| import Mathlib |
| import OAI.RepresentationTheory.Saxl.Main |
| import OAI.RepresentationTheory.UniversalSquare.Band.BandCriterion |
|
|
| namespace OAI |
|
|
| /-! Intermediate. -/ |
|
|
| section |
|
|
| noncomputable section |
| namespace UniversalTensorSquare |
| open Saxl |
|
|
| def testWidth (M f : ℕ) : ℕ := min 8 ((2*M-8-f)/f) |
|
|
| lemma testWidth_valid {M f : ℕ} (hM : 9 ≤ M) (hf : 1 ≤ f) (hf' : f ≤ 4) : |
| testWidth M f ≤ 8 ∧ f * testWidth M f + 7 + f ≤ 2*M-1 := by |
| refine ⟨min_le_left _ _, ?_⟩ |
| have h := Nat.mul_le_mul_left f (min_le_right 8 ((2*M-8-f)/f)) |
| have h' := Nat.div_mul_le_self (2*M-8-f) f |
| unfold testWidth |
| nlinarith [Nat.sub_add_cancel (show f ≤ 2*M-8 by omega), |
| Nat.sub_add_cancel (show 8 ≤ 2*M by omega), |
| Nat.sub_add_cancel (show 1 ≤ 2*M by omega)] |
|
|
| def prefixBounds (M r d e : ℕ) : List (ℕ × ℕ) := |
| (if 1 < d then [(d-1,2*M-2)] else []) ++ |
| ((List.range 4).filterMap fun k => |
| let f := k+1 |
| let q := testWidth M f |
| if e ≤ f ∧ 0 < q then some (q,2*r-2+q*(f+1)) else none) |
|
|
| lemma prefixBounds_sound (μ : YoungDiagram) {M r d e : ℕ} |
| (hM : 9 ≤ M) (hr : 0 < r) |
| (hd : 1 < d → colPrefix μ (d-1) < 2*M-1) |
| (he : 2*M-1 ≤ rowPrefix μ e) (hfail : ¬BandTest M r μ) : |
| ∀ p ∈ prefixBounds M r d e, 0 < p.1 ∧ colPrefix μ p.1 ≤ p.2 := by |
| intro p hp |
| simp only [prefixBounds, List.mem_append] at hp |
| rcases hp with hp | hp |
| · split_ifs at hp with hd' |
| · simp only [List.mem_cons, List.not_mem_nil, or_false] at hp |
| subst p |
| exact ⟨by omega, by have := hd hd'; dsimp; omega⟩ |
| · simp at hp |
| · obtain ⟨k,hk,hkp⟩ := List.mem_filterMap.mp hp |
| simp only [List.mem_range] at hk |
| split_ifs at hkp with hke |
| · have hp' := Option.some.inj hkp |
| subst p |
| have hw := testWidth_valid hM (by omega : 1 ≤ k+1) (by omega : k+1 ≤ 4) |
| have hc : bandCapacity μ (k+1) (testWidth M (k+1)) < r := by |
| by_contra hh |
| apply hfail |
| exact ⟨k+1, by omega, by omega, testWidth M (k+1), hw.1, |
| he.trans (rowPrefix_mono μ hke.1), hw.2, by omega⟩ |
| have hb := prefix_le_capacity μ (k+1) (testWidth M (k+1)) |
| exact ⟨hke.2, by dsimp; omega⟩ |
|
|
| def exceptionalPair (M r : ℕ) : Prop := |
| (M = 10 ∧ 8 ≤ r ∧ r ≤ 17) ∨ (M = 12 ∧ 14 ≤ r ∧ r ≤ 20) ∨ |
| (M = 14 ∧ 22 ≤ r ∧ r ≤ 23) |
|
|
| instance (M r : ℕ) : Decidable (exceptionalPair M r) := inferInstanceAs |
| (Decidable ((M = 10 ∧ 8 ≤ r ∧ r ≤ 17) ∨ (M = 12 ∧ 14 ≤ r ∧ r ≤ 20) ∨ |
| (M = 14 ∧ 22 ≤ r ∧ r ≤ 23))) |
|
|
| def admissibleR (M r : ℕ) : Prop := 0 < r ∧ |
| (if M % 2 = 1 then 2*r ≤ M-1 else 2*r ≤ 3*M+4) ∧ ¬exceptionalPair M r |
|
|
| instance (M r : ℕ) : Decidable (admissibleR M r) := inferInstanceAs |
| (Decidable (0 < r ∧ (if M % 2 = 1 then 2*r ≤ M-1 else 2*r ≤ 3*M+4) ∧ |
