openAI-math-dataset / lean /OAI /RepresentationTheory /UniversalSquare /Capacity /RefinedCapacity.lean
| import Mathlib | |
| import OAI.RepresentationTheory.Saxl.Main | |
| import OAI.RepresentationTheory.UniversalSquare.Capacity.Intermediate | |
| namespace OAI | |
| /-! Refined Capacity. -/ | |
| section | |
| noncomputable section | |
| namespace UniversalTensorSquare | |
| open Saxl | |
| lemma prefix_last_after_first (μ : YoungDiagram) {j : ℕ} (hj : 0 < j) : | |
| μ.rowLen 0 + (j-1)*μ.rowLen (j-1) ≤ rowPrefix μ j := by | |
| have he : j = (j-1)+1 := by omega | |
| unfold rowPrefix | |
| conv_rhs => rw [he, Finset.sum_range_succ'] | |
| have hh := Finset.sum_le_sum (s := Finset.range (j-1)) | |
| (f := fun _ => μ.rowLen (j-1)) (g := fun a => μ.rowLen (a+1)) (by | |
| intro a ha | |
| exact μ.rowLen_anti _ _ (by have := Finset.mem_range.mp ha; omega)) | |
| simp only [Finset.sum_const, Finset.card_range, smul_eq_mul] at hh | |
| omega | |
| def endpointBound (i U H : ℕ) : ℕ := | |
| if i = 1 then U else min (U/i) ((U-H)/(i-1)) | |
| lemma endpointBound_sound (μ : YoungDiagram) {i U H : ℕ} (hi : 0 < i) | |
| (hU : rowPrefix μ i ≤ U) (hH : H ≤ μ.rowLen 0) : | |
| μ.rowLen (i-1) ≤ endpointBound i U H := by | |
| unfold endpointBound | |
| split_ifs with he | |
| · subst i | |
| simpa [rowPrefix] using hU | |
| · apply le_min | |
| · apply (Nat.le_div_iff_mul_le hi).mpr | |
| simpa [Nat.mul_comm] using (prefix_last_mul_le μ hi).trans hU | |
| · apply (Nat.le_div_iff_mul_le (by omega : 0 < i-1)).mpr | |
| have h := prefix_last_after_first μ hi | |
| have hx : (i-1)*μ.rowLen (i-1) ≤ U-H := by omega | |
| simpa [Nat.mul_comm] using hx | |
| lemma prefix_intersection_hook (μ : YoungDiagram) {i j H W : ℕ} | |
| (hi : 0 < i) (hj : 0 < j) (hH : H ≤ μ.colLen 0) (hW : W ≤ μ.rowLen 0) | |
| (hHp : 0 < H) (hWp : 0 < W) : | |
| min i W + min j H - 1 ≤ | |
| ((rowTruncate μ j).cells ∩ (colTruncate μ i).cells).card := by | |
| let A := (rectangle 1 (min i W)).cells | |
| let B := (rectangle (min j H) 1).cells | |
| have hA : A ⊆ (rowTruncate μ j).cells ∩ (colTruncate μ i).cells := by | |
| rintro ⟨a,b⟩ hp | |
| have hx : a < 1 ∧ b < min i W := mem_rectangle.mp hp | |
| have ha : a = 0 := by omega | |
| subst a | |
| have hc : (0,b) ∈ μ := YoungDiagram.mem_iff_lt_rowLen.mpr (by omega) | |
| simpa only [Finset.mem_inter,YoungDiagram.mem_cells,mem_rowTruncate,mem_colTruncate] | |
| using And.intro ⟨hc,hj⟩ ⟨hc,lt_of_lt_of_le hx.2 (min_le_left _ _)⟩ | |
| have hB : B ⊆ (rowTruncate μ j).cells ∩ (colTruncate μ i).cells := by | |
| rintro ⟨a,b⟩ hp | |
| have hx : a < min j H ∧ b < 1 := mem_rectangle.mp hp | |
| have hb : b = 0 := by omega | |
| subst b | |
| have hc : (a,0) ∈ μ := YoungDiagram.mem_iff_lt_colLen.mpr (by omega) | |
| simpa only [Finset.mem_inter,YoungDiagram.mem_cells,mem_rowTruncate,mem_colTruncate] | |
| using And.intro ⟨hc,lt_of_lt_of_le hx.1 (min_le_left _ _)⟩ ⟨hc,hi⟩ | |
| have hAB : A ∩ B = {(0,0)} := by | |
| ext ⟨a,b⟩ | |
| simp only [Finset.mem_inter,Finset.mem_singleton,Prod.mk.injEq] | |
| simp only [A,B,rectangle,Finset.mem_product,Finset.mem_range] | |
| omega | |
| have he := Finset.card_union_add_card_inter A B | |
