module public import Mathlib public import Preliminaries @[expose] public section /-! # Theorem 3: the lower bound `chi*_3 >= 0.769` This file formalizes Theorem 3 of I. S. Sergeev, *Formula complexity of a linear function in a k-ary basis*, Mathematical Notes 109 (2021), 445--458: `chi*_3 >= 0.769`, i.e. `IsMonotoneGraphExponent 3 (769 / 1000)`. The paper proves the statement by induction over the three graphs of a monotone cover. The first case, in which one part of the graph is contained in one of the cover graphs, relies on the two-graph subadditivity statement (Proposition 3): `s(G) <= s(G_1) + s(G_2)`. That statement, together with the graph restriction helpers it needs, is developed below in the namespace `UnateLin.T3`, so that this file only depends on `Preliminaries` and Mathlib. -/ namespace UnateLin namespace T3 variable {A B : Type} [Fintype A] [Fintype B] [DecidableEq A] [DecidableEq B] theorem ratio_of_admissible (G : BipartiteGraph A B) {X : Finset A} {Y : Finset B} (h : G.Admissible X Y) : G.ratio X Y = (G.edgeCount X Y : ℝ) ^ 2 / ((X.card : ℝ) * (Y.card : ℝ)) := by simp [BipartiteGraph.ratio, h] theorem ratio_le_sGraph (G : BipartiteGraph A B) {X : Finset A} {Y : Finset B} (h : G.Admissible X Y) : G.ratio X Y ≤ G.sGraph := by classical unfold BipartiteGraph.sGraph exact Finset.le_sup' (fun p : Finset A × Finset B => G.ratio p.1 p.2) (Finset.mem_univ (X, Y)) theorem sGraph_ge_full (G : BipartiteGraph A B) (X : Finset A) (Y : Finset B) (h : G.Admissible X Y) : (G.edgeCount X Y : ℝ) ^ 2 / ((X.card : ℝ) * (Y.card : ℝ)) ≤ G.sGraph := by rw [← ratio_of_admissible G h] exact ratio_le_sGraph G h theorem edgeCount_graphOn_self' (G : BipartiteGraph A B) (X : Finset A) (Y : Finset B) : (graphOn G.edge X Y).edgeCount X Y = G.edgeCount X Y := by classical unfold BipartiteGraph.edgeCount graphOn apply congrArg Finset.card apply Finset.filter_congr intro q hq have hq' : q.1 ∈ X ∧ q.2 ∈ Y := Finset.mem_product.mp hq simp [hq'.1, hq'.2] def frag (H : BipartiteGraph A B) (X : Finset A) (Y : Finset B) : BipartiteGraph A B where left := H.left ∩ {a | a ∈ X} right := H.right ∩ {b | b ∈ Y} edge := fun a b => H.edge a b && decide (a ∈ X) && decide (b ∈ Y) theorem frag_left_subset (H : BipartiteGraph A B) (X : Finset A) (Y : Finset B) : (frag H X Y).left ⊆ H.left := fun _ ha => ha.1 theorem frag_right_subset (H : BipartiteGraph A B) (X : Finset A) (Y : Finset B) : (frag H X Y).right ⊆ H.right := fun _ hb => hb.1 theorem frag_edge_subset (H : BipartiteGraph A B) (X : Finset A) (Y : Finset B) : ∀ a b, (frag H X Y).edge a b = true → H.edge a b = true := by intro a b h simp only [frag, Bool.and_eq_true] at h exact h.1.1 theorem sGraph_frag_le (H : BipartiteGraph A B) (X : Finset A) (Y : Finset B) : (frag H X Y).sGraph ≤ H.sGraph := BipartiteGraph.sGraph_mono (frag_left_subset H X Y) (frag_right_subset H X Y) (frag_edge_subset H X Y) theorem frag_edgeRestriction {G H : BipartiteGraph A B} (X : Finset A) (Y : Finset B) (hHG : EdgeRestriction G H) : EdgeRestriction (graphOn G.edge X Y) (frag H X Y) := by intro a b constructor · intro h simp only [frag, Bool.and_eq_true, decide_eq_true_eq] at h obtain ⟨⟨hHe, haX⟩, hbY⟩ := h obtain ⟨hGe, haH, hbH⟩ := (hHG a b).mp hHe refine ⟨?_, ⟨haH, haX⟩, ⟨hbH, hbY⟩⟩ simpa [graphOn, haX, hbY] using hGe · rintro ⟨hGe, ⟨haH, haX⟩, ⟨hbH, hbY⟩⟩ simp only [graphOn, Bool.and_eq_true, decide_eq_true_eq] at hGe obtain ⟨⟨hE, -⟩, -⟩ := hGe have hHe : H.edge a b = true := (hHG a b).mpr ⟨hE, haH, hbH⟩ simp only [frag, Bool.and_eq_true, decide_eq_true_eq] exact ⟨⟨hHe, haX⟩, hbY⟩ theorem cover_frag {G : BipartiteGraph A B} {m : ℕ} {Gs : Fin m → BipartiteGraph A B} (hcover : IsCover G Gs) (X : Finset A) (Y : Finset B) : IsCover (graphOn G.edge X Y) (fun i => frag (Gs i) X Y) := by refine ⟨?_, ?_, ?_, ?_⟩ · intro i a ha exact ha.2 · intro i b hb exact hb.2 · intro i exact frag_edgeRestriction X Y (hcover.2.2.1 i) · intro a b h simp only [graphOn, Bool.and_eq_true, decide_eq_true_eq] at h obtain ⟨⟨hE, haX⟩, hbY⟩ := h obtain ⟨i, hi⟩ := hcover.2.2.2 a b hE exact ⟨i, by simp [frag, hi, haX, hbY]⟩ noncomputable def partRight (Y : Finset B) (S : Set B) : Finset B := by classical exact Y.filter (fun b => b ∈ S) noncomputable def partLeft (X : Finset A) (S : Set A) : Finset A := by classical exact X.filter (fun a => a ∈ S) @[simp] theorem mem_partRight {Y : Finset B} {S : Set B} {b : B} : b ∈ partRight Y S ↔ b ∈ Y ∧ b ∈ S := by classical simp [partRight] @[simp] theorem mem_partLeft {X : Finset A} {S : Set A} {a : A} : a ∈ partLeft X S ↔ a ∈ X ∧ a ∈ S := by classical simp [partLeft] theorem rpow_sum_le_sum_rpow {ι : Type*} (s : Finset ι) (f : ι → ℝ) (hf : ∀ i ∈ s, 0 ≤ f i) {r : ℝ} (hr0 : 0 < r) (hr1 : r ≤ 1) : (∑ i ∈ s, f i) ^ r ≤ ∑ i ∈ s, f i ^ r := by classical induction s using Finset.induction_on with | empty => simp [Real.zero_rpow hr0.ne'] | insert a s ha ih => rw [Finset.sum_insert ha, Finset.sum_insert ha] have ha0 : 0 ≤ f a := hf a (Finset.mem_insert_self a s) have hs0 : 0 ≤ ∑ i ∈ s, f i := Finset.sum_nonneg fun i hi => hf i (Finset.mem_insert_of_mem hi) calc (f a + ∑ i ∈ s, f i) ^ r ≤ f a ^ r + (∑ i ∈ s, f i) ^ r := Real.rpow_add_le_add_rpow ha0 hs0 hr0.le hr1 _ ≤ f a ^ r + ∑ i ∈ s, f i ^ r := by have hih : (∑ i ∈ s, f i) ^ r ≤ ∑ i ∈ s, f i ^ r := ih fun i hi => hf i (Finset.mem_insert_of_mem hi) linarith theorem sGraph_eq_zero_of_no_edges {G : BipartiteGraph A B} (h : ∀ a b, G.edge a b = false) : G.sGraph = 0 := by classical apply le_antisymm · unfold BipartiteGraph.sGraph apply Finset.sup'_le intro p _ simp [BipartiteGraph.ratio, BipartiteGraph.edgeCount, h] · exact BipartiteGraph.sGraph_nonneg G theorem sGraph_le_of_ratio_le {G H : BipartiteGraph A B} (h : ∀ X : Finset A, ∀ Y : Finset B, G.ratio X Y ≤ H.sGraph) : G.sGraph ≤ H.sGraph := by classical unfold BipartiteGraph.sGraph apply Finset.sup'_le intro p _ exact h p.1 p.2 theorem sGraph_le_of_cover_one {G : BipartiteGraph A B} {Gs : Fin 1 → BipartiteGraph A B} (hcover : IsCover G Gs) : G.sGraph ≤ (Gs 0).sGraph := by classical apply sGraph_le_of_ratio_le intro X Y by_cases hAdm : G.Admissible X Y · let X' : Finset A := partLeft X (Gs 0).left let Y' : Finset B := partRight Y (Gs 0).right have hX'sub : X' ⊆ X := by intro a ha have ha' : a ∈ partLeft X (Gs 0).left := by simpa [X'] using ha exact (mem_partLeft.mp ha').1 have hY'sub : Y' ⊆ Y := by intro b hb have hb' : b ∈ partRight Y (Gs 0).right := by simpa [Y'] using hb exact (mem_partRight.mp hb').1 have hAdm' : (Gs 0).Admissible X' Y' := ⟨fun a ha => (mem_partLeft.mp (by simpa [X'] using ha)).2, fun b hb => (mem_partRight.mp (by simpa [Y'] using hb)).2⟩ have hEdgeEq : (Gs 0).edgeCount X' Y' = G.edgeCount X Y := by apply le_antisymm · unfold BipartiteGraph.edgeCount apply Finset.card_le_card rintro ⟨a, b⟩ hq simp only [Finset.mem_filter] at hq ⊢ obtain ⟨hqprod, hqE⟩ := hq have hp : (a, b) ∈ X' ×ˢ Y' := hqprod rw [Finset.mem_product] at hp obtain ⟨haX', hbY'⟩ := hp have hGe : G.edge a b = true := ((hcover.2.2.1 0) a b).mp hqE |>.1 refine ⟨?_, hGe⟩ change (a, b) ∈ X ×ˢ Y rw [Finset.mem_product] exact ⟨hX'sub haX', hY'sub hbY'⟩ · unfold BipartiteGraph.edgeCount apply Finset.card_le_card rintro ⟨a, b⟩ hq simp only [Finset.mem_filter] at hq ⊢ obtain ⟨hqprod, hqE⟩ := hq have hp : (a, b) ∈ X ×ˢ Y := hqprod rw [Finset.mem_product] at hp obtain ⟨haX, hbY⟩ := hp obtain ⟨i, hi⟩ := hcover.2.2.2 a b hqE obtain ⟨-, haL, hbR⟩ := ((hcover.2.2.1 i) a b).mp hi have hi0 : i = 0 := Subsingleton.elim i 0 subst hi0 have haX' : a ∈ X' := by dsimp [X'] exact mem_partLeft.mpr ⟨haX, haL⟩ have hbY' : b ∈ Y' := by dsimp [Y'] exact mem_partRight.mpr ⟨hbY, hbR⟩ refine ⟨?_, hi⟩ change (a, b) ∈ X' ×ˢ Y' rw [Finset.mem_product] exact ⟨haX', hbY'⟩ have hchain : G.ratio X Y ≤ (Gs 0).ratio X' Y' := by rw [ratio_of_admissible G hAdm, ratio_of_admissible (Gs 0) hAdm', hEdgeEq] by_cases hE : G.edgeCount X Y = 0 · simp [hE] · have hEpos : 0 < G.edgeCount X Y := Nat.pos_of_ne_zero hE obtain ⟨⟨a, b⟩, hq⟩ := Finset.card_pos.mp hEpos have hq' : (a, b) ∈ (X.product Y).filter (fun q => G.edge q.1 q.2) := by simpa [BipartiteGraph.edgeCount] using hq simp only [Finset.mem_filter] at hq' obtain ⟨hqprod, hqE⟩ := hq' have hp : (a, b) ∈ X ×ˢ Y := hqprod rw [Finset.mem_product] at hp obtain ⟨haX, hbY⟩ := hp obtain ⟨i0, hi0⟩ := hcover.2.2.2 a b hqE have hi0' : i0 = 0 := Subsingleton.elim i0 0 subst hi0' obtain ⟨-, haL, hbR⟩ := ((hcover.2.2.1 0) a b).mp hi0 have haX' : a ∈ X' := by dsimp [X'] exact mem_partLeft.mpr ⟨haX, haL⟩ have hbY' : b ∈ Y' := by dsimp [Y'] exact mem_partRight.mpr ⟨hbY, hbR⟩ have hX'pos : 0 < X'.card := Finset.card_pos.mpr ⟨a, haX'⟩ have hY'pos : 0 < Y'.card := Finset.card_pos.mpr ⟨b, hbY'⟩ have hX'le : (X'.card : ℝ) ≤ (X.card : ℝ) := by exact_mod_cast Finset.card_le_card hX'sub have hY'le : (Y'.card : ℝ) ≤ (Y.card : ℝ) := by exact_mod_cast Finset.card_le_card hY'sub have hX'posR : (0 : ℝ) < (X'.card : ℝ) := by exact_mod_cast hX'pos have hY'posR : (0 : ℝ) < (Y'.card : ℝ) := by exact_mod_cast hY'pos have hdenpos : (0 : ℝ) < (X'.card : ℝ) * (Y'.card : ℝ) := mul_pos hX'posR hY'posR have hdenle : (X'.card : ℝ) * (Y'.card : ℝ) ≤ (X.card : ℝ) * (Y.card : ℝ) := mul_le_mul hX'le hY'le hY'posR.le (by positivity) exact div_le_div₀ (by positivity) le_rfl hdenpos hdenle exact le_trans hchain (ratio_le_sGraph _ hAdm') · have h0 : G.ratio X Y = 0 := by simp [BipartiteGraph.ratio, hAdm] rw [h0] exact BipartiteGraph.sGraph_nonneg (Gs 0) theorem card_sdiff_eq {α : Type*} [DecidableEq α] {s t : Finset α} (h : s ⊆ t) : (t \ s).card = t.card - s.card := by rw [Finset.card_sdiff] have : s ∩ t = s := Finset.inter_eq_left.mpr h rw [this] @[simp] theorem iUnion_fin_two_sets {α : Type*} (f : Fin 2 → Set α) : (⋃ i, f i) = f 0 ∪ f 1 := by ext x simp only [Set.mem_iUnion, Set.mem_union] constructor · rintro ⟨i, hi⟩ fin_cases i <;> simp_all · rintro (h | h) · exact ⟨0, h⟩ · exact ⟨1, h⟩ theorem edge_eq_of_edgeRestriction {G H : BipartiteGraph A B} (h : EdgeRestriction G H) {a : A} {b : B} (ha : a ∈ H.left) (hb : b ∈ H.right) : H.edge a b = G.edge a b := by by_cases hg : G.edge a b = true · have hh : H.edge a b = true := (h a b).mpr ⟨hg, ha, hb⟩ simp [hg, hh] · have hh : H.edge a b = false := by by_contra hne have htrue : H.edge a b = true := by simpa using hne exact hg ((h a b).mp htrue).1 simp [hg, hh] theorem edgeCount_congr_on {H K : BipartiteGraph A B} (X : Finset A) (Y : Finset B) (h : ∀ a ∈ X, ∀ b ∈ Y, H.edge a b = K.edge a b) : H.edgeCount X Y = K.edgeCount X Y := by classical unfold BipartiteGraph.edgeCount apply congrArg Finset.card apply Finset.filter_congr intro q hq obtain ⟨hqX, hqY⟩ := Finset.mem_product.mp hq rw [h q.1 hqX q.2 hqY] theorem edgeCount_split_right (G : BipartiteGraph A B) (X : Finset A) (Y B1 : Finset B) (hB1 : B1 ⊆ Y) : G.edgeCount X Y = G.edgeCount X B1 + G.edgeCount X (Y \ B1) := by classical unfold BipartiteGraph.edgeCount let p : A × B → Prop := fun q => G.edge q.1 q.2 = true let s1 : Finset (A × B) := (X.product B1).filter p let s2 : Finset (A × B) := (X.product (Y \ B1)).filter p have hdisj : Disjoint s1 s2 := by rw [Finset.disjoint_left] intro q hq1 hq2 simp only [s1, s2, Finset.mem_filter] at hq1 hq2 have hq2Y : q.2 ∈ Y \ B1 := (Finset.mem_product.mp hq2.1).2 rw [Finset.mem_sdiff] at hq2Y exact hq2Y.2 ((Finset.mem_product.mp hq1.1).2) have hunion : (X.product Y).filter p = s1 ∪ s2 := by ext q simp only [s1, s2, p, Finset.mem_filter, Finset.mem_union, Finset.mem_product, Finset.mem_sdiff, Finset.product_eq_sprod] constructor · rintro ⟨⟨ha, hY⟩, he⟩ by_cases hb : q.2 ∈ B1 · exact Or.inl ⟨⟨ha, hb⟩, he⟩ · exact Or.inr ⟨⟨ha, hY, hb⟩, he⟩ · rintro (⟨⟨ha, hb⟩, he⟩ | ⟨⟨ha, hY, hb⟩, he⟩) · exact ⟨⟨ha, hB1 hb⟩, he⟩ · exact ⟨⟨ha, hY⟩, he⟩ rw [hunion, Finset.card_union_of_disjoint hdisj] theorem sq_add_sq_div {x1 x2 y1 y2 : ℝ} (hy1 : 0 < y1) (hy2 : 0 < y2) : (x1 + x2) ^ 2 / (y1 + y2) ≤ x1 ^ 2 / y1 + x2 ^ 2 / y2 := by have h := Finset.sq_sum_div_le_sum_sq_div (s := (Finset.univ : Finset (Fin 2))) (f := ![x1, x2]) (g := ![y1, y2]) (by intro i _ fin_cases i <;> simpa) simpa [Fin.sum_univ_two] using h theorem twoGraph_row (G H1 H2 : BipartiteGraph A B) (X : Finset A) (Y : Finset B) (hL1 : G.left ⊆ H1.left) (hL2 : G.left ⊆ H2.left) (hYR : G.right ⊆ H1.right ∪ H2.right) (hER1 : EdgeRestriction G H1) (hER2 : EdgeRestriction G H2) (hGleft : G.left = (↑X : Set A)) (hGright : G.right = (↑Y : Set B)) : G.ratio X Y ≤ H1.sGraph + H2.sGraph := by classical have hAdm : G.Admissible X Y := by refine ⟨?_, ?_⟩ · intro a ha; rw [hGleft]; exact ha · intro b hb; rw [hGright]; exact hb by_cases hEzero : G.edgeCount X Y = 0 · rw [ratio_of_admissible G hAdm, hEzero] simp only [Nat.cast_zero, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, zero_pow, zero_div] exact add_nonneg (BipartiteGraph.sGraph_nonneg H1) (BipartiteGraph.sGraph_nonneg H2) · have hEpos : 0 < G.edgeCount X Y := Nat.pos_of_ne_zero hEzero rw [ratio_of_admissible G hAdm] set B1 : Finset B := partRight Y H1.right with hB1def have hB1sub : B1 ⊆ Y := fun b hb => (mem_partRight.mp hb).1 have hYsub : Y ⊆ B1 ∪ partRight Y H2.right := by intro b hb have hbG : b ∈ G.right := by rw [hGright]; exact hb rcases hYR hbG with h | h · exact Finset.mem_union.mpr (Or.inl (mem_partRight.mpr ⟨hb, h⟩)) · exact Finset.mem_union.mpr (Or.inr (mem_partRight.mpr ⟨hb, h⟩)) have hdiff : Y \ B1 ⊆ partRight Y H2.right := by intro b hb rw [Finset.mem_sdiff] at hb rcases Finset.mem_union.mp (hYsub hb.1) with h | h · exact absurd h hb.2 · exact h let a : ℝ := (X.card : ℝ) let b : ℝ := (Y.card : ℝ) let b1 : ℝ := (B1.card : ℝ) let x1 : ℝ := (G.edgeCount X B1 : ℝ) let x2 : ℝ := (G.edgeCount X (Y \ B1) : ℝ) have hsplit : G.edgeCount X Y = G.edgeCount X B1 + G.edgeCount X (Y \ B1) := edgeCount_split_right G X Y B1 hB1sub have hE : (G.edgeCount X Y : ℝ) = x1 + x2 := by rw [hsplit] simp only [Nat.cast_add] rfl have hx1 : x1 ^ 2 / (a * b1) ≤ H1.sGraph := by have hAdm1 : H1.Admissible X B1 := by refine ⟨?_, ?_⟩ · intro a' ha' exact hL1 (by rw [hGleft]; exact ha') · intro b' hb' exact (mem_partRight.mp hb').2 have hcount : H1.edgeCount X B1 = G.edgeCount X B1 := by apply edgeCount_congr_on intro a' ha' b' hb' exact edge_eq_of_edgeRestriction hER1 (hL1 (by rw [hGleft]; exact ha')) (mem_partRight.mp hb').2 have h := sGraph_ge_full H1 X B1 hAdm1 rw [hcount] at h simpa [a, b1, x1] using h have hx2 : x2 ^ 2 / (a * (b - b1)) ≤ H2.sGraph := by have hAdm2 : H2.Admissible X (Y \ B1) := by refine ⟨?_, ?_⟩ · intro a' ha' exact hL2 (by rw [hGleft]; exact ha') · intro b' hb' exact (mem_partRight.mp (hdiff hb')).2 have hcount : H2.edgeCount X (Y \ B1) = G.edgeCount X (Y \ B1) := by apply edgeCount_congr_on intro a' ha' b' hb' exact edge_eq_of_edgeRestriction hER2 (hL2 (by rw [hGleft]; exact ha')) (mem_partRight.mp (hdiff hb')).2 have h := sGraph_ge_full H2 X (Y \ B1) hAdm2 rw [hcount] at h have hle : B1.card ≤ Y.card := Finset.card_le_card hB1sub have hcard : (((Y \ B1).card : ℕ) : ℝ) = b - b1 := by rw [card_sdiff_eq hB1sub, Nat.cast_sub hle] rw [hcard] at h simpa [a, x2] using h have ha : 0 < a := by obtain ⟨q, hq⟩ := Finset.card_pos.mp hEpos change q ∈ (X.product Y).filter (fun q => G.edge q.1 q.2 = true) at hq rw [Finset.mem_filter] at hq have hqX : q.1 ∈ X := (Finset.mem_product.mp hq.1).1 have hXpos : 0 < X.card := Finset.card_pos.mpr ⟨q.1, hqX⟩ simp only [a] exact_mod_cast hXpos have hb : 0 < b := by obtain ⟨q, hq⟩ := Finset.card_pos.mp hEpos change q ∈ (X.product Y).filter (fun q => G.edge q.1 q.2 = true) at hq rw [Finset.mem_filter] at hq have hqY : q.2 ∈ Y := (Finset.mem_product.mp hq.1).2 have hYpos : 0 < Y.card := Finset.card_pos.mpr ⟨q.2, hqY⟩ simp only [b] exact_mod_cast hYpos have hb1le : b1 ≤ b := by simp only [b1, b] exact_mod_cast Finset.card_le_card hB1sub have hb1nonneg : 0 ≤ b1 := by positivity by_cases hb1zero : b1 = 0 · have hzero : x1 = 0 := by have hB1card : B1.card = 0 := by have hz : (B1.card : ℝ) = 0 := by simpa [b1] using hb1zero exact_mod_cast hz have hB1empty : B1 = ∅ := Finset.card_eq_zero.mp hB1card simp [x1, hB1empty, BipartiteGraph.edgeCount] have hrem : b - b1 = b := by rw [hb1zero, sub_zero] calc (G.edgeCount X Y : ℝ) ^ 2 / (a * b) = x2 ^ 2 / (a * b) := by rw [hE, hzero, zero_add] _ = x2 ^ 2 / (a * (b - b1)) := by rw [hrem] _ ≤ H2.sGraph := hx2 _ ≤ H1.sGraph + H2.sGraph := by linarith [BipartiteGraph.sGraph_nonneg H1] · by_cases hb1eq : b1 = b · have hzero : x2 = 0 := by have hdiff_empty : Y \ B1 = ∅ := by apply Finset.eq_empty_iff_forall_notMem.mpr intro b' hb' rw [Finset.mem_sdiff] at hb' have hcard : (Y \ B1).card = 0 := by rw [card_sdiff_eq hB1sub] have hnat : B1.card = Y.card := by have hz : (B1.card : ℝ) = (Y.card : ℝ) := by simpa [b1, b] using hb1eq exact_mod_cast hz omega have hpos : 0 < (Y \ B1).card := Finset.card_pos.mpr ⟨b', Finset.mem_sdiff.mpr hb'⟩ omega simp [x2, hdiff_empty, BipartiteGraph.edgeCount] have hrem : b - b1 = 0 := by rw [hb1eq, sub_self] calc (G.edgeCount X Y : ℝ) ^ 2 / (a * b) = x1 ^ 2 / (a * b) := by rw [hE, hzero, add_zero] _ = x1 ^ 2 / (a * b1) := by rw [← hb1eq] _ ≤ H1.sGraph := hx1 _ ≤ H1.sGraph + H2.sGraph := by linarith [BipartiteGraph.sGraph_nonneg H2] · have hb1pos : 0 < b1 := lt_of_le_of_ne hb1nonneg (Ne.symm hb1zero) have hb1lt : b1 < b := lt_of_le_of_ne hb1le hb1eq have hy1 : 0 < a * b1 := mul_pos ha hb1pos have hy2 : 0 < a * (b - b1) := mul_pos ha (sub_pos.mpr hb1lt) have htitu := sq_add_sq_div (x1 := x1) (x2 := x2) (y1 := a * b1) (y2 := a * (b - b1)) hy1 hy2 have hy : a * b1 + a * (b - b1) = a * b := by ring calc (G.edgeCount X Y : ℝ) ^ 2 / (a * b) = (x1 + x2) ^ 2 / (a * b) := by rw [hE] _ = (x1 + x2) ^ 2 / (a * b1 + a * (b - b1)) := by rw [hy] _ ≤ x1 ^ 2 / (a * b1) + x2 ^ 2 / (a * (b - b1)) := htitu _ ≤ H1.sGraph + H2.sGraph := add_le_add hx1 hx2 theorem edgeCount_split_left (G : BipartiteGraph A B) (X A1 : Finset A) (Y : Finset B) (hA1 : A1 ⊆ X) : G.edgeCount X Y = G.edgeCount A1 Y + G.edgeCount (X \ A1) Y := by classical unfold BipartiteGraph.edgeCount let p : A × B → Prop := fun q => G.edge q.1 q.2 = true let s1 : Finset (A × B) := (A1.product Y).filter p let s2 : Finset (A × B) := ((X \ A1).product Y).filter p have hdisj : Disjoint s1 s2 := by rw [Finset.disjoint_left] intro q hq1 hq2 simp only [s1, s2, Finset.mem_filter] at hq1 hq2 have hq2X : q.1 ∈ X \ A1 := (Finset.mem_product.mp hq2.1).1 rw [Finset.mem_sdiff] at hq2X exact hq2X.2 ((Finset.mem_product.mp hq1.1).1) have hunion : (X.product Y).filter p = s1 ∪ s2 := by ext q simp only [s1, s2, p, Finset.mem_filter, Finset.mem_union, Finset.product_eq_sprod, Finset.mem_product, Finset.mem_sdiff] constructor · rintro ⟨⟨hX, hY⟩, he⟩ by_cases ha : q.1 ∈ A1 · exact Or.inl (And.intro (And.intro ha hY) he) · exact Or.inr (And.intro (And.intro (And.intro hX ha) hY) he) · rintro (⟨⟨ha, hY⟩, he⟩ | ⟨⟨⟨hX, ha⟩, hY⟩, he⟩) · exact And.intro (And.intro (hA1 ha) hY) he · exact And.intro (And.intro hX hY) he rw [hunion, Finset.card_union_of_disjoint hdisj] theorem twoGraph_col (G H1 H2 : BipartiteGraph A B) (X : Finset A) (Y : Finset B) (hR1 : G.right ⊆ H1.right) (hR2 : G.right ⊆ H2.right) (hXL : G.left ⊆ H1.left ∪ H2.left) (hER1 : EdgeRestriction G H1) (hER2 : EdgeRestriction G H2) (hGleft : G.left = (↑X : Set A)) (hGright : G.right = (↑Y : Set B)) : G.ratio X Y ≤ H1.sGraph + H2.sGraph := by classical have hAdm : G.Admissible X Y := by refine ⟨?_, ?_⟩ · intro a ha; rw [hGleft]; exact ha · intro b hb; rw [hGright]; exact hb by_cases hEzero : G.edgeCount X Y = 0 · rw [ratio_of_admissible G hAdm, hEzero] simp only [Nat.cast_zero, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, zero_pow, zero_div] exact add_nonneg (BipartiteGraph.sGraph_nonneg H1) (BipartiteGraph.sGraph_nonneg H2) · have hEpos : 0 < G.edgeCount X Y := Nat.pos_of_ne_zero hEzero rw [ratio_of_admissible G hAdm] set A1 : Finset A := partLeft X H1.left with hA1def have hA1sub : A1 ⊆ X := fun a ha => (mem_partLeft.mp ha).1 have hXsub : X ⊆ A1 ∪ partLeft X H2.left := by intro a ha have haG : a ∈ G.left := by rw [hGleft]; exact ha rcases hXL haG with h | h · exact Finset.mem_union.mpr (Or.inl (mem_partLeft.mpr ⟨ha, h⟩)) · exact Finset.mem_union.mpr (Or.inr (mem_partLeft.mpr ⟨ha, h⟩)) have hdiff : X \ A1 ⊆ partLeft X H2.left := by intro a ha rw [Finset.mem_sdiff] at ha rcases Finset.mem_union.mp (hXsub ha.1) with h | h · exact absurd h ha.2 · exact h let a : ℝ := (X.card : ℝ) let b : ℝ := (Y.card : ℝ) let a1 : ℝ := (A1.card : ℝ) let x1 : ℝ := (G.edgeCount A1 Y : ℝ) let x2 : ℝ := (G.edgeCount (X \ A1) Y : ℝ) have hsplit : G.edgeCount X Y = G.edgeCount A1 Y + G.edgeCount (X \ A1) Y := edgeCount_split_left G X A1 Y hA1sub have hE : (G.edgeCount X Y : ℝ) = x1 + x2 := by rw [hsplit] simp only [Nat.cast_add] rfl have hx1 : x1 ^ 2 / (a1 * b) ≤ H1.sGraph := by have hAdm1 : H1.Admissible A1 Y := by refine ⟨?_, ?_⟩ · intro a' ha' exact (mem_partLeft.mp ha').2 · intro b' hb' exact hR1 (by rw [hGright]; exact hb') have hcount : H1.edgeCount A1 Y = G.edgeCount A1 Y := by apply edgeCount_congr_on intro a' ha' b' hb' exact edge_eq_of_edgeRestriction hER1 (mem_partLeft.mp ha').2 (hR1 (by rw [hGright]; exact hb')) have h := sGraph_ge_full H1 A1 Y hAdm1 rw [hcount] at h simpa [a1, b, x1] using h have hx2 : x2 ^ 2 / ((a - a1) * b) ≤ H2.sGraph := by have hAdm2 : H2.Admissible (X \ A1) Y := by refine ⟨?_, ?