openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /JointExtraction /CoarseAssignment.lean
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.Extraction | |
| namespace OAI | |
| /-! Joint tensor extraction, compatibility and entropy estimates. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.JointCoarseAssignment | |
| open MatrixMultiplication.Foundation JointExtraction | |
| attribute [local instance] Classical.propDecidable | |
| variable {F X Y Z A B C I V : Type*} [CommSemiring F] | |
| abbrev Triple (A B C : Type*) := A × B × C | |
| def survives (hx : A → Prop) (hy : B → Prop) (hz : C → Prop) | |
| (e : Triple A B C) : Prop := hx e.1 ∧ hy e.2.1 ∧ hz e.2.2 | |
| def eligible (ambient targets : Finset (Triple A B C)) | |
| (hx : A → Prop) (hy : B → Prop) (hz : C → Prop) : Finset (Triple A B C) := | |
| ambient.filter fun e => e ∈ targets ∧ survives hx hy hz e ∧ | |
| ∀ f ∈ ambient, survives hx hy hz f → f.1 = e.1 → f = e | |
| theorem mem_eligible_iff (ambient targets : Finset (Triple A B C)) | |
| (hx : A → Prop) (hy : B → Prop) (hz : C → Prop) (e : Triple A B C) : | |
| e ∈ eligible ambient targets hx hy hz ↔ | |
| e ∈ ambient ∧ e ∈ targets ∧ survives hx hy hz e ∧ | |
| ∀ f ∈ ambient, survives hx hy hz f → f.1 = e.1 → f = e := by | |
| simp [eligible] | |
| def assignUseful (eligible : Finset I) (compatible useful : I → V → Prop) | |
| (v : V) : Option I := | |
| retainAssignment (uniqueAssignment eligible compatible) | |
| (fun v => ∃ i, uniqueAssignment eligible compatible v = some i ∧ useful i v) v | |
| theorem assignUseful_spec (eligible : Finset I) (compatible useful : I → V → Prop) | |
| (v : V) (i : I) (h : assignUseful eligible compatible useful v = some i) : | |
| i ∈ eligible ∧ compatible i v ∧ useful i v ∧ | |
| ∀ j, j ∈ eligible → compatible j v → j = i := by | |
| classical | |
| have hu : uniqueAssignment eligible compatible v = some i := | |
| retainAssignment_some _ _ v i h | |
| have hh := uniqueAssignment_spec eligible compatible v i hu | |
| have huse : useful i v := by | |
| unfold assignUseful retainAssignment at h | |
| split_ifs at h with hk | |
| obtain ⟨j, hj, hju⟩ := hk | |
| have hji : j = i := Option.some.inj (hj.symm.trans hu) | |
| exact hji ▸ hju | |
| exact ⟨hh.1, hh.2.1, huse, hh.2.2⟩ | |
| def assignments (ambient targets : Finset (Triple A B C)) | |
| (hx : A → Prop) (hy : B → Prop) (hz : C → Prop) | |
| (cx : X → A) (cy : Y → B) (cz : Z → C) | |
| (compatibleY : Triple A B C → Y → Prop) | |
| (compatibleZ : Triple A B C → Z → Prop) | |
| (usefulX : Triple A B C → X → Prop) | |
| (usefulY : Triple A B C → Y → Prop) | |
| (usefulZ : Triple A B C → Z → Prop) : Assignment X Y Z (Triple A B C) where | |
| x := assignUseful (eligible ambient targets hx hy hz) (fun e x => cx x = e.1) usefulX | |
| y := assignUseful (eligible ambient targets hx hy hz) | |
| (fun e y => cy y = e.2.1 ∧ compatibleY e y) usefulY | |
| z := assignUseful (eligible ambient targets hx hy hz) | |
| (fun e z => cz z = e.2.2 ∧ compatibleZ e z) usefulZ | |
| theorem assignments_coherent | |
| (T : Tensor F X Y Z) (ambient targets : Finset (Triple A B C)) | |
| (hx : A → Prop) (hy : B → Prop) (hz : C → Prop) | |
| (cx : X → A) (cy : Y → B) (cz : Z → C) | |
| (compatibleY : Triple A B C → Y → Prop) | |
| (compatibleZ : Triple A B C → Z → Prop) | |
| (usefulX : Triple A B C → X → Prop) | |
| (usefulY : Triple A B C → Y → Prop) | |
| (usefulZ : Triple A B C → Z → Prop) | |
| (support : ∀ x y z, T x y z ≠ 0 → (cx x, cy y, cz z) ∈ ambient) | |
| (transferY : ∀ x y z e, T x y z ≠ 0 → (cx x, cy y, cz z) = e → | |
| usefulX e x → compatibleY e y) | |
| (transferZ : ∀ x y z e, T x y z ≠ 0 → (cx x, cy y, cz z) = e → | |
| usefulX e x → usefulY e y → compatibleZ e z) : | |
| Coherent T (assignments ambient targets hx hy hz cx cy cz | |
| compatibleY compatibleZ usefulX usefulY usefulZ) := by | |
| intro x y z i j k hT hxi hyj hzk | |
| have hi := assignUseful_spec (eligible ambient targets hx hy hz) | |
| (fun e x => cx x = e.1) usefulX x i hxi | |
| have hj := assignUseful_spec (eligible ambient targets hx hy hz) | |
| (fun e y => cy y = e.2.1 ∧ compatibleY e y) usefulY y j hyj | |
| have hk := assignUseful_spec (eligible ambient targets hx hy hz) | |
| (fun e z => cz z = e.2.2 ∧ compatibleZ e z) usefulZ z k hzk | |
| have hie := (mem_eligible_iff ambient targets hx hy hz i).mp hi.1 | |
| have hje := (mem_eligible_iff ambient targets hx hy hz j).mp hj.1 | |
| have hke := (mem_eligible_iff ambient targets hx hy hz k).mp hk.1 | |
| have hs : survives hx hy hz (cx x, cy y, cz z) := | |
| ⟨hi.2.1.symm ▸ hie.2.2.1.1, hj.2.1.1.symm ▸ hje.2.2.1.2.1, | |
| hk.2.1.1.symm ▸ hke.2.2.1.2.2⟩ | |
| have he : (cx x, cy y, cz z) = i := | |
| hie.2.2.2 _ (support x y z hT) hs hi.2.1 | |
| have hyi : cy y = i.2.1 := congrArg (fun e : Triple A B C => e.2.1) he | |
| have hzi : cz z = i.2.2 := congrArg (fun e : Triple A B C => e.2.2) he | |
| have hij : i = j := hj.2.2.2 i hi.1 | |
| ⟨hyi, transferY x y z i hT he hi.2.2.1⟩ | |
| have huy : usefulY i y := hij.symm ▸ hj.2.2.1 | |
| have hik : i = k := hk.2.2.2 i hi.1 | |
| ⟨hzi, transferZ x y z i hT he hi.2.2.1 huy⟩ | |
| exact ⟨hij, hik⟩ | |
| end MatrixMultiplication.JointCoarseAssignment | |
| end | |
| end OAI | |