| ¬exceptionalPair M r)) |
|
|
| def intermediateCheck (M r d e : ℕ) : Prop := |
| ∃ p ∈ prefixBounds M r d e, ∃ q ∈ prefixBounds M r e d, |
| max (p.2+q.2-1) (p.2*q.2/(p.1*q.1)) < M*(M+1)/2+2*r |
|
|
| instance (M r d e : ℕ) : Decidable (intermediateCheck M r d e) := |
| inferInstanceAs (Decidable (∃ p ∈ prefixBounds M r d e, |
| ∃ q ∈ prefixBounds M r e d, |
| max (p.2+q.2-1) (p.2*q.2/(p.1*q.1)) < M*(M+1)/2+2*r)) |
|
|
| lemma intermediate9 : ∀ r ∈ List.range 34, admissibleR 9 r → |
| ∀ d ∈ List.range 4, ∀ e ∈ List.range 4, intermediateCheck 9 r (d+1) (e+1) := by decide |
|
|
| lemma intermediate10 : ∀ r ∈ List.range 34, admissibleR 10 r → |
| ∀ d ∈ List.range 4, ∀ e ∈ List.range 4, intermediateCheck 10 r (d+1) (e+1) := by decide |
|
|
| lemma intermediate11 : ∀ r ∈ List.range 34, admissibleR 11 r → |
| ∀ d ∈ List.range 4, ∀ e ∈ List.range 4, intermediateCheck 11 r (d+1) (e+1) := by decide |
|
|
| lemma intermediate12 : ∀ r ∈ List.range 34, admissibleR 12 r → |
| ∀ d ∈ List.range 4, ∀ e ∈ List.range 4, intermediateCheck 12 r (d+1) (e+1) := by decide |
|
|
| lemma intermediate13 : ∀ r ∈ List.range 34, admissibleR 13 r → |
| ∀ d ∈ List.range 4, ∀ e ∈ List.range 4, intermediateCheck 13 r (d+1) (e+1) := by decide |
|
|
| lemma intermediate14 : ∀ r ∈ List.range 34, admissibleR 14 r → |
| ∀ d ∈ List.range 4, ∀ e ∈ List.range 4, intermediateCheck 14 r (d+1) (e+1) := by decide |
|
|
| lemma intermediate15 : ∀ r ∈ List.range 34, admissibleR 15 r → |
| ∀ d ∈ List.range 4, ∀ e ∈ List.range 4, intermediateCheck 15 r (d+1) (e+1) := by decide |
|
|
| lemma intermediate16 : ∀ r ∈ List.range 34, admissibleR 16 r → |
| ∀ d ∈ List.range 4, ∀ e ∈ List.range 4, intermediateCheck 16 r (d+1) (e+1) := by decide |
|
|
| lemma intermediate17 : ∀ r ∈ List.range 34, admissibleR 17 r → |
| ∀ d ∈ List.range 4, ∀ e ∈ List.range 4, intermediateCheck 17 r (d+1) (e+1) := by decide |
|
|
| lemma intermediate18 : ∀ r ∈ List.range 34, admissibleR 18 r → |
| ∀ d ∈ List.range 4, ∀ e ∈ List.range 4, intermediateCheck 18 r (d+1) (e+1) := by decide |
|
|
| lemma intermediate19 : ∀ r ∈ List.range 34, admissibleR 19 r → |
| ∀ d ∈ List.range 4, ∀ e ∈ List.range 4, intermediateCheck 19 r (d+1) (e+1) := by decide |
|
|
| lemma intermediate20 : ∀ r ∈ List.range 34, admissibleR 20 r → |
| ∀ d ∈ List.range 4, ∀ e ∈ List.range 4, intermediateCheck 20 r (d+1) (e+1) := by decide |
|
|
| lemma intermediate21 : ∀ r ∈ List.range 34, admissibleR 21 r → |
| ∀ d ∈ List.range 4, ∀ e ∈ List.range 4, intermediateCheck 21 r (d+1) (e+1) := by decide |
|
|
| lemma capacity_intermediate_numeric {M r d e : ℕ} |
| (hM : 9 ≤ M) (hM' : M ≤ 21) (hr : admissibleR M r) |
| (hd : 1 ≤ d) (hd' : d ≤ 4) (he : 1 ≤ e) (he' : e ≤ 4) : |