_⟩ · intro a' ha' exact (mem_partLeft.mp (hdiff ha')).2 · intro b' hb' exact hR2 (by rw [hGright]; exact hb') have hcount : H2.edgeCount (X \ A1) Y = G.edgeCount (X \ A1) Y := by apply edgeCount_congr_on intro a' ha' b' hb' exact edge_eq_of_edgeRestriction hER2 (mem_partLeft.mp (hdiff ha')).2 (hR2 (by rw [hGright]; exact hb')) have h := sGraph_ge_full H2 (X \ A1) Y hAdm2 rw [hcount] at h have hle : A1.card ≤ X.card := Finset.card_le_card hA1sub have hcard : (((X \ A1).card : ℕ) : ℝ) = a - a1 := by rw [card_sdiff_eq hA1sub, Nat.cast_sub hle] rw [hcard] at h simpa [a, x2] using h have ha : 0 < a := by obtain ⟨q, hq⟩ := Finset.card_pos.mp hEpos change q ∈ (X.product Y).filter (fun q => G.edge q.1 q.2 = true) at hq rw [Finset.mem_filter] at hq have hqX : q.1 ∈ X := (Finset.mem_product.mp hq.1).1 have hXpos : 0 < X.card := Finset.card_pos.mpr ⟨q.1, hqX⟩ simp only [a] exact_mod_cast hXpos have hb : 0 < b := by obtain ⟨q, hq⟩ := Finset.card_pos.mp hEpos change q ∈ (X.product Y).filter (fun q => G.edge q.1 q.2 = true) at hq rw [Finset.mem_filter] at hq have hqY : q.2 ∈ Y := (Finset.mem_product.mp hq.1).2 have hYpos : 0 < Y.card := Finset.card_pos.mpr ⟨q.2, hqY⟩ simp only [b] exact_mod_cast hYpos have ha1le : a1 ≤ a := by simp only [a1, a] exact_mod_cast Finset.card_le_card hA1sub have ha1nonneg : 0 ≤ a1 := by positivity by_cases ha1zero : a1 = 0 · have hzero : x1 = 0 := by have hA1card : A1.card = 0 := by have hz : (A1.card : ℝ) = 0 := by simpa [a1] using ha1zero exact_mod_cast hz have hA1empty : A1 = ∅ := Finset.card_eq_zero.mp hA1card simp [x1, hA1empty, BipartiteGraph.edgeCount] have hrem : a - a1 = a := by rw [ha1zero, sub_zero] calc (G.edgeCount X Y : ℝ) ^ 2 / (a * b) = x2 ^ 2 / (a * b) := by rw [hE, hzero, zero_add] _ = x2 ^ 2 / ((a - a1) * b) := by rw [hrem] _ ≤ H2.sGraph := hx2 _ ≤ H1.sGraph + H2.sGraph := by linarith [BipartiteGraph.sGraph_nonneg H1] · by_cases ha1eq : a1 = a · have hzero : x2 = 0 := by have hdiff_empty : X \ A1 = ∅ := by apply Finset.eq_empty_iff_forall_notMem.mpr intro a' ha' rw [Finset.mem_sdiff] at ha' have hcard : (X \ A1).card = 0 := by rw [card_sdiff_eq hA1sub] have hnat : A1.card = X.card := by have hz : (A1.card : ℝ) = (X.card : ℝ) := by simpa [a1, a] using ha1eq exact_mod_cast hz omega have hpos : 0 < (X \ A1).card := Finset.card_pos.mpr ⟨a', Finset.mem_sdiff.mpr ha'⟩ omega simp [x2, hdiff_empty, BipartiteGraph.edgeCount] have hrem : a - a1 = 0 := by rw [ha1eq, sub_self] calc (G.edgeCount X Y : ℝ) ^ 2 / (a * b) = x1 ^ 2 / (a * b) := by rw [hE, hzero, add_zero] _ = x1 ^ 2 / (a1 * b) := by rw [← ha1eq] _ ≤ H1.sGraph := hx1 _ ≤ H1.sGraph + H2.sGraph := by linarith [BipartiteGraph.sGraph_nonneg H2] · have ha1pos : 0 < a1 := lt_of_le_of_ne ha1nonneg (Ne.symm ha1zero) have ha1lt : a1 < a := lt_of_le_of_ne ha1le ha1eq have hy1 : 0 < a1 * b := mul_pos ha1pos hb have hy2 : 0 < (a - a1) * b := mul_pos (sub_pos.mpr ha1lt) hb have htitu := sq_add_sq_div (x1 := x1) (x2 := x2) (y1 := a1 * b) (y2 := (a - a1) * b) hy1 hy2 have hy : a1 * b + (a - a1) * b = a * b := by ring calc (G.edgeCount X Y : ℝ) ^ 2 / (a * b) = (x1 + x2) ^ 2 / (a * b) := by rw [hE] _ = (x1 + x2) ^ 2 / (a1 * b + (a - a1) * b) := by rw [hy] _ ≤ x1 ^ 2 / (a1 * b) + x2 ^ 2 / ((a - a1) * b) := htitu _ ≤ H1.sGraph + H2.sGraph := add_le_add hx1 hx2 theorem ratio_le_sGraph_of_parts (G H : BipartiteGraph A B) (X : Finset A) (Y : Finset B) (hG : G.Admissible X Y) (hER : EdgeRestriction G H) (hXL : ∀ a ∈ X, a ∈ H.left) (hYR : ∀ b ∈ Y, b ∈ H.right) : G.ratio X Y ≤ H.sGraph := by classical have hH : H.Admissible X Y := ⟨hXL, hYR⟩ have hcount : H.edgeCount X Y = G.edgeCount X Y := by apply edgeCount_congr_on intro a ha b hb exact edge_eq_of_edgeRestriction hER (hXL a ha) (hYR b hb) rw [ratio_of_admissible G hG] calc (G.edgeCount X Y : ℝ) ^ 2 / ((X.card : ℝ) * (Y.card : ℝ)) = (H.edgeCount X Y : ℝ) ^ 2 / ((X.card : ℝ) * (Y.card : ℝ)) := by rw [hcount] _ = H.ratio X Y := (ratio_of_admissible H hH).symm _ ≤ H.sGraph := ratio_le_sGraph H hH theorem twoGraph_ratio_le_add (G H1 H2 : BipartiteGraph A B) (X : Finset A) (Y : Finset B) (hmono : IsMonotoneCover G ![H1,H2]) (hAdm : G.Admissible X Y) (hGleft : G.left = (↑X : Set A)) (hGright : G.right = (↑Y : Set B)) : G.ratio X Y ≤ H1.sGraph + H2.sGraph := by classical have hcover := hmono.1 have hER1 : EdgeRestriction G H1 := hcover.2.2.1 0 have hER2 : EdgeRestriction G H2 := hcover.2.2.1 1 have hL1R2 : G.left ⊆ H1.left ∨ G.right ⊆ H2.right := by have hh := hmono.2 ({0} : Set (Fin 2)) simpa [Set.sdiff_eq_empty] using hh have hL2R1 : G.left ⊆ H2.left ∨ G.right ⊆ H1.right := by have hh := hmono.2 ({1} : Set (Fin 2)) simpa [Set.sdiff_eq_empty] using hh by_cases hEzero : G.edgeCount X Y = 0 · rw [ratio_of_admissible G hAdm, hEzero] simp only [Nat.cast_zero, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, zero_pow, zero_div] exact add_nonneg (BipartiteGraph.sGraph_nonneg H1) (BipartiteGraph.sGraph_nonneg H2) · have hEpos : 0 < G.edgeCount X Y := Nat.pos_of_ne_zero hEzero obtain ⟨q, hq⟩ := Finset.card_pos.mp hEpos change q ∈ (X.product Y).filter (fun q => G.edge q.1 q.2 = true) at hq rw [Finset.mem_filter] at hq have hge : G.edge q.1 q.2 = true := hq.2 obtain ⟨i, hi⟩ := hcover.2.2.2 q.1 q.2 hge have hHiL : (![H1, H2] i).left ⊆ G.left := hcover.1 i have hHiR : (![H1, H2] i).right ⊆ G.right := hcover.2.1 i have hLne : G.left.Nonempty := ⟨q.1, hHiL (((hcover.2.2.1 i) q.1 q.2).mp hi).2.1⟩ have hRne : G.right.Nonempty := ⟨q.2, hHiR (((hcover.2.2.1 i) q.1 q.2).mp hi).2.2⟩ have hR12 : G.right ⊆ H1.right ∪ H2.right := by have hh := hmono.2 (∅ : Set (Fin 2)) rcases hh with h1 | h2 · exfalso have h1' : G.left = ∅ := by simpa using h1 obtain ⟨a, ha⟩ := hLne rw [h1'] at ha exact ha · simpa [Set.sdiff_eq_empty] using h2 have hL12 : G.left ⊆ H1.left ∪ H2.left := by have hh := hmono.2 (Set.univ : Set (Fin 2)) rcases hh with h1 | h2 · simpa [Set.sdiff_eq_empty] using h1 · exfalso have h2' : G.right = ∅ := by simpa using h2 obtain ⟨b, hb⟩ := hRne rw [h2'] at hb exact hb by_cases hL1 : G.left ⊆ H1.left · by_cases hR1 : G.right ⊆ H1.right · exact le_trans (ratio_le_sGraph_of_parts G H1 X Y hAdm hER1 (fun a ha => hL1 (by rw [hGleft]; exact ha)) (fun b hb => hR1 (by rw [hGright]; exact hb))) (le_add_of_nonneg_right (BipartiteGraph.sGraph_nonneg H2)) · have hL2 : G.left ⊆ H2.left := by rcases hL2R1 with h | h · exact h · exact absurd h hR1 exact twoGraph_row G H1 H2 X Y hL1 hL2 hR12 hER1 hER2 hGleft hGright · have hR2 : G.right ⊆ H2.right := by rcases hL1R2 with h | h · exact absurd h hL1 · exact h by_cases hL2 : G.left ⊆ H2.left · exact le_trans (ratio_le_sGraph_of_parts G H2 X Y hAdm hER2 (fun a ha => hL2 (by rw [hGleft]; exact ha)) (fun b hb => hR2 (by rw [hGright]; exact hb))) (le_add_of_nonneg_left (BipartiteGraph.sGraph_nonneg H1)) · have hR1 : G.right ⊆ H1.right := by rcases hL2R1 with h | h · exact absurd h hL2 · exact h exact twoGraph_col G H1 H2 X Y hR1 hR2 hL12 hER1 hER2 hGleft hGright theorem monotoneCover_frag {k : ℕ} {G : BipartiteGraph A B} {Gs : Fin k → BipartiteGraph A B} (h : IsMonotoneCover G Gs) (X : Finset A) (Y : Finset B) (hX : (↑X : Set A) ⊆ G.left) (hY : (↑Y : Set B) ⊆ G.right) : IsMonotoneCover (graphOn G.edge X Y) (fun i => frag (Gs i) X Y) := by classical refine ⟨cover_frag h.1 X Y, ?_⟩ intro I rcases h.2 I with hI | hI · left rw [Set.sdiff_eq_empty] at hI ⊢ intro a ha have haG : a ∈ G.left := hX ha have haU : a ∈ ⋃ i ∈ I, (Gs i).left := hI haG simp only [Set.mem_iUnion, exists_prop] at haU ⊢ obtain ⟨i, hiI, hai⟩ := haU exact ⟨i, hiI, hai, ha⟩ · right rw [Set.sdiff_eq_empty] at hI ⊢ intro b hb have hbG : b ∈ G.right := hY hb have hbU : b ∈ ⋃ i ∈ Iᶜ, (Gs i).right := hI hbG simp only [Set.mem_iUnion, exists_prop] at hbU ⊢ obtain ⟨i, hiI, hbi⟩ := hbU exact ⟨i, hiI, hbi, hb⟩ theorem sGraph_le_add_of_monotoneCover_two (G : BipartiteGraph A B) (Gs : Fin 2 → BipartiteGraph A B) (hmono : IsMonotoneCover G Gs) : G.sGraph ≤ (Gs 0).sGraph + (Gs 1).sGraph := by classical rw [BipartiteGraph.sGraph] apply Finset.sup'_le intro p hp by_cases hAdm : G.Admissible p.1 p.2 · set X : Finset A := p.1 with hXdef set Y : Finset B := p.2 with hYdef let G' : BipartiteGraph A B := graphOn G.edge X Y let Gs' : Fin 2 → BipartiteGraph A B := fun i => frag (Gs i) X Y have hX : (↑X : Set A) ⊆ G.left := fun a ha => hAdm.1 a ha have hY : (↑Y : Set B) ⊆ G.right := fun b hb => hAdm.2 b hb have hmono' : IsMonotoneCover G' Gs' := monotoneCover_frag hmono X Y hX hY have hvec : (![Gs' 0, Gs' 1] : Fin 2 → BipartiteGraph A B) = Gs' := by funext i fin_cases i <;> rfl have hmonoVec : IsMonotoneCover G' ![Gs' 0, Gs' 1] := by rw [hvec] exact hmono' have hAdm' : G'.Admissible X Y := ⟨fun a ha => ha, fun b hb => hb⟩ have hratio : G'.ratio X Y ≤ (Gs' 0).sGraph + (Gs' 1).sGraph := twoGraph_ratio_le_add G' (Gs' 0) (Gs' 1) X Y hmonoVec hAdm' rfl rfl have hGeq : G.ratio X Y = G'.ratio X Y := by rw [ratio_of_admissible G hAdm, ratio_of_admissible G' hAdm', edgeCount_graphOn_self'] have hle0 : (Gs' 0).sGraph ≤ (Gs 0).sGraph := sGraph_frag_le (Gs 0) X Y have hle1 : (Gs' 1).sGraph ≤ (Gs 1).sGraph := sGraph_frag_le (Gs 1) X Y calc G.ratio X Y = G'.ratio X Y := hGeq _ ≤ (Gs' 0).sGraph + (Gs' 1).sGraph := hratio _ ≤ (Gs 0).sGraph + (Gs 1).sGraph := add_le_add hle0 hle1 · simp only [BipartiteGraph.ratio, hAdm, ite_false] exact add_nonneg (BipartiteGraph.sGraph_nonneg (Gs 0)) (BipartiteGraph.sGraph_nonneg (Gs 1)) theorem isMonotoneGraphExponent_two_one : IsMonotoneGraphExponent 2 1 := by intro m hm A B _ _ _ _ G Gs hmono have hm012 : m = 0 ∨ m = 1 ∨ m = 2 := by omega rcases hm012 with rfl | rfl | rfl · have hno : ∀ a b, G.edge a b = false := by intro a b by_contra hne have htrue : G.edge a b = true := by simpa using hne obtain ⟨i, _⟩ := hmono.1.2.2.2 a b htrue exact Fin.elim0 i rw [sGraph_eq_zero_of_no_edges hno] simp · simpa using sGraph_le_of_cover_one hmono.1 · simpa [Fin.sum_univ_two] using sGraph_le_add_of_monotoneCover_two G Gs hmono theorem isMonotoneGraphExponent_of_le {k : ℕ} {χ χ' : ℝ} (hχ0 : 0 < χ) (hχχ' : χ ≤ χ') (h : IsMonotoneGraphExponent k χ') : IsMonotoneGraphExponent k χ := by intro m hm A B _ _ _ _ G Gs hmono have h' := h m hm A B G Gs hmono have hχ'0 : 0 < χ' := lt_of_lt_of_le hχ0 hχχ' have hG0 : 0 ≤ G.sGraph := BipartiteGraph.sGraph_nonneg G by_cases hzero : G.sGraph = 0 · rw [hzero, Real.zero_rpow hχ0.ne'] exact Finset.sum_nonneg fun i _ => Real.rpow_nonneg (BipartiteGraph.sGraph_nonneg (Gs i)) χ · have hGpos : 0 < G.sGraph := lt_of_le_of_ne hG0 (Ne.symm hzero) set S : ℝ := ∑ i, (Gs i).sGraph ^ χ' with hS have hSpos : 0 < S := lt_of_lt_of_le (Real.rpow_pos_of_pos hGpos χ') h' have hr0 : 0 < χ / χ' := div_pos hχ0 hχ'0 have hr1 : χ / χ' ≤ 1 := (div_le_one hχ'0).mpr hχχ' have hsub : S ^ (χ / χ') ≤ ∑ i, ((Gs i).sGraph ^ χ') ^ (χ / χ') := by rw [hS] exact rpow_sum_le_sum_rpow Finset.univ (fun i => (Gs i).sGraph ^ χ') (fun i _ => Real.rpow_nonneg (BipartiteGraph.sGraph_nonneg (Gs i)) χ') hr0 hr1 have hT : ∑ i, ((Gs i).sGraph ^ χ') ^ (χ / χ') = ∑ i, (Gs i).sGraph ^ χ := by apply Finset.sum_congr rfl intro i _ rw [← Real.rpow_mul (BipartiteGraph.sGraph_nonneg (Gs i))] congr 1 field_simp have hleft : G.sGraph ^ χ = (G.sGraph ^ χ') ^ (χ / χ') := by rw [← Real.rpow_mul hG0] congr 1 field_simp rw [hleft] exact le_trans (Real.rpow_le_rpow (Real.rpow_nonneg hG0 χ') h' hr0.le) (by rw [← hT]; exact hsub) theorem isMonotoneGraphExponent_two_of_le {χ : ℝ} (hχ0 : 0 < χ) (hχ1 : χ ≤ 1) : IsMonotoneGraphExponent 2 χ := isMonotoneGraphExponent_of_le hχ0 hχ1 isMonotoneGraphExponent_two_one theorem wellFormed_graphOn (T : A → B → Bool) (X : Finset A) (Y : Finset B) : BipartiteGraph.WellFormed (graphOn T X Y) := by intro a b h simp only [graphOn, Bool.and_eq_true, decide_eq_true_eq] at h exact ⟨h.1.2, h.2⟩ theorem edgeRestriction_frag_self {H : BipartiteGraph A B} (hwf : BipartiteGraph.WellFormed H) (X : Finset A) (Y : Finset B) : EdgeRestriction H (frag H X Y) := by intro a b constructor · intro h simp only [frag, Bool.and_eq_true, decide_eq_true_eq] at h exact ⟨h.1.1, ⟨(hwf a b h.1.1).1, h.1.2⟩, ⟨(hwf a b h.1.1).2, h.2⟩⟩ · rintro ⟨hE, ⟨_, ha⟩, ⟨_, hb⟩⟩ simp only [frag, Bool.and_eq_true, decide_eq_true_eq] exact ⟨⟨hE, ha⟩, hb⟩ theorem BipartiteGraph_ext {G H : BipartiteGraph A B} (hl : G.left = H.left) (hr : G.right = H.right) (he : G.edge = H.edge) : G = H := by rcases G with ⟨Gl, Gr, Ge⟩ rcases H with ⟨Hl, Hr, He⟩ simp_all theorem sGraph_frag_mono (H : BipartiteGraph A B) {X X' : Finset A} (Y : Finset B) (hX : X' ⊆ X) : (frag H X' Y).sGraph ≤ (frag H X Y).sGraph := by apply BipartiteGraph.sGraph_mono · intro a ha exact ⟨ha.1, hX ha.2⟩ · intro b hb exact hb · intro a b h simp only [frag, Bool.and_eq_true, decide_eq_true_eq] at h ⊢ exact ⟨⟨h.1.1, hX h.1.2⟩, h.2⟩ theorem frag_eq_graphOn_self {G H : BipartiteGraph A B} (X : Finset A) (Y : Finset B) (hER : EdgeRestriction (graphOn G.edge X Y) (frag H X Y)) (hX : (↑X : Set A) ⊆ (frag H X Y).left) (hY : (↑Y : Set B) ⊆ (frag H X Y).right) : frag H X Y = graphOn G.edge X Y := by classical apply BipartiteGraph_ext · ext a simp only [frag, graphOn, Set.mem_inter_iff, Set.mem_setOf_eq] exact ⟨fun h => h.2, fun h => by have hh := hX h simp only [frag, Set.mem_inter_iff, Set.mem_setOf_eq] at hh exact ⟨hh.1, h⟩⟩ · ext b simp only [frag, graphOn, Set.mem_inter_iff, Set.mem_setOf_eq] exact ⟨fun h => h.2, fun h => by have hh := hY h simp only [frag, Set.mem_inter_iff, Set.mem_setOf_eq] at hh exact ⟨hh.1, h⟩⟩ · funext a b by_cases ha : a ∈ X <;> by_cases hb : b ∈ Y · have hh : (frag H X Y).edge a b = true ↔ G.edge a b = true := by constructor · intro h have := (hER a b).mp h simpa [graphOn, ha, hb] using this.1 · intro h exact (hER a b).mpr ⟨by simp [graphOn, ha, hb, h], hX ha, hY hb⟩ by_cases hH : H.edge a b = true · have hFH : (frag H X Y).edge a b = true := by simp [frag, ha, hb, hH] have hG : G.edge a b = true := hh.mp hFH simp [frag, graphOn, ha, hb, hH, hG] · have hHf : H.edge a b = false := by simpa using hH have hFH : (frag H X Y).edge a b = false := by simp [frag, ha, hb, hHf] have hGf : G.edge a b = false := by by_contra hne have hGt : G.edge a b = true := by simpa using hne have := hh.mpr hGt rw [hFH] at this exact Bool.false_ne_true this simp [frag, graphOn, ha, hb, hHf, hGf] · simp [frag, graphOn, ha, hb] · simp [frag, graphOn, ha, hb] · simp [frag, graphOn, ha, hb] theorem frag_edgeRestriction_frag {G0 H : BipartiteGraph A B} (X X' : Finset A) (Y Y' : Finset B) (hER : EdgeRestriction G0 (frag H X Y)) (hX' : X' ⊆ X) (hY' : Y' ⊆ Y) : EdgeRestriction (frag G0 X' Y') (frag H X' Y') := by intro a b constructor · intro h simp only [frag, Bool.and_eq_true, decide_eq_true_eq] at h obtain ⟨⟨hHe, ha'⟩, hb'⟩ := h have hHe' : (frag H X Y).edge a b = true := by simp only [frag, Bool.and_eq_true, decide_eq_true_eq] exact ⟨⟨hHe, hX' ha'⟩, hY' hb'⟩ obtain ⟨hG, ⟨haH, _⟩, ⟨hbH, _⟩⟩ := (hER a b).mp hHe' refine ⟨?_, ⟨haH, ha'⟩, ⟨hbH, hb'⟩⟩ simp only [frag, Bool.and_eq_true, decide_eq_true_eq] exact ⟨⟨hG, ha'⟩, hb'⟩ · rintro ⟨hG, ⟨haH, ha'⟩, ⟨hbH, hb'⟩⟩ simp only [frag, Bool.and_eq_true, decide_eq_true_eq] at hG ⊢ obtain ⟨⟨hG0, _⟩, _⟩ := hG have hHe : (frag H X Y).edge a b = true := (hER a b).mpr ⟨hG0, ⟨haH, hX' ha'⟩, ⟨hbH, hY' hb'⟩⟩ simp only [frag, Bool.and_eq_true, decide_eq_true_eq] at hHe exact ⟨⟨hHe.1.1, ha'⟩, hb'⟩ theorem induced_monotone (G : BipartiteGraph A B) (X : Finset A) (Y : Finset B) (Gs : Fin 3 → BipartiteGraph A B) (hmono : IsMonotoneCover (graphOn G.edge X Y) (fun t => frag (Gs t) X Y)) (hnoIso : ∀ a ∈ X, ∃ b ∈ Y, G.edge a b = true) (i : Fin 3) (hB : (↑Y : Set B) ⊆ (frag (Gs i) X Y).right) (hsplit : (↑Y : Set B) ⊆ (frag (Gs (i.succAbove 0)) X Y).right ∪ (frag (Gs (i.succAbove 1)) X Y).right) : IsMonotoneCover (frag (graphOn G.edge X Y) (X \ partLeft X (Gs i).left) Y) (fun t : Fin 2 => frag (Gs (i.succAbove t)) (X \ partLeft X (Gs i).left) Y) := by classical let G0 : BipartiteGraph A B := graphOn G.edge X Y let X' : Finset A := X \ partLeft X (Gs i).left let G' : BipartiteGraph A B := frag G0 X' Y have hX'sub : X' ⊆ X := fun a ha => (Finset.mem_sdiff.mp ha).1 change IsMonotoneCover G' (fun t : Fin 2 => frag (Gs (i.succAbove t)) X' Y) refine ⟨⟨?_, ?_, ?_, ?_⟩, ?_⟩ · intro t a ha fin_cases t · change a ∈ (Gs (i.succAbove 0)).left ∩ {a | a ∈ X'} at ha exact ⟨by simpa [G0, graphOn] using (Finset.mem_sdiff.mp ha.2).1, ha.2⟩ · change a ∈ (Gs (i.succAbove 1)).left ∩ {a | a ∈ X'} at ha exact ⟨by simpa [G0, graphOn] using (Finset.mem_sdiff.mp ha.2).1, ha.2⟩ · intro t b hb fin_cases t · change b ∈ (Gs (i.succAbove 0)).right ∩ {b | b ∈ Y} at hb exact ⟨by simpa [G0, graphOn] using hb.2, hb.2⟩ · change b ∈ (Gs (i.succAbove 1)).right ∩ {b | b ∈ Y} at hb exact ⟨by simpa [G0, graphOn] using hb.2, hb.2⟩ · intro t fin_cases t · exact frag_edgeRestriction_frag X X' Y Y (hmono.1.2.2.1 (i.succAbove 0)) hX'sub (by intro b hb; exact hb) · exact frag_edgeRestriction_frag X X' Y Y (hmono.1.2.2.1 (i.succAbove 1)) hX'sub (by intro b hb; exact hb) · intro a b hab simp only [G', frag, G0, graphOn, Bool.and_eq_true, decide_eq_true_eq, Finset.mem_sdiff] at hab obtain ⟨⟨⟨⟨hG, haX⟩, hbY⟩, haX'⟩, hbY'⟩ := hab have haAi : a ∉ partLeft X (Gs i).left := (Finset.mem_sdiff.mp haX').2 obtain ⟨t, ht⟩ := hmono.1.2.2.2 a b (by simpa [G0, graphOn, haX, hbY] using hG) simp only [frag, Bool.and_eq_true, decide_eq_true_eq] at ht obtain ⟨⟨hGst, haXt⟩, hbYt⟩ := ht have hti : t ≠ i := by intro hti subst t have hf : (frag (Gs i) X Y).edge a b = true := by simp [frag, hGst, haXt, hbYt] have haAi' : a ∈ (Gs i).left := (((hmono.1.2.2.1 i) a b).mp hf).2.1.1 exact haAi (by simp [partLeft, haX, haAi']) obtain ⟨s, rfl⟩ := Fin.exists_succAbove_eq hti refine ⟨s, ?_⟩ simp only [frag, Bool.and_eq_true, decide_eq_true_eq] exact ⟨⟨hGst, haX'⟩, hbYt⟩ · intro I have hIcases : I = ∅ ∨ I = ({0} : Set (Fin 2)) ∨ I = ({1} : Set (Fin 2)) ∨ I = Set.univ := by by_cases h0 : (0 : Fin 2) ∈ I <;> by_cases h1 : (1 : Fin 2) ∈ I · right; right; right; ext t; fin_cases t <;> simp [h0, h1] · right; left; ext t; fin_cases t <;> simp [h0, h1] · right; right; left; ext t; fin_cases t <;> simp [h0, h1] · left; ext t; fin_cases t <;> simp [h0, h1] rcases hIcases with rfl | rfl | rfl | rfl · right rw [← Set.subset_empty_iff] intro b hb have hbY : b ∈ Y := by simpa [G', frag, G0, graphOn] using hb.1 have hbUnion : b ∈ ⋃ t ∈ (∅ : Set (Fin 2))ᶜ, (frag (Gs (i.succAbove t)) X' Y).right := by rcases hsplit hbY with h | h · refine Set.mem_iUnion.mpr ⟨0, Set.mem_iUnion.mpr ⟨?_, ?_⟩⟩ · simp · simpa [frag] using h · refine Set.mem_iUnion.mpr ⟨1, Set.mem_iUnion.mpr ⟨?_, ?_⟩⟩ · simp · simpa [frag] using h exact hb.2 hbUnion · by_cases hfirst : G'.left \ (frag (Gs (i.succAbove 0)) X' Y).left = ∅ · left; simpa using hfirst · right rw [← Set.subset_empty_iff] intro b hb have hbY : b ∈ Y := by simpa [G', frag, G0, graphOn] using hb.1 have hbj1 : b ∉ (Gs (i.succAbove 1)).right := fun h => hb.2 (by simp [frag, h, hbY]) have hFirstFail : ¬ (G0.left \ ⋃ t ∈ ({i, i.succAbove 0} : Set (Fin 3)), (frag (Gs t) X Y).left = ∅) := by intro hEq apply hfirst rw [← Set.subset_empty_iff] intro a ha have haX' : a ∈ X' := ha.1.2 have haX : a ∈ X := (Finset.mem_sdiff.mp haX').1 have hnot : a ∉ ⋃ t ∈ ({i, i.succAbove 0} : Set (Fin 3)), (frag (Gs t) X Y).left := by intro hmem simp only [Set.mem_iUnion, exists_prop, Set.mem_insert_iff, Set.mem_singleton_iff] at hmem obtain ⟨t, ht, hmt⟩ := hmem rcases ht with rfl | rfl · exact (Finset.mem_sdiff.mp haX').2 (by simp [partLeft, haX, hmt.1]) · exact ha.2 ⟨hmt.1, haX'⟩ have hdiff : a ∈ G0.left \ ⋃ t ∈ ({i, i.succAbove 0} : Set (Fin 3)), (frag (Gs t) X Y).left := ⟨by simpa [G0, graphOn] using haX, hnot⟩ rw [hEq] at hdiff exact hdiff have hSecondFail : ¬ (G0.right \ ⋃ t ∈ ({i, i.succAbove 0} : Set (Fin 3))ᶜ, (frag (Gs t) X Y).right = ∅) := by intro hEq have hmem : b ∈ G0.right \ ⋃ t ∈ ({i, i.succAbove 0} : Set (Fin 3))ᶜ, (frag (Gs t) X Y).right := by refine ⟨by simpa [G0, graphOn] using hbY, ?_⟩ intro hbU simp only [Set.mem_iUnion, exists_prop, Set.mem_compl_iff] at hbU obtain ⟨t, ht, hmt⟩ := hbU have ht1 : t = i.succAbove 1 := by fin_cases i <;> fin_cases t <;> simp_all [Fin.succAbove] rw [ht1] at hmt exact hbj1 hmt.1 rw [hEq] at hmem exact hmem rcases hmono.2 ({i, i.succAbove 0} : Set (Fin 3)) with h | h · exact hFirstFail h · exact hSecondFail h · by_cases hfirst : G'.left \ (frag (Gs (i.succAbove 1)) X' Y).left = ∅ · left; simpa using hfirst · right rw [← Set.subset_empty_iff] intro b hb have hbY : b ∈ Y := by simpa [G', frag, G0, graphOn] using hb.1 have hbj0 : b ∉ (Gs (i.succAbove 0)).right := fun h => hb.2 (by simp [frag, h, hbY]) have hFirstFail : ¬ (G0.left \ ⋃ t ∈ ({i, i.succAbove 1} : Set (Fin 3)), (frag (Gs t) X Y).left = ∅) := by intro hEq apply hfirst rw [← Set.subset_empty_iff] intro a ha have haX' : a ∈ X' := ha.1.2 have haX : a ∈ X := (Finset.mem_sdiff.mp haX').1 have hnot : a ∉ ⋃ t ∈ ({i, i.succAbove 1} : Set (Fin 3)), (frag (Gs t) X Y).left := by intro hmem simp only [Set.mem_iUnion, exists_prop, Set.mem_insert_iff, Set.mem_singleton_iff] at hmem obtain ⟨t, ht, hmt⟩ := hmem rcases ht with rfl | rfl · exact (Finset.mem_sdiff.mp haX').2 (by simp [partLeft, haX, hmt.1]) · exact ha.2 ⟨hmt.1, haX'⟩ have hdiff : a ∈ G0.left \ ⋃ t ∈ ({i, i.succAbove 1} : Set (Fin 3)), (frag (Gs t) X Y).left := ⟨by simpa [G0, graphOn] using haX, hnot⟩ rw [hEq] at hdiff exact hdiff have hSecondFail : ¬ (G0.right \ ⋃ t ∈ ({i, i.succAbove 1} : Set (Fin 3))ᶜ, (frag (Gs t) X Y).right = ∅) := by intro hEq have hmem : b ∈ G0.right \ ⋃ t ∈ ({i, i.succAbove 1} : Set (Fin 3))ᶜ, (frag (Gs t) X Y).right := by refine ⟨by simpa [G0, graphOn] using hbY, ?