| have ha : A.card = min i W := by simp [A,rectangle] | |
| have hb : B.card = min j H := by simp [B,rectangle] | |
| rw [hAB,Finset.card_singleton,ha,hb] at he | |
| have h := Finset.card_le_card (Finset.union_subset hA hB) | |
| omega | |
| def refinedArea (i j U V X Y H W : ℕ) : ℕ := | |
| if j ≤ X ∧ i ≤ Y then | |
| max (U+V-(min i W + min j H - 1)) (U+V-i*j+(X-j)*(Y-i)) | |
| else U+V-(min i W + min j H - 1) | |
| lemma area_le_refined (μ : YoungDiagram) {i j U V X Y H W : ℕ} | |
| (hi : 0 < i) (hj : 0 < j) (hU : colPrefix μ i ≤ U) (hV : rowPrefix μ j ≤ V) | |
| (hX : μ.colLen (i-1) ≤ X) (hY : μ.rowLen (j-1) ≤ Y) | |
| (hH : H ≤ μ.colLen 0) (hW : W ≤ μ.rowLen 0) (hHp : 0 < H) (hWp : 0 < W) : | |
| μ.card ≤ refinedArea i j U V X Y H W := by | |
| classical | |
| have hid := strip_area_identity μ i j | |
| have hh := prefix_intersection_hook μ hi hj hH hW hHp hWp | |
| by_cases hcell : (j-1,i-1) ∈ μ | |
| · have heq : (rowTruncate μ j).cells ∩ (colTruncate μ i).cells = | |
| (rectangle j i).cells := by | |
| ext ⟨a,b⟩ | |
| simp only [Finset.mem_inter, YoungDiagram.mem_cells, mem_rowTruncate, | |
| mem_colTruncate, mem_rectangle] | |
| constructor | |
| · tauto | |
| · rintro ⟨ha,hb⟩ | |
| have hx := μ.up_left_mem (by omega : a ≤ j-1) (by omega : b ≤ i-1) hcell | |
| tauto | |
| have hint : ((rowTruncate μ j).cells ∩ (colTruncate μ i).cells).card = j*i := by | |
| rw [heq] | |
| simp only [rectangle,Finset.card_product,Finset.card_range] | |
| have hcor := Finset.card_le_card (corner_subset_rectangle μ hi hj) | |
| simp only [Finset.card_product,Nat.card_Ico] at hcor | |
| have hmul := Nat.mul_le_mul (Nat.sub_le_sub_right hX j) (Nat.sub_le_sub_right hY i) | |
| have hx : j ≤ μ.colLen (i-1) := by | |
| have := YoungDiagram.mem_iff_lt_colLen.mp hcell; omega | |
| have hy : i ≤ μ.rowLen (j-1) := by | |
| have := YoungDiagram.mem_iff_lt_rowLen.mp hcell; omega | |
| rw [hint] at hid | |
| have hup : i*j ≤ U+V := by | |
| have ht := prefix_last_mul_le μ.transpose hi | |
| simp only [YoungDiagram.rowLen_transpose] at ht | |
| change i * μ.colLen (i-1) ≤ colPrefix μ i at ht | |
| nlinarith | |
| have hsub := Nat.sub_add_cancel hup | |
| have ha : μ.card ≤ U+V-i*j+(X-j)*(Y-i) := by | |
| rw [Nat.mul_comm j i] at hid | |
| omega | |
| unfold refinedArea | |
| rw [ite_eq_left ⟨hx.trans hX,hy.trans hY⟩] | |
| exact ha.trans (le_max_right _ _) | |
| · have hz : μ.cells \ ((rowTruncate μ j).cells ∪ (colTruncate μ i).cells) = ∅ := by | |
| apply Finset.eq_empty_iff_forall_notMem.mpr | |
| rintro ⟨a,b⟩ hp | |
| simp only [Finset.mem_sdiff,Finset.mem_union,YoungDiagram.mem_cells, | |
| mem_rowTruncate,mem_colTruncate,not_or] at hp | |
| have ha : j ≤ a := Nat.le_of_not_gt fun ha => hp.2.1 ⟨hp.1,ha⟩ | |
| have hb : i ≤ b := Nat.le_of_not_gt fun hb => hp.2.2 ⟨hp.1,hb⟩ | |
| exact hcell (μ.up_left_mem (by omega) (by omega) hp.1) | |
| rw [hz,Finset.card_empty,add_zero] at hid | |
| have ha : μ.card ≤ U+V-(min i W+min j H-1) := by omega | |
| unfold refinedArea | |
| split_ifs | |
| · exact ha.trans (le_max_left _ _) | |
| · exact ha | |
| end UniversalTensorSquare | |
| end | |
| end | |
| end OAI | |