openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /JointExtraction /AssignmentLoss.lean
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.CoarseAssignment | |
| namespace OAI | |
| /-! Joint tensor extraction, compatibility and entropy estimates. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.JointAssignmentLoss | |
| open JointExtraction JointCoarseAssignment | |
| attribute [local instance] Classical.propDecidable | |
| variable {A B C I V Ω : Type*} | |
| theorem uniqueAssignment_of_unique (candidates : Finset I) | |
| (compatible : I → V → Prop) (v : V) (i : I) | |
| (hi : i ∈ candidates) (hc : compatible i v) | |
| (hu : ∀ j, j ∈ candidates → compatible j v → j = i) : | |
| uniqueAssignment candidates compatible v = some i := by | |
| classical | |
| have hex : ∃ j, j ∈ candidates ∧ compatible j v ∧ | |
| ∀ k, k ∈ candidates → compatible k v → k = j := ⟨i, hi, hc, hu⟩ | |
| unfold uniqueAssignment | |
| rw [dite_eq_left hex] | |
| exact congrArg some (hu _ (Classical.choose_spec hex).1 (Classical.choose_spec hex).2.1) | |
| theorem assignUseful_of_unique (candidates : Finset I) | |
| (compatible useful : I → V → Prop) (v : V) (i : I) | |
| (hi : i ∈ candidates) (hc : compatible i v) (huse : useful i v) | |
| (hu : ∀ j, j ∈ candidates → compatible j v → j = i) : | |
| assignUseful candidates compatible useful v = some i := by | |
| have hassign := uniqueAssignment_of_unique candidates compatible v i hi hc hu | |
| have hkeep : ∃ j, uniqueAssignment candidates compatible v = some j ∧ useful j v := | |
| ⟨i, hassign, huse⟩ | |
| simp only [assignUseful, retainAssignment, ite_eq_left hkeep] | |
| exact hassign | |
| def ambientXCollision (ambient : Finset (Triple A B C)) | |
| (hx : A → Prop) (hy : B → Prop) (hz : C → Prop) (e : Triple A B C) : Prop := | |
| ∃ f ∈ ambient, f ≠ e ∧ survives hx hy hz f ∧ f.1 = e.1 | |
| def targetCompatibleCollision (targets : Finset (Triple A B C)) | |
| (hx : A → Prop) (hy : B → Prop) (hz : C → Prop) | |
| (compatible : Triple A B C → V → Prop) (e : Triple A B C) (v : V) : Prop := | |
| ∃ f ∈ targets, f ≠ e ∧ survives hx hy hz f ∧ compatible f v | |
| theorem eligible_of_no_ambientXCollision | |
| (ambient targets : Finset (Triple A B C)) | |
| (hx : A → Prop) (hy : B → Prop) (hz : C → Prop) (e : Triple A B C) | |
| (ha : e ∈ ambient) (ht : e ∈ targets) (hs : survives hx hy hz e) | |
| (hno : ¬ ambientXCollision ambient hx hy hz e) : | |
| e ∈ eligible ambient targets hx hy hz := by | |
| apply (mem_eligible_iff ambient targets hx hy hz e).2 | |
| refine ⟨ha, ht, hs, ?_⟩ | |
| intro f hf hfs hfx | |
| by_contra hne | |
| exact hno ⟨f, hf, hne, hfs, hfx⟩ | |
| theorem not_eligible_iff_ambientXCollision | |
| (ambient targets : Finset (Triple A B C)) | |
| (hx : A → Prop) (hy : B → Prop) (hz : C → Prop) (e : Triple A B C) | |
| (ha : e ∈ ambient) (ht : e ∈ targets) (hs : survives hx hy hz e) : | |
| e ∉ eligible ambient targets hx hy hz ↔ ambientXCollision ambient hx hy hz e := by | |
| constructor | |
| · intro h | |
| by_contra hno | |
| exact h (eligible_of_no_ambientXCollision ambient targets hx hy hz e ha ht hs hno) | |
| · rintro ⟨f, hf, hne, hfs, hfx⟩ he | |
| exact hne (((mem_eligible_iff ambient targets hx hy hz e).1 he).2.2.2 f hf hfs hfx) | |
| theorem assignUseful_of_no_targetCollision | |
| (ambient targets : Finset (Triple A B C)) | |
| (hx : A → Prop) (hy : B → Prop) (hz : C → Prop) | |
| (compatible useful : Triple A B C → V → Prop) (e : Triple A B C) (v : V) | |
| (he : e ∈ eligible ambient targets hx hy hz) | |
