H complete last).val := by | |
| constructor | |
| · rintro ⟨word, hword⟩ | |
| rw [terminalWord_of_represents H complete last u v w word hword] | |
| exact ⟨hword.1, hword.2.1, hword.2.2.1⟩ | |
| · rintro ⟨hu, hv, hw⟩ | |
| refine ⟨(terminalWord H complete last).val, hu, hv, hw, ?_⟩ | |
| intro i | |
| exact (terminalWord_record H complete last i).symm | |
| end MatrixMultiplication.CompletionHierarchyWords | |
| end | |
| end OAI | |