| intermediateCheck M r d e := by |
| have hr' : r ∈ List.range 34 := by |
| have hb := hr.2.1 |
| split_ifs at hb <;> simp only [List.mem_range] <;> omega |
| have hdd : d-1 ∈ List.range 4 := by simp only [List.mem_range]; omega |
| have hee : e-1 ∈ List.range 4 := by simp only [List.mem_range]; omega |
| have hde : d = (d-1)+1 := by omega |
| have hee' : e = (e-1)+1 := by omega |
| rw [hde, hee'] |
| interval_cases M |
| · exact intermediate9 r hr' hr _ hdd _ hee |
| · exact intermediate10 r hr' hr _ hdd _ hee |
| · exact intermediate11 r hr' hr _ hdd _ hee |
| · exact intermediate12 r hr' hr _ hdd _ hee |
| · exact intermediate13 r hr' hr _ hdd _ hee |
| · exact intermediate14 r hr' hr _ hdd _ hee |
| · exact intermediate15 r hr' hr _ hdd _ hee |
| · exact intermediate16 r hr' hr _ hdd _ hee |
| · exact intermediate17 r hr' hr _ hdd _ hee |
| · exact intermediate18 r hr' hr _ hdd _ hee |
| · exact intermediate19 r hr' hr _ hdd _ hee |
| · exact intermediate20 r hr' hr _ hdd _ hee |
| · exact intermediate21 r hr' hr _ hdd _ hee |
|
|
| lemma first_prefix_height (μ : YoungDiagram) {K : ℕ} (hK : 0 < K) |
| (h : K ≤ rowPrefix μ 4) : |
| ∃ d, 1 ≤ d ∧ d ≤ 4 ∧ K ≤ rowPrefix μ d ∧ |
| (1 < d → rowPrefix μ (d-1) < K) := by |
| let h' : ∃ d, K ≤ rowPrefix μ d := ⟨4,h⟩ |
| refine ⟨Nat.find h', ?_, Nat.find_min' h' h, Nat.find_spec h', ?_⟩ |
| · have hx := Nat.find_spec h' |
| by_contra hh |
| have hd : Nat.find h' = 0 := by omega |
| change K ≤ rowPrefix μ (Nat.find h') at hx |
| rw [hd] at hx |
| simp only [rowPrefix, Finset.range_zero, Finset.sum_empty] at hx |
| omega |
| · intro hd |
| exact Nat.lt_of_not_ge (Nat.find_min h' (by omega)) |
|
|
| theorem capacity_intermediate (μ : YoungDiagram) {M r : ℕ} |
| (hM : 9 ≤ M) (hM' : M ≤ 21) (hr : admissibleR M r) |
| (hn : μ.card = (staircase M).card + 2*r) |
| (hH : 2*M-1 ≤ colPrefix μ 4) (hW : 2*M-1 ≤ rowPrefix μ 4) : |
| BandTest M r μ ∨ BandTest M r μ.transpose := by |
| classical |
| by_contra hh |
| push Not at hh |
| obtain ⟨d,hd,hd',hHd,hHd'⟩ := first_prefix_height μ.transpose (by omega) hH |
| obtain ⟨e,he,he',hWe,hWe'⟩ := first_prefix_height μ (by omega) hW |
| obtain ⟨p,hp,q,hq,hpq⟩ := capacity_intermediate_numeric hM hM' hr hd hd' he he' |
| have hp' := prefixBounds_sound μ hM hr.1 hHd' hWe hh.1 p hp |
| have hq' := prefixBounds_sound μ.transpose hM hr.1 |
| (by simpa only [colPrefix, YoungDiagram.transpose_transpose] using hWe') |
| hHd hh.2 q hq |
| have hb := capacity_rectangle μ hp'.1 hq'.1 hp'.2 |
| (by simpa only [colPrefix, YoungDiagram.transpose_transpose] using hq'.2) |
| have hn' : μ.card = M*(M+1)/2+2*r := by rw [hn, staircase_card] |
| omega |
|
|
| end UniversalTensorSquare |
| end |
| end |
|
|
| end OAI |
|
|