| (hc : compatible e v) (hu : useful e v) | |
| (hno : ¬ targetCompatibleCollision targets hx hy hz compatible e v) : | |
| assignUseful (eligible ambient targets hx hy hz) compatible useful v = some e := by | |
| apply assignUseful_of_unique _ _ _ _ _ he hc hu | |
| intro f hf hcf | |
| have hfe := (mem_eligible_iff ambient targets hx hy hz f).1 hf | |
| by_contra hne | |
| exact hno ⟨f, hfe.2.1, hne, hfe.2.2.1, hcf⟩ | |
| theorem assignment_failure_implies_collision | |
| (ambient targets : Finset (Triple A B C)) | |
| (hx : A → Prop) (hy : B → Prop) (hz : C → Prop) | |
| (compatible useful : Triple A B C → V → Prop) (e : Triple A B C) (v : V) | |
| (ha : e ∈ ambient) (ht : e ∈ targets) (hs : survives hx hy hz e) | |
| (hc : compatible e v) (hu : useful e v) | |
| (hfail : assignUseful (eligible ambient targets hx hy hz) compatible useful v ≠ some e) : | |
| ambientXCollision ambient hx hy hz e ∨ | |
| targetCompatibleCollision targets hx hy hz compatible e v := by | |
| by_cases hX : ambientXCollision ambient hx hy hz e | |
| · exact Or.inl hX | |
| apply Or.inr | |
| by_contra hC | |
| have he := eligible_of_no_ambientXCollision ambient targets hx hy hz e ha ht hs hX | |
| exact hfail (assignUseful_of_no_targetCollision ambient targets hx hy hz | |
| compatible useful e v he hc hu hC) | |
| theorem retained_assignment_of_no_collision | |
| (ambient targets : Finset (Triple A B C)) | |
| (hx : A → Prop) (hy : B → Prop) (hz : C → Prop) | |
| (compatible useful : Triple A B C → V → Prop) (keep : V → Prop) | |
| (e : Triple A B C) (v : V) | |
| (ha : e ∈ ambient) (ht : e ∈ targets) (hs : survives hx hy hz e) | |
| (hc : compatible e v) (hu : useful e v) (hk : keep v) | |
| (hX : ¬ ambientXCollision ambient hx hy hz e) | |
| (hC : ¬ targetCompatibleCollision targets hx hy hz compatible e v) : | |
| retainAssignment (assignUseful (eligible ambient targets hx hy hz) compatible useful) | |
| keep v = some e := by | |
| have he := eligible_of_no_ambientXCollision ambient targets hx hy hz e ha ht hs hX | |
| have hassign := assignUseful_of_no_targetCollision ambient targets hx hy hz | |
| compatible useful e v he hc hu hC | |
| simpa only [retainAssignment, ite_eq_left hk] using hassign | |
| theorem passing_assignment_failure_subset | |
| (samples : Finset Ω) (ambient targets : Finset (Triple A B C)) | |
| (hx : Ω → A → Prop) (hy : Ω → B → Prop) (hz : Ω → C → Prop) | |
| (compatible useful : Triple A B C → V → Prop) (keep : V → Prop) | |
| (e : Triple A B C) (v : V) (ha : e ∈ ambient) (ht : e ∈ targets) | |
| (hc : compatible e v) (hu : useful e v) (hk : keep v) : | |
| samples.filter (fun ω => survives (hx ω) (hy ω) (hz ω) e ∧ | |
| retainAssignment | |
| (assignUseful (eligible ambient targets (hx ω) (hy ω) (hz ω)) compatible useful) | |
| keep v ≠ some e) ⊆ | |
| samples.filter (fun ω => survives (hx ω) (hy ω) (hz ω) e ∧ | |
| (ambientXCollision ambient (hx ω) (hy ω) (hz ω) e ∨ | |
| targetCompatibleCollision targets (hx ω) (hy ω) (hz ω) compatible e v)) := by | |
| intro ω hω | |
| obtain ⟨hωs, hs, hfail⟩ := Finset.mem_filter.1 hω | |
| apply Finset.mem_filter.2 | |
| refine ⟨hωs, hs, ?_⟩ | |
| have hfail' : assignUseful (eligible ambient targets (hx ω) (hy ω) (hz ω)) | |
| compatible useful v ≠ some e := by | |
| simpa only [retainAssignment, ite_eq_left hk] using hfail | |
| exact assignment_failure_implies_collision ambient targets (hx ω) (hy ω) (hz ω) | |
| compatible useful e v ha ht hs hc hu hfail' | |
| end MatrixMultiplication.JointAssignmentLoss | |
| end | |
| end OAI | |