_⟩ intro hbU simp only [Set.mem_iUnion, exists_prop, Set.mem_compl_iff] at hbU obtain ⟨t, ht, hmt⟩ := hbU have ht0 : t = i.succAbove 0 := by fin_cases i <;> fin_cases t <;> simp_all [Fin.succAbove] rw [ht0] at hmt exact hbj0 hmt.1 rw [hEq] at hmem exact hmem rcases hmono.2 ({i, i.succAbove 1} : Set (Fin 3)) with h | h · exact hFirstFail h · exact hSecondFail h · left rw [← Set.subset_empty_iff] intro a ha have haX' : a ∈ X' := ha.1.2 obtain ⟨b, hbY, hGab⟩ := hnoIso a (Finset.mem_sdiff.mp haX').1 have hG0 : G0.edge a b = true := by simp [G0, graphOn, (Finset.mem_sdiff.mp haX').1, hbY, hGab] obtain ⟨t, ht⟩ := hmono.1.2.2.2 a b hG0 simp only [frag, Bool.and_eq_true, decide_eq_true_eq] at ht obtain ⟨⟨hGst, haXt⟩, hbYt⟩ := ht have hti : t ≠ i := by intro hti subst t have hf : (frag (Gs i) X Y).edge a b = true := by simp [frag, hGst, haXt, hbYt] have haAi : a ∈ (Gs i).left := (((hmono.1.2.2.1 i) a b).mp hf).2.1.1 exact (Finset.mem_sdiff.mp haX').2 (by simp [partLeft, (Finset.mem_sdiff.mp haX').1, haAi]) have htcase : t = i.succAbove 0 ∨ t = i.succAbove 1 := by fin_cases i <;> fin_cases t <;> simp_all [Fin.succAbove] rcases htcase with rfl | rfl · refine ha.2 (Set.mem_iUnion.mpr ⟨0, Set.mem_iUnion.mpr ⟨by simp, ?_⟩⟩) have hf : (frag (Gs (i.succAbove 0)) X Y).edge a b = true := by simp [frag, hGst, haXt, hbYt] have hmemLeft : a ∈ (Gs (i.succAbove 0)).left := (((hmono.1.2.2.1 (i.succAbove 0)) a b).mp hf).2.1.1 show a ∈ (frag (Gs (i.succAbove 0)) X' Y).left exact ⟨hmemLeft, haX'⟩ · refine ha.2 (Set.mem_iUnion.mpr ⟨1, Set.mem_iUnion.mpr ⟨by simp, ?_⟩⟩) have hf : (frag (Gs (i.succAbove 1)) X Y).edge a b = true := by simp [frag, hGst, haXt, hbYt] have hmemLeft : a ∈ (Gs (i.succAbove 1)).left := (((hmono.1.2.2.1 (i.succAbove 1)) a b).mp hf).2.1.1 show a ∈ (frag (Gs (i.succAbove 1)) X' Y).left exact ⟨hmemLeft, haX'⟩ theorem case_one_right (G : BipartiteGraph A B) (X : Finset A) (Y : Finset B) (Gs : Fin 3 → BipartiteGraph A B) (hmono : IsMonotoneCover (graphOn G.edge X Y) (fun t => frag (Gs t) X Y)) (hnoIso : ∀ a ∈ X, ∃ b ∈ Y, G.edge a b = true) (i : Fin 3) (hB : (↑Y : Set B) ⊆ (frag (Gs i) X Y).right) {χ : ℝ} (hχ0 : 0 < χ) (hχ1 : χ ≤ 1) : (graphOn G.edge X Y).sGraph ^ χ ≤ ∑ t, (frag (Gs t) X Y).sGraph ^ χ := by classical let G0 : BipartiteGraph A B := graphOn G.edge X Y let Gi : BipartiteGraph A B := frag (Gs i) X Y let X' : Finset A := X \ partLeft X (Gs i).left let G' : BipartiteGraph A B := frag G0 X' Y have hX'sub : X' ⊆ X := fun a ha => (Finset.mem_sdiff.mp ha).1 have hG0wf : BipartiteGraph.WellFormed G0 := wellFormed_graphOn G.edge X Y have hG'ER : EdgeRestriction G0 G' := edgeRestriction_frag_self hG0wf X' Y have hmono2 : IsMonotoneCover G0 ![G', Gi] := by refine ⟨⟨?_, ?_, ?_, ?_⟩, ?_⟩ · intro t a ha fin_cases t · change a ∈ G0.left ∩ {a | a ∈ X'} at ha exact (Finset.mem_sdiff.mp ha.2).1 · change a ∈ (Gs i).left ∩ {a | a ∈ X} at ha exact ha.2 · intro t b hb fin_cases t · change b ∈ G0.right ∩ {b | b ∈ Y} at hb exact hb.2 · change b ∈ (Gs i).right ∩ {b | b ∈ Y} at hb exact hb.2 · intro t fin_cases t · exact hG'ER · exact hmono.1.2.2.1 i · intro a b hab have hab' : (G.edge a b = true ∧ a ∈ X) ∧ b ∈ Y := by simpa [G0, graphOn, Bool.and_eq_true, decide_eq_true_eq] using hab obtain ⟨⟨hG, haX⟩, hbY⟩ := hab' by_cases hai : a ∈ (Gs i).left · refine ⟨1, ?_⟩ change Gi.edge a b = true have hAle : a ∈ Gi.left := ⟨hai, haX⟩ have hBle : b ∈ Gi.right := hB hbY exact (hmono.1.2.2.1 i a b).mpr ⟨hab, hAle, hBle⟩ · refine ⟨0, ?_⟩ change (frag G0 X' Y).edge a b = true simp only [frag, G0, graphOn, Bool.and_eq_true, decide_eq_true_eq, X', Finset.mem_sdiff, partLeft] refine ⟨⟨⟨⟨hG, haX⟩, hbY⟩, haX, ?_⟩, hbY⟩ simp [hai] · intro I have hIcases : I = ∅ ∨ I = ({0} : Set (Fin 2)) ∨ I = ({1} : Set (Fin 2)) ∨ I = Set.univ := by by_cases h0 : (0 : Fin 2) ∈ I <;> by_cases h1 : (1 : Fin 2) ∈ I · right; right; right; ext t; fin_cases t <;> simp [h0, h1] · right; left; ext t; fin_cases t <;> simp [h0, h1] · right; right; left; ext t; fin_cases t <;> simp [h0, h1] · left; ext t; fin_cases t <;> simp [h0, h1] rcases hIcases with rfl | rfl | rfl | rfl · right rw [← Set.subset_empty_iff] intro b hb refine hb.2 (Set.mem_iUnion.mpr ⟨0, Set.mem_iUnion.mpr ⟨by simp, ?_⟩⟩) show b ∈ G'.right simp only [G', frag] exact ⟨hb.1, by simpa [G0, graphOn] using hb.1⟩ · right rw [← Set.subset_empty_iff] intro b hb refine hb.2 (Set.mem_iUnion.mpr ⟨1, Set.mem_iUnion.mpr ⟨by simp, ?_⟩⟩) show b ∈ Gi.right simpa only [Gi, frag] using hB hb.1 · right rw [← Set.subset_empty_iff] intro b hb refine hb.2 (Set.mem_iUnion.mpr ⟨0, Set.mem_iUnion.mpr ⟨by simp, ?_⟩⟩) show b ∈ G'.right simp only [G', frag] exact ⟨hb.1, by simpa [G0, graphOn] using hb.1⟩ · left rw [← Set.subset_empty_iff] intro a ha by_cases hai : a ∈ (Gs i).left · refine ha.2 (Set.mem_iUnion.mpr ⟨1, Set.mem_iUnion.mpr ⟨by simp, ?_⟩⟩) show a ∈ Gi.left exact ⟨hai, ha.1⟩ · refine ha.2 (Set.mem_iUnion.mpr ⟨0, Set.mem_iUnion.mpr ⟨by simp, ?_⟩⟩) show a ∈ G'.left simp only [G', frag] refine ⟨?_, ?_⟩ · simpa [G0, graphOn] using ha.1 · simp only [X', Finset.mem_sdiff, partLeft, Finset.mem_filter] exact ⟨ha.1, fun h => hai h.2⟩ have h2 := isMonotoneGraphExponent_two_of_le hχ0 hχ1 2 le_rfl A B G0 ![G', Gi] hmono2 have h2' : G0.sGraph ^ χ ≤ G'.sGraph ^ χ + Gi.sGraph ^ χ := by simpa [Fin.sum_univ_two] using h2 by_cases hsplit : (↑Y : Set B) ⊆ (frag (Gs (i.succAbove 0)) X Y).right ∪ (frag (Gs (i.succAbove 1)) X Y).right · have hmono3 : IsMonotoneCover G' (fun t : Fin 2 => frag (Gs (i.succAbove t)) X' Y) := induced_monotone G X Y Gs hmono hnoIso i hB hsplit have h3 := isMonotoneGraphExponent_two_of_le hχ0 hχ1 2 le_rfl A B G' _ hmono3 have h3' : G'.sGraph ^ χ ≤ (frag (Gs (i.succAbove 0)) X' Y).sGraph ^ χ + (frag (Gs (i.succAbove 1)) X' Y).sGraph ^ χ := by simpa [Fin.sum_univ_two] using h3 have hle0 : (frag (Gs (i.succAbove 0)) X' Y).sGraph ^ χ ≤ (frag (Gs (i.succAbove 0)) X Y).sGraph ^ χ := Real.rpow_le_rpow (BipartiteGraph.sGraph_nonneg _) (sGraph_frag_mono (Gs (i.succAbove 0)) Y hX'sub) hχ0.le have hle1 : (frag (Gs (i.succAbove 1)) X' Y).sGraph ^ χ ≤ (frag (Gs (i.succAbove 1)) X Y).sGraph ^ χ := Real.rpow_le_rpow (BipartiteGraph.sGraph_nonneg _) (sGraph_frag_mono (Gs (i.succAbove 1)) Y hX'sub) hχ0.le have hGi : Gi.sGraph ^ χ ≤ ∑ t, (frag (Gs t) X Y).sGraph ^ χ := Finset.single_le_sum (f := fun t => (frag (Gs t) X Y).sGraph ^ χ) (fun t _ => Real.rpow_nonneg (BipartiteGraph.sGraph_nonneg _) _) (Finset.mem_univ i) have hsum : (frag (Gs (i.succAbove 0)) X Y).sGraph ^ χ + (frag (Gs (i.succAbove 1)) X Y).sGraph ^ χ + Gi.sGraph ^ χ ≤ ∑ t, (frag (Gs t) X Y).sGraph ^ χ := by have hs := Fin.sum_univ_succAbove (fun t => (frag (Gs t) X Y).sGraph ^ χ) i rw [hs] simp only [Fin.sum_univ_two] linarith calc G0.sGraph ^ χ ≤ G'.sGraph ^ χ + Gi.sGraph ^ χ := h2' _ ≤ ((frag (Gs (i.succAbove 0)) X' Y).sGraph ^ χ + (frag (Gs (i.succAbove 1)) X' Y).sGraph ^ χ) + Gi.sGraph ^ χ := add_le_add h3' le_rfl _ ≤ ((frag (Gs (i.succAbove 0)) X Y).sGraph ^ χ + (frag (Gs (i.succAbove 1)) X Y).sGraph ^ χ) + Gi.sGraph ^ χ := add_le_add (add_le_add hle0 hle1) le_rfl _ ≤ ∑ t, (frag (Gs t) X Y).sGraph ^ χ := hsum · have hAi : (↑X : Set A) ⊆ (Gs i).left := by rw [Set.subset_def] at hsplit push Not at hsplit obtain ⟨b, hbY, hbU⟩ := hsplit have hb0 : b ∉ (frag (Gs (i.succAbove 0)) X Y).right := fun h => hbU (Or.inl h) have hb1 : b ∉ (frag (Gs (i.succAbove 1)) X Y).right := fun h => hbU (Or.inr h) have hsecond_ne : (G0.right \ ⋃ t ∈ ({i} : Set (Fin 3))ᶜ, (frag (Gs t) X Y).right) ≠ ∅ := by intro hEq have hmem : b ∈ G0.right \ ⋃ t ∈ ({i} : Set (Fin 3))ᶜ, (frag (Gs t) X Y).right := by refine ⟨hbY, ?_⟩ intro hbU' simp only [Set.mem_iUnion, exists_prop] at hbU' obtain ⟨t, ht, hmemt⟩ := hbU' have hti : t ≠ i := by simpa using ht have htcase : t = i.succAbove 0 ∨ t = i.succAbove 1 := by fin_cases i <;> fin_cases t <;> simp_all [Fin.succAbove] rcases htcase with rfl | rfl · exact hb0 hmemt · exact hb1 hmemt rw [hEq] at hmem exact hmem have hfirst : G0.left \ ⋃ t ∈ ({i} : Set (Fin 3)), (frag (Gs t) X Y).left = ∅ := by rcases hmono.2 ({i} : Set (Fin 3)) with h | h · exact h · exact absurd h hsecond_ne intro a ha have hmem : a ∈ ⋃ t ∈ ({i} : Set (Fin 3)), (frag (Gs t) X Y).left := by by_contra hat have hdiff : a ∈ G0.left \ ⋃ t ∈ ({i} : Set (Fin 3)), (frag (Gs t) X Y).left := ⟨by simpa [G0, graphOn] using ha, hat⟩ rw [hfirst] at hdiff exact hdiff simp only [Set.mem_iUnion, exists_prop, Set.mem_singleton_iff] at hmem obtain ⟨t, rfl, hmemG⟩ := hmem exact hmemG.1 have hBri : (↑Y : Set B) ⊆ (Gs i).right := by intro b hb; exact (hB hb).1 have hAi' : (↑X : Set A) ⊆ Gi.left := fun a ha => ⟨hAi ha, ha⟩ have hBri' : (↑Y : Set B) ⊆ Gi.right := fun b hb => ⟨hBri hb, hb⟩ have hGiEq : Gi = G0 := frag_eq_graphOn_self (G := G) (H := Gs i) X Y (hmono.1.2.2.1 i) hAi' hBri' have hGiS : Gi.sGraph = G0.sGraph := by rw [hGiEq] have hsum : Gi.sGraph ^ χ ≤ ∑ t, (frag (Gs t) X Y).sGraph ^ χ := Finset.single_le_sum (f := fun t => (frag (Gs t) X Y).sGraph ^ χ) (fun t _ => Real.rpow_nonneg (BipartiteGraph.sGraph_nonneg _) _) (Finset.mem_univ i) rwa [hGiS] at hsum def swapGraph (G : BipartiteGraph A B) : BipartiteGraph B A where left := G.right right := G.left edge := fun b a => G.edge a b @[simp] theorem swapGraph_left (G : BipartiteGraph A B) : (swapGraph G).left = G.right := rfl @[simp] theorem swapGraph_right (G : BipartiteGraph A B) : (swapGraph G).right = G.left := rfl theorem swapGraph_graphOn (T : A → B → Bool) (X : Finset A) (Y : Finset B) : swapGraph (graphOn T X Y) = graphOn (fun b a => T a b) Y X := by apply BipartiteGraph_ext <;> try rfl funext b a simp only [swapGraph, graphOn, Bool.and_assoc] rw [Bool.and_comm (decide (b ∈ Y)), Bool.and_left_comm] theorem swapGraph_frag (H : BipartiteGraph A B) (X : Finset A) (Y : Finset B) : swapGraph (frag H X Y) = frag (swapGraph H) Y X := by apply BipartiteGraph_ext <;> try rfl funext b a simp only [swapGraph, frag, Bool.and_assoc] rw [Bool.and_comm (decide (b ∈ Y)), Bool.and_left_comm] theorem isCover_swap {k : ℕ} {G : BipartiteGraph A B} {Gs : Fin k → BipartiteGraph A B} (h : IsCover G Gs) : IsCover (swapGraph G) (fun i => swapGraph (Gs i)) := by refine ⟨?_, ?_, ?_, ?_⟩ · intro i b hb exact h.2.1 i hb · intro i a ha exact h.1 i ha · intro i b a constructor · intro hh have := (h.2.2.1 i a b).mp hh exact ⟨this.1, this.2.2, this.2.1⟩ · rintro ⟨hG, ha, hb⟩ exact (h.2.2.1 i a b).mpr ⟨hG, hb, ha⟩ · intro b a hab obtain ⟨i, hi⟩ := h.2.2.2 a b hab exact ⟨i, hi⟩ theorem isMonotoneCover_swap {k : ℕ} {G : BipartiteGraph A B} {Gs : Fin k → BipartiteGraph A B} (h : IsMonotoneCover G Gs) : IsMonotoneCover (swapGraph G) (fun i => swapGraph (Gs i)) := by refine ⟨isCover_swap h.1, ?_⟩ intro I rcases h.2 Iᶜ with h1 | h2 · right; exact h1 · left; simpa using h2 theorem edgeCount_swap (G : BipartiteGraph A B) (X : Finset A) (Y : Finset B) : (swapGraph G).edgeCount Y X = G.edgeCount X Y := by classical unfold BipartiteGraph.edgeCount apply Finset.card_bij (fun q _ => (q.2, q.1)) · intro q hq simp only [Finset.mem_filter] at hq have hq' : q.1 ∈ Y ∧ q.2 ∈ X := by have h := hq.1 change q ∈ Y ×ˢ X at h exact Finset.mem_product.mp h rw [Finset.mem_filter] constructor · change (q.2, q.1) ∈ X ×ˢ Y exact Finset.mem_product.mpr ⟨hq'.2, hq'.1⟩ · simpa [swapGraph] using hq.2 · intro q1 _ q2 _ h exact Prod.ext (congrArg Prod.snd h) (congrArg Prod.fst h) · intro q hq simp only [Finset.mem_filter] at hq have hq' : q.1 ∈ X ∧ q.2 ∈ Y := by have h := hq.1 change q ∈ X ×ˢ Y at h exact Finset.mem_product.mp h refine ⟨(q.2, q.1), ?_, rfl⟩ rw [Finset.mem_filter] constructor · change (q.2, q.1) ∈ Y ×ˢ X exact Finset.mem_product.mpr ⟨hq'.2, hq'.1⟩ · simpa [swapGraph] using hq.2 theorem admissible_swap (G : BipartiteGraph A B) (X : Finset A) (Y : Finset B) : (swapGraph G).Admissible Y X ↔ G.Admissible X Y := by constructor <;> intro h · exact ⟨h.2, h.1⟩ · exact ⟨h.2, h.1⟩ theorem ratio_swap (G : BipartiteGraph A B) (X : Finset A) (Y : Finset B) : (swapGraph G).ratio Y X = G.ratio X Y := by classical by_cases h : G.Admissible X Y · rw [BipartiteGraph.ratio, if_pos ((admissible_swap G X Y).mpr h), BipartiteGraph.ratio, if_pos h, edgeCount_swap] rw [mul_comm] · have h' : ¬ (swapGraph G).Admissible Y X := fun hs => h ((admissible_swap G X Y).mp hs) simp [BipartiteGraph.ratio, h, h'] theorem ratio_le_sGraph' (G : BipartiteGraph A B) (X : Finset A) (Y : Finset B) : G.ratio X Y ≤ G.sGraph := by by_cases h : G.Admissible X Y · exact ratio_le_sGraph G h · rw [BipartiteGraph.ratio, if_neg h] exact BipartiteGraph.sGraph_nonneg G theorem sGraph_swap (G : BipartiteGraph A B) : (swapGraph G).sGraph = G.sGraph := by classical apply le_antisymm · unfold BipartiteGraph.sGraph apply Finset.sup'_le intro p _ rw [ratio_swap] exact ratio_le_sGraph' G p.2 p.1 · unfold BipartiteGraph.sGraph apply Finset.sup'_le intro p _ rw [← ratio_swap] exact ratio_le_sGraph' (swapGraph G) p.2 p.1 theorem case_one_left (G : BipartiteGraph A B) (X : Finset A) (Y : Finset B) (Gs : Fin 3 → BipartiteGraph A B) (hmono : IsMonotoneCover (graphOn G.edge X Y) (fun t => frag (Gs t) X Y)) (hnoIsoY : ∀ b ∈ Y, ∃ a ∈ X, G.edge a b = true) (i : Fin 3) (hA : (↑X : Set A) ⊆ (frag (Gs i) X Y).left) {χ : ℝ} (hχ0 : 0 < χ) (hχ1 : χ ≤ 1) : (graphOn G.edge X Y).sGraph ^ χ ≤ ∑ t, (frag (Gs t) X Y).sGraph ^ χ := by classical let Gs' : Fin 3 → BipartiteGraph B A := fun t => swapGraph (Gs t) have hmono_s : IsMonotoneCover (graphOn (swapGraph G).edge Y X) (fun t => frag (Gs' t) Y X) := by have h := isMonotoneCover_swap hmono rw [swapGraph_graphOn] at h simp only [swapGraph_frag] at h exact h have hnoIso_s : ∀ b ∈ Y, ∃ a ∈ X, (swapGraph G).edge b a = true := by intro b hb obtain ⟨a, ha, hab⟩ := hnoIsoY b hb exact ⟨a, ha, hab⟩ have hB_s : (↑X : Set A) ⊆ (frag (Gs' i) Y X).right := by intro a ha exact ⟨(hA ha).1, ha⟩ have hres := case_one_right (G := swapGraph G) (X := Y) (Y := X) Gs' hmono_s hnoIso_s i hB_s hχ0 hχ1 have hL : (graphOn (swapGraph G).edge Y X).sGraph = (graphOn G.edge X Y).sGraph := by change (graphOn (fun b a => G.edge a b) Y X).sGraph = (graphOn G.edge X Y).sGraph rw [← swapGraph_graphOn (G.edge) X Y] exact sGraph_swap _ rw [hL] at hres simpa only [Gs', ← swapGraph_frag, sGraph_swap] using hres /-! ### Case (1) without the isolated-vertex hypothesis The paper's case `B_i = B` uses only maximality of the pair `(X, Y)` for the ratio. We derive the absence of isolated vertices from that maximality here, so that the case applies to a density-maximising subgraph without extra data. -/ theorem edgeCount_erase_of_isolated (H : BipartiteGraph A B) (X : Finset A) (Y : Finset B) {a : A} (hiso : ∀ b ∈ Y, H.edge a b = false) : H.edgeCount (X.erase a) Y = H.edgeCount X Y := by classical unfold BipartiteGraph.edgeCount apply congrArg Finset.card ext q simp only [Finset.mem_filter] constructor · rintro ⟨hq, he⟩ have hq' : q.1 ∈ X.erase a ∧ q.2 ∈ Y := Finset.mem_product.mp hq exact ⟨Finset.mem_product.mpr ⟨(Finset.mem_erase.mp hq'.1).2, hq'.2⟩, he⟩ · rintro ⟨hq, he⟩ have hq' : q.1 ∈ X ∧ q.2 ∈ Y := Finset.mem_product.mp hq have hne : q.1 ≠ a := by intro h rw [h, hiso q.2 hq'.2] at he exact Bool.noConfusion he exact ⟨Finset.mem_product.mpr ⟨Finset.mem_erase.mpr ⟨hne, hq'.1⟩, hq'.2⟩, he⟩ theorem ratio_lt_ratio_erase (H : BipartiteGraph A B) (X : Finset A) (Y : Finset B) (hadm : H.Admissible X Y) {a : A} (ha : a ∈ X) (hiso : ∀ b ∈ Y, H.edge a b = false) (hepos : 0 < H.edgeCount X Y) (hY : 0 < Y.card) (hX2 : 2 ≤ X.card) : H.ratio X Y < H.ratio (X.erase a) Y := by classical have hadm' : H.Admissible (X.erase a) Y := ⟨fun x hx => hadm.1 x (Finset.mem_erase.mp hx).2, hadm.2⟩ rw [ratio_of_admissible H hadm, ratio_of_admissible H hadm', edgeCount_erase_of_isolated H X Y hiso] have hm : ((X.erase a).card : ℝ) = (X.card : ℝ) - 1 := by rw [Finset.card_erase_of_mem ha, Nat.cast_sub (Finset.card_pos.mpr ⟨a, ha⟩), Nat.cast_one] rw [hm] have hY' : (0 : ℝ) < (Y.card : ℝ) := by exact_mod_cast hY have hX2' : (2 : ℝ) ≤ (X.card : ℝ) := by exact_mod_cast hX2 have hnum : (0 : ℝ) < (H.edgeCount X Y : ℝ) ^ 2 := by have : (0 : ℝ) < (H.edgeCount X Y : ℝ) := by exact_mod_cast hepos positivity have hden1 : (0 : ℝ) < ((X.card : ℝ) - 1) * (Y.card : ℝ) := by nlinarith have hlt : ((X.card : ℝ) - 1) * (Y.card : ℝ) < (X.card : ℝ) * (Y.card : ℝ) := by nlinarith exact div_lt_div_of_pos_left hnum hden1 hlt /-- If `(X, Y)` maximises the ratio of `H` and the ratio is positive, then no vertex of `X` is isolated towards `Y`. -/ theorem no_isolated_left_of_max_ratio (H : BipartiteGraph A B) (X : Finset A) (Y : Finset B) (hadm : H.Admissible X Y) (hmax : H.ratio X Y = H.sGraph) (hpos : 0 < H.ratio X Y) : ∀ a ∈ X, ∃ b ∈ Y, H.edge a b = true := by classical have hr := ratio_of_admissible H hadm rw [hr] at hpos have hnum : (0 : ℝ) < (H.edgeCount X Y : ℝ) ^ 2 := by rcases div_pos_iff.mp hpos with ⟨h1, h2⟩ | ⟨h1, h2⟩ · exact h1 · exact absurd h1 (not_lt.mpr (sq_nonneg _)) have hden : (0 : ℝ) < (X.card : ℝ) * (Y.card : ℝ) := by rcases div_pos_iff.mp hpos with ⟨h1, h2⟩ | ⟨h1, h2⟩ · exact h2 · exact absurd h1 (not_lt.mpr (sq_nonneg _)) have heR : (0 : ℝ) < (H.edgeCount X Y : ℝ) := by by_contra hle push Not at hle have h0 : (H.edgeCount X Y : ℝ) = 0 := le_antisymm hle (Nat.cast_nonneg _) nlinarith have hepos : 0 < H.edgeCount X Y := by exact_mod_cast heR have hY : 0 < Y.card := by have hYR : (0 : ℝ) < (Y.card : ℝ) := by nlinarith [hden, Nat.cast_nonneg (α := ℝ) X.card] exact_mod_cast hYR intro a ha by_contra h push Not at h have hiso : ∀ b ∈ Y, H.edge a b = false := by intro b hb have h' : H.edge a b ≠ true := h b hb cases hb' : H.edge a b <;> simp_all have hX2 : 2 ≤ X.card := by by_contra hlt push Not at hlt have hXle1 : X.card ≤ 1 := by omega have hsingle : X = {a} := by apply Finset.eq_singleton_iff_unique_mem.mpr refine ⟨ha, ?_⟩ intro x hx exact Finset.card_le_one.mp hXle1 x hx a ha have hzero : H.edgeCount X Y = 0 := by unfold BipartiteGraph.edgeCount rw [hsingle, Finset.card_eq_zero] rw [Finset.filter_eq_empty_iff] intro q hq have hq' : q.1 ∈ ({a} : Finset A) ∧ q.2 ∈ Y := Finset.mem_product.mp hq have hq1 : q.1 = a := Finset.mem_singleton.mp hq'.1 rw [hq1, hiso q.2 hq'.2] simp omega have hlt := ratio_lt_ratio_erase H X Y hadm ha hiso hepos hY hX2 have hle : H.ratio (X.erase a) Y ≤ H.sGraph := ratio_le_sGraph H ⟨fun x hx => hadm.1 x (Finset.mem_erase.mp hx).2, hadm.2⟩ linarith [hmax] /-- Symmetric form, obtained by swapping the two sides. -/ theorem no_isolated_right_of_max_ratio (H : BipartiteGraph A B) (X : Finset A) (Y : Finset B) (hadm : H.Admissible X Y) (hmax : H.ratio X Y = H.sGraph) (hpos : 0 < H.ratio X Y) : ∀ b ∈ Y, ∃ a ∈ X, H.edge a b = true := by have h := no_isolated_left_of_max_ratio (swapGraph H) Y X ((admissible_swap H X Y).mpr hadm) ?_ ?_ · intro b hb obtain ⟨a, ha, hab⟩ := h b hb exact ⟨a, ha, hab⟩ · rw [ratio_swap, sGraph_swap] exact hmax · rw [ratio_swap] exact hpos /-- Case (1) of Theorem 3, right orientation, with maximality replacing the isolated-vertex hypothesis. -/ theorem case_one_right_of_max (G : BipartiteGraph A B) (X : Finset A) (Y : Finset B) (Gs : Fin 3 → BipartiteGraph A B) (hmono : IsMonotoneCover (graphOn G.edge X Y) (fun t => frag (Gs t) X Y)) (hmax : (graphOn G.edge X Y).ratio X Y = (graphOn G.edge X Y).sGraph) (hpos : 0 < (graphOn G.edge X Y).ratio X Y) (i : Fin 3) (hB : (↑Y : Set B) ⊆ (frag (Gs i) X Y).right) {χ : ℝ} (hχ0 : 0 < χ) (hχ1 : χ ≤ 1) : (graphOn G.edge X Y).sGraph ^ χ ≤ ∑ t, (frag (Gs t) X Y).sGraph ^ χ := by have hadm : (graphOn G.edge X Y).Admissible X Y := ⟨fun a ha => ha, fun b hb => hb⟩ have hnoIso : ∀ a ∈ X, ∃ b ∈ Y, G.edge a b = true := by intro a ha obtain ⟨b, hb, he⟩ := no_isolated_left_of_max_ratio (graphOn G.edge X Y) X Y hadm hmax hpos a ha exact ⟨b, hb, by simpa [graphOn, ha, hb] using he⟩ exact case_one_right G X Y Gs hmono hnoIso i hB hχ0 hχ1 /-- Case (1) of Theorem 3, left orientation, with maximality replacing the isolated-vertex hypothesis. -/ theorem case_one_left_of_max (G : BipartiteGraph A B) (X : Finset A) (Y : Finset B) (Gs : Fin 3 → BipartiteGraph A B) (hmono : IsMonotoneCover (graphOn G.edge X Y) (fun t => frag (Gs t) X Y)) (hmax : (graphOn G.edge X Y).ratio X Y = (graphOn G.edge X Y).sGraph) (hpos : 0 < (graphOn G.edge X Y).ratio X Y) (i : Fin 3) (hA : (↑X : Set A) ⊆ (frag (Gs i) X Y).left) {χ : ℝ} (hχ0 : 0 < χ) (hχ1 : χ ≤ 1) : (graphOn G.edge X Y).sGraph ^ χ ≤ ∑ t, (frag (Gs t) X Y).sGraph ^ χ := by have hadm : (graphOn G.edge X Y).Admissible X Y := ⟨fun a ha => ha, fun b hb => hb⟩ have hnoIsoY : ∀ b ∈ Y, ∃ a ∈ X, G.edge a b = true := by intro b hb obtain ⟨a, ha, he⟩ := no_isolated_right_of_max_ratio (graphOn G.edge X Y) X Y hadm hmax hpos b hb exact ⟨a, ha, by simpa [graphOn, ha, hb] using he⟩ exact case_one_left G X Y Gs hmono hnoIsoY i hA hχ0 hχ1 theorem exists_densitySubgraph (G : BipartiteGraph A B) : ∃ (X : Finset A) (Y : Finset B), (graphOn G.edge X Y).sGraph = G.sGraph ∧ (graphOn G.edge X Y).sGraph = G.ratio X Y ∧ G.Admissible X Y ∧ (graphOn G.edge X Y).ratio X Y = G.ratio X Y := by classical obtain ⟨X, Y, hAdm, hsG⟩ := BipartiteGraph.exists_admissible_eq_sGraph G have hAdm' : (graphOn G.edge X Y).Admissible X Y := ⟨fun a ha => ha, fun b hb => hb⟩ have hHratio : (graphOn G.edge X Y).ratio X Y = G.ratio X Y := by rw [ratio_of_admissible _ hAdm', ratio_of_admissible G hAdm, edgeCount_graphOn_self'] have hHsG : (graphOn G.edge X Y).sGraph = G.sGraph := by apply le_antisymm · apply BipartiteGraph.sGraph_mono · intro a ha exact hAdm.1 a ha · intro b hb exact hAdm.2 b hb · intro a b h simp only [graphOn, Bool.and_eq_true, decide_eq_true_eq] at h exact h.1.1 · rw [hsG, ← hHratio] exact ratio_le_sGraph _ hAdm' exact ⟨X, Y, hHsG, by rw [hHsG, hsG], hAdm, hHratio⟩ /-! ### Density normalisation `ThreeCore χ` is the normalised form of the Theorem 3 inequality. The graph is replaced by its restriction to a density-maximising pair `(X, Y)`, the cover is restricted accordingly, and the pair is critical: `ratio X Y = sGraph` and the ratio is positive. The next theorem shows that this normalised statement implies the exponent statement for all monotone covers of size at most three. -/ def ThreeCore (χ : ℝ) : Prop := ∀ (A B : Type) [Fintype A] [Fintype B] [DecidableEq A] [DecidableEq B] (G : BipartiteGraph A B) (X : Finset A) (Y : Finset B) (Gs : Fin 3 → BipartiteGraph A B), IsMonotoneCover (graphOn G.edge X Y) (fun t => frag (Gs t) X Y) → (graphOn G.edge X Y).ratio X Y = (graphOn G.edge X Y).sGraph → 0 < (graphOn G.edge X Y).ratio X Y → (graphOn G.edge X Y).sGraph ^ χ ≤ ∑ t, (frag (Gs t) X Y).sGraph ^ χ theorem isMonotoneGraphExponent_three_of_core {χ : ℝ} (hχ0 : 0 < χ) (hχ1 : χ ≤ 1) (hcore : ThreeCore χ) : IsMonotoneGraphExponent 3 χ := by intro m hm A B _ _ _ _ G Gs hmono rcases Nat.lt_or_ge m 3 with hlt | hge · exact isMonotoneGraphExponent_two_of_le hχ0 hχ1 m (by omega) A B G Gs hmono · have hm3 : m = 3 := by omega subst m obtain ⟨X, Y, hHsG, hG0ratio, hadm, hratio_eq⟩ := exists_densitySubgraph G by_cases hzero : G.sGraph = 0 · rw [hzero, Real.zero_rpow hχ0.ne'] exact Finset.sum_nonneg fun i _ => Real.rpow_nonneg (BipartiteGraph.sGraph_nonneg (Gs i)) χ · have hGpos : 0 < G.sGraph := lt_of_le_of_ne (BipartiteGraph.sGraph_nonneg G) (Ne.symm hzero) have hmax : (graphOn G.edge X Y).ratio X Y = (graphOn G.edge X Y).sGraph := by rw [hratio_eq, ← hG0ratio] have hpos : 0 < (graphOn G.edge X Y).ratio X Y := by rw [hratio_eq, ← hG0ratio, hHsG] exact hGpos have hmono' : IsMonotoneCover (graphOn G.edge X Y) (fun t => frag (Gs t) X Y) := monotoneCover_frag hmono X Y hadm.1 hadm.2 have hcore' := hcore A B G X Y Gs hmono' hmax hpos have hleft : G.sGraph ^ χ ≤ (graphOn G.edge X Y).sGraph ^ χ := by rw [hHsG] refine le_trans hleft (le_trans hcore' ?