openAI-math-dataset / lean /OAI /RepresentationTheory /UniversalSquare /Support /UnionRowChecker.lean
| import Mathlib | |
| import OAI.RepresentationTheory.Saxl.Main | |
| import OAI.RepresentationTheory.UniversalSquare.Support.FastRowCheck | |
| namespace OAI | |
| /-! Union Row Checker. -/ | |
| section | |
| namespace UniversalTensorSquare | |
| def liveRegions (fuel n upper : ℕ) (regions : List PrefixRegion) : List PrefixRegion := | |
| regions.filter fun R => !quickRejected fuel n upper R.1 R.2 | |
| def nextRegions (a : ℕ) (regions : List PrefixRegion) : List PrefixRegion := | |
| regions.filterMap fun R => if quickAllowed a R.1 R.2 then | |
| some (clipBudget R.1 a,rowBudget R.2 a) else none | |
| def checkRegionPrefixes (P : List ℕ → Bool) : ℕ → ℕ → ℕ → | |
| List PrefixRegion → List ℕ → Bool | |
| | fuel,n,upper,regions,pref => | |
| let alive := liveRegions fuel n upper regions | |
| if alive.isEmpty then true else | |
| match fuel with | |
| | 0 => if n = 0 then P pref.reverse else true | |
| | k+1 => | |
| let lo := max 1 (n/(k+1)) | |
| (List.range' lo (min n upper+1-lo)).all fun a => | |
| if n ≤ (k+1)*a then | |
| checkRegionPrefixes P k (n-a) a (nextRegions a alive) (a::pref) | |
| else true | |
| lemma checkRegionPrefixes_sound (P : List ℕ → Bool) (fuel n upper : ℕ) | |
| (regions : List PrefixRegion) (pref : List ℕ) | |
| (hc : checkRegionPrefixes P fuel n upper regions pref = true) : | |
| ∀ A B, (A,B) ∈ regions → ∀ rs ∈ constrainedRows fuel n upper A B, | |
| P (pref.reverse ++ rs) = true := by | |
| induction fuel generalizing n upper regions pref with | |
| | zero => | |
| intro A B hR rs hrs | |
| rw [constrainedRows] at hrs | |
| split_ifs at hrs with hrej hn | |
| · simp at hrs | |
| · simp only [List.mem_singleton] at hrs | |
| subst rs | |
| have hmem : (A,B) ∈ liveRegions 0 n upper regions := by | |
| apply List.mem_filter.mpr | |
| exact ⟨hR,by simpa only [Bool.not_eq_true',quickRejected_iff, | |
| Bool.eq_false_iff,ne_eq] using hrej⟩ | |
| have hne : ¬(liveRegions 0 n upper regions).isEmpty := by | |
| simp only [List.isEmpty_iff] | |
| intro he | |
| simp only [he,List.not_mem_nil] at hmem | |
| simpa only [checkRegionPrefixes,ite_eq_right hne,ite_eq_left hn,List.append_nil] using hc | |
| · simp at hrs | |
| | succ k ih => | |
| intro A B hR rs hrs | |
| rw [constrainedRows] at hrs | |
| split_ifs at hrs with hrej | |
| · simp at hrs | |
| · have hmem : (A,B) ∈ liveRegions (k+1) n upper regions := by | |
| apply List.mem_filter.mpr | |
| exact ⟨hR,by simpa only [Bool.not_eq_true',quickRejected_iff, | |
| Bool.eq_false_iff,ne_eq] using hrej⟩ | |
| have hne : ¬(liveRegions (k+1) n upper regions).isEmpty := by | |
| simp only [List.isEmpty_iff] | |
| intro he | |
| simp only [he,List.not_mem_nil] at hmem | |
| obtain ⟨a,ha,hr⟩ := List.mem_flatMap.mp hrs | |
