openAI-math-dataset / lean /OAI /RepresentationTheory /UniversalSquare /Support /NumericalSupport.lean
| import Mathlib | |
| import OAI.RepresentationTheory.Saxl.Main | |
| import OAI.RepresentationTheory.UniversalSquare.Capacity.SentinelExhaustion | |
| namespace OAI | |
| /-! Numerical Support. -/ | |
| section | |
| noncomputable section | |
| namespace UniversalTensorSquare | |
| open Saxl Saxl.Balance Saxl.Columns | |
| def transposeRows (rs : List ℕ) : List ℕ := | |
| (List.range (rs.headD 0)).map fun j => rs.countP (fun a => decide (j < a)) | |
| lemma headD_rowLens (μ : YoungDiagram) : μ.rowLens.headD 0 = μ.rowLen 0 := by | |
| have h (xs : List ℕ) : xs.headD 0 = xs[0]?.getD 0 := by cases xs <;> rfl | |
| rw [h, ← ofRowLens_rowLen μ.rowLens μ.rowLens_sorted, | |
| YoungDiagram.ofRowLens_to_rowLens_eq_self] | |
| lemma transposeRows_rowLens (μ : YoungDiagram) : | |
| transposeRows μ.rowLens = μ.transpose.rowLens := by | |
| unfold transposeRows | |
| rw [headD_rowLens] | |
| conv_rhs => rw [YoungDiagram.rowLens,YoungDiagram.colLen_transpose] | |
| apply List.map_congr_left | |
| intro j _ | |
| rw [YoungDiagram.rowLen_transpose] | |
| have he := congrArg (fun θ : YoungDiagram => θ.colLen j) (rowDiagram_rowLens μ) | |
| simpa only [rowDiagram,YoungDiagram.colLen_transpose,columnShape_rowLen] using he | |
| lemma transposeRows_eq {rs : List ℕ} (h : GoodRows rs) : | |
| transposeRows rs = (rowDiagram rs).transpose.rowLens := by | |
| conv_lhs => rw [← rowDiagram_rows h,transposeRows_rowLens] | |
| lemma rowDiagram_transposeRows {rs : List ℕ} (h : GoodRows rs) : | |
| rowDiagram (transposeRows rs) = (rowDiagram rs).transpose := by | |
| rw [transposeRows_eq h,rowDiagram_rowLens] | |
| def RowsBand (M r : ℕ) (rs : List ℕ) : Prop := | |
| ∃ k ∈ List.range 4, | |
| let d := k+1 | |
| let q := testWidth M d | |
| d*q+7+d ≤ 2*M-1 ∧ 2*M-1 ≤ (rs.take d).sum ∧ | |
| r ≤ ((List.range q).map fun j => (rs.countP (fun a => decide (j < a))-d)/2).sum | |
| instance (M r : ℕ) (rs : List ℕ) : Decidable (RowsBand M r rs) := by | |
| unfold RowsBand; infer_instance | |
| lemma rowsBand_sound {M r : ℕ} {rs : List ℕ} (hr : GoodRows rs) | |
| (h : RowsBand M r rs) : BandTest M r (rowDiagram rs) := by | |
| obtain ⟨k,hk,hclear,hprefix,hcap⟩ := h | |
| have hk' : k < 4 := List.mem_range.mp hk | |
| refine ⟨k+1,by omega,by omega,testWidth M (k+1),min_le_left _ _,?_,hclear,?_⟩ | |
| · rw [rowPrefix_eq_take,rowDiagram_rows hr] | |
| exact hprefix | |
| · simpa only [bandCapacity,rowDiagram,YoungDiagram.colLen_transpose,columnShape_rowLen, | |
| ← List.sum_toFinset _ List.nodup_range,List.toFinset_range] using hcap | |
| def ResidualProperty (M r : ℕ) (qs : List (List ℕ)) (ls : List (List ℕ)) | |
| (rs : List ℕ) : Prop := | |
| RowsBand M r rs ∨ RowsBand M r (transposeRows rs) ∨ | |
| (∃ q ∈ qs, RowsDom rs q ∨ RowsDom (transposeRows rs) q) ∨ | |
| rs ∈ ls ∨ transposeRows rs ∈ ls | |
| instance (M r : ℕ) (qs ls : List (List ℕ)) : DecidablePred (ResidualProperty M r qs ls) := by | |
| intro rs; unfold ResidualProperty; infer_instance | |
| lemma SquareOccurs.of_packing {n d : ℕ} (p : PackingPlan) {lam μ : YoungDiagram} | |
| (hv : p.Valid d) (hb : BandsValid p.bands) (hh : lam.colLen 0 = d) | |