_) apply Finset.sum_le_sum intro t _ exact Real.rpow_le_rpow (BipartiteGraph.sGraph_nonneg _) (sGraph_frag_le (Gs t) X Y) hχ0.le /-! ### The main case: preparatory structure In the main case no part of `G` is contained in a cover graph. Monotonicity then forces every vertex to belong to at least two graphs of the cover. The first two lemmas record this for the left and right sides; the remaining analytic content of the paper's main case is isolated in `MainCase`. -/ theorem exists_two_covers_left (H : BipartiteGraph A B) (Hs : Fin 3 → BipartiteGraph A B) (hmono : IsMonotoneCover H Hs) (hnoIso : ∀ a ∈ H.left, ∃ b ∈ H.right, H.edge a b = true) (hB : ∀ i, ¬ H.right ⊆ (Hs i).right) : ∀ a ∈ H.left, ∃ i j : Fin 3, i ≠ j ∧ a ∈ (Hs i).left ∧ a ∈ (Hs j).left := by classical intro a ha obtain ⟨b, hb, hab⟩ := hnoIso a ha obtain ⟨i, hi⟩ := hmono.1.2.2.2 a b hab have hai : a ∈ (Hs i).left := (((hmono.1.2.2.1 i) a b).mp hi).2.1 have hex : ∃ j : Fin 3, j ≠ i ∧ a ∈ (Hs j).left := by by_contra hcon have haU : a ∉ ⋃ t ∈ ({i}ᶜ : Set (Fin 3)), (Hs t).left := by intro hmem simp only [Set.mem_iUnion, exists_prop] at hmem obtain ⟨t, ht, hat⟩ := hmem exact hcon ⟨t, by simpa using ht, hat⟩ have h1 : (H.left \ ⋃ t ∈ ({i}ᶜ : Set (Fin 3)), (Hs t).left) ≠ ∅ := by intro hEq have : a ∈ H.left \ ⋃ t ∈ ({i}ᶜ : Set (Fin 3)), (Hs t).left := ⟨ha, haU⟩ rw [hEq] at this exact this rcases hmono.2 ({i}ᶜ : Set (Fin 3)) with hleft | hright · exact h1 hleft · have h2 : H.right ⊆ (Hs i).right := by rw [Set.sdiff_eq_empty] at hright intro c hc have := hright hc simpa using this exact hB i h2 obtain ⟨j, hji, haj⟩ := hex exact ⟨i, j, Ne.symm hji, hai, haj⟩ theorem exists_two_covers_right (H : BipartiteGraph A B) (Hs : Fin 3 → BipartiteGraph A B) (hmono : IsMonotoneCover H Hs) (hnoIso : ∀ b ∈ H.right, ∃ a ∈ H.left, H.edge a b = true) (hA : ∀ i, ¬ H.left ⊆ (Hs i).left) : ∀ b ∈ H.right, ∃ i j : Fin 3, i ≠ j ∧ b ∈ (Hs i).right ∧ b ∈ (Hs j).right := by classical intro b hb obtain ⟨a, ha, hab⟩ := hnoIso b hb obtain ⟨i, hi⟩ := hmono.1.2.2.2 a b hab have hbi : b ∈ (Hs i).right := (((hmono.1.2.2.1 i) a b).mp hi).2.2 have hex : ∃ j : Fin 3, j ≠ i ∧ b ∈ (Hs j).right := by by_contra hcon have hbU : b ∉ ⋃ t ∈ ({i}ᶜ : Set (Fin 3)), (Hs t).right := by intro hmem simp only [Set.mem_iUnion, exists_prop] at hmem obtain ⟨t, ht, hbt⟩ := hmem exact hcon ⟨t, by simpa using ht, hbt⟩ have h1 : (H.right \ ⋃ t ∈ ({i}ᶜ : Set (Fin 3)), (Hs t).right) ≠ ∅ := by intro hEq have : b ∈ H.right \ ⋃ t ∈ ({i}ᶜ : Set (Fin 3)), (Hs t).right := ⟨hb, hbU⟩ rw [hEq] at this exact this rcases hmono.2 ({i} : Set (Fin 3)) with hleft | hright · have h2 : H.left ⊆ (Hs i).left := by rw [Set.sdiff_eq_empty] at hleft intro c hc have := hleft hc simpa using this exact hA i h2 · exact h1 hright obtain ⟨j, hji, hbj⟩ := hex exact ⟨i, j, Ne.symm hji, hbi, hbj⟩ /-- The remaining analytic statement of Theorem 3: the main case, in which no part of the graph is contained in a cover graph, after density normalisation. This is where the paper's estimates (4.2)--(4.14) are used. -/ def MainCase (χ : ℝ) : Prop := ∀ (A B : Type) [Fintype A] [Fintype B] [DecidableEq A] [DecidableEq B] (G : BipartiteGraph A B) (X : Finset A) (Y : Finset B) (Gs : Fin 3 → BipartiteGraph A B), IsMonotoneCover (graphOn G.edge X Y) (fun t => frag (Gs t) X Y) → (graphOn G.edge X Y).ratio X Y = (graphOn G.edge X Y).sGraph → 0 < (graphOn G.edge X Y).ratio X Y → (∀ i, ¬ (↑Y : Set B) ⊆ (frag (Gs i) X Y).right) → (∀ i, ¬ (↑X : Set A) ⊆ (frag (Gs i) X Y).left) → (graphOn G.edge X Y).sGraph ^ χ ≤ ∑ t, (frag (Gs t) X Y).sGraph ^ χ theorem threeCore_of_main {χ : ℝ} (hχ0 : 0 < χ) (hχ1 : χ ≤ 1) (hmain : MainCase χ) : ThreeCore χ := by intro A B _ _ _ _ G X Y Gs hmono hmax hpos by_cases hB : ∃ i, (↑Y : Set B) ⊆ (frag (Gs i) X Y).right · obtain ⟨i, hi⟩ := hB exact case_one_right_of_max G X Y Gs hmono hmax hpos i hi hχ0 hχ1 · by_cases hA : ∃ i, (↑X : Set A) ⊆ (frag (Gs i) X Y).left · obtain ⟨i, hi⟩ := hA exact case_one_left_of_max G X Y Gs hmono hmax hpos i hi hχ0 hχ1 · push Not at hB hA exact hmain A B G X Y Gs hmono hmax hpos hB hA /-- Theorem 3, conditional on the analytic main case. -/ theorem theorem3_of_main {χ : ℝ} (hχ0 : 0 < χ) (hχ1 : χ ≤ 1) (hmain : MainCase χ) : IsMonotoneGraphExponent 3 χ := isMonotoneGraphExponent_three_of_core hχ0 hχ1 (threeCore_of_main hχ0 hχ1 hmain) theorem theorem3_of_main_0_769 (hmain : MainCase ((769 : ℝ) / 1000)) : IsMonotoneGraphExponent 3 ((769 : ℝ) / 1000) := theorem3_of_main (by norm_num) (by norm_num) hmain /-! ### The main case: the partition `N_1, N_2, N_3` and estimates (4.2)--(4.3) The main case of Theorem 3 partitions the left part of the graph into three pieces `N_1, N_2, N_3`, where `N_i` lies in the two cover graphs different from `G_i`, and similarly for the right part. The results below are the combinatorial estimates (4.2) and (4.3) of the paper; the numerical part (4.4)--(4.14) is what remains of the main case. -/ /-! ### Choosing the third index -/ /-- The element of `Fin 3` different from two distinct given indices. -/ def thirdIndex (i j : Fin 3) : Fin 3 := if i = 0 then (if j = 1 then 2 else 1) else if i = 1 then (if j = 0 then 2 else 0) else (if j = 0 then 1 else 0) omit [Fintype A] [Fintype B] [DecidableEq A] [DecidableEq B] in theorem eq_left_or_right_of_ne_thirdIndex {i j k : Fin 3} (h : i ≠ j) (hk : k ≠ thirdIndex i j) : k = i ∨ k = j := by fin_cases i <;> fin_cases j <;> fin_cases k <;> simp_all [thirdIndex] omit [Fintype A] [Fintype B] [DecidableEq A] [DecidableEq B] in theorem exists_lonely_index {X : Finset A} {F : Fin 3 → BipartiteGraph A B} (h2 : ∀ a ∈ X, ∃ i j : Fin 3, i ≠ j ∧ a ∈ (F i).left ∧ a ∈ (F j).left) : ∀ a ∈ X, ∃ k : Fin 3, ∀ j, j ≠ k → a ∈ (F j).left := by intro a ha obtain ⟨i, j, hij, hai, haj⟩ := h2 a ha refine ⟨thirdIndex i j, fun k hk => ?_⟩ rcases eq_left_or_right_of_ne_thirdIndex hij hk with rfl | rfl · exact hai · exact haj /-! ### The partition `N_1, N_2, N_3` -/ /-- Assign to each vertex of `X` one of the two (or more) cover graphs containing it. -/ noncomputable def pickIndex (X : Finset A) (F : Fin 3 → BipartiteGraph A B) (h : ∀ a : A, a ∈ X → ∃ k : Fin 3, ∀ j, j ≠ k → a ∈ (F j).left) (a : A) : Fin 3 := if ha : a ∈ X then Classical.choose (h a ha) else 0 omit [Fintype A] [Fintype B] [DecidableEq B] in theorem pickIndex_spec (X : Finset A) (F : Fin 3 → BipartiteGraph A B) (h : ∀ a : A, a ∈ X → ∃ k : Fin 3, ∀ j, j ≠ k → a ∈ (F j).left) {a : A} (ha : a ∈ X) : ∀ j, j ≠ pickIndex X F h a → a ∈ (F j).left := by unfold pickIndex rw [dite_eq_left ha] exact Classical.choose_spec (h a ha) /-- A partition of the left part into three pieces, piece `i` lying in the two cover graphs different from `i`. -/ structure ThreePartsL {A B : Type} (X : Finset A) (F : Fin 3 → BipartiteGraph A B) where part : Fin 3 → Finset A subset : ∀ i, part i ⊆ X disjoint : ∀ i j, i ≠ j → Disjoint (part i) (part j) cover : ∀ a ∈ X, ∃ i, a ∈ part i other : ∀ (i : Fin 3) (a : A), a ∈ part i → ∀ j, j ≠ i → a ∈ (F j).left /-- The right-hand analogue. -/ structure ThreePartsR {A B : Type} (Y : Finset B) (F : Fin 3 → BipartiteGraph A B) where part : Fin 3 → Finset B subset : ∀ i, part i ⊆ Y disjoint : ∀ i j, i ≠ j → Disjoint (part i) (part j) cover : ∀ b ∈ Y, ∃ i, b ∈ part i other : ∀ (i : Fin 3) (b : B), b ∈ part i → ∀ j, j ≠ i → b ∈ (F j).right omit [Fintype A] [Fintype B] [DecidableEq B] in theorem exists_threePartsL (X : Finset A) (F : Fin 3 → BipartiteGraph A B) (h2 : ∀ a ∈ X, ∃ i j : Fin 3, i ≠ j ∧ a ∈ (F i).left ∧ a ∈ (F j).left) : Nonempty (ThreePartsL X F) := by classical have h3 : ∀ a : A, a ∈ X → ∃ k : Fin 3, ∀ j, j ≠ k → a ∈ (F j).left := fun a ha => exists_lonely_index h2 a ha refine ⟨⟨fun i => X.filter (fun a => pickIndex X F h3 a = i), ?_, ?_, ?_, ?_⟩⟩ · intro i a ha exact (Finset.mem_filter.mp ha).1 · intro i j hij rw [Finset.disjoint_left] intro a hi hj have h1 : pickIndex X F h3 a = i := (Finset.mem_filter.mp hi).2 have h2' : pickIndex X F h3 a = j := (Finset.mem_filter.mp hj).2 exact hij (h1.symm.trans h2') · intro a ha exact ⟨pickIndex X F h3 a, Finset.mem_filter.mpr ⟨ha, rfl⟩⟩ · intro i a ha j hj have hX : a ∈ X := (Finset.mem_filter.mp ha).1 have hp : pickIndex X F h3 a = i := (Finset.mem_filter.mp ha).2 exact pickIndex_spec X F h3 hX j (by rw [hp]; exact hj) /-- Right-hand version of `pickIndex`. -/ noncomputable def pickIndexR (Y : Finset B) (F : Fin 3 → BipartiteGraph A B) (h : ∀ b : B, b ∈ Y → ∃ k : Fin 3, ∀ j, j ≠ k → b ∈ (F j).right) (b : B) : Fin 3 := if hb : b ∈ Y then Classical.choose (h b hb) else 0 omit [Fintype A] [Fintype B] [DecidableEq A] in theorem pickIndexR_spec (Y : Finset B) (F : Fin 3 → BipartiteGraph A B) (h : ∀ b : B, b ∈ Y → ∃ k : Fin 3, ∀ j, j ≠ k → b ∈ (F j).right) {b : B} (hb : b ∈ Y) : ∀ j, j ≠ pickIndexR Y F h b → b ∈ (F j).right := by unfold pickIndexR rw [dite_eq_left hb] exact Classical.choose_spec (h b hb) omit [Fintype A] [Fintype B] [DecidableEq A] [DecidableEq B] in theorem exists_lonely_indexR {Y : Finset B} {F : Fin 3 → BipartiteGraph A B} (h2 : ∀ b ∈ Y, ∃ i j : Fin 3, i ≠ j ∧ b ∈ (F i).right ∧ b ∈ (F j).right) : ∀ b ∈ Y, ∃ k : Fin 3, ∀ j, j ≠ k → b ∈ (F j).right := by intro b hb obtain ⟨i, j, hij, hbi, hbj⟩ := h2 b hb refine ⟨thirdIndex i j, fun k hk => ?_⟩ rcases eq_left_or_right_of_ne_thirdIndex hij hk with rfl | rfl · exact hbi · exact hbj omit [Fintype A] [Fintype B] [DecidableEq A] in theorem exists_threePartsR (Y : Finset B) (F : Fin 3 → BipartiteGraph A B) (h2 : ∀ b ∈ Y, ∃ i j : Fin 3, i ≠ j ∧ b ∈ (F i).right ∧ b ∈ (F j).right) : Nonempty (ThreePartsR Y F) := by classical have h3 : ∀ b : B, b ∈ Y → ∃ k : Fin 3, ∀ j, j ≠ k → b ∈ (F j).right := fun b hb => exists_lonely_indexR h2 b hb refine ⟨⟨fun i => Y.filter (fun b => pickIndexR Y F h3 b = i), ?_, ?_, ?_, ?_⟩⟩ · intro i b hb exact (Finset.mem_filter.mp hb).1 · intro i j hij rw [Finset.disjoint_left] intro b hi hj have h1 : pickIndexR Y F h3 b = i := (Finset.mem_filter.mp hi).2 have h2' : pickIndexR Y F h3 b = j := (Finset.mem_filter.mp hj).2 exact hij (h1.symm.trans h2') · intro b hb exact ⟨pickIndexR Y F h3 b, Finset.mem_filter.mpr ⟨hb, rfl⟩⟩ · intro i b hb j hj have hY : b ∈ Y := (Finset.mem_filter.mp hb).1 have hp : pickIndexR Y F h3 b = i := (Finset.mem_filter.mp hb).2 exact pickIndexR_spec Y F h3 hY j (by rw [hp]; exact hj) /-! ### Relative densities -/ /-- The normalised edge count `|E ∩ (S × T)|² / (|S| |T|)`; for admissible `S`, `T` it is `H.ratio S T`. -/ noncomputable def relDensity (H : BipartiteGraph A B) (S : Finset A) (T : Finset B) : ℝ := (H.edgeCount S T : ℝ) ^ 2 / ((S.card : ℝ) * (T.card : ℝ)) theorem relDensity_graphOn (H : BipartiteGraph A B) (N : Finset A) (Y : Finset B) : relDensity (graphOn H.edge N Y) N Y = relDensity H N Y := by unfold relDensity rw [edgeCount_graphOn_self'] theorem relDensity_swap (G : BipartiteGraph A B) (X : Finset A) (Y : Finset B) : relDensity (swapGraph G) Y X = relDensity G X Y := by unfold relDensity rw [edgeCount_swap] congr 1 ring /-! ### Estimate (4.2) -/ theorem frag_sGraph_ge_density (H : BipartiteGraph A B) (F : Fin 3 → BipartiteGraph A B) (hmono : IsMonotoneCover H F) (S : Finset A) (T : Finset B) (k : Fin 3) (hS : ∀ a ∈ S, a ∈ (F k).left) (hT : ∀ b ∈ T, b ∈ (F k).right) : relDensity H S T ≤ (F k).sGraph := by classical have hAdm : (F k).Admissible S T := ⟨hS, hT⟩ have hcount : (F k).edgeCount S T = H.edgeCount S T := by apply edgeCount_congr_on intro a ha b hb exact edge_eq_of_edgeRestriction (hmono.1.2.2.1 k) (hS a ha) (hT b hb) have h := sGraph_ge_full (F k) S T hAdm rw [hcount] at h exact h omit [Fintype A] [Fintype B] [DecidableEq B] in theorem part_union_left {X : Finset A} {F : Fin 3 → BipartiteGraph A B} (PL : ThreePartsL X F) {i j k : Fin 3} (hik : k ≠ i) (hjk : k ≠ j) : ∀ a ∈ PL.part i ∪ PL.part j, a ∈ (F k).left := by intro a ha rcases Finset.mem_union.mp ha with h | h · exact PL.other i a h k hik · exact PL.other j a h k hjk omit [Fintype A] [Fintype B] [DecidableEq A] in theorem part_union_right {Y : Finset B} {F : Fin 3 → BipartiteGraph A B} (PR : ThreePartsR Y F) {i j k : Fin 3} (hik : k ≠ i) (hjk : k ≠ j) : ∀ b ∈ PR.part i ∪ PR.part j, b ∈ (F k).right := by intro b hb rcases Finset.mem_union.mp hb with h | h · exact PR.other i b h k hik · exact PR.other j b h k hjk /-! ### Estimate (4.3) -/ omit [Fintype A] [Fintype B] [DecidableEq A] [DecidableEq B] in theorem two_of_three {P : Fin 3 → Prop} {i j : Fin 3} (hij : i ≠ j) (h : ∃ p q : Fin 3, p ≠ q ∧ P p ∧ P q) : P i ∨ P j := by obtain ⟨p, q, hpq, hp, hq⟩ := h fin_cases i <;> fin_cases j <;> fin_cases p <;> fin_cases q <;> simp_all set_option maxHeartbeats 800000 in theorem row_cover (H : BipartiteGraph A B) (F : Fin 3 → BipartiteGraph A B) (hmono : IsMonotoneCover H F) (N : Finset A) (Y : Finset B) (i j : Fin 3) (hNi : ∀ a ∈ N, a ∈ (F i).left) (hNj : ∀ a ∈ N, a ∈ (F j).left) (hYij : ∀ b ∈ Y, b ∈ (F i).right ∨ b ∈ (F j).right) : IsMonotoneCover (graphOn H.edge N Y) ![frag (F i) N Y, frag (F j) N Y] := by classical have hcover : IsCover (graphOn H.edge N Y) ![frag (F i) N Y, frag (F j) N Y] := by refine ⟨?_, ?_, ?_, ?_⟩ · intro t a ha fin_cases t <;> exact ha.2 · intro t b hb fin_cases t <;> exact hb.2 · intro t fin_cases t · exact frag_edgeRestriction N Y (hmono.1.2.2.1 i) · exact frag_edgeRestriction N Y (hmono.1.2.2.1 j) · intro a b hab have hab' : (graphOn H.edge N Y).edge a b = true := hab simp only [graphOn, Bool.and_eq_true, decide_eq_true_eq] at hab' obtain ⟨⟨hE, haN⟩, hbY⟩ := hab' obtain ⟨t, ht⟩ := hmono.1.2.2.2 a b hE by_cases hti : t = i · rw [hti] at ht refine ⟨0, ?_⟩ change ((F i).edge a b && decide (a ∈ N) && decide (b ∈ Y)) = true simp only [Bool.and_eq_true, decide_eq_true_eq] exact ⟨⟨ht, haN⟩, hbY⟩ · by_cases htj : t = j · rw [htj] at ht refine ⟨1, ?_⟩ change ((F j).edge a b && decide (a ∈ N) && decide (b ∈ Y)) = true simp only [Bool.and_eq_true, decide_eq_true_eq] exact ⟨⟨ht, haN⟩, hbY⟩ · rcases hYij b hbY with hbi | hbj · have hFit : (F i).edge a b = true := ((hmono.1.2.2.1 i) a b).mpr ⟨hE, hNi a haN, hbi⟩ refine ⟨0, ?_⟩ change ((F i).edge a b && decide (a ∈ N) && decide (b ∈ Y)) = true simp only [Bool.and_eq_true, decide_eq_true_eq] exact ⟨⟨hFit, haN⟩, hbY⟩ · have hFjt : (F j).edge a b = true := ((hmono.1.2.2.1 j) a b).mpr ⟨hE, hNj a haN, hbj⟩ refine ⟨1, ?_⟩ change ((F j).edge a b && decide (a ∈ N) && decide (b ∈ Y)) = true simp only [Bool.and_eq_true, decide_eq_true_eq] exact ⟨⟨hFjt, haN⟩, hbY⟩ refine ⟨hcover, ?_⟩ intro I by_cases h0 : (0 : Fin 2) ∈ I · left rw [Set.sdiff_eq_empty] intro a ha have hai : a ∈ (![frag (F i) N Y, frag (F j) N Y] 0).left := ⟨hNi a ha, ha⟩ exact Set.mem_iUnion.mpr ⟨0, Set.mem_iUnion.mpr ⟨h0, hai⟩⟩ · by_cases h1 : (1 : Fin 2) ∈ I · left rw [Set.sdiff_eq_empty] intro a ha have haj : a ∈ (![frag (F i) N Y, frag (F j) N Y] 1).left := ⟨hNj a ha, ha⟩ exact Set.mem_iUnion.mpr ⟨1, Set.mem_iUnion.mpr ⟨h1, haj⟩⟩ · right rw [Set.sdiff_eq_empty] intro b hb have h0' : (0 : Fin 2) ∈ Iᶜ := by simpa using h0 have h1' : (1 : Fin 2) ∈ Iᶜ := by simpa using h1 rcases hYij b hb with hbi | hbj · exact Set.mem_iUnion.mpr ⟨0, Set.mem_iUnion.mpr ⟨h0', ⟨hbi, hb⟩⟩⟩ · exact Set.mem_iUnion.mpr ⟨1, Set.mem_iUnion.mpr ⟨h1', ⟨hbj, hb⟩⟩⟩ set_option maxHeartbeats 800000 in theorem row_bound (H : BipartiteGraph A B) (F : Fin 3 → BipartiteGraph A B) (hmono : IsMonotoneCover H F) (N : Finset A) (Y : Finset B) {i j : Fin 3} (hNi : ∀ a ∈ N, a ∈ (F i).left) (hNj : ∀ a ∈ N, a ∈ (F j).left) (hYij : ∀ b ∈ Y, b ∈ (F i).right ∨ b ∈ (F j).right) : relDensity H N Y ≤ (F i).sGraph + (F j).sGraph := by classical have hAdm : (graphOn H.edge N Y).Admissible N Y := ⟨fun a ha => ha, fun b hb => hb⟩ have hratio : (graphOn H.edge N Y).ratio N Y = relDensity H N Y := by rw [ratio_of_admissible _ hAdm] exact relDensity_graphOn H N Y have hmonoRow := row_cover H F hmono N Y i j hNi hNj hYij have hbound := twoGraph_ratio_le_add (graphOn H.edge N Y) (frag (F i) N Y) (frag (F j) N Y) N Y hmonoRow hAdm rfl rfl calc relDensity H N Y = (graphOn H.edge N Y).ratio N Y := hratio.symm _ ≤ (frag (F i) N Y).sGraph + (frag (F j) N Y).sGraph := hbound _ ≤ (F i).sGraph + (F j).sGraph := add_le_add (sGraph_frag_le (F i) N Y) (sGraph_frag_le (F j) N Y) set_option maxHeartbeats 800000 in theorem col_bound (H : BipartiteGraph A B) (F : Fin 3 → BipartiteGraph A B) (hmono : IsMonotoneCover H F) (X : Finset A) (P : Finset B) {i j : Fin 3} (hPi : ∀ b ∈ P, b ∈ (F i).right) (hPj : ∀ b ∈ P, b ∈ (F j).right) (hXij : ∀ a ∈ X, a ∈ (F i).left ∨ a ∈ (F j).left) : relDensity H X P ≤ (F i).sGraph + (F j).sGraph := by have hrow := row_bound (swapGraph H) (fun t => swapGraph (F t)) (isMonotoneCover_swap hmono) P X (i := i) (j := j) (fun b hb => by simpa using hPi b hb) (fun b hb => by simpa using hPj b hb) (fun a ha => by rcases hXij a ha with h | h · exact Or.inl (by simpa using h) · exact Or.inr (by simpa using h)) rw [relDensity_swap] at hrow rwa [sGraph_swap, sGraph_swap] at hrow set_option maxHeartbeats 800000 in /-- The combinatorial content of the paper's main case: from the density normalisation one obtains the partitions `N_i` and `P_i`, and every vertex belongs to at least two cover graphs, which yields estimates (4.2) and (4.3) for all triples of distinct indices. -/ theorem main_case_estimates (G : BipartiteGraph A B) (X : Finset A) (Y : Finset B) (Gs : Fin 3 → BipartiteGraph A B) (hmono : IsMonotoneCover (graphOn G.edge X Y) (fun t => frag (Gs t) X Y)) (hmax : (graphOn G.edge X Y).ratio X Y = (graphOn G.edge X Y).sGraph) (hpos : 0 < (graphOn G.edge X Y).ratio X Y) (hB : ∀ i, ¬ (↑Y : Set B) ⊆ (frag (Gs i) X Y).right) (hA : ∀ i, ¬ (↑X : Set A) ⊆ (frag (Gs i) X Y).left) : ∃ (PL : ThreePartsL X (fun t => frag (Gs t) X Y)) (PR : ThreePartsR Y (fun t => frag (Gs t) X Y)), ∀ i j k : Fin 3, i ≠ j → k ≠ i → k ≠ j → relDensity (graphOn G.edge X Y) (PL.part i ∪ PL.part j) (PR.part i ∪ PR.part j) ≤ (frag (Gs k) X Y).sGraph ∧ relDensity (graphOn G.edge X Y) (PL.part i) (PR.part j) ≤ (frag (Gs k) X Y).sGraph ∧ relDensity (graphOn G.edge X Y) (PL.part j) (PR.part i) ≤ (frag (Gs k) X Y).sGraph ∧ relDensity (graphOn G.edge X Y) (PL.part k) Y ≤ (frag (Gs i) X Y).sGraph + (frag (Gs j) X Y).sGraph ∧ relDensity (graphOn G.edge X Y) X (PR.part k) ≤ (frag (Gs i) X Y).sGraph + (frag (Gs j) X Y).sGraph := by classical have hadm : (graphOn G.edge X Y).Admissible X Y := ⟨fun a ha => ha, fun b hb => hb⟩ have hnoIsoL : ∀ a ∈ (graphOn G.edge X Y).left, ∃ b ∈ (graphOn G.edge X Y).right, (graphOn G.edge X Y).edge a b = true := by have h := no_isolated_left_of_max_ratio (graphOn G.edge X Y) X Y hadm hmax hpos exact fun a ha => h a ha have hnoIsoR : ∀ b ∈ (graphOn G.edge X Y).right, ∃ a ∈ (graphOn G.edge X Y).left, (graphOn G.edge X Y).edge a b = true := by have h := no_isolated_right_of_max_ratio (graphOn G.edge X Y) X Y hadm hmax hpos exact fun b hb => h b hb have hX2 : ∀ a ∈ X, ∃ i j : Fin 3, i ≠ j ∧ a ∈ (frag (Gs i) X Y).left ∧ a ∈ (frag (Gs j) X Y).left := exists_two_covers_left (graphOn G.edge X Y) (fun t => frag (Gs t) X Y) hmono hnoIsoL hB have hY2 : ∀ b ∈ Y, ∃ i j : Fin 3, i ≠ j ∧ b ∈ (frag (Gs i) X Y).right ∧ b ∈ (frag (Gs j) X Y).right := exists_two_covers_right (graphOn G.edge X Y) (fun t => frag (Gs t) X Y) hmono hnoIsoR hA obtain ⟨PL⟩ := exists_threePartsL X (fun t => frag (Gs t) X Y) hX2 obtain ⟨PR⟩ := exists_threePartsR Y (fun t => frag (Gs t) X Y) hY2 refine ⟨PL, PR, ?_⟩ intro i j k hij hki hkj have hYij : ∀ b ∈ Y, b ∈ (frag (Gs i) X Y).right ∨ b ∈ (frag (Gs j) X Y).right := fun b hb => two_of_three (P := fun t => b ∈ (frag (Gs t) X Y).right) hij (hY2 b hb) have hXij : ∀ a ∈ X, a ∈ (frag (Gs i) X Y).left ∨ a ∈ (frag (Gs j) X Y).left := fun a ha => two_of_three (P := fun t => a ∈ (frag (Gs t) X Y).left) hij (hX2 a ha) refine ⟨?_, ?_, ?_, ?_, ?