| split_ifs at hr with hallowed | |
| · obtain ⟨ys,hy,rfl⟩ := List.mem_map.mp hr | |
| have hlo : max 1 (n/(k+1)) ≤ a := by | |
| have hh := Nat.div_le_of_le_mul hallowed.2 | |
| have ha' := (List.mem_range'_1.mp ha).1 | |
| exact max_le ha' hh | |
| have hai : a ∈ List.range' (max 1 (n/(k+1))) | |
| (min n upper+1-max 1 (n/(k+1))) := by | |
| have ha' := List.mem_range'_1.mp ha | |
| apply List.mem_range'_1.mpr | |
| refine ⟨hlo, ?_⟩ | |
| have he : max 1 (n/(k+1)) + (min n upper+1-max 1 (n/(k+1))) = min n upper+1 := by | |
| apply Nat.add_sub_of_le | |
| omega | |
| rw [he] | |
| omega | |
| rw [checkRegionPrefixes] at hc | |
| simp only [ite_eq_right hne] at hc | |
| have hh := List.all_eq_true.mp hc a hai | |
| rw [ite_eq_left hallowed.2] at hh | |
| have hnext : (clipBudget A a,rowBudget B a) ∈ | |
| nextRegions a (liveRegions (k+1) n upper regions) := by | |
| apply List.mem_filterMap.mpr | |
| refine ⟨(A,B),hmem,?_⟩ | |
| simp only [quickAllowed_iff,ite_eq_left hallowed.1] | |
| have hi := ih (n-a) a (nextRegions a (liveRegions (k+1) n upper regions)) | |
| (a::pref) hh _ _ hnext ys hy | |
| simpa only [List.reverse_cons,List.append_assoc,List.singleton_append] using hi | |
| · simp at hr | |
| lemma checkedRegion_diagram {n M r H W : ℕ} {qs : List (List ℕ)} {t : RowTree} | |
| {regions : List PrefixRegion} | |
| (check : ∀ h ≤ H, checkRegionPrefixes (quickResidual M r qs t) h n W regions [] = true) | |
| (μ : YoungDiagram) (hn : μ.card = n) {A B : List (ℕ × ℕ)} | |
| (hR : (A,B) ∈ regions) (hA : RowBounds μ.transpose A) (hB : RowBounds μ B) | |
| (hH : firstBound n A ≤ H) (hW : firstBound n B ≤ W) : | |
| TreeResidual M r qs t μ.rowLens := by | |
| have hu : ∀ a ∈ μ.rowLens, a ≤ W := by | |
| intro a ha | |
| obtain ⟨i,hi,he⟩ := List.getElem_of_mem ha | |
| rw [← he, YoungDiagram.get_rowLens] | |
| exact (μ.rowLen_anti _ _ (Nat.zero_le _)).trans ((firstBound_sound μ hn hB).trans hW) | |
| have hl : μ.rowLens.length ≤ H := by | |
| rw [YoungDiagram.length_rowLens] | |
| have hh := (firstBound_sound μ.transpose ((Saxl.transpose_card μ).trans hn) hA).trans hH | |
| simpa only [YoungDiagram.rowLen_transpose] using hh | |
| have hmem : μ.rowLens ∈ constrainedRows μ.rowLens.length n W A B := by | |
| apply mem_constrainedRows _ _ _ _ _ _ rfl | |
| · exact (Saxl.card_eq_sum_rowLens μ).symm.trans hn | |
| · exact μ.pos_of_mem_rowLens | |
| · exact hu | |
| · exact μ.rowLens_sorted | |
| · intro p hp | |
| rw [clippedSum_rowLens] | |
| exact (hA p hp).2 | |
| · intro p hp | |
| rw [← rowPrefix_eq_take] | |
| exact (hB p hp).2 | |
| apply quickResidual_sound | |
| have hh := checkRegionPrefixes_sound (quickResidual M r qs t) μ.rowLens.length n W | |
| regions [] (check _ hl) A B hR μ.rowLens hmem | |
| simpa only [List.reverse_nil,List.nil_append] using hh | |
| end UniversalTensorSquare | |
| end | |
| end OAI | |