| (hc : p.columns.Perm lam.transpose.rowLens) (hp : ∀ x ∈ p.parts, 0 < x) | |
| (hlam : lam.card = n) (hμ : μ.card = n) | |
| (hd : Dominates μ (rowDiagram p.parts)) : SquareOccurs lam μ := by | |
| exact ⟨n,canonicalTableau lam hlam,canonicalTableau μ hμ, | |
| packingPlan_pos p hv hb hh hc hp _ _ hd⟩ | |
| lemma residualProperty_sound {n M b δ r : ℕ} (hM : 4 ≤ M) (hδ : δ ≤ 1) | |
| (hn : (candidate M b δ).card = n) (hre : r = 2*b+δ) | |
| {qs ls : List (List ℕ)} | |
| (hcones : ∀ q ∈ qs, ∀ μ : YoungDiagram, μ.card = n → | |
| RowsDom μ.rowLens q → SquareOccurs (candidate M b δ) μ) | |
| (hlists : ∀ rs ∈ ls, SquareOccurs (candidate M b δ) (rowDiagram rs)) | |
| {rs : List ℕ} (hgood : GoodRows rs) (hsum : rs.sum = n) | |
| (h : ResidualProperty M r qs ls rs) : SquareOccurs (candidate M b δ) (rowDiagram rs) := by | |
| have hc : (rowDiagram rs).card = n := (rowDiagram_card _).trans hsum | |
| have ht : (rowDiagram rs).transpose.card = n := (transpose_card _).trans hc | |
| have hgood' : GoodRows (transposeRows rs) := by | |
| rw [transposeRows_eq hgood] | |
| exact ⟨YoungDiagram.rowLens_sorted _,YoungDiagram.pos_of_mem_rowLens _⟩ | |
| have back : SquareOccurs (candidate M b δ) (rowDiagram rs).transpose → | |
| SquareOccurs (candidate M b δ) (rowDiagram rs) := by | |
| intro hh | |
| simpa only [YoungDiagram.transpose_transpose] using hh.target_transpose (candidate_transpose ..) | |
| rcases h with h | h | ⟨q,hq,h | h⟩ | h | h | |
| · exact ⟨n,canonicalTableau _ hn,canonicalTableau _ hc, | |
| band_kronecker_pos hM hδ hre _ _ _ (rowsBand_sound hgood h)⟩ | |
| · have hh := rowsBand_sound hgood' h | |
| rw [rowDiagram_transposeRows hgood] at hh | |
| exact back ⟨n,canonicalTableau _ hn,canonicalTableau _ ht, | |
| band_kronecker_pos hM hδ hre _ _ _ hh⟩ | |
| · exact hcones q hq _ hc (by simpa only [rowDiagram_rows hgood] using h) | |
| · exact back (hcones q hq _ ht (by simpa only [transposeRows_eq hgood] using h)) | |
| · exact hlists rs h | |
| · apply back | |
| simpa only [rowDiagram_transposeRows hgood] using hlists _ h | |
| end UniversalTensorSquare | |
| namespace UniversalTensorSquare | |
| open Saxl Saxl.Balance Saxl.Columns | |
| lemma candidate_cone {n M b δ : ℕ} (hM : 4 ≤ M) | |
| (hn : (candidate M b δ).card = n) (p : PackingPlan) (q : List ℕ) | |
| (hq : GoodRows q) (hqn : q.sum = n) | |
| (hv : p.Valid (M+b+δ)) (hb : BandsValid p.bands) | |
| (hc : p.columns.Perm ([M+b+δ,M-1+b] ++ | |
| (List.range' 3 (M-4)).reverse ++ List.replicate (b+1) 2 ++ List.replicate (1+δ) 1)) | |
| (hp : p.parts.Perm q) (μ : YoungDiagram) (hμ : μ.card = n) | |
| (hd : RowsDom μ.rowLens q) : SquareOccurs (candidate M b δ) μ := by | |
| have hh : (candidate M b δ).colLen 0 = M+b+δ := by | |
| rw [candidate_colLen _ _ _ _ hM]; simp [candidateLengths] | |
| apply SquareOccurs.of_packing p hv hb hh | |
| · rw [candidate_transpose,candidate_rowLens hM]; exact hc | |
| · exact fun a ha => hq.2 a (hp.mem_iff.mp ha) | |
| · exact hn | |
| · exact hμ | |
| · have he : rowDiagram p.parts = rowDiagram q := by | |
| unfold rowDiagram; rw [columnShape_perm hp] | |
| rw [he] | |
| apply dominates_of_rowsDom | |
| · rw [rowDiagram_card,hqn,hμ] | |
| · simpa only [rowDiagram_rows hq] using hd | |
| end UniversalTensorSquare | |
| end | |
| end | |
| end OAI | |