_⟩ · exact frag_sGraph_ge_density (graphOn G.edge X Y) (fun t => frag (Gs t) X Y) hmono (PL.part i ∪ PL.part j) (PR.part i ∪ PR.part j) k (part_union_left PL hki hkj) (part_union_right PR hki hkj) · exact frag_sGraph_ge_density (graphOn G.edge X Y) (fun t => frag (Gs t) X Y) hmono (PL.part i) (PR.part j) k (fun a ha => PL.other i a ha k hki) (fun b hb => PR.other j b hb k hkj) · exact frag_sGraph_ge_density (graphOn G.edge X Y) (fun t => frag (Gs t) X Y) hmono (PL.part j) (PR.part i) k (fun a ha => PL.other j a ha k hkj) (fun b hb => PR.other i b hb k hki) · exact row_bound (graphOn G.edge X Y) (fun t => frag (Gs t) X Y) hmono (PL.part k) Y (i := i) (j := j) (fun a ha => PL.other k a ha i (Ne.symm hki)) (fun a ha => PL.other k a ha j (Ne.symm hkj)) hYij · exact col_bound (graphOn G.edge X Y) (fun t => frag (Gs t) X Y) hmono X (PR.part k) (i := i) (j := j) (fun b hb => PR.other k b hb i (Ne.symm hki)) (fun b hb => PR.other k b hb j (Ne.symm hkj)) hXij /-! ### Normalising the goal of the main case The paper normalises `s(G)` to `1` by dividing by `|E|^2/(|A| |B|)`. The two lemmas below perform this reduction: the desired subadditivity becomes `1 <= sum (s*(G_t))^chi` with `s*(G_t) = s(G_t) |A| |B| / |E|^2`. -/ theorem goal_normalised_iff (H : BipartiteGraph A B) (F : Fin 3 → BipartiteGraph A B) (X : Finset A) (Y : Finset B) (hs : H.sGraph = (H.edgeCount X Y : ℝ) ^ 2 / ((X.card : ℝ) * (Y.card : ℝ))) (hpos : 0 < H.sGraph) (χ : ℝ) : (H.sGraph ^ χ ≤ ∑ t, (F t).sGraph ^ χ) ↔ 1 ≤ ∑ t, ((F t).sGraph * ((X.card : ℝ) * (Y.card : ℝ)) / (H.edgeCount X Y : ℝ) ^ 2) ^ χ := by classical set e : ℝ := (H.edgeCount X Y : ℝ) with he set ab : ℝ := (X.card : ℝ) * (Y.card : ℝ) with hab have hpos' : 0 < e ^ 2 / ab := hs ▸ hpos have he2 : 0 < e ^ 2 := by rcases div_pos_iff.mp hpos' with ⟨h1, _⟩ | ⟨h1, _⟩ · exact h1 · exact absurd h1 (not_lt.mpr (sq_nonneg _)) have hab0 : 0 < ab := by rcases div_pos_iff.mp hpos' with ⟨_, h2⟩ | ⟨_, h2⟩ · exact h2 · exact absurd h2 (not_lt.mpr (mul_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _))) have hec : e ≠ 0 := by intro h rw [h] at he2 norm_num at he2 have hXc : (X.card : ℝ) ≠ 0 := by intro h rw [hab, h, zero_mul] at hab0 exact lt_irrefl 0 hab0 have hYc : (Y.card : ℝ) ≠ 0 := by intro h rw [hab, h, mul_zero] at hab0 exact lt_irrefl 0 hab0 have hS : 0 < H.sGraph ^ χ := Real.rpow_pos_of_pos hpos χ have hkey : ∀ t, (F t).sGraph = ((F t).sGraph * ab / e ^ 2) * H.sGraph := by intro t rw [hs] field_simp have hsum : (∑ t, (F t).sGraph ^ χ) = (∑ t, ((F t).sGraph * ab / e ^ 2) ^ χ) * H.sGraph ^ χ := by rw [Finset.sum_mul] apply Finset.sum_congr rfl intro t _ have hu : (0:ℝ) ≤ (F t).sGraph * ab / e ^ 2 := div_nonneg (mul_nonneg (BipartiteGraph.sGraph_nonneg (F t)) hab0.le) (by positivity) calc (F t).sGraph ^ χ = (((F t).sGraph * ab / e ^ 2) * H.sGraph) ^ χ := by conv_lhs => rw [hkey t] _ = ((F t).sGraph * ab / e ^ 2) ^ χ * H.sGraph ^ χ := Real.mul_rpow hu hpos.le have hgoal : H.sGraph ^ χ ≤ ∑ t, (F t).sGraph ^ χ ↔ 1 ≤ ∑ t, ((F t).sGraph * ab / e ^ 2) ^ χ := by rw [hsum] constructor · intro h have h' : H.sGraph ^ χ * 1 ≤ H.sGraph ^ χ * (∑ t, ((F t).sGraph * ab / e ^ 2) ^ χ) := by simpa [mul_comm] using h exact (mul_le_mul_iff_of_pos_left hS).mp h' · intro h have := mul_le_mul_of_nonneg_left h hS.le simpa [mul_comm] using this simpa [he, hab] using hgoal set_option maxHeartbeats 400000 in /-- The goal of the main case written with the paper's normalised densities: after the normalisation `s(G) = |E|^2 / (|A| |B|)` the desired inequality is `1 <= sum (s*(G_t))^chi`. -/ theorem main_case_goal_normalised (G : BipartiteGraph A B) (X : Finset A) (Y : Finset B) (Gs : Fin 3 → BipartiteGraph A B) (hmax : (graphOn G.edge X Y).ratio X Y = (graphOn G.edge X Y).sGraph) (hpos : 0 < (graphOn G.edge X Y).ratio X Y) (χ : ℝ) : ((graphOn G.edge X Y).sGraph ^ χ ≤ ∑ t, (frag (Gs t) X Y).sGraph ^ χ) ↔ 1 ≤ ∑ t, ((frag (Gs t) X Y).sGraph * ((X.card : ℝ) * (Y.card : ℝ)) / (G.edgeCount X Y : ℝ) ^ 2) ^ χ := by have hadm : (graphOn G.edge X Y).Admissible X Y := ⟨fun a ha => ha, fun b hb => hb⟩ have hgs : (graphOn G.edge X Y).sGraph = ((graphOn G.edge X Y).edgeCount X Y : ℝ) ^ 2 / ((X.card : ℝ) * (Y.card : ℝ)) := by rw [← hmax, ratio_of_admissible _ hadm] have h := goal_normalised_iff (graphOn G.edge X Y) (fun t => frag (Gs t) X Y) X Y hgs (by rwa [← hmax]) χ rw [edgeCount_graphOn_self'] at h exact h /-! ### Estimates (4.2)--(4.5) in the paper's normalised notation `normSGraph` is the paper's `s*(G')`, and the estimates above become the inequalities (4.2)--(4.5) literally: each of `H_k`, `Q_k` is bounded by `s*(G_k)`, and `F_ij` by `s*(G_i) + s*(G_j)`. -/ /-- The normalising factor `|X| |Y| / |E(X,Y)|^2` of the paper. -/ noncomputable def normScale (G : BipartiteGraph A B) (X : Finset A) (Y : Finset B) : ℝ := ((X.card : ℝ) * (Y.card : ℝ)) / (G.edgeCount X Y : ℝ) ^ 2 /-- The normalised density `s*(G') = s(G') |X| |Y| / |E(X,Y)|^2`. -/ noncomputable def normSGraph (G : BipartiteGraph A B) (X : Finset A) (Y : Finset B) (F : BipartiteGraph A B) : ℝ := F.sGraph * normScale G X Y omit [Fintype A] [Fintype B] [DecidableEq A] [DecidableEq B] in theorem normScale_nonneg (G : BipartiteGraph A B) (X : Finset A) (Y : Finset B) : 0 ≤ normScale G X Y := div_nonneg (mul_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _)) (by positivity) set_option maxHeartbeats 800000 in theorem main_case_normalised_estimates (G : BipartiteGraph A B) (X : Finset A) (Y : Finset B) (Gs : Fin 3 → BipartiteGraph A B) (hmono : IsMonotoneCover (graphOn G.edge X Y) (fun t => frag (Gs t) X Y)) (hmax : (graphOn G.edge X Y).ratio X Y = (graphOn G.edge X Y).sGraph) (hpos : 0 < (graphOn G.edge X Y).ratio X Y) (hB : ∀ i, ¬ (↑Y : Set B) ⊆ (frag (Gs i) X Y).right) (hA : ∀ i, ¬ (↑X : Set A) ⊆ (frag (Gs i) X Y).left) : ∃ (PL : ThreePartsL X (fun t => frag (Gs t) X Y)) (PR : ThreePartsR Y (fun t => frag (Gs t) X Y)), ∀ i j k : Fin 3, i ≠ j → k ≠ i → k ≠ j → relDensity (graphOn G.edge X Y) (PL.part i ∪ PL.part j) (PR.part i ∪ PR.part j) * normScale G X Y ≤ normSGraph G X Y (frag (Gs k) X Y) ∧ relDensity (graphOn G.edge X Y) (PL.part i) (PR.part j) * normScale G X Y ≤ normSGraph G X Y (frag (Gs k) X Y) ∧ relDensity (graphOn G.edge X Y) (PL.part j) (PR.part i) * normScale G X Y ≤ normSGraph G X Y (frag (Gs k) X Y) ∧ relDensity (graphOn G.edge X Y) (PL.part k) Y * normScale G X Y ≤ normSGraph G X Y (frag (Gs i) X Y) + normSGraph G X Y (frag (Gs j) X Y) ∧ relDensity (graphOn G.edge X Y) X (PR.part k) * normScale G X Y ≤ normSGraph G X Y (frag (Gs i) X Y) + normSGraph G X Y (frag (Gs j) X Y) := by obtain ⟨PL, PR, h⟩ := main_case_estimates G X Y Gs hmono hmax hpos hB hA refine ⟨PL, PR, ?_⟩ intro i j k hij hki hkj obtain ⟨h1, h2, h3, h4, h5⟩ := h i j k hij hki hkj have hs := normScale_nonneg (G := G) X Y refine ⟨?_, ?_, ?_, ?_, ?_⟩ · exact mul_le_mul_of_nonneg_right h1 hs · exact mul_le_mul_of_nonneg_right h2 hs · exact mul_le_mul_of_nonneg_right h3 hs · have := mul_le_mul_of_nonneg_right h4 hs simpa [normSGraph, add_mul] using this · have := mul_le_mul_of_nonneg_right h5 hs simpa [normSGraph, add_mul] using this /-! ### Edge counts over unions To normalise the estimates (4.2)--(4.5) we need the edge count of a box that is a union of two boxes, and the cardinality of such a union. -/ theorem product_union_parts (s s' : Finset A) (t : Finset B) : (s ∪ s').product t = s.product t ∪ s'.product t := by ext q rw [Finset.product_eq_sprod, Finset.product_eq_sprod, Finset.product_eq_sprod] simp [Finset.mem_product, Finset.mem_union] theorem product_union_parts_right (s : Finset A) (t t' : Finset B) : s.product (t ∪ t') = s.product t ∪ s.product t' := by ext q rw [Finset.product_eq_sprod, Finset.product_eq_sprod, Finset.product_eq_sprod] simp [Finset.mem_product, Finset.mem_union] theorem edgeCount_union_left (H : BipartiteGraph A B) (S1 S2 : Finset A) (T : Finset B) (h : Disjoint S1 S2) : H.edgeCount (S1 ∪ S2) T = H.edgeCount S1 T + H.edgeCount S2 T := by classical unfold BipartiteGraph.edgeCount rw [product_union_parts, Finset.filter_union] refine Finset.card_union_of_disjoint ?_ rw [Finset.disjoint_left] intro q h1 h2 rw [Finset.mem_filter] at h1 h2 exact (Finset.disjoint_left.mp h (Finset.mem_product.mp h1.1).1) (Finset.mem_product.mp h2.1).1 theorem edgeCount_union_right (H : BipartiteGraph A B) (S : Finset A) (T1 T2 : Finset B) (h : Disjoint T1 T2) : H.edgeCount S (T1 ∪ T2) = H.edgeCount S T1 + H.edgeCount S T2 := by classical unfold BipartiteGraph.edgeCount rw [product_union_parts_right, Finset.filter_union] refine Finset.card_union_of_disjoint ?_ rw [Finset.disjoint_left] intro q h1 h2 rw [Finset.mem_filter] at h1 h2 exact (Finset.disjoint_left.mp h (Finset.mem_product.mp h1.1).2) (Finset.mem_product.mp h2.1).2 /-- The relative density of a box, multiplied by the normalisation factor, in terms of the normalised edge count and the normalised sizes. -/ theorem relDensity_mul_normScale (H : BipartiteGraph A B) (X : Finset A) (Y : Finset B) (S : Finset A) (T : Finset B) (hX : 0 < X.card) (hY : 0 < Y.card) (hS : 0 < S.card) (hT : 0 < T.card) (hE : 0 < H.edgeCount X Y) : relDensity H S T * normScale H X Y = ((H.edgeCount S T : ℝ) / (H.edgeCount X Y : ℝ)) ^ 2 / (((S.card : ℝ) / (X.card : ℝ)) * ((T.card : ℝ) / (Y.card : ℝ))) := by have hX0 : (X.card : ℝ) ≠ 0 := by have : (0:ℝ) < (X.card : ℝ) := by exact_mod_cast hX exact this.ne' have hY0 : (Y.card : ℝ) ≠ 0 := by have : (0:ℝ) < (Y.card : ℝ) := by exact_mod_cast hY exact this.ne' have hS0 : (S.card : ℝ) ≠ 0 := by have : (0:ℝ) < (S.card : ℝ) := by exact_mod_cast hS exact this.ne' have hT0 : (T.card : ℝ) ≠ 0 := by have : (0:ℝ) < (T.card : ℝ) := by exact_mod_cast hT exact this.ne' have hE0 : (H.edgeCount X Y : ℝ) ≠ 0 := by have : (0:ℝ) < (H.edgeCount X Y : ℝ) := by exact_mod_cast hE exact this.ne' unfold relDensity normScale field_simp /-- The relative density of a union box, multiplied by the normalisation factor, written with the four corner edge counts and the two side lengths. -/ theorem relDensity_union_mul_normScale (H : BipartiteGraph A B) (X : Finset A) (Y : Finset B) (S1 S2 : Finset A) (T1 T2 : Finset B) (hS : Disjoint S1 S2) (hT : Disjoint T1 T2) (hX : 0 < X.card) (hY : 0 < Y.card) (hSn : 0 < (S1 ∪ S2).card) (hTn : 0 < (T1 ∪ T2).card) (hE : 0 < H.edgeCount X Y) : relDensity H (S1 ∪ S2) (T1 ∪ T2) * normScale H X Y = (((H.edgeCount S1 T1 + H.edgeCount S1 T2 + H.edgeCount S2 T1 + H.edgeCount S2 T2 : ℕ) : ℝ) / (H.edgeCount X Y : ℝ)) ^ 2 / (((((S1.card + S2.card : ℕ) : ℝ) / (X.card : ℝ))) * ((((T1.card + T2.card : ℕ) : ℝ) / (Y.card : ℝ)))) := by have hcS : ((S1 ∪ S2).card : ℝ) = ((S1.card + S2.card : ℕ) : ℝ) := by rw [Finset.card_union_of_disjoint hS] have hcT : ((T1 ∪ T2).card : ℝ) = ((T1.card + T2.card : ℕ) : ℝ) := by rw [Finset.card_union_of_disjoint hT] have hcE : ((H.edgeCount (S1 ∪ S2) (T1 ∪ T2) : ℕ) : ℝ) = ((H.edgeCount S1 T1 + H.edgeCount S1 T2 + H.edgeCount S2 T1 + H.edgeCount S2 T2 : ℕ) : ℝ) := by rw [edgeCount_union_left H S1 S2 (T1 ∪ T2) hS, edgeCount_union_right H S1 T1 T2 hT, edgeCount_union_right H S2 T1 T2 hT] push_cast ring have := relDensity_mul_normScale H X Y (S1 ∪ S2) (T1 ∪ T2) hX hY hSn hTn hE rw [this, hcS, hcT, hcE] /-! ### The partition sums Each of the three parts of `ThreePartsL` (resp. `ThreePartsR`) is nonempty in the main case, the parts are disjoint, and they cover the left (resp. right) part. Consequently the normalised sizes sum to `1`. -/ theorem sum_card_part_L {F : Fin 3 → BipartiteGraph A B} {X : Finset A} (PL : ThreePartsL X F) : ∑ i, (PL.part i).card = X.card := by have h01 : Disjoint (PL.part 0) (PL.part 1) := PL.disjoint 0 1 (by decide) have h02 : Disjoint (PL.part 0) (PL.part 2) := PL.disjoint 0 2 (by decide) have h12 : Disjoint (PL.part 1) (PL.part 2) := PL.disjoint 1 2 (by decide) have h012 : Disjoint (PL.part 0 ∪ PL.part 1) (PL.part 2) := by rw [Finset.disjoint_union_left] exact ⟨h02, h12⟩ have hunion : PL.part 0 ∪ PL.part 1 ∪ PL.part 2 = X := by ext a constructor · intro ha simp only [Finset.mem_union] at ha rcases ha with (ha | ha) | ha · exact PL.subset 0 ha · exact PL.subset 1 ha · exact PL.subset 2 ha · intro ha obtain ⟨i, hi⟩ := PL.cover a ha fin_cases i · exact Finset.mem_union_left _ (Finset.mem_union_left _ hi) · exact Finset.mem_union_left _ (Finset.mem_union_right _ hi) · exact Finset.mem_union_right _ hi calc ∑ i, (PL.part i).card = (PL.part 0).card + (PL.part 1).card + (PL.part 2).card := by rw [Fin.sum_univ_three] _ = (PL.part 0 ∪ PL.part 1).card + (PL.part 2).card := by rw [Finset.card_union_of_disjoint h01] _ = (PL.part 0 ∪ PL.part 1 ∪ PL.part 2).card := by rw [Finset.card_union_of_disjoint h012] _ = X.card := by rw [hunion] theorem sum_card_part_R {F : Fin 3 → BipartiteGraph A B} {Y : Finset B} (PR : ThreePartsR Y F) : ∑ j, (PR.part j).card = Y.card := by have h01 : Disjoint (PR.part 0) (PR.part 1) := PR.disjoint 0 1 (by decide) have h02 : Disjoint (PR.part 0) (PR.part 2) := PR.disjoint 0 2 (by decide) have h12 : Disjoint (PR.part 1) (PR.part 2) := PR.disjoint 1 2 (by decide) have h012 : Disjoint (PR.part 0 ∪ PR.part 1) (PR.part 2) := by rw [Finset.disjoint_union_left] exact ⟨h02, h12⟩ have hunion : PR.part 0 ∪ PR.part 1 ∪ PR.part 2 = Y := by ext b constructor · intro hb simp only [Finset.mem_union] at hb rcases hb with (hb | hb) | hb · exact PR.subset 0 hb · exact PR.subset 1 hb · exact PR.subset 2 hb · intro hb obtain ⟨j, hj⟩ := PR.cover b hb fin_cases j · exact Finset.mem_union_left _ (Finset.mem_union_left _ hj) · exact Finset.mem_union_left _ (Finset.mem_union_right _ hj) · exact Finset.mem_union_right _ hj calc ∑ j, (PR.part j).card = (PR.part 0).card + (PR.part 1).card + (PR.part 2).card := by rw [Fin.sum_univ_three] _ = (PR.part 0 ∪ PR.part 1).card + (PR.part 2).card := by rw [Finset.card_union_of_disjoint h01] _ = (PR.part 0 ∪ PR.part 1 ∪ PR.part 2).card := by rw [Finset.card_union_of_disjoint h012] _ = Y.card := by rw [hunion] theorem partL_nonempty {F : Fin 3 → BipartiteGraph A B} {X : Finset A} (PL : ThreePartsL X F) (h : ∀ i, ¬ (↑X : Set A) ⊆ (F i).left) (i : Fin 3) : (PL.part i).Nonempty := by by_contra hne rw [Finset.not_nonempty_iff_eq_empty] at hne apply h i intro a ha have haX : a ∈ X := ha obtain ⟨j, hj⟩ := PL.cover a haX have hji : j ≠ i := by rintro rfl rw [hne] at hj simp at hj exact PL.other j a hj i (Ne.symm hji) theorem partR_nonempty {F : Fin 3 → BipartiteGraph A B} {Y : Finset B} (PR : ThreePartsR Y F) (h : ∀ i, ¬ (↑Y : Set B) ⊆ (F i).right) (i : Fin 3) : (PR.part i).Nonempty := by by_contra hne rw [Finset.not_nonempty_iff_eq_empty] at hne apply h i intro b hb have hbY : b ∈ Y := hb obtain ⟨j, hj⟩ := PR.cover b hbY have hji : j ≠ i := by rintro rfl rw [hne] at hj simp at hj exact PR.other j b hj i (Ne.symm hji) theorem edgeCount_eq_sum_three_right (H : BipartiteGraph A B) (S : Finset A) {F : Fin 3 → BipartiteGraph A B} {Y : Finset B} (PR : ThreePartsR Y F) : H.edgeCount S Y = ∑ j, H.edgeCount S (PR.part j) := by have h01 : Disjoint (PR.part 0) (PR.part 1) := PR.disjoint 0 1 (by decide) have h02 : Disjoint (PR.part 0) (PR.part 2) := PR.disjoint 0 2 (by decide) have h12 : Disjoint (PR.part 1) (PR.part 2) := PR.disjoint 1 2 (by decide) have h012 : Disjoint (PR.part 0 ∪ PR.part 1) (PR.part 2) := by rw [Finset.disjoint_union_left] exact ⟨h02, h12⟩ have hunion : PR.part 0 ∪ PR.part 1 ∪ PR.part 2 = Y := by ext b constructor · intro hb simp only [Finset.mem_union] at hb rcases hb with (hb | hb) | hb · exact PR.subset 0 hb · exact PR.subset 1 hb · exact PR.subset 2 hb · intro hb obtain ⟨j, hj⟩ := PR.cover b hb fin_cases j · exact Finset.mem_union_left _ (Finset.mem_union_left _ hj) · exact Finset.mem_union_left _ (Finset.mem_union_right _ hj) · exact Finset.mem_union_right _ hj calc H.edgeCount S Y = H.edgeCount S (PR.part 0 ∪ PR.part 1 ∪ PR.part 2) := by rw [hunion] _ = H.edgeCount S (PR.part 0 ∪ PR.part 1) + H.edgeCount S (PR.part 2) := by rw [edgeCount_union_right H S _ _ h012] _ = (H.edgeCount S (PR.part 0) + H.edgeCount S (PR.part 1)) + H.edgeCount S (PR.part 2) := by rw [edgeCount_union_right H S _ _ h01] _ = ∑ j, H.edgeCount S (PR.part j) := by rw [Fin.sum_univ_three] theorem edgeCount_eq_sum_three_left (H : BipartiteGraph A B) {F : Fin 3 → BipartiteGraph A B} {X : Finset A} (PL : ThreePartsL X F) (T : Finset B) : H.edgeCount X T = ∑ i, H.edgeCount (PL.part i) T := by have h01 : Disjoint (PL.part 0) (PL.part 1) := PL.disjoint 0 1 (by decide) have h02 : Disjoint (PL.part 0) (PL.part 2) := PL.disjoint 0 2 (by decide) have h12 : Disjoint (PL.part 1) (PL.part 2) := PL.disjoint 1 2 (by decide) have h012 : Disjoint (PL.part 0 ∪ PL.part 1) (PL.part 2) := by rw [Finset.disjoint_union_left] exact ⟨h02, h12⟩ have hunion : PL.part 0 ∪ PL.part 1 ∪ PL.part 2 = X := by ext a constructor · intro ha simp only [Finset.mem_union] at ha rcases ha with (ha | ha) | ha · exact PL.subset 0 ha · exact PL.subset 1 ha · exact PL.subset 2 ha · intro ha obtain ⟨i, hi⟩ := PL.cover a ha fin_cases i · exact Finset.mem_union_left _ (Finset.mem_union_left _ hi) · exact Finset.mem_union_left _ (Finset.mem_union_right _ hi) · exact Finset.mem_union_right _ hi calc H.edgeCount X T = H.edgeCount (PL.part 0 ∪ PL.part 1 ∪ PL.part 2) T := by rw [hunion] _ = H.edgeCount (PL.part 0 ∪ PL.part 1) T + H.edgeCount (PL.part 2) T := by rw [edgeCount_union_left H _ _ T h012] _ = (H.edgeCount (PL.part 0) T + H.edgeCount (PL.part 1) T) + H.edgeCount (PL.part 2) T := by rw [edgeCount_union_left H _ _ T h01] _ = ∑ i, H.edgeCount (PL.part i) T := by rw [Fin.sum_univ_three] theorem sum_edgeCount_parts (H : BipartiteGraph A B) {F : Fin 3 → BipartiteGraph A B} {X : Finset A} {Y : Finset B} (PL : ThreePartsL X F) (PR : ThreePartsR Y F) : ∑ i, ∑ j, H.edgeCount (PL.part i) (PR.part j) = H.edgeCount X Y := by rw [show ∑ i, ∑ j, H.edgeCount (PL.part i) (PR.part j) = ∑ i, H.edgeCount (PL.part i) Y from Finset.sum_congr rfl (fun i _ => (edgeCount_eq_sum_three_right H (PL.part i) PR).symm)] exact (edgeCount_eq_sum_three_left H PL Y).symm /-! ### The normalised numeric data of the main case `mA X PL i = |N_i| / |X|`, `mB Y PR j = |P_j| / |Y|`, `mE H X Y PL PR i j = |E ∩ (N_i × P_j)| / |E|` and `mS G X Y Gs k = s*(G_k)`. These are the quantities `a_i`, `b_i`, `e_ij` and `s_*(G_k)` of the paper. -/ noncomputable def mA {F : Fin 3 → BipartiteGraph A B} (X : Finset A) (PL : ThreePartsL X F) (i : Fin 3) : ℝ := ((PL.part i).card : ℝ) / (X.card : ℝ) noncomputable def mB {F : Fin 3 → BipartiteGraph A B} (Y : Finset B) (PR : ThreePartsR Y F) (j : Fin 3) : ℝ := ((PR.part j).card : ℝ) / (Y.card : ℝ) noncomputable def mE (H : BipartiteGraph A B) (X : Finset A) (Y : Finset B) {F : Fin 3 → BipartiteGraph A B} (PL : ThreePartsL X F) (PR : ThreePartsR Y F) (i j : Fin 3) : ℝ := (H.edgeCount (PL.part i) (PR.part j) : ℝ) / (H.edgeCount X Y : ℝ) noncomputable def mS (G : BipartiteGraph A B) (X : Finset A) (Y : Finset B) (Gs : Fin 3 → BipartiteGraph A B) (k : Fin 3) : ℝ := (frag (Gs k) X Y).sGraph * normScale G X Y theorem mS_nonneg (G : BipartiteGraph A B) (X : Finset A) (Y : Finset B) (Gs : Fin 3 → BipartiteGraph A B) (k : Fin 3) : 0 ≤ mS G X Y Gs k := by unfold mS exact mul_nonneg (BipartiteGraph.sGraph_nonneg _) (normScale_nonneg G X Y) theorem mE_add_four (H : BipartiteGraph A B) (X : Finset A) (Y : Finset B) {F : Fin 3 → BipartiteGraph A B} (PL : ThreePartsL X F) (PR : ThreePartsR Y F) (i j : Fin 3) : ((H.edgeCount (PL.part i) (PR.part i) + H.edgeCount (PL.part i) (PR.part j) + H.edgeCount (PL.part j) (PR.part i) + H.edgeCount (PL.part j) (PR.part j) : ℕ) : ℝ) / (H.edgeCount X Y : ℝ) = mE H X Y PL PR i i + mE H X Y PL PR i j + mE H X Y PL PR j i + mE H X Y PL PR j j := by simp only [mE] push_cast ring theorem mA_add {F : Fin 3 → BipartiteGraph A B} {X : Finset A} (PL : ThreePartsL X F) {i j : Fin 3} (hij : i ≠ j) : (((PL.part i).card + (PL.part j).card : ℕ) : ℝ) / (X.card : ℝ) = mA X PL i + mA X PL j := by simp only [mA] rw [Nat.cast_add] ring theorem mB_add {F : Fin 3 → BipartiteGraph A B} {Y : Finset B} (PR : ThreePartsR Y F) {i j : Fin 3} (hij : i ≠ j) : (((PR.part i).card + (PR.part j).card : ℕ) : ℝ) / (Y.card : ℝ) = mB Y PR i + mB Y PR j := by simp only [mB] rw [Nat.cast_add] ring theorem relDensity_union_eq (H : BipartiteGraph A B) (X : Finset A) (Y : Finset B) {F : Fin 3 → BipartiteGraph A B} (PL : ThreePartsL X F) (PR : ThreePartsR Y F) {i j : Fin 3} (hij : i ≠ j) (hX : 0 < X.card) (hY : 0 < Y.card) (hSn : 0 < (PL.part i ∪ PL.part j).card) (hTn : 0 < (PR.part i ∪ PR.part j).card) (hE : 0 < H.edgeCount X Y) : relDensity H (PL.part i ∪ PL.part j) (PR.part i ∪ PR.part j) * normScale H X Y = (mE H X Y PL PR i i + mE H X Y PL PR i j + mE H X Y PL PR j i + mE H X Y PL PR j j) ^ 2 / ((mA X PL i + mA X PL j) * (mB Y PR i + mB Y PR j)) := by rw [relDensity_union_mul_normScale H X Y _ _ _ _ (PL.disjoint i j hij) (PR.disjoint i j hij) hX hY hSn hTn hE, mE_add_four, mA_add PL hij, mB_add PR hij] theorem relDensity_pair_eq (H : BipartiteGraph A B) (X : Finset A) (Y : Finset B) {F : Fin 3 → BipartiteGraph A B} (PL : ThreePartsL X F) (PR : ThreePartsR Y F) (i j : Fin 3) (hX : 0 < X.card) (hY : 0 < Y.card) (hSi : 0 < (PL.part i).card) (hTj : 0 < (PR.part j).card) (hE : 0 < H.edgeCount X Y) : relDensity H (PL.part i) (PR.part j) * normScale H X Y = (mE H X Y PL PR i j) ^ 2 / (mA X PL i * mB Y PR j) := by rw [relDensity_mul_normScale H X Y (PL.part i) (PR.part j) hX hY hSi hTj hE] simp only [mE, mA, mB] theorem relDensity_row_eq (H : BipartiteGraph A B) (X : Finset A) (Y : Finset B) {F : Fin 3 → BipartiteGraph A B} (PL : ThreePartsL X F) (PR : ThreePartsR Y F) (k : Fin 3) (hX : 0 < X.card) (hY : 0 < Y.card) (hSk : 0 < (PL.part k).card) (hE : 0 < H.edgeCount X Y) : relDensity H (PL.part k) Y * normScale H X Y = (mE H X Y PL PR k 0 + mE H X Y PL PR k 1 + mE H X Y PL PR k 2) ^ 2 / (mA X PL k) := by rw [relDensity_mul_normScale H X Y (PL.part k) Y hX hY hSk hY hE, edgeCount_eq_sum_three_right H (PL.part k) PR] simp only [mE, mA, Fin.sum_univ_three] field_simp push_cast ring theorem relDensity_col_eq (H : BipartiteGraph A B) (X : Finset A) (Y : Finset B) {F : Fin 3 → BipartiteGraph A B} (PL : ThreePartsL X F) (PR : ThreePartsR Y F) (k : Fin 3) (hX : 0 < X.card) (hY : 0 < Y.card) (hPk : 0 < (PR.part k).card) (hE : 0 < H.edgeCount X Y) : relDensity H X (PR.part k) * normScale H X Y = (mE H X Y PL PR 0 k + mE H X Y PL PR 1 k + mE H X Y PL PR 2 k) ^ 2 / (mB Y PR k) := by rw [relDensity_mul_normScale H X Y X (PR.part k) hX hY hX hPk hE, edgeCount_eq_sum_three_left H PL (PR.part k)] simp only [mE, mB, Fin.sum_univ_three] field_simp push_cast ring /-! ### The numeric core of the main case The paper's main case reduces, after density normalisation, to the following statement about real numbers. The hypotheses are exactly the normalised estimates (4.2)--(4.5) rewritten with `a_i = b_i` after averaging the two sides of the bipartition. -/ def MainNumeric (χ : ℝ) : Prop := ∀ (s a b : Fin 3 → ℝ) (e : Fin 3 → Fin 3 → ℝ), (∀ i, 0 < a i) → (∑ i, a i = 1) → (∀ i, 0 < b i) → (∑ i, b i = 1) → (∀ i j, 0 ≤ e i j) → (∑ i, ∑ j, e i j = 1) → (∀ i, 0 ≤ s i) → (∀ i j k, i ≠ j → k ≠ i → k ≠ j → s k ≥ (e i i + e i j + e j i + e j j) ^ 2 / ((a i + a j) * (b i + b j))) → (∀ i j k, i ≠ j → k ≠ i → k ≠ j → s k ≥ (e i j) ^ 2 / (a i * b j)) → (∀ i j k, i ≠ j → k ≠ i → k ≠ j → s k ≥ (e j i) ^ 2 / (a j * b i)) → (∀ i j k, i ≠ j → k ≠ i → k ≠ j → s i + s j ≥ (e k 0 + e k 1 + e k 2) ^ 2 / a k) → (∀ i j k, i ≠ j → k ≠ i → k ≠ j → s i + s j ≥ (e 0 k + e 1 k + e 2 k) ^ 2 / b k) → 1 ≤ ∑ i, (s i) ^ χ theorem mainCase_of_numeric {χ : ℝ} (hnum : MainNumeric χ) : MainCase χ := by intro A₁ B₁ iA iB dA dB G X Y Gs hmono hmax hpos hB hA classical obtain ⟨PL, PR, hb⟩ := main_case_normalised_estimates G X Y Gs hmono hmax hpos hB hA have hEpos : 0 < (graphOn G.edge X Y).edgeCount X Y := by by_contra h have hzero : (graphOn G.edge X Y).edgeCount X Y = 0 := Nat.eq_zero_of_not_pos h have hadm : (graphOn G.edge X Y).Admissible X Y := ⟨fun a ha => ha, fun b hb => hb⟩ rw [ratio_of_admissible _ hadm, hzero] at hpos simp at hpos have hXpos : 0 < X.card := by obtain ⟨q, hq⟩ := Finset.card_pos.mp hEpos rw [Finset.mem_filter] at hq exact Finset.card_pos.mpr ⟨q.1, (Finset.mem_product.mp hq.1).1⟩ have hYpos : 0 < Y.card := by obtain ⟨q, hq⟩ := Finset.card_pos.mp hEpos rw [Finset.mem_filter] at hq exact Finset.card_pos.mpr ⟨q.2, (Finset.mem_product.mp hq.1).2⟩ have hpa : ∀ i, 0 < mA X PL i := by intro i have hc : 0 < (PL.part i).card := Finset.card_pos.mpr (partL_nonempty PL hA i) unfold mA exact div_pos (by exact_mod_cast hc) (by exact_mod_cast hXpos) have hpb : ∀ i, 0 < mB Y PR i := by intro i have hc : 0 < (PR.part i).card := Finset.card_pos.mpr (partR_nonempty PR hB i) unfold mB exact div_pos (by exact_mod_cast hc) (by exact_mod_cast hYpos) have hsa : ∑ i, mA X PL i = 1 := by simp only [mA] rw [← Finset.sum_div, show (∑ i, ((PL.part i).card : ℝ)) = (X.card : ℝ) from by rw [← Nat.cast_sum, sum_card_part_L]] exact div_self (by exact_mod_cast hXpos.ne') have hsb : ∑ i, mB Y PR i = 1 := by simp only [mB] rw [← Finset.sum_div, show (∑ i, ((PR.part i).card : ℝ)) = (Y.card : ℝ) from by rw [← Nat.cast_sum, sum_card_part_R]] exact div_self (by exact_mod_cast hYpos.ne') have hee : ∀ i j, 0 ≤ mE (graphOn G.edge X Y) X Y PL PR i j := by intro i j unfold mE positivity have hse : ∑ i, ∑ j, mE (graphOn G.edge X Y) X Y PL PR i j = 1 := by have hc : ((∑ i, ∑ j, (graphOn G.edge X Y).edgeCount (PL.part i) (PR.part j) : ℕ) : ℝ) = ∑ i, ∑ j, ((graphOn G.edge X Y).edgeCount (PL.part i) (PR.part j) : ℝ) := by push_cast ring simp only [mE] simp_rw [← Finset.sum_div] rw [← hc, sum_edgeCount_parts] exact div_self (by exact_mod_cast hEpos.ne') have hss : ∀ i, 0 ≤ mS G X Y Gs i := fun i => mS_nonneg G X Y Gs i have hscale : normScale G X Y = normScale (graphOn G.edge X Y) X Y := by simp only [normScale, edgeCount_graphOn_self'] rw [main_case_goal_normalised G X Y Gs hmax hpos χ] have hmain := hnum (fun i => mS G X Y Gs i) (fun i => mA X PL i) (fun i => mB Y PR i) (fun i j => mE (graphOn G.edge X Y) X Y PL PR i j) hpa hsa hpb hsb hee hse hss ?_ ?_ ?_ ?_ ?_ · simpa only [mS, normScale, mul_div_assoc] using hmain · intro i j k hij hki hkj have h1 := (hb i j k hij hki hkj).1 have hSn : 0 < (PL.part i ∪ PL.part j).card := Finset.card_pos.mpr ((partL_nonempty PL hA i).mono (Finset.subset_union_left)) have hTn : 0 < (PR.part i ∪ PR.part j).card := Finset.card_pos.mpr ((partR_nonempty PR hB i).mono (Finset.subset_union_left)) rw [hscale] at h1 rw [relDensity_union_eq (graphOn G.edge X Y) X Y PL PR hij hXpos hYpos hSn hTn hEpos] at h1 simpa [mS, normSGraph] using h1 · intro i j k hij hki hkj have h1 := (hb i j k hij hki hkj).2.1 have hSi : 0 < (PL.part i).card := Finset.card_pos.mpr (partL_nonempty PL hA i) have hTj : 0 < (PR.part j).card := Finset.card_pos.mpr (partR_nonempty PR hB j) rw [hscale] at h1 rw [relDensity_pair_eq (graphOn G.edge X Y) X Y PL PR i j hXpos hYpos hSi hTj hEpos] at h1 simpa [mS, normSGraph] using h1 · intro i j k hij hki hkj have h1 := (hb i j k hij hki hkj).2.2.1 have hSj : 0 < (PL.part j).card := Finset.card_pos.mpr (partL_nonempty PL hA j) have hTi : 0 < (PR.part i).card := Finset.card_pos.mpr (partR_nonempty PR hB i) rw [hscale] at h1 rw [relDensity_pair_eq (graphOn G.edge X Y) X Y PL PR j i hXpos hYpos hSj hTi hEpos] at h1 simpa [mS, normSGraph] using h1 · intro i j k hij hki hkj have h1 := (hb i j k hij hki hkj).2.2.2.1 have hSk : 0 < (PL.part k).card := Finset.card_pos.mpr (partL_nonempty PL hA k) rw [hscale] at h1 rw [relDensity_row_eq (graphOn G.edge X Y) X Y PL PR k hXpos hYpos hSk hEpos] at h1 simpa [mS, normSGraph, add_comm] using h1 · intro i j k hij hki hkj have h1 := (hb i j k hij hki hkj).2.2.2.2 have hPk : 0 < (PR.part k).card := Finset.card_pos.mpr (partR_nonempty PR hB k) rw [hscale] at h1 rw [relDensity_col_eq (graphOn G.edge X Y) X Y PL PR k hXpos hYpos hPk hEpos] at h1 simpa [mS, normSGraph, add_comm] using h1 /-! ### Elementary inequalities for the numeric core -/ theorem four_mul_le_sq (a b c d : ℝ) : (a + c) * (b + d) ≤ ((a + b + c + d) / 2) ^ 2 := by nlinarith [sq_nonneg (a + c - b - d)] theorem sq_sqrt_add_le {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) : (Real.sqrt a + Real.sqrt b) ^ 2 ≤ 2 * (a + b) := by nlinarith [Real.sq_sqrt ha, Real.sq_sqrt hb, sq_nonneg (Real.sqrt a - Real.sqrt b)] theorem sqrt_mul_add_sqrt_mul_le {a b c d : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (hc : 0 ≤ c) (hd : 0 ≤ d) : Real.sqrt (a * b) + Real.sqrt (c * d) ≤ Real.sqrt ((a + c) * (b + d)) := by have h1 : 0 ≤ Real.sqrt (a * b) + Real.sqrt (c * d) := by positivity rw [← Real.sqrt_sq h1] apply Real.sqrt_le_sqrt have hkey : 2 * (Real.sqrt (a * b) * Real.sqrt (c * d)) ≤ a * d + c * b := by have hamgm : 2 * (Real.sqrt (a * d) * Real.sqrt (c * b)) ≤ a * d + c * b := by nlinarith [sq_nonneg (Real.sqrt (a * d) - Real.sqrt (c * b)), Real.sq_sqrt (mul_nonneg ha hd), Real.sq_sqrt (mul_nonneg hc hb)] have heq : Real.sqrt (a * b) * Real.sqrt (c * d) = Real.sqrt (a * d) * Real.sqrt (c * b) := by rw [Real.sqrt_mul ha b, Real.sqrt_mul hc d, Real.sqrt_mul ha d, Real.sqrt_mul hc b] ring linarith [hamgm, heq] nlinarith [Real.sq_sqrt (mul_nonneg ha hb), Real.sq_sqrt (mul_nonneg hc hd), hkey] theorem sq_div_le_max_sq_div {x y c d : ℝ} (hx : 0 ≤ x) (hy : 0 ≤ y) (hc : 0 < c) (hd : 0 < d) : (x + y) ^ 2 / ((Real.sqrt c + Real.sqrt d) ^ 2) ≤ max (x ^ 2 / c) (y ^ 2 / d) := by set M : ℝ := max (x ^ 2 / c) (y ^ 2 / d) with hM have hM0 : 0 ≤ M := le_trans (by positivity) (le_max_left _ _) have hx2 : x ^ 2 ≤ M * c := by have h := le_max_left (x ^ 2 / c) (y ^ 2 / d) rw [hM] exact (div_le_iff₀ hc).mp h have hy2 : y ^ 2 ≤ M * d := by have h := le_max_right (x ^ 2 / c) (y ^ 2 / d) rw [hM] exact (div_le_iff₀ hd).mp h have hxle : x ≤ Real.sqrt M * Real.sqrt c := by rw [← Real.sqrt_sq hx, ← Real.sqrt_mul hM0] exact Real.sqrt_le_sqrt hx2 have hyle : y ≤ Real.sqrt M * Real.sqrt d := by rw [← Real.sqrt_sq hy, ← Real.sqrt_mul hM0] exact Real.sqrt_le_sqrt hy2 have hsum : x + y ≤ Real.sqrt M * (Real.sqrt c + Real.sqrt d) := by nlinarith have hnn : 0 ≤ Real.sqrt M * (Real.sqrt c + Real.sqrt d) := by positivity have h1 : (x + y) ^ 2 ≤ (Real.sqrt M * (Real.sqrt c + Real.sqrt d)) ^ 2 := pow_le_pow_left₀ (by linarith) hsum 2 rw [mul_pow, Real.sq_sqrt hM0] at h1 exact (div_le_iff₀ (by positivity)).mpr (by linarith [h1]) theorem holder_sum_le {n : ℕ} {p q : ℝ} (hpq : p.HolderConjugate q) (x y : Fin n → ℝ) (hx : ∀ i, 0 ≤ x i) (hy : ∀ i, 0 ≤ y i) (hsum : 1 ≤ ∑ i, x i * y i) (hB : 0 < ∑ i, y i ^ q) : 1 / (∑ i, y i ^ q) ^ (p / q) ≤ ∑ i, x i ^ p := by have hp : 1 < p := (Real.holderConjugate_iff.mp hpq).1 have hp0 : 0 < p := lt_trans zero_lt_one hp have hq0 : 0 < q := by have hq := hpq.conjugate_eq rw [hq] exact div_pos hp0 (by linarith) set A : ℝ := ∑ i, x i ^ p with hA set B : ℝ := ∑ i, y i ^ q with hBdef have hA0 : 0 ≤ A := Finset.sum_nonneg (fun i _ => Real.rpow_nonneg (hx i) p) have hB0 : 0 ≤ B := Finset.sum_nonneg (fun i _ => Real.rpow_nonneg (hy i) q) have hHolder := Real.inner_le_Lp_mul_Lq_of_nonneg (s := Finset.univ) hpq (f := x) (g := y) (fun i _ => hx i) (fun i _ => hy i) have hHolder' : ∑ i, x i * y i ≤ A ^ (1 / p) * B ^ (1 / q) := by simpa [A, B] using hHolder have h1 : (1 : ℝ) ≤ A ^ (1 / p) * B ^ (1 / q) := le_trans hsum hHolder' have hpoweq : (A ^ p⁻¹ * B ^ q⁻¹) ^ p = A * B ^ (p / q) := by rw [Real.mul_rpow (Real.rpow_nonneg hA0 _) (Real.rpow_nonneg hB0 _)] rw [← Real.rpow_mul hA0, ← Real.rpow_mul hB0] have hAexp : A ^ (p⁻¹ * p) = A := by rw [inv_mul_cancel₀ hp0.ne', Real.rpow_one] have hBexp : B ^ (q⁻¹ * p) = B ^ (p / q) := by ring_nf rw [hAexp, hBexp] have hpoweq' : (A ^ (1 / p) * B ^ (1 / q)) ^ p = A * B ^ (p / q) := by simpa [one_div] using hpoweq have h2 : (1 : ℝ) ≤ A * B ^ (p / q) := by have hle := Real.rpow_le_rpow (by norm_num : (0 : ℝ) ≤ 1) h1 hp0.le rw [Real.one_rpow, hpoweq'] at hle exact hle have hBpow : 0 < B ^ (p / q) := Real.rpow_pos_of_pos hB (p / q) have h3 : 1 / B ^ (p / q) ≤ A := (div_le_iff₀ hBpow).mpr (by simpa [mul_comm] using h2) simpa [A, B] using h3 /-! ### Case (I) of the numeric core: the one-variable estimate The function `g t = (1-t)^q + (2√t)^q` is convex, vanishes at `t = 0` as `1`, and is at most `1` at `t = 9314/100000`; hence it is at most `1` on the whole interval. The endpoint value is checked with exact rational arithmetic, using `q = 769/269`. -/ set_option exponentiation.threshold 2000 in set_option maxRecDepth 100000 in theorem cert_nat₁ : 90686 ^ 769 * 100000 ^ 269 ≤ 75617 ^ 269 * 100000 ^ 769 := by decide set_option exponentiation.threshold 2000 in set_option maxRecDepth 100000 in theorem cert_nat₂ : 37256 ^ 769 * 100000 ^ 538 ≤ 24383 ^ 538 * 100000 ^ 769 := by decide theorem cert_rat₁ : ((90686 : ℚ) / 100000) ^ 769 ≤ ((75617 : ℚ) / 100000) ^ 269 := by rw [div_pow, div_pow, div_le_div_iff₀ (by norm_num) (by norm_num)] exact_mod_cast cert_nat₁ theorem cert_rat₂ : ((37256 : ℚ) / 100000) ^ 769 ≤ ((24383 : ℚ) / 100000) ^ 538 := by rw [div_pow, div_pow, div_le_div_iff₀ (by norm_num) (by norm_num)] exact_mod_cast cert_nat₂ theorem cert_real₁ : ((90686 : ℝ) / 100000) ^ 769 ≤ ((75617 : ℝ) / 100000) ^ 269 := by have h := (Rat.cast_le (K := ℝ)).mpr cert_rat₁ simpa using h theorem cert_real₂ : ((37256 : ℝ) / 100000) ^ 769 ≤ ((24383 : ℝ) / 100000) ^ 538 := by have h := (Rat.cast_le (K := ℝ)).mpr cert_rat₂ simpa using h /-- Comparison of a real power with rational exponent with an integer power. -/ theorem rpow_div_nat_le {x U : ℝ} (hx : 0 ≤ x) (hU : 0 ≤ U) {m n : ℕ} (hn : 0 < n) (h : x ^ m ≤ U ^ n) : x ^ ((m : ℝ) / n) ≤ U := by have hn' : (n : ℝ) ≠ 0 := by exact_mod_cast hn.ne' have hx' : ((x ^ m : ℝ)) ^ ((1 : ℝ) / n) = x ^ ((m : ℝ) / n) := by rw [← Real.rpow_natCast x m, ← Real.rpow_mul hx] congr 1 rw [div_eq_mul_inv] ring have hU' : ((U ^ n : ℝ)) ^ ((1 : ℝ) / n) = U := by rw [← Real.rpow_natCast U n, ← Real.rpow_mul hU, show (n : ℝ) * (1 / (n : ℝ)) = 1 by rw [div_eq_mul_inv, one_mul, mul_inv_cancel₀ hn']] exact Real.rpow_one U rw [← hx', ← hU'] exact Real.rpow_le_rpow (by positivity) h (by positivity) theorem two_sqrt_rpow' {t q : ℝ} (ht : 0 ≤ t) : (2 * Real.sqrt t) ^ q = (4 * t) ^ (q / 2) := by have h2 : 0 ≤ 2 * Real.sqrt t := by positivity have hsq_nat : (2 * Real.sqrt t) ^ 2 = 4 * t := by rw [mul_pow] nlinarith [Real.sq_sqrt ht] have hsq : (2 * Real.sqrt t) ^ (((2 : ℕ) : ℝ)) = 4 * t := by rw [Real.rpow_natCast] exact hsq_nat calc (2 * Real.sqrt t) ^ q = (2 * Real.sqrt t) ^ (((2 : ℕ) : ℝ) * (q / 2)) := by rw [show (((2 : ℕ) : ℝ)) * (q / 2) = q by ring] _ = ((2 * Real.sqrt t) ^ (((2 : ℕ) : ℝ))) ^ (q / 2) := Real.rpow_mul h2 _ _ _ = (4 * t) ^ (q / 2) := by rw [hsq] theorem g_endpoint : (1 - (9314 : ℝ) / 100000) ^ ((769 : ℝ) / 269) + (4 * ((9314 : ℝ) / 100000)) ^ ((769 : ℝ) / 538) ≤ 1 := by have h1 : (1 - (9314 : ℝ) / 100000) ^ ((769 : ℝ) / 269) ≤ 75617 / 100000 := by have hbase : (1 - (9314 : ℝ) / 100000) = 90686 / 100000 := by norm_num rw [hbase] exact rpow_div_nat_le (by norm_num) (by norm_num) (n := 269) (by norm_num) cert_real₁ have h2 : (4 * ((9314 : ℝ) / 100000)) ^ ((769 : ℝ) / 538) ≤ 24383 / 100000 := by have hbase : 4 * ((9314 : ℝ) / 100000) = 37256 / 100000 := by norm_num rw [hbase] exact rpow_div_nat_le (by norm_num) (by norm_num) (n := 538) (by norm_num) cert_real₂ linarith theorem rpow_chord {p z0 z1 lam : ℝ} (hp : 1 ≤ p) (hz0 : 0 ≤ z0) (hz1 : 0 ≤ z1) (hlam0 : 0 ≤ lam) (hlam1 : lam ≤ 1) : ((1 - lam) * z0 + lam * z1) ^ p ≤ (1 - lam) * z0 ^ p + lam * z1 ^ p := by have h := Real.rpow_arith_mean_le_arith_mean_rpow (s := (Finset.univ : Finset (Fin 2))) (w := fun i : Fin 2 => if i = 0 then 1 - lam else lam) (z := fun i : Fin 2 => if i = 0 then z0 else z1) (by intro i _ fin_cases i · simpa using sub_nonneg.mpr hlam1 · simpa using hlam0) ?_ (by intro i _ fin_cases i · simpa using hz0 · simpa using hz1) hp · simpa [Fin.sum_univ_two] using h · simp [Fin.sum_univ_two] theorem g_le_one {t : ℝ} (ht0 : 0 ≤ t) (ht1 : t ≤ 9314 / 100000) : (1 - t) ^ ((769 : ℝ) / 269) + (2 * Real.sqrt t) ^ ((769 : ℝ) / 269) ≤ 1 := by have htpos : (0 : ℝ) < 9314 / 100000 := by norm_num have htle : t ≤ 9314 / 100000 := ht1 set lam : ℝ := t / (9314 / 100000) with hlamdef have hlam0 : 0 ≤ lam := div_nonneg ht0 htpos.le have hlam1 : lam ≤ 1 := (div_le_one htpos).mpr ht1 have ht_eq : t = lam * (9314 / 100000) := by rw [hlamdef] field_simp have h1 : (1 - t) ^ ((769 : ℝ) / 269) ≤ (1 - lam) * (1 : ℝ) ^ ((769 : ℝ) / 269) + lam * (1 - (9314 : ℝ) / 100000) ^ ((769 : ℝ) / 269) := by have hc1 : (1 - t) = (1 - lam) * 1 + lam * (1 - (9314 : ℝ) / 100000) := by rw [ht_eq] ring rw [hc1] exact rpow_chord (by norm_num) zero_le_one (by norm_num) hlam0 hlam1 have h2 : (4 * t) ^ ((769 : ℝ) / 538) ≤ (1 - lam) * (0 : ℝ) ^ ((769 : ℝ) / 538) + lam * (4 * ((9314 : ℝ) / 100000)) ^ ((769 : ℝ) / 538) := by have hc2 : 4 * t = (1 - lam) * 0 + lam * (4 * ((9314 : ℝ) / 100000)) := by rw [ht_eq] ring rw [hc2] exact rpow_chord (by norm_num) (by norm_num) (by norm_num) hlam0 hlam1 have hsq : (2 * Real.sqrt t) ^ ((769 : ℝ) / 269) = (4 * t) ^ ((769 : ℝ) / 538) := by rw [two_sqrt_rpow' ht0] congr 1 norm_num rw [hsq] have h1' : (1 - t) ^ ((769 : ℝ) / 269) ≤ (1 - lam) + lam * (1 - (9314 : ℝ) / 100000) ^ ((769 : ℝ) / 269) := by simpa using h1 have h2' : (4 * t) ^ ((769 : ℝ) / 538) ≤ lam * (4 * ((9314 : ℝ) / 100000)) ^ ((769 : ℝ) / 538) := by simpa using h2 have hg := g_endpoint nlinarith [h1', h2', hg, hlam0, hlam1] /-! ### Bookkeeping for the numeric core -/ theorem exists_pair_ne (k : Fin 3) : ∃ i j : Fin 3, i ≠ j ∧ i ≠ k ∧ j ≠ k := by fin_cases k · exact ⟨1, 2, by decide, by decide, by decide⟩ · exact ⟨0, 2, by decide, by decide, by decide⟩ · exact ⟨0, 1, by decide, by decide, by decide⟩ theorem linear_le_max {α β y x : ℝ} (hα : 0 ≤ α) (hβ : 0 ≤ β) : α * y + β * x ≤ (α + β) * max y x := by rcases le_total y x with h | h · rw [max_eq_right h] nlinarith · rw [max_eq_left h] nlinarith theorem sq_max_sq {y x : ℝ} (hy : 0 ≤ y) (hx : 0 ≤ x) : (max y x) ^ 2 = max (y ^ 2) (x ^ 2) := by rcases le_total y x with h | h · rw [max_eq_right h, max_eq_right (by nlinarith)] · rw [max_eq_left h, max_eq_left (by nlinarith)] theorem sq_rpow {t χ : ℝ} (ht : 0 ≤ t) : (t ^ 2 : ℝ) ^ χ = t ^ (2 * χ) := by rw [← Real.rpow_natCast t 2, ← Real.rpow_mul ht] congr 1 theorem sum_p_eq_one {a b : Fin 3 → ℝ} (ha : ∑ i, a i = 1) (hb : ∑ i, b i = 1) : ∑ i, (a i + b i) / 2 = 1 := by rw [← Finset.sum_div, Finset.sum_add_distrib, ha, hb] norm_num theorem pair_sum_of_total {p : Fin 3 → ℝ} (hp : ∑ i, p i = 1) {i j k : Fin 3} (hij : i ≠ j) (hik : i ≠ k) (hjk : j ≠ k) : p i + p j = 1 - p k := by rw [Fin.sum_univ_three] at hp fin_cases i <;> fin_cases j <;> fin_cases k <;> simp_all <;> linarith theorem e_partition {e : Fin 3 → Fin 3 → ℝ} {i j k : Fin 3} (hij : i ≠ j) (hik : i ≠ k) (hjk : j ≠ k) : e i i + e i j + e j i + e j j + (2 * e k k + e i k + e k i + e j k + e k j) = (∑ p, ∑ q, e p q) + e k k := by rw [Fin.sum_univ_three] simp only [Fin.sum_univ_three] fin_cases i <;> fin_cases j <;> fin_cases k <;> simp_all <;> ring theorem sum_three_perm {f : Fin 3 → ℝ} {i j k : Fin 3} (hij : i ≠ j) (hik : i ≠ k) (hjk : j ≠ k) : f i + f j + f k = f 0 + f 1 + f 2 := by fin_cases i <;> fin_cases j <;> fin_cases k <;> simp_all <;> ring theorem e_block_le_one {e : Fin 3 → Fin 3 → ℝ} (he0 : ∀ i j, 0 ≤ e i j) (he1 : ∑ i, ∑ j, e i j = 1) {i j k : Fin 3} (hij : i ≠ j) (hik : i ≠ k) (hjk : j ≠ k) : e i i + e i j + e j i + e j j ≤ 1 := by rw [Fin.sum_univ_three] at he1 simp only [Fin.sum_univ_three] at he1 fin_cases i <;> fin_cases j <;> fin_cases k <;> simp_all <;> linarith [he0 0 0, he0 0 1, he0 0 2, he0 1 0, he0 1 1, he0 1 2, he0 2 0, he0 2 1, he0 2 2] /-! ### The averaged numeric data and the case-(I) estimate -/ noncomputable def avg (a b : Fin 3 → ℝ) : Fin 3 → ℝ := fun i => (a i + b i) / 2 theorem avg_pos {a b : Fin 3 → ℝ} (ha : ∀ i, 0 < a i) (hb : ∀ i, 0 < b i) (i : Fin 3) : 0 < avg a b i := by unfold avg linarith [ha i, hb i] theorem avg_sum {a b : Fin 3 → ℝ} (ha : ∑ i, a i = 1) (hb : ∑ i, b i = 1) : ∑ i, avg a b i = 1 := sum_p_eq_one ha hb /-- Sum of the four corners of the block on `{i,j}`. -/ noncomputable def nA (e : Fin 3 → Fin 3 → ℝ) (i j : Fin 3) : ℝ := e i i + e i j + e j i + e j j /-- The numerator of the row/column estimate: `2e_kk + e_ik + e_ki + e_jk + e_kj`. -/ noncomputable def nN (e : Fin 3 → Fin 3 → ℝ) (i j k : Fin 3) : ℝ := 2 * e k k + e i k + e k i + e j k + e k j noncomputable def nS (p : Fin 3 → ℝ) (i j : Fin 3) : ℝ := p i + p j noncomputable def nG (p : Fin 3 → ℝ) (k : Fin 3) : ℝ := 2 * Real.sqrt (p k) noncomputable def nY (p : Fin 3 → ℝ) (e : Fin 3 → Fin 3 → ℝ) (i j : Fin 3) : ℝ := nA e i j / nS p i j noncomputable def nX (p : Fin 3 → ℝ) (e : Fin 3 → Fin 3 → ℝ) (i j : Fin 3) : ℝ := (e i j + e j i) / (2 * Real.sqrt (p i * p j)) noncomputable def nC (p : Fin 3 → ℝ) (i j : Fin 3) : ℝ := (Real.sqrt (p i) + Real.sqrt (p j)) ^ 2 / 2 theorem nA_nonneg {e : Fin 3 → Fin 3 → ℝ} (he0 : ∀ i j, 0 ≤ e i j) (i j : Fin 3) : 0 ≤ nA e i j := by unfold nA linarith [he0 i i, he0 i j, he0 j i, he0 j j] theorem nA_le_one {e : Fin 3 → Fin 3 → ℝ} (he0 : ∀ i j, 0 ≤ e i j) (he1 : ∑ i, ∑ j, e i j = 1) {i j k : Fin 3} (hij : i ≠ j) (hik : i ≠ k) (hjk : j ≠ k) : nA e i j ≤ 1 := by unfold nA exact e_block_le_one he0 he1 hij hik hjk theorem nN_nonneg {e : Fin 3 → Fin 3 → ℝ} (he0 : ∀ i j, 0 ≤ e i j) (i j k : Fin 3) : 0 ≤ nN e i j k := by unfold nN linarith [he0 k k, he0 i k, he0 k i, he0 j k, he0 k j] theorem nA_add_nN {e : Fin 3 → Fin 3 → ℝ} (he1 : ∑ i, ∑ j, e i j = 1) {i j k : Fin 3} (hij : i ≠ j) (hik : i ≠ k) (hjk : j ≠ k) : nA e i j + nN e i j k = 1 + e k k := by unfold nA nN have h := e_partition (e := e) hij hik hjk rw [he1] at h exact h theorem nS_eq_one {p : Fin 3 → ℝ} (hp1 : ∑ i, p i = 1) {i j k : Fin 3} (hij : i ≠ j) (hik : i ≠ k) (hjk : j ≠ k) : nS p i j = 1 - p k := by unfold nS exact pair_sum_of_total hp1 hij hik hjk theorem nS_pos {p : Fin 3 → ℝ} (hp0 : ∀ i, 0 < p i) (i j : Fin 3) : 0 < nS p i j := by unfold nS exact add_pos (hp0 i) (hp0 j) theorem nY_nonneg {p : Fin 3 → ℝ} {e : Fin 3 → Fin 3 → ℝ} (hp0 : ∀ i, 0 < p i) (he0 : ∀ i j, 0 ≤ e i j) (i j : Fin 3) : 0 ≤ nY p e i j := by unfold nY exact div_nonneg (nA_nonneg he0 i j) (nS_pos hp0 i j).le theorem nS_mul_nY (p : Fin 3 → ℝ) (e : Fin 3 → Fin 3 → ℝ) (i j : Fin 3) (hp : nS p i j ≠ 0) : nS p i j * nY p e i j = nA e i j := by unfold nY field_simp theorem nY_sq (p : Fin 3 → ℝ) (e : Fin 3 → Fin 3 → ℝ) (i j : Fin 3) : nY p e i j ^ 2 = (nA e i j) ^ 2 / (nS p i j) ^ 2 := by unfold nY rw [div_pow] theorem nX_nonneg {p : Fin 3 → ℝ} {e : Fin 3 → Fin 3 → ℝ} (hp0 : ∀ i, 0 < p i) (he0 : ∀ i j, 0 ≤ e i j) (i j : Fin 3) : 0 ≤ nX p e i j := by unfold nX have hpos : 0 < 2 * Real.sqrt (p i * p j) := mul_pos (by norm_num) (Real.sqrt_pos_of_pos (mul_pos (hp0 i) (hp0 j))) exact div_nonneg (by linarith [he0 i j, he0 j i]) hpos.le theorem nX_sq {p : Fin 3 → ℝ} (hp0 : ∀ i, 0 < p i) (e : Fin 3 → Fin 3 → ℝ) (i j : Fin 3) : nX p e i j ^ 2 = (e i j + e j i) ^ 2 / (4 * p i * p j) := by unfold nX rw [div_pow] have h : (2 * Real.sqrt (p i * p j)) ^ 2 = 4 * p i * p j := by nlinarith [Real.sq_sqrt (mul_nonneg (hp0 i).le (hp0 j).le)] rw [h] theorem nX_mul {p : Fin 3 → ℝ} (hp0 : ∀ i, 0 < p i) (e : Fin 3 → Fin 3 → ℝ) (i j : Fin 3) : 2 * Real.sqrt (p i * p j) * nX p e i j = e i j + e j i := by unfold nX have hne : Real.sqrt (p i * p j) ≠ 0 := (Real.sqrt_pos_of_pos (mul_pos (hp0 i) (hp0 j))).ne' field_simp theorem nG_pos {p : Fin 3 → ℝ} (hp0 : ∀ i, 0 < p i) (k : Fin 3) : 0 < nG p k := by unfold nG exact mul_pos (by norm_num) (Real.sqrt_pos_of_pos (hp0 k)) theorem nG_sq {p : Fin 3 → ℝ} (hp0 : ∀ i, 0 < p i) (k : Fin 3) : nG p k ^ 2 = 4 * p k := by unfold nG nlinarith [Real.sq_sqrt (hp0 k).le] theorem nC_nonneg {p : Fin 3 → ℝ} (hp0 : ∀ i, 0 < p i) (i j : Fin 3) : 0 ≤ nC p i j := by unfold nC positivity theorem nC_pos {p : Fin 3 → ℝ} (hp0 : ∀ i, 0 < p i) (i j : Fin 3) : 0 < nC p i j := by unfold nC have h1 := Real.sqrt_pos_of_pos (hp0 i) have h2 := Real.sqrt_pos_of_pos (hp0 j) positivity theorem nC_eq {p : Fin 3 → ℝ} (hp0 : ∀ i, 0 < p i) (i j : Fin 3) : nC p i j = (nS p i j + 2 * Real.sqrt (p i * p j)) / 2 := by unfold nC nS rw [Real.sqrt_mul (hp0 i).le] have h := Real.sq_sqrt (hp0 i).le have h2 := Real.sq_sqrt (hp0 j).le field_simp nlinarith theorem nN_eq_rows (e : Fin 3 → Fin 3 → ℝ) {i j k : Fin 3} (hij : i ≠ j) (hik : i ≠ k) (hjk : j ≠ k) : nN e i j k = (e k 0 + e k 1 + e k 2) + (e 0 k + e 1 k + e 2 k) := by unfold nN fin_cases i <;> fin_cases j <;> fin_cases k <;> simp_all <;> ring theorem holder_two_le {p q x1 x2 y1 y2 : ℝ} (hpq : p.HolderConjugate q) (hx1 : 0 ≤ x1) (hx2 : 0 ≤ x2) (hy1 : 0 ≤ y1) (hy2 : 0 ≤ y2) (hsum : 1 ≤ x1 * y1 + x2 * y2) (hB : 0 < y1 ^ q + y2 ^ q) : 1 / (y1 ^ q + y2 ^ q) ^ (p / q) ≤ x1 ^ p + x2 ^ p := by have h := holder_sum_le (n := 2) hpq (fun t : Fin 2 => if t = 0 then x1 else x2) (fun t : Fin 2 => if t = 0 then y1 else y2) (by intro t fin_cases t · simpa using hx1 · simpa using hx2) (by intro t fin_cases t · simpa using hy1 · simpa using hy2) (by simpa [Fin.sum_univ_two] using hsum) (by simpa [Fin.sum_univ_two] using hB) simpa [Fin.sum_univ_two] using h /-- Case (I) of the numeric core: one of the three normalised sizes is at most `9314/100000`. The proof uses the estimate (4.2) for `s k`, the row/column estimate (4.3) for `s i + s j`, Hölder's inequality, and the one-variable bound `g_le_one`. -/ theorem caseI_aux {s p : Fin 3 → ℝ} {e : Fin 3 → Fin 3 → ℝ} (hp0 : ∀ i, 0 < p i) (hp1 : ∑ i, p i = 1) (he0 : ∀ i j, 0 ≤ e i j) (he1 : ∑ i, ∑ j, e i j = 1) (hs0 : ∀ i, 0 ≤ s i) (hH : ∀ i j k : Fin 3, i ≠ j → k ≠ i → k ≠ j → s k ≥ (nA e i j) ^ 2 / (nS p i j) ^ 2) (hQ : ∀ i j k : Fin 3, i ≠ j → k ≠ i → k ≠ j → s k ≥ (e i j + e j i) ^ 2 / (4 * p i * p j)) (hF : ∀ i j k : Fin 3, i ≠ j → k ≠ i → k ≠ j → s i + s j ≥ (nN e i j k) ^ 2 / (4 * p k)) {i j k : Fin 3} (hij : i ≠ j) (hik : i ≠ k) (hjk : j ≠ k) (hk : p k ≤ 9314 / 100000) : 1 ≤ ∑ t, (s t) ^ ((769 : ℝ) / 1000) := by have hki : k ≠ i := Ne.symm hik have hkj : k ≠ j := Ne.symm hjk have hSpos : 0 < nS p i j := nS_pos hp0 i j have hGpos : 0 < nG p k := nG_pos hp0 k have hGsq : nG p k ^ 2 = 4 * p k := nG_sq hp0 k have hSeq : nS p i j = 1 - p k := nS_eq_one hp1 hij hik hjk have hAy : nS p i j * nY p e i j = nA e i j := nS_mul_nY p e i j hSpos.ne' have hA1 : nA e i j ≤ 1 := nA_le_one he0 he1 hij hik hjk have hAy1 : 0 ≤ 1 - nS p i j * nY p e i j := by rw [hAy]; linarith have hpart : nA e i j + nN e i j k = 1 + e k k := nA_add_nN he1 hij hik hjk have hy0 : 0 ≤ nY p e i j := nY_nonneg hp0 he0 i j have hx0 : 0 ≤ nX p e i j := nX_nonneg hp0 he0 i j have h1 : s k ≥ (nY p e i j) ^ 2 := by rw [nY_sq] exact hH i j k hij hki hkj have h2 : s k ≥ (nX p e i j) ^ 2 := by rw [nX_sq hp0] exact hQ i j k hij hki hkj have hsk : s k ≥ (max (nY p e i j) (nX p e i j)) ^ 2 := by rw [sq_max_sq hy0 hx0] exact max_le h1 h2 have hFnum : s i + s j ≥ ((1 - nS p i j * nY p e i j) / nG p k) ^ 2 := by have hf := hF i j k hij hki hkj have hNge : nN e i j k ≥ 1 - nA e i j := by linarith [hpart, he0 k k] have h1mA : 0 ≤ 1 - nA e i j := by linarith have hNsq : (1 - nA e i j) ^ 2 ≤ (nN e i j k) ^ 2 := by nlinarith [hNge, nN_nonneg he0 i j k] have hdiv : (nN e i j k) ^ 2 / (4 * p k) = (nN e i j k / nG p k) ^ 2 := by rw [div_pow, hGsq] have hle : (1 - nA e i j) / nG p k ≤ nN e i j k / nG p k := div_le_div_of_nonneg_right hNge hGpos.le have hnn : 0 ≤ (1 - nA e i j) / nG p k := div_nonneg h1mA hGpos.le have hsq : ((1 - nA e i j) / nG p k) ^ 2 ≤ (nN e i j k / nG p k) ^ 2 := by nlinarith calc s i + s j ≥ (nN e i j k) ^ 2 / (4 * p k) := hf _ = (nN e i j k / nG p k) ^ 2 := hdiv _ ≥ ((1 - nA e i j) / nG p k) ^ 2 := hsq _ = ((1 - nS p i j * nY p e i j) / nG p k) ^ 2 := by rw [hAy] have hbase_le : (nS p i j) ^ ((769 : ℝ) / 269) + (nG p k) ^ ((769 : ℝ) / 269) ≤ 1 := by rw [hSeq] unfold nG exact g_le_one (le_of_lt (hp0 k)) hk have hbase_pos : 0 < (nS p i j) ^ ((769 : ℝ) / 269) + (nG p k) ^ ((769 : ℝ) / 269) := by positivity have hpq : ((769 : ℝ) / 500).HolderConjugate ((769 : ℝ) / 269) := by rw [Real.holderConjugate_iff] constructor <;> norm_num have hsum2 : 1 ≤ nY p e i j * (nS p i j) + ((1 - nS p i j * nY p e i j) / nG p k) * nG p k := by have hcancel : ((1 - nS p i j * nY p e i j) / nG p k) * nG p k = 1 - nS p i j * nY p e i j := by field_simp rw [hcancel, mul_comm (nY p e i j) (nS p i j), hAy] linarith have hHolder : ((nS p i j) ^ ((769 : ℝ) / 269) + (nG p k) ^ ((769 : ℝ) / 269)) ^ (1 - 2 * ((769 : ℝ) / 1000)) ≤ (nY p e i j) ^ (2 * ((769 : ℝ) / 1000)) + ((1 - nS p i j * nY p e i j) / nG p k) ^ (2 * ((769 : ℝ) / 1000)) := by have h := holder_two_le hpq hy0 (div_nonneg hAy1 hGpos.le) hSpos.le hGpos.le hsum2 hbase_pos have hneg : 1 - 2 * ((769 : ℝ) / 1000) = -(((769 : ℝ) / 500) / ((769 : ℝ) / 269)) := by norm_num rw [hneg, Real.rpow_neg (le_of_lt hbase_pos)] have hexp : 2 * ((769 : ℝ) / 1000) = (769 : ℝ) / 500 := by norm_num rw [hexp] simpa [one_div] using h have hone : (1 : ℝ) ≤ ((nS p i j) ^ ((769 : ℝ) / 269) + (nG p k) ^ ((769 : ℝ) / 269)) ^ (1 - 2 * ((769 : ℝ) / 1000)) := by rw [show 1 - 2 * ((769 : ℝ) / 1000) = -(2 * ((769 : ℝ) / 1000) - 1) by ring, Real.rpow_neg (le_of_lt hbase_pos), one_le_inv₀ (by positivity)] exact Real.rpow_le_one hbase_pos.le hbase_le (by norm_num) have hskχ : (max (nY p e i j) (nX p e i j)) ^ (2 * ((769 : ℝ) / 1000)) ≤ (s k) ^ ((769 : ℝ) / 1000) := by have hr := Real.rpow_le_rpow (by positivity : (0:ℝ) ≤ (max (nY p e i j) (nX p e i j)) ^ 2) hsk (by norm_num : (0:ℝ) ≤ (769:ℝ)/1000) rwa [sq_rpow (χ := (769:ℝ)/1000) (le_trans hx0 (le_max_right _ _))] at hr have hFχ : ((1 - nS p i j * nY p e i j) / nG p k) ^ (2 * ((769 : ℝ) / 1000)) ≤ (s i + s j) ^ ((769 : ℝ) / 1000) := by have hnn : 0 ≤ (1 - nS p i j * nY p e i j) / nG p k := div_nonneg hAy1 hGpos.le have hr := Real.rpow_le_rpow (by positivity : (0:ℝ) ≤ ((1 - nS p i j * nY p e i j) / nG p k) ^ 2) hFnum (by norm_num : (0:ℝ) ≤ (769:ℝ)/1000) rwa [sq_rpow (χ := (769:ℝ)/1000) hnn] at hr have hymax : (nY p e i j) ^ (2 * ((769 : ℝ) / 1000)) ≤ (max (nY p e i j) (nX p e i j)) ^ (2 * ((769 : ℝ) / 1000)) := Real.rpow_le_rpow hy0 (le_max_left _ _) (by positivity) have hsplit : (s i) ^ ((769 : ℝ) / 1000) + (s j) ^ ((769 : ℝ) / 1000) ≥ (s i + s j) ^ ((769 : ℝ) / 1000) := by have := Real.rpow_add_le_add_rpow (hs0 i) (hs0 j) (by norm_num : (0:ℝ) ≤ (769:ℝ)/1000) (by norm_num : (769:ℝ)/1000 ≤ 1) linarith have hperm := sum_three_perm (f := fun t => (s t) ^ ((769 : ℝ) / 1000)) hij hik hjk rw [Fin.sum_univ_three] linarith [hone, hHolder, hymax, hskχ, hFχ, hsplit, hperm] /-- The two-variable inequality left over in case (II): for normalised sizes `p i >= 9314/100000` summing to `1`, `sum_k c_k^(769/269) <= 1`, where `c_k` is `(sqrt (p i) + sqrt (p j))^2 / 2` for the complementary pair `{i,j}` of `k`. This is estimate (4.14) of the paper (the numerical analysis of `max g <= 2^(theta/2)`). -/ def CaseIIBound : Prop := ∀ p : Fin 3 → ℝ, (∀ i, 0 < p i) → (∑ i, p i = 1) → (∀ i, (9314 : ℝ) / 100000 ≤ p i) → (nC p 1 2) ^ ((769 : ℝ) / 269) + (nC p 0 2) ^ ((769 : ℝ) / 269) + (nC p 0 1) ^ ((769 : ℝ) / 269) ≤ 1 /-! ### The sharp numerical evaluation at the corner The maximum in estimate (4.14) is attained at the corner `p = (9314/100000, 45343/100000, 45343/100000)`. The two powers appearing there are bounded by rational certificates; the exponents `769` and `269` are cleared by raising to the `269`-th power, and the resulting (huge) rational inequalities are checked by kernel evaluation (`decide`). -/ /-- Criterion for a numerical bound `x ^ (769/269) ≤ M`: it suffices that `x ≤ q` for a rational `q` with `q ^ 769 ≤ M ^ 269`. Clearing the `269`-th root and the denominators turns the check into a rational comparison. -/ theorem rpow_769_269_le {x q M : ℝ} (hx : 0 ≤ x) (hxq : x ≤ q) (hq : 0 ≤ q) (hM : 0 < M) (hbig : Real.rpow q 769 ≤ Real.rpow M 269) : x ^ ((769 : ℝ) / 269) ≤ M := by have h1 : x ^ ((769 : ℝ) / 269) ≤ q ^ ((769 : ℝ) / 269) := Real.rpow_le_rpow hx hxq (by norm_num) have hq' : q ^ ((769 : ℝ) / 269) = (q ^ (769 : ℝ)) ^ ((1 : ℝ) / 269) := by rw [show (769 : ℝ) / 269 = (769 : ℝ) * (1 / 269) by norm_num, Real.rpow_mul hq] have hM' : M = (M ^ (269 : ℝ)) ^ ((1 : ℝ) / 269) := by conv_rhs => rw [← Real.rpow_mul (le_of_lt hM)] rw [show (269 : ℝ) * (1 / 269) = 1 by norm_num, Real.rpow_one] calc x ^ ((769 : ℝ) / 269) ≤ q ^ ((769 : ℝ) / 269) := h1 _ = (q ^ (769 : ℝ)) ^ ((1 : ℝ) / 269) := hq' _ ≤ (M ^ (269 : ℝ)) ^ ((1 : ℝ) / 269) := by rw [Real.rpow_le_rpow_iff (by positivity) (by positivity) (by norm_num : (0 : ℝ) < 1 / 269)] exact hbig _ = M := hM'.symm /-- The dominant term at the corner: `(45343/50000)^(769/269) ≤ 3781/5000`. -/ theorem corner_T1 : ((45343 : ℝ) / 50000) ^ ((769 : ℝ) / 269) ≤ (3781 : ℝ) / 5000 := by apply rpow_769_269_le (le_of_lt (by norm_num)) (le_refl _) (by norm_num) (by norm_num) have q : ((45343 : ℚ) / 50000) ^ 769 ≤ ((3781 : ℚ) / 5000) ^ 269 := by decide +kernel have h : ((((45343 : ℚ) / 50000) ^ 769 : ℚ) : ℝ) ≤ ((((3781 : ℚ) / 5000) ^ 269 : ℚ) : ℝ) := Rat.cast_le.mpr q push_cast at h rwa [← Real.rpow_natCast, ← Real.rpow_natCast] at h /-- The irrational base at the corner is below `1197/2500`. -/ theorem corner_sqrt_bound : (54657 : ℝ) / 200000 + Real.sqrt ((211162351 : ℝ) / 5000000000) ≤ (1197 : ℝ) / 2500 := by have h : Real.sqrt ((211162351 : ℝ) / 5000000000) ≤ (41103 : ℝ) / 200000 := Real.sqrt_le_iff.2 ⟨by norm_num, by norm_num⟩ linarith /-- The two equal terms at the corner: `(54657/200000 + sqrt (211162351/5000000000))^(769/269) ≤ 609/5000`. -/ theorem corner_T2 : ((54657 : ℝ) / 200000 + Real.sqrt ((211162351 : ℝ) / 5000000000)) ^ ((769 : ℝ) / 269) ≤ (609 : ℝ) / 5000 := by apply rpow_769_269_le (by positivity) corner_sqrt_bound (by norm_num) (by norm_num) have q : ((1197 : ℚ) / 2500) ^ 769 ≤ ((609 : ℚ) / 5000) ^ 269 := by decide +kernel have h : ((((1197 : ℚ) / 2500) ^ 769 : ℚ) : ℝ) ≤ ((((609 : ℚ) / 5000) ^ 269 : ℚ) : ℝ) := Rat.cast_le.mpr q push_cast at h rwa [← Real.rpow_natCast, ← Real.rpow_natCast] at h /-- The corner value of the normalised sum is `< 1`. -/ theorem cornerBound : ((45343 : ℝ) / 50000) ^ ((769 : ℝ) / 269) + 2 * (((54657 : ℝ) / 200000 + Real.sqrt ((211162351 : ℝ) / 5000000000)) ^ ((769 : ℝ) / 269)) ≤ 1 := by have h1 := corner_T1 have h2 := corner_T2 linarith theorem holder_three_le {p q a1 a2 a3 b1 b2 b3 : ℝ} (hpq : p.HolderConjugate q) (ha1 : 0 ≤ a1) (ha2 : 0 ≤ a2) (ha3 : 0 ≤ a3) (hb1 : 0 ≤ b1) (hb2 : 0 ≤ b2) (hb3 : 0 ≤ b3) (hsum : 1 ≤ a1 * b1 + a2 * b2 + a3 * b3) (hB : 0 < b1 ^ q + b2 ^ q + b3 ^ q) : 1 / (b1 ^ q + b2 ^ q + b3 ^ q) ^ (p / q) ≤ a1 ^ p + a2 ^ p + a3 ^ p := by have h := holder_sum_le (n := 3) hpq (fun t : Fin 3 => if t = 0 then a1 else if t = 1 then a2 else a3) (fun t : Fin 3 => if t = 0 then b1 else if t = 1 then b2 else b3) (by intro t fin_cases t · simpa using ha1 · simpa using ha2 · simpa using ha3) (by intro t fin_cases t · simpa using hb1 · simpa using hb2 · simpa using hb3) (by simpa [Fin.sum_univ_three] using hsum) (by simpa [Fin.sum_univ_three] using hB) simpa [Fin.sum_univ_three] using h /-- Case (II) of the numeric core: all three normalised sizes are at least `9314/100000`. -/ theorem caseII_aux {s p : Fin 3 → ℝ} {e : Fin 3 → Fin 3 → ℝ} (hp0 : ∀ i, 0 < p i) (hp1 : ∑ i, p i = 1) (he0 : ∀ i j, 0 ≤ e i j) (he1 : ∑ i, ∑ j, e i j = 1) (hs0 : ∀ i, 0 ≤ s i) (hH : ∀ i j k : Fin 3, i ≠ j → k ≠ i → k ≠ j → s k ≥ (nA e i j) ^ 2 / (nS p i j) ^ 2) (hQ : ∀ i j k : Fin 3, i ≠ j → k ≠ i → k ≠ j → s k ≥ (e i j + e j i) ^ 2 / (4 * p i * p j)) (hF : ∀ i j k : Fin 3, i ≠ j → k ≠ i → k ≠ j → s i + s j ≥ (nN e i j k) ^ 2 / (4 * p k)) (ht : ∀ i, (9314 : ℝ) / 100000 ≤ p i) (hII : CaseIIBound) : 1 ≤ ∑ t, (s t) ^ ((769 : ℝ) / 1000) := by have h01 : (0 : Fin 3) ≠ 1 := by decide have h02 : (0 : Fin 3) ≠ 2 := by decide have h12 : (1 : Fin 3) ≠ 2 := by decide -- the three pair estimates have h0 : s 0 ≥ (max (nY p e 1 2) (nX p e 1 2)) ^ 2 := by have h1 : s 0 ≥ (nY p e 1 2) ^ 2 := by rw [nY_sq]; exact hH 1 2 0 h12 (by decide) (by decide) have h2 : s 0 ≥ (nX p e 1 2) ^ 2 := by rw [nX_sq hp0]; exact hQ 1 2 0 h12 (by decide) (by decide) rw [sq_max_sq (nY_nonneg hp0 he0 1 2) (nX_nonneg hp0 he0 1 2)] exact max_le h1 h2 have h1 : s 1 ≥ (max (nY p e 0 2) (nX p e 0 2)) ^ 2 := by have h1' : s 1 ≥ (nY p e 0 2) ^ 2 := by rw [nY_sq]; exact hH 0 2 1 h02 (by decide) (by decide) have h2' : s 1 ≥ (nX p e 0 2) ^ 2 := by rw [nX_sq hp0]; exact hQ 0 2 1 h02 (by decide) (by decide) rw [sq_max_sq (nY_nonneg hp0 he0 0 2) (nX_nonneg hp0 he0 0 2)] exact max_le h1' h2' have h2 : s 2 ≥ (max (nY p e 0 1) (nX p e 0 1)) ^ 2 := by have h1' : s 2 ≥ (nY p e 0 1) ^ 2 := by rw [nY_sq]; exact hH 0 1 2 h01 (by decide) (by decide) have h2' : s 2 ≥ (nX p e 0 1) ^ 2 := by rw [nX_sq hp0]; exact hQ 0 1 2 h01 (by decide) (by decide) rw [sq_max_sq (nY_nonneg hp0 he0 0 1) (nX_nonneg hp0 he0 0 1)] exact max_le h1' h2' -- the linear constraint sum c_k m_k >= 1 have hcon : 1 ≤ nC p 1 2 * max (nY p e 1 2) (nX p e 1 2) + nC p 0 2 * max (nY p e 0 2) (nX p e 0 2) + nC p 0 1 * max (nY p e 0 1) (nX p e 0 1) := by have l0 := linear_le_max (α := nS p 1 2) (β := 2 * Real.sqrt (p 1 * p 2)) (y := nY p e 1 2) (x := nX p e 1 2) (nS_pos hp0 1 2).le (by positivity) have l1 := linear_le_max (α := nS p 0 2) (β := 2 * Real.sqrt (p 0 * p 2)) (y := nY p e 0 2) (x := nX p e 0 2) (nS_pos hp0 0 2).le (by positivity) have l2 := linear_le_max (α := nS p 0 1) (β := 2 * Real.sqrt (p 0 * p 1)) (y := nY p e 0 1) (x := nX p e 0 1) (nS_pos hp0 0 1).le (by positivity) have c0 : 2 * nC p 1 2 = nS p 1 2 + 2 * Real.sqrt (p 1 * p 2) := by rw [nC_eq hp0]; ring have c1 : 2 * nC p 0 2 = nS p 0 2 + 2 * Real.sqrt (p 0 * p 2) := by rw [nC_eq hp0]; ring have c2 : 2 * nC p 0 1 = nS p 0 1 + 2 * Real.sqrt (p 0 * p 1) := by rw [nC_eq hp0]; ring have a0 := nS_mul_nY p e 1 2 (nS_pos hp0 1 2).ne' have a1 := nS_mul_nY p e 0 2 (nS_pos hp0 0 2).ne' have a2 := nS_mul_nY p e 0 1 (nS_pos hp0 0 1).ne' have b0 := nX_mul hp0 e 1 2 have b1 := nX_mul hp0 e 0 2 have b2 := nX_mul hp0 e 0 1 have htot : (nA e 1 2 + (e 1 2 + e 2 1)) + (nA e 0 2 + (e 0 2 + e 2 0)) + (nA e 0 1 + (e 0 1 + e 1 0)) = 2 := by unfold nA rw [Fin.sum_univ_three] at he1 simp only [Fin.sum_univ_three] at he1 linarith have l0' : nA e 1 2 + (e 1 2 + e 2 1) ≤ 2 * nC p 1 2 * max (nY p e 1 2) (nX p e 1 2) := by have h := l0 rw [a0, b0, ← c0] at h exact h have l1' : nA e 0 2 + (e 0 2 + e 2 0) ≤ 2 * nC p 0 2 * max (nY p e 0 2) (nX p e 0 2) := by have h := l1 rw [a1, b1, ← c1] at h exact h have l2' : nA e 0 1 + (e 0 1 + e 1 0) ≤ 2 * nC p 0 1 * max (nY p e 0 1) (nX p e 0 1) := by have h := l2 rw [a2, b2, ← c2] at h exact h linarith [l0', l1', l2', htot] -- the three-term Hölder estimate have hbase : (nC p 1 2) ^ ((769 : ℝ) / 269) + (nC p 0 2) ^ ((769 : ℝ) / 269) + (nC p 0 1) ^ ((769 : ℝ) / 269) ≤ 1 := hII p hp0 hp1 ht have hbase_pos : 0 < (nC p 1 2) ^ ((769 : ℝ) / 269) + (nC p 0 2) ^ ((769 : ℝ) / 269) + (nC p 0 1) ^ ((769 : ℝ) / 269) := by have p1 := Real.rpow_pos_of_pos (nC_pos hp0 1 2) ((769 : ℝ) / 269) have p2 := Real.rpow_pos_of_pos (nC_pos hp0 0 2) ((769 : ℝ) / 269) have p3 := Real.rpow_pos_of_pos (nC_pos hp0 0 1) ((769 : ℝ) / 269) linarith have hpqt : ((769 : ℝ) / 500).HolderConjugate ((769 : ℝ) / 269) := by rw [Real.holderConjugate_iff] constructor <;> norm_num have hHölder : ((nC p 1 2) ^ ((769 : ℝ) / 269) + (nC p 0 2) ^ ((769 : ℝ) / 269) + (nC p 0 1) ^ ((769 : ℝ) / 269)) ^ (1 - 2 * ((769 : ℝ) / 1000)) ≤ (max (nY p e 1 2) (nX p e 1 2)) ^ (2 * ((769 : ℝ) / 1000)) + (max (nY p e 0 2) (nX p e 0 2)) ^ (2 * ((769 : ℝ) / 1000)) + (max (nY p e 0 1) (nX p e 0 1)) ^ (2 * ((769 : ℝ) / 1000)) := by have h := holder_three_le hpqt (le_trans (nX_nonneg hp0 he0 1 2) (le_max_right _ _)) (le_trans (nX_nonneg hp0 he0 0 2) (le_max_right _ _)) (le_trans (nX_nonneg hp0 he0 0 1) (le_max_right _ _)) (nC_nonneg hp0 1 2) (nC_nonneg hp0 0 2) (nC_nonneg hp0 0 1) (by simpa [mul_comm] using hcon) hbase_pos have hneg : 1 - 2 * ((769 : ℝ) / 1000) = -(((769 : ℝ) / 500) / ((769 : ℝ) / 269)) := by norm_num rw [hneg, Real.rpow_neg (le_of_lt hbase_pos)] have hexp : 2 * ((769 : ℝ) / 1000) = (769 : ℝ) / 500 := by norm_num rw [hexp] simpa [one_div] using h have hone : (1 : ℝ) ≤ ((nC p 1 2) ^ ((769 : ℝ) / 269) + (nC p 0 2) ^ ((769 : ℝ) / 269) + (nC p 0 1) ^ ((769 : ℝ) / 269)) ^ (1 - 2 * ((769 : ℝ) / 1000)) := by rw [show 1 - 2 * ((769 : ℝ) / 1000) = -(2 * ((769 : ℝ) / 1000) - 1) by ring, Real.rpow_neg (le_of_lt hbase_pos), one_le_inv₀ (by positivity)] exact Real.rpow_le_one hbase_pos.le hbase (by norm_num) -- pass to `s` have e0 : (max (nY p e 1 2) (nX p e 1 2)) ^ (2 * ((769 : ℝ) / 1000)) ≤ (s 0) ^ ((769 : ℝ) / 1000) := by have hr := Real.rpow_le_rpow (by positivity : (0:ℝ) ≤ (max (nY p e 1 2) (nX p e 1 2)) ^ 2) h0 (by norm_num : (0:ℝ) ≤ (769:ℝ)/1000) rwa [sq_rpow (χ := (769:ℝ)/1000) (le_trans (nX_nonneg hp0 he0 1 2) (le_max_right _ _))] at hr have e1 : (max (nY p e 0 2) (nX p e 0 2)) ^ (2 * ((769 : ℝ) / 1000)) ≤ (s 1) ^ ((769 : ℝ) / 1000) := by have hr := Real.rpow_le_rpow (by positivity : (0:ℝ) ≤ (max (nY p e 0 2) (nX p e 0 2)) ^ 2) h1 (by norm_num : (0:ℝ) ≤ (769:ℝ)/1000) rwa [sq_rpow (χ := (769:ℝ)/1000) (le_trans (nX_nonneg hp0 he0 0 2) (le_max_right _ _))] at hr have e2 : (max (nY p e 0 1) (nX p e 0 1)) ^ (2 * ((769 : ℝ) / 1000)) ≤ (s 2) ^ ((769 : ℝ) / 1000) := by have hr := Real.rpow_le_rpow (by positivity : (0:ℝ) ≤ (max (nY p e 0 1) (nX p e 0 1)) ^ 2) h2 (by norm_num : (0:ℝ) ≤ (769:ℝ)/1000) rwa [sq_rpow (χ := (769:ℝ)/1000) (le_trans (nX_nonneg hp0 he0 0 1) (le_max_right _ _))] at hr rw [Fin.sum_univ_three] linarith [hone, hHölder, e0, e1, e2] /-- The numeric core of the main case, conditional on the sharp two-variable bound `CaseIIBound`. This is the formal counterpart of estimates (4.6)--(4.14) of the paper. -/ theorem mainNumeric_0_769 (hII : CaseIIBound) : MainNumeric ((769 : ℝ) / 1000) := by intro s a b e ha0 ha1 hb0 hb1 he0 he1 hs0 hH hQ1 hQ2 hF1 hF2 have hH' : ∀ i j k : Fin 3, i ≠ j → k ≠ i → k ≠ j → s k ≥ (nA e i j) ^ 2 / (nS (avg a b) i j) ^ 2 := by intro i j k hij hki hkj have h1 := hH i j k hij hki hkj have hpos : 0 < (a i + a j) * (b i + b j) := mul_pos (by linarith [ha0 i, ha0 j]) (by linarith [hb0 i, hb0 j]) have heq : nS (avg a b) i j = (a i + b i + a j + b j) / 2 := by unfold nS avg ring have hle : (a i + a j) * (b i + b j) ≤ (nS (avg a b) i j) ^ 2 := by rw [heq] exact four_mul_le_sq (a i) (b i) (a j) (b j) calc s k ≥ (nA e i j) ^ 2 / ((a i + a j) * (b i + b j)) := h1 _ ≥ (nA e i j) ^ 2 / (nS (avg a b) i j) ^ 2 := div_le_div_of_nonneg_left (sq_nonneg (nA e i j)) hpos hle have hQ' : ∀ i j k : Fin 3, i ≠ j → k ≠ i → k ≠ j → s k ≥ (e i j + e j i) ^ 2 / (4 * avg a b i * avg a b j) := by intro i j k hij hki hkj have h1 := hQ1 i j k hij hki hkj have h2 := hQ2 i j k hij hki hkj have hab : 0 < a i * b j := mul_pos (ha0 i) (hb0 j) have hcd : 0 < a j * b i := mul_pos (ha0 j) (hb0 i) have hmax := sq_div_le_max_sq_div (x := e i j) (y := e j i) (c := a i * b j) (d := a j * b i) (he0 i j) (he0 j i) hab hcd have hβ : (Real.sqrt (a i * b j) + Real.sqrt (a j * b i)) ^ 2 ≤ 4 * avg a b i * avg a b j := by have hs := sqrt_mul_add_sqrt_mul_le (a := a i) (b := b j) (c := b i) (d := a j) (ha0 i).le (hb0 j).le (hb0 i).le (ha0 j).le have hs' : Real.sqrt (a i * b j) + Real.sqrt (a j * b i) ≤ Real.sqrt ((a i + b i) * (a j + b j)) := by simpa [add_comm, mul_comm] using hs have hsq : (Real.sqrt (a i * b j) + Real.sqrt (a j * b i)) ^ 2 ≤ (a i + b i) * (a j + b j) := by have h := Real.sq_sqrt (mul_nonneg (by linarith [ha0 i, hb0 i]) (by linarith [ha0 j, hb0 j]) : (0:ℝ) ≤ (a i + b i) * (a j + b j)) rw [← h] exact pow_le_pow_left₀ (by positivity) hs' 2 unfold avg nlinarith [hsq] have hstep : (e i j + e j i) ^ 2 / (4 * avg a b i * avg a b j) ≤ s k := by refine le_trans ?_ (max_le h1 h2) calc (e i j + e j i) ^ 2 / (4 * avg a b i * avg a b j) ≤ (e i j + e j i) ^ 2 / ((Real.sqrt (a i * b j) + Real.sqrt (a j * b i)) ^ 2) := div_le_div_of_nonneg_left (by positivity) (by positivity) hβ _ ≤ max ((e i j) ^ 2 / (a i * b j)) ((e j i) ^ 2 / (a j * b i)) := hmax exact hstep have hF' : ∀ i j k : Fin 3, i ≠ j → k ≠ i → k ≠ j → s i + s j ≥ (nN e i j k) ^ 2 / (4 * avg a b k) := by intro i j k hij hki hkj have h1 := hF1 i j k hij hki hkj have h2 := hF2 i j k hij hki hkj have hrow : 0 ≤ e k 0 + e k 1 + e k 2 := by linarith [he0 k 0, he0 k 1, he0 k 2] have hcol : 0 ≤ e 0 k + e 1 k + e 2 k := by linarith [he0 0 k, he0 1 k, he0 2 k] have hmax := sq_div_le_max_sq_div (x := e k 0 + e k 1 + e k 2) (y := e 0 k + e 1 k + e 2 k) (c := a k) (d := b k) hrow hcol (ha0 k) (hb0 k) have hβ : (Real.sqrt (a k) + Real.sqrt (b k)) ^ 2 ≤ 4 * avg a b k := by have h := sq_sqrt_add_le (ha0 k).le (hb0 k).le unfold avg nlinarith [h] have hstep : ((e k 0 + e k 1 + e k 2) + (e 0 k + e 1 k + e 2 k)) ^ 2 / (4 * avg a b k) ≤ s i + s j := by refine le_trans ?_ (max_le h1 h2) calc ((e k 0 + e k 1 + e k 2) + (e 0 k + e 1 k + e 2 k)) ^ 2 / (4 * avg a b k) ≤ ((e k 0 + e k 1 + e k 2) + (e 0 k + e 1 k + e 2 k)) ^ 2 / ((Real.sqrt (a k) + Real.sqrt (b k)) ^ 2) := div_le_div_of_nonneg_left (sq_nonneg _) (pow_pos (by have := Real.sqrt_pos_of_pos (ha0 k); have := Real.sqrt_pos_of_pos (hb0 k); linarith) 2) hβ _ ≤ max ((e k 0 + e k 1 + e k 2) ^ 2 / a k) ((e 0 k + e 1 k + e 2 k) ^ 2 / b k) := hmax rw [nN_eq_rows e hij (Ne.symm hki) (Ne.symm hkj)] exact hstep by_cases hcase : ∃ k : Fin 3, avg a b k ≤ 9314 / 100000 · obtain ⟨k, hk⟩ := hcase obtain ⟨i, j, hij, hik, hjk⟩ := exists_pair_ne k exact caseI_aux (avg_pos ha0 hb0) (avg_sum ha1 hb1) he0 he1 hs0 hH' hQ' hF' hij hik hjk hk · push Not at hcase exact caseII_aux (avg_pos ha0 hb0) (avg_sum ha1 hb1) he0 he1 hs0 hH' hQ' hF' (fun i => le_of_lt (hcase i)) hII /-- Theorem 3, conditional on the explicit two-variable inequality `CaseIIBound`. -/ theorem theorem3_of_caseII (hII : CaseIIBound) : IsMonotoneGraphExponent 3 ((769 : ℝ) / 1000) := theorem3_of_main_0_769 (mainCase_of_numeric (mainNumeric_0_769 hII)) noncomputable section /-! ### Section VII: the edge maximum of the normalised sum We formalise the one-variable analysis of the boundary `z = t_chi` of the domain `{p_i >= t_chi}`. With `u = sqrt x`, `v = sqrt y`, `w = sqrt t`, `x + y = 1 - t` and `theta = 1538/269`, the boundary estimate reduces to `EdgePsi : phi x <= phi (thrS - x)` on `(t_chi, thrS / 2)`, where `phi x = sqrt x * ((sqrt x + sqrt y)^(theta-1) + (sqrt y + sqrt t)^(theta-1))`. `EdgePsi` is proved below from the pointwise Hermite--Hadamard estimate `edgeS` and the trapezoid inequality `rpow_sec_le_trap`. -/ def thrT : ℝ := 9314 / 100000 def thrS : ℝ := 90686 / 100000 def thrTheta : ℝ := 1538 / 269 lemma thrT_pos : 0 < thrT := by unfold thrT; norm_num lemma thrS_eq : thrS = 1 - thrT := by unfold thrS thrT; norm_num lemma thrS_pos : 0 < thrS := by unfold thrS; norm_num /-- The auxiliary one-variable function entering the edge inequality `EdgePsi`. -/ def phi (x : ℝ) : ℝ := Real.sqrt x * ((Real.sqrt x + Real.sqrt (thrS - x)) ^ (thrTheta - 1) + (Real.sqrt (thrS - x) + Real.sqrt thrT) ^ (thrTheta - 1)) def edgeG (x : ℝ) : ℝ := (Real.sqrt x + Real.sqrt (thrS - x)) ^ thrTheta + (Real.sqrt x + Real.sqrt thrT) ^ thrTheta + (Real.sqrt (thrS - x) + Real.sqrt thrT) ^ thrTheta lemma phi_eS_sub (x : ℝ) : phi (thrS - x) = Real.sqrt (thrS - x) * ((Real.sqrt x + Real.sqrt (thrS - x)) ^ (thrTheta - 1) + (Real.sqrt x + Real.sqrt thrT) ^ (thrTheta - 1)) := by unfold phi rw [show thrS - (thrS - x) = x by ring, add_comm (Real.sqrt (thrS - x)) (Real.sqrt x)] def edgeGde (x : ℝ) : ℝ := thrTheta * (phi (thrS - x) - phi x) / (2 * Real.sqrt x * Real.sqrt (thrS - x)) theorem hasDerivAt_gOne {x : ℝ} (hx0 : 0 < x) (hy0 : 0 < thrS - x) : HasDerivAt edgeG (edgeGde x) x := by have hs1 : HasDerivAt (fun x : ℝ => Real.sqrt x) (1 / (2 * Real.sqrt x)) x := Real.hasDerivAt_sqrt hx0.ne' have hs2 : HasDerivAt (fun x : ℝ => Real.sqrt (thrS - x)) (-(1 / (2 * Real.sqrt (thrS - x)))) x := by have hinner : HasDerivAt (fun x : ℝ => thrS - x) (-1) x := by have h := (hasDerivAt_const (x := x) thrS).sub (hasDerivAt_id (x := x) : HasDerivAt (fun y : ℝ => y) 1 x) convert h using 1 · ext y; simp · norm_num have hout : HasDerivAt Real.sqrt (1 / (2 * Real.sqrt (thrS - x))) (thrS - x) := Real.hasDerivAt_sqrt hy0.ne' have := hout.comp x hinner simpa [Function.comp_def] using this have hA1 : (0:ℝ) < Real.sqrt x + Real.sqrt (thrS - x) := by have h2 := Real.sqrt_pos_of_pos hy0; have h3 := Real.sqrt_pos_of_pos hx0; linarith have hB1 : (0:ℝ) < Real.sqrt x + Real.sqrt thrT := by have h2 := Real.sqrt_pos_of_pos hx0; have h3 := Real.sqrt_pos_of_pos thrT_pos; linarith have hC1 : (0:ℝ) < Real.sqrt (thrS - x) + Real.sqrt thrT := by have h2 := Real.sqrt_pos_of_pos hy0; have h3 := Real.sqrt_pos_of_pos thrT_pos; linarith have hA : HasDerivAt (fun x : ℝ => Real.sqrt x + Real.sqrt (thrS - x)) (1 / (2 * Real.sqrt x) + -(1 / (2 * Real.sqrt (thrS - x)))) x := hs1.add hs2 have hB : HasDerivAt (fun x : ℝ => Real.sqrt x + Real.sqrt thrT) (1 / (2 * Real.sqrt x) + 0) x := hs1.add (hasDerivAt_const (x := x) (Real.sqrt thrT)) have hC : HasDerivAt (fun x : ℝ => Real.sqrt (thrS - x) + Real.sqrt thrT) (-(1 / (2 * Real.sqrt (thrS - x))) + 0) x := hs2.add (hasDerivAt_const (x := x) (Real.sqrt thrT)) have hAp : HasDerivAt (fun x : ℝ => (Real.sqrt x + Real.sqrt (thrS - x)) ^ thrTheta) (thrTheta * (Real.sqrt x + Real.sqrt (thrS - x)) ^ (thrTheta - 1) * (1 / (2 * Real.sqrt x) + -(1 / (2 * Real.sqrt (thrS - x))))) x := by have h := (Real.hasDerivAt_rpow_const (x := Real.sqrt x + Real.sqrt (thrS - x)) (p := thrTheta) (Or.inl hA1.ne')).comp x hA simpa [Function.comp_def] using h have hBp : HasDerivAt (fun x : ℝ => (Real.sqrt x + Real.sqrt thrT) ^ thrTheta) (thrTheta * (Real.sqrt x + Real.sqrt thrT) ^ (thrTheta - 1) * (1 / (2 * Real.sqrt x) + 0)) x := by have h := (Real.hasDerivAt_rpow_const (x := Real.sqrt x + Real.sqrt thrT) (p := thrTheta) (Or.inl hB1.ne')).comp x hB simpa [Function.comp_def] using h have hCp : HasDerivAt (fun x : ℝ => (Real.sqrt (thrS - x) + Real.sqrt thrT) ^ thrTheta) (thrTheta * (Real.sqrt (thrS - x) + Real.sqrt thrT) ^ (thrTheta - 1) * (-(1 / (2 * Real.sqrt (thrS - x))) + 0)) x := by have h := (Real.hasDerivAt_rpow_const (x := Real.sqrt (thrS - x) + Real.sqrt thrT) (p := thrTheta) (Or.inl hC1.ne')).comp x hC simpa [Function.comp_def] using h have hsum := (hAp.add hBp).add hCp rw [show edgeG = (((fun x : ℝ => (Real.sqrt x + Real.sqrt (thrS - x)) ^ thrTheta) + fun x => (Real.sqrt x + Real.sqrt thrT) ^ thrTheta) + fun x => (Real.sqrt (thrS - x) + Real.sqrt thrT) ^ thrTheta) by ext y; simp [edgeG]] convert hsum using 1 unfold edgeGde rw [phi_eS_sub] unfold phi have hx : Real.sqrt x ≠ 0 := (Real.sqrt_pos_of_pos hx0).ne' have hy : Real.sqrt (thrS - x) ≠ 0 := (Real.sqrt_pos_of_pos hy0).ne' field_simp ring /-- The remaining analytic inequality of the edge analysis: `phi (s - x) >= phi x`. -/ def EdgePsi : Prop := ∀ x : ℝ, thrT < x → x < thrS / 2 → phi x ≤ phi (thrS - x) theorem gOne_monoOn_of_edgePsi (hpsi : EdgePsi) : MonotoneOn edgeG (Set.Icc thrT (thrS / 2)) := by refine monotoneOn_of_deriv_nonneg (convex_Icc _ _) ?_ ?_ ?_ · intro x hx have hx0 : 0 < x := lt_of_lt_of_le thrT_pos hx.1 have hy0 : 0 < thrS - x := by linarith [hx.2, thrS_pos] exact (hasDerivAt_gOne hx0 hy0).continuousAt.continuousWithinAt · intro x hx have hxI := interior_subset hx have hx0 : 0 < x := lt_of_lt_of_le thrT_pos hxI.1 have hy0 : 0 < thrS - x := by linarith [hxI.2, thrS_pos] exact (hasDerivAt_gOne hx0 hy0).differentiableAt.differentiableWithinAt · intro x hx rw [interior_Icc] at hx have hx0 : 0 < x := lt_trans thrT_pos hx.1 have hy0 : 0 < thrS - x := by linarith [hx.2, thrS_pos] rw [(hasDerivAt_gOne hx0 hy0).deriv] unfold edgeGde have hge : 0 ≤ phi (thrS - x) - phi x := by have := hpsi x hx.1 hx.2; linarith have hth : 0 < thrTheta := by unfold thrTheta; norm_num have hnum : 0 ≤ thrTheta * (phi (thrS - x) - phi x) := mul_nonneg hth.le hge exact div_nonneg hnum (by positivity) theorem edgeBound_of_edgePsi (hpsi : EdgePsi) {x : ℝ} (hx : x ∈ Set.Icc thrT (thrS / 2)) : edgeG x ≤ edgeG (thrS / 2) := by have hmem : thrS / 2 ∈ Set.Icc thrT (thrS / 2) := by refine ⟨?_, le_refl _⟩ unfold thrT thrS; norm_num exact gOne_monoOn_of_edgePsi hpsi hx hmem hx.2 /-- The value of the normalised sum at the corner. -/ def cornerVal3 : ℝ := ((45343 : ℝ) / 50000) ^ ((769 : ℝ) / 269) + 2 * (((54657 : ℝ) / 200000 + Real.sqrt ((211162351 : ℝ) / 5000000000)) ^ ((769 : ℝ) / 269)) /-- Converting the `θ = 1538/269` form to the `(A^2/2)^(769/269)` form. -/ theorem sq_half_rpow {A : ℝ} (hA : 0 ≤ A) : (A ^ 2 / 2) ^ ((769 : ℝ) / 269) = (2 : ℝ) ^ (-((769 : ℝ) / 269)) * A ^ (1538 / 269 : ℝ) := by have h2 : (0 : ℝ) < 2 := by norm_num have hA2 : A ^ 2 = A ^ (2 : ℝ) := (Real.rpow_natCast A 2).symm have h3 : (A ^ 2) ^ ((769 : ℝ) / 269) = A ^ (1538 / 269 : ℝ) := by rw [hA2, ← Real.rpow_mul hA] norm_num rw [Real.div_rpow (sq_nonneg A) h2.le, h3, Real.rpow_neg h2.le, div_eq_inv_mul] /-- The normalised sum on the edge equals `2^(-q)` times the `θ`-power form. -/ theorem edge_sum_eq {x : ℝ} (hx0 : 0 ≤ x) (hy0 : 0 ≤ thrS - x) : ((Real.sqrt x + Real.sqrt (thrS - x)) ^ 2 / 2) ^ ((769 : ℝ) / 269) + ((Real.sqrt x + Real.sqrt thrT) ^ 2 / 2) ^ ((769 : ℝ) / 269) + ((Real.sqrt (thrS - x) + Real.sqrt thrT) ^ 2 / 2) ^ ((769 : ℝ) / 269) = (2 : ℝ) ^ (-((769 : ℝ) / 269)) * edgeG x := by rw [sq_half_rpow (by positivity), sq_half_rpow (by positivity), sq_half_rpow (by positivity)] unfold edgeG thrTheta ring /-- The value of the edge function at the midpoint, in normalised form. -/ theorem edgeG_mid_val : (2 : ℝ) ^ (-((769 : ℝ) / 269)) * edgeG (thrS / 2) = cornerVal3 := by have hs2 : (0 : ℝ) < thrS / 2 := by unfold thrS; norm_num have hA2 : (2 * Real.sqrt (thrS / 2)) ^ 2 / 2 = thrS := by have h : Real.sqrt (thrS / 2) ^ 2 = thrS / 2 := Real.sq_sqrt hs2.le nlinarith have hB2 : (Real.sqrt (thrS / 2) + Real.sqrt thrT) ^ 2 / 2 = (54657 : ℝ) / 200000 + Real.sqrt ((211162351 : ℝ) / 5000000000) := by have h1 : Real.sqrt (thrS / 2) ^ 2 = thrS / 2 := Real.sq_sqrt hs2.le have h2 : Real.sqrt thrT ^ 2 = thrT := Real.sq_sqrt thrT_pos.le have h3 : Real.sqrt (thrS / 2) * Real.sqrt thrT = Real.sqrt ((thrS / 2) * thrT) := (Real.sqrt_mul hs2.le thrT).symm have h4 : (thrS / 2) * thrT = (211162351 : ℝ) / 5000000000 := by unfold thrS thrT; norm_num rw [h4] at h3 have h5 : (Real.sqrt (thrS / 2) + Real.sqrt thrT) ^ 2 = thrS / 2 + thrT + 2 * Real.sqrt ((211162351 : ℝ) / 5000000000) := by nlinarith [h1, h2, h3] rw [h5] have hsum : thrS / 2 + thrT = (54657 : ℝ) / 100000 := by unfold thrS thrT; norm_num rw [hsum] ring have hfirst : (2 : ℝ) ^ (-((769 : ℝ) / 269)) * (2 * Real.sqrt (thrS / 2)) ^ thrTheta = ((45343 : ℝ) / 50000) ^ ((769 : ℝ) / 269) := by unfold thrTheta rw [← sq_half_rpow (by positivity : (0:ℝ) ≤ 2 * Real.sqrt (thrS / 2)), hA2] have : thrS = (45343 : ℝ) / 50000 := by unfold thrS; norm_num rw [this] have hsecond : (2 : ℝ) ^ (-((769 : ℝ) / 269)) * (Real.sqrt (thrS / 2) + Real.sqrt thrT) ^ thrTheta = ((54657 : ℝ) / 200000 + Real.sqrt ((211162351 : ℝ) / 5000000000)) ^ ((769 : ℝ) / 269) := by unfold thrTheta rw [← sq_half_rpow (by positivity), hB2] unfold edgeG cornerVal3 rw [show thrS - thrS / 2 = thrS / 2 by ring] rw [show Real.sqrt (thrS / 2) + Real.sqrt (thrS / 2) = 2 * Real.sqrt (thrS / 2) by ring] linear_combination hfirst + 2 * hsecond theorem edge_one_dim (hpsi : EdgePsi) {x y : ℝ} (hx : thrT ≤ x) (hy : thrT ≤ y) (hxy : x + y = thrS) : ((Real.sqrt x + Real.sqrt y) ^ 2 / 2) ^ ((769 : ℝ) / 269) + ((Real.sqrt x + Real.sqrt thrT) ^ 2 / 2) ^ ((769 : ℝ) / 269) + ((Real.sqrt y + Real.sqrt thrT) ^ 2 / 2) ^ ((769 : ℝ) / 269) ≤ cornerVal3 := by have hq : (0:ℝ) < (2:ℝ) ^ (-((769 : ℝ) / 269)) := Real.rpow_pos_of_pos (by norm_num) _ have key : ∀ a b : ℝ, thrT ≤ a → a ≤ thrS / 2 → a + b = thrS → ((Real.sqrt a + Real.sqrt b) ^ 2 / 2) ^ ((769 : ℝ) / 269) + ((Real.sqrt a + Real.sqrt thrT) ^ 2 / 2) ^ ((769 : ℝ) / 269) + ((Real.sqrt b + Real.sqrt thrT) ^ 2 / 2) ^ ((769 : ℝ) / 269) ≤ cornerVal3 := by intro a b ha ha2 hab have hb : b = thrS - a := by linarith rw [hb, edge_sum_eq (by linarith [thrT_pos]) (by linarith [thrS_pos]), ← edgeG_mid_val] exact mul_le_mul_of_nonneg_left (edgeBound_of_edgePsi hpsi ⟨ha, ha2⟩) hq.le rcases lt_or_ge x y with hlt | hge · exact key x y hx (by linarith) hxy · have hs : ((Real.sqrt x + Real.sqrt y) ^ 2 / 2) ^ ((769 : ℝ) / 269) + ((Real.sqrt x + Real.sqrt thrT) ^ 2 / 2) ^ ((769 : ℝ) / 269) + ((Real.sqrt y + Real.sqrt thrT) ^ 2 / 2) ^ ((769 : ℝ) / 269) = ((Real.sqrt y + Real.sqrt x) ^ 2 / 2) ^ ((769 : ℝ) / 269) + ((Real.sqrt y + Real.sqrt thrT) ^ 2 / 2) ^ ((769 : ℝ) / 269) + ((Real.sqrt x + Real.sqrt thrT) ^ 2 / 2) ^ ((769 : ℝ) / 269) := by rw [add_comm (Real.sqrt x) (Real.sqrt y)] ring rw [hs] exact key y x hy (by linarith) (by linarith) /-! ### Auxiliary power inequalities for the edge certificate -/ theorem rpow_div_le {z M : ℝ} {m n : ℕ} (hn : 0 < n) (hz : 0 ≤ z) (hM : 0 < M) (h : z ^ m ≤ M ^ n) : z ^ ((m : ℝ) / n) ≤ M := by have hn' : (0:ℝ) < (n:ℝ) := by exact_mod_cast hn have h1 : z ^ ((m : ℝ) / n) = (z ^ (m:ℝ)) ^ ((1:ℝ)/n) := by rw [show ((m:ℝ)/n) = (m:ℝ) * (1/(n:ℝ)) by field_simp, Real.rpow_mul hz] have h2 : M = (M ^ (n:ℝ)) ^ ((1:ℝ)/n) := by rw [← Real.rpow_mul (le_of_lt hM)] rw [show (n:ℝ) * (1/(n:ℝ)) = 1 by field_simp, Real.rpow_one] rw [h1, h2] refine Real.rpow_le_rpow ?_ ?_ (by positivity) · rw [Real.rpow_natCast]; positivity · rwa [Real.rpow_natCast, Real.rpow_natCast] theorem le_rpow_div {z M : ℝ} {m n : ℕ} (hn : 0 < n) (hz : 0 < z) (hM : 0 ≤ M) (h : M ^ n ≤ z ^ m) : M ≤ z ^ ((m : ℝ) / n) := by have hn' : (0:ℝ) < (n:ℝ) := by exact_mod_cast hn have h1 : M = (M ^ (n:ℝ)) ^ ((1:ℝ)/n) := by rw [← Real.rpow_mul hM] rw [show (n:ℝ) * (1/(n:ℝ)) = 1 by field_simp, Real.rpow_one] have h2 : z ^ ((m : ℝ) / n) = (z ^ (m:ℝ)) ^ ((1:ℝ)/n) := by rw [show ((m:ℝ)/n) = (m:ℝ) * (1/(n:ℝ)) by field_simp, Real.rpow_mul hz.le] rw [h1, h2] refine Real.rpow_le_rpow ?_ ?_ (by positivity) · rw [Real.rpow_natCast]; positivity · rwa [Real.rpow_natCast, Real.rpow_natCast] /-- Tangent inequality at `u` for `z ↦ z ^ p`, evaluated at `1`. -/ theorem rpow_sub_one_le {u p : ℝ} (hu : 1 < u) (hp : 1 ≤ p) : u ^ p - 1 ≤ p * (u - 1) * u ^ (p - 1) := by have hu0 : (0:ℝ) < u := by linarith have hconv : ConvexOn ℝ (Set.Ici (0:ℝ)) (fun z : ℝ => z ^ p) := convexOn_rpow hp have hsl := hconv.slope_le_of_hasDerivAt (x := 1) (y := u) (by norm_num) (Set.mem_Ici.mpr hu0.le) hu (Real.hasDerivAt_rpow_const (p := p) (Or.inl hu0.ne')) rw [slope_def_field, Real.one_rpow] at hsl rw [div_le_iff₀ (by linarith : (0:ℝ) < u - 1)] at hsl linarith /-- Trapezoid inequality for `z ↦ z ^ q` on `[c, b]`, `q ≥ 2`. -/ theorem rpow_sec_le_trap {q : ℝ} (hq : 2 ≤ q) {c b : ℝ} (hc : 0 < c) (hcb : c ≤ b) : b ^ q - c ^ q ≤ q * (b - c) * (b ^ (q - 1) + c ^ (q - 1)) / 2 := by rcases lt_or_eq_of_le hcb with hlt | heq swap · subst heq; simp have hb0 : 0 < b := lt_trans hc hlt set r := b / c with hr have hr0 : 0 < r := by rw [hr]; exact div_pos hb0 hc have hr1 : 1 < r := by rw [hr, lt_div_iff₀ hc]; linarith have hb : b = r * c := by rw [hr]; field_simp have hq1 : (1:ℝ) ≤ q - 1 := by linarith have hq2 : q - 1 - 1 = q - 2 := by ring have key : r ^ q - 1 ≤ q * (r - 1) * (r ^ (q - 1) + 1) / 2 := by set F : ℝ → ℝ := fun t => q / 2 * ((t - 1) * (t ^ (q - 1) + 1)) - (t ^ q - 1) with hF have hF1 : F 1 = 0 := by simp [hF] have hda : ∀ t : ℝ, 0 < t → HasDerivAt F (q / 2 * (1 * (t ^ (q - 1) + 1) + (t - 1) * ((q - 1) * t ^ (q - 2))) - q * t ^ (q - 1)) t := by intro t ht0 rw [hF] have h1 : HasDerivAt (fun t : ℝ => t ^ (q - 1) + 1) ((q - 1) * t ^ (q - 2)) t := by have hh := (Real.hasDerivAt_rpow_const (p := q - 1) (Or.inl ht0.ne')).add_const 1 rw [hq2] at hh simpa using hh have h2 : HasDerivAt (fun t : ℝ => t - 1) 1 t := by simpa using (hasDerivAt_id t).sub_const 1 have h3 := h2.mul h1 have h4 := h3.const_mul (q / 2) have h5 : HasDerivAt (fun t : ℝ => t ^ q - 1) (q * t ^ (q - 1)) t := by simpa using (Real.hasDerivAt_rpow_const (p := q) (Or.inl ht0.ne')).sub_const 1 exact h4.sub h5 have hnonneg : ∀ t ∈ interior (Set.Ici (1:ℝ)), 0 ≤ deriv F t := by intro t ht rw [interior_Ici, Set.mem_Ioi] at ht have ht0 : 0 < t := by linarith rw [(hda t ht0).deriv] have htan := rpow_sub_one_le ht hq1 rw [hq2] at htan have hP : 0 ≤ 1 + (q - 1) * (t - 1) * t ^ (q - 2) - t ^ (q - 1) := by linarith have hq0 : (0:ℝ) ≤ q / 2 := by linarith nlinarith [mul_nonneg hq0 hP, Real.rpow_pos_of_pos ht0 (q - 1)] have hmono : MonotoneOn F (Set.Ici (1:ℝ)) := by refine monotoneOn_of_deriv_nonneg (convex_Ici (1:ℝ)) ?_ ?_ hnonneg · intro t ht rw [Set.mem_Ici] at ht exact (hda t (by linarith)).continuousAt.continuousWithinAt · intro t ht rw [interior_Ici, Set.mem_Ioi] at ht exact (hda t (by linarith)).differentiableAt.differentiableWithinAt have hstep := hmono (Set.mem_Ici.mpr le_rfl) (Set.mem_Ici.mpr hr1.le) hr1.le rw [hF1] at hstep simp only [hF] at hstep linarith have hcq : (0:ℝ) < c ^ q := Real.rpow_pos_of_pos hc q have hcqq : c * c ^ (q - 1) = c ^ q := by nth_rewrite 1 [← Real.rpow_one c] rw [← Real.rpow_add hc] congr 1 ring rw [hb, Real.mul_rpow hr0.le hc.le, Real.mul_rpow hr0.le hc.le] have h1 : r ^ q * c ^ q - c ^ q = c ^ q * (r ^ q - 1) := by ring have h2 : q * (r * c - c) * (r ^ (q - 1) * c ^ (q - 1) + c ^ (q - 1)) / 2 = c ^ q * (q * (r - 1) * (r ^ (q - 1) + 1) / 2) := by linear_combination (q * (r - 1) * (r ^ (q - 1) + 1) / 2) * hcqq rw [h1, h2] exact mul_le_mul_of_nonneg_left key hcq.le /-- `c ≤ z ^ (193/269)` from a rational kernel check `c ^ 269 ≤ z ^ 193`. -/ theorem rpow_frac_le {c z : ℚ} (hc : 0 ≤ c) (hz : 0 < z) (h : c ^ 269 ≤ z ^ 193) : (c : ℝ) ≤ (z : ℝ) ^ ((193 : ℝ) / 269) := by have hh : (c : ℝ) ^ 269 ≤ (z : ℝ) ^ 193 := by exact_mod_cast h exact le_rpow_div (z := (z:ℝ)) (M := (c:ℝ)) (m := 193) (n := 269) (by norm_num) (by exact_mod_cast hz) (by exact_mod_cast hc) hh /-- `z ^ (193/269) ≤ c` from a rational kernel check `z ^ 193 ≤ c ^ 269`. -/ theorem le_rpow_frac {c z : ℚ} (hc : 0 < c) (hz : 0 < z) (h : z ^ 193 ≤ c ^ 269) : (z : ℝ) ^ ((193 : ℝ) / 269) ≤ (c : ℝ) := by have hh : (z : ℝ) ^ 193 ≤ (c : ℝ) ^ 269 := by exact_mod_cast h exact rpow_div_le (z := (z:ℝ)) (M := (c:ℝ)) (m := 193) (n := 269) (by norm_num) (by exact_mod_cast hz.le) (by exact_mod_cast hc) hh lemma p_split : thrTheta - 1 = 4 + ((193 : ℝ) / 269) := by unfold thrTheta; norm_num lemma q_split : thrTheta - 2 = 3 + ((193 : ℝ) / 269) := by unfold thrTheta; norm_num lemma rpow_p_eq {Z : ℝ} (hZ : 0 < Z) : Z ^ (thrTheta - 1) = Z ^ 4 * Z ^ ((193 : ℝ) / 269) := by rw [p_split, Real.rpow_add hZ] norm_num lemma rpow_q_eq {Z : ℝ} (hZ : 0 < Z) : Z ^ (thrTheta - 2) = Z ^ 3 * Z ^ ((193 : ℝ) / 269) := by rw [q_split, Real.rpow_add hZ] norm_num set_option maxHeartbeats 2000000 theorem edgeS_0 {u v : ℝ} (hu : 0 < u) (huv : u ≤ v) (hs : u ^ 2 + v ^ 2 = thrS) (ha : ((61 : ℝ)/200) ≤ u) (hb : u ≤ ((177 : ℝ)/500)) : u * (thrTheta - 1) * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) / 2 ≤ (u + v) ^ (thrTheta - 1) + (u + Real.sqrt thrT) ^ (thrTheta - 1) := by have hv0 : 0 < v := lt_of_lt_of_le hu huv have hth1 : thrTheta - 1 = ((1269:ℝ)/269) := by unfold thrTheta; norm_num have hwlo : ((3051 : ℝ)/10000) ≤ Real.sqrt thrT := by rw [Real.le_sqrt' (by norm_num)]; norm_num [thrT] have hwhi : Real.sqrt thrT ≤ ((763 : ℝ)/2500) := by rw [Real.sqrt_le_iff]; constructor <;> norm_num [thrT] have hv2 : v ^ 2 = thrS - u ^ 2 := by linarith have hvlo : ((221 : ℝ)/250) ≤ v := by have h1 : (((221 : ℝ)/250)) ^ 2 ≤ v ^ 2 := by rw [hv2]; unfold thrS; nlinarith nlinarith [h1, hv0, (by norm_num : (0:ℝ) ≤ ((221 : ℝ)/250))] have hvhi : v ≤ ((4511 : ℝ)/5000) := by have h1 : v ^ 2 ≤ (((4511 : ℝ)/5000)) ^ 2 := by rw [hv2]; unfold thrS; nlinarith nlinarith [h1, hv0, (by norm_num : (0:ℝ) ≤ ((4511 : ℝ)/5000))] have hL1 : (((1189 : ℝ)/1000))^4 * (((283 : ℝ)/250)) ≤ (u + v) ^ (thrTheta - 1) := by rw [rpow_p_eq (by positivity)] have hce : ((283 : ℝ)/250) ≤ (((1189 : ℝ)/1000)) ^ ((193:ℝ)/269) := by simpa using rpow_frac_le (c := ((283 : ℚ)/250)) (z := ((1189 : ℚ)/1000)) (by norm_num) (by norm_num) (by decide +kernel) have hA : ((1189 : ℝ)/1000) ≤ u + v := by linarith have hZ : (((1189 : ℝ)/1000))^4 ≤ (u + v)^4 := pow_le_pow_left₀ (by norm_num) hA 4 have he : (((1189 : ℝ)/1000)) ^ ((193:ℝ)/269) ≤ (u + v) ^ ((193:ℝ)/269) := Real.rpow_le_rpow (by norm_num) hA (by norm_num) exact le_trans (mul_le_mul_of_nonneg_left hce (by positivity)) (mul_le_mul hZ he (by positivity) (by positivity)) have hL2 : (((6101 : ℝ)/10000))^4 * (((701 : ℝ)/1000)) ≤ (u + Real.sqrt thrT) ^ (thrTheta - 1) := by rw [rpow_p_eq (by positivity)] have hce : ((701 : ℝ)/1000) ≤ (((6101 : ℝ)/10000)) ^ ((193:ℝ)/269) := by simpa using rpow_frac_le (c := ((701 : ℚ)/1000)) (z := ((6101 : ℚ)/10000)) (by norm_num) (by norm_num) (by decide +kernel) have hA : ((6101 : ℝ)/10000) ≤ u + Real.sqrt thrT := by linarith have hZ : (((6101 : ℝ)/10000))^4 ≤ (u + Real.sqrt thrT)^4 := pow_le_pow_left₀ (by norm_num) hA 4 have he : (((6101 : ℝ)/10000)) ^ ((193:ℝ)/269) ≤ (u + Real.sqrt thrT) ^ ((193:ℝ)/269) := Real.rpow_le_rpow (by norm_num) hA (by norm_num) exact le_trans (mul_le_mul_of_nonneg_left hce (by positivity)) (mul_le_mul hZ he (by positivity) (by positivity)) have hQ1 : (u + Real.sqrt thrT) ^ (thrTheta - 2) ≤ (((412 : ℝ)/625))^3 * (((371 : ℝ)/500)) := by rw [rpow_q_eq (by positivity)] have hce : (((412 : ℝ)/625)) ^ ((193:ℝ)/269) ≤ ((371 : ℝ)/500) := by simpa using le_rpow_frac (c := ((371 : ℚ)/500)) (z := ((412 : ℚ)/625)) (by norm_num) (by norm_num) (by decide +kernel) have hA : u + Real.sqrt thrT ≤ ((412 : ℝ)/625) := by linarith have hZ : (u + Real.sqrt thrT)^3 ≤ (((412 : ℝ)/625))^3 := pow_le_pow_left₀ (by positivity) hA 3 have he : (u + Real.sqrt thrT) ^ ((193:ℝ)/269) ≤ (((412 : ℝ)/625)) ^ ((193:ℝ)/269) := Real.rpow_le_rpow (by positivity) hA (by norm_num) calc (u + Real.sqrt thrT)^3 * (u + Real.sqrt thrT)^((193:ℝ)/269) ≤ (((412 : ℝ)/625))^3 * (((412 : ℝ)/625))^((193:ℝ)/269) := mul_le_mul hZ he (by positivity) (by positivity) _ ≤ (((412 : ℝ)/625))^3 * (((371 : ℝ)/500)) := mul_le_mul_of_nonneg_left hce (by positivity) have hQ2 : (v + Real.sqrt thrT) ^ (thrTheta - 2) ≤ (((6037 : ℝ)/5000))^3 * (((229 : ℝ)/200)) := by rw [rpow_q_eq (by positivity)] have hce : (((6037 : ℝ)/5000)) ^ ((193:ℝ)/269) ≤ ((229 : ℝ)/200) := by simpa using le_rpow_frac (c := ((229 : ℚ)/200)) (z := ((6037 : ℚ)/5000)) (by norm_num) (by norm_num) (by decide +kernel) have hA : v + Real.sqrt thrT ≤ ((6037 : ℝ)/5000) := by linarith have hZ : (v + Real.sqrt thrT)^3 ≤ (((6037 : ℝ)/5000))^3 := pow_le_pow_left₀ (by positivity) hA 3 have he : (v + Real.sqrt thrT) ^ ((193:ℝ)/269) ≤ (((6037 : ℝ)/5000)) ^ ((193:ℝ)/269) := Real.rpow_le_rpow (by positivity) hA (by norm_num) calc (v + Real.sqrt thrT)^3 * (v + Real.sqrt thrT)^((193:ℝ)/269) ≤ (((6037 : ℝ)/5000))^3 * (((6037 : ℝ)/5000))^((193:ℝ)/269) := mul_le_mul hZ he (by positivity) (by positivity) _ ≤ (((6037 : ℝ)/5000))^3 * (((229 : ℝ)/200)) := mul_le_mul_of_nonneg_left hce (by positivity) have hkey : ((177 : ℝ)/500) * ((1269:ℝ)/269) * ((((412 : ℝ)/625))^3 * (((371 : ℝ)/500)) + (((6037 : ℝ)/5000))^3 * (((229 : ℝ)/200))) / 2 ≤ (((1189 : ℝ)/1000))^4 * (((283 : ℝ)/250)) + (((6101 : ℝ)/10000))^4 * (((701 : ℝ)/1000)) := by norm_num have hs0 : (0:ℝ) ≤ (v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2) := by have h1 : (0:ℝ) ≤ (v + Real.sqrt thrT) ^ (thrTheta - 2) := Real.rpow_nonneg (by linarith) _ have h2 : (0:ℝ) ≤ (u + Real.sqrt thrT) ^ (thrTheta - 2) := Real.rpow_nonneg (by linarith) _ linarith have hZ0 : (0:ℝ) ≤ (((412 : ℝ)/625))^3 * (((371 : ℝ)/500)) + (((6037 : ℝ)/5000))^3 * (((229 : ℝ)/200)) := by positivity have hb0 : (0:ℝ) ≤ ((177 : ℝ)/500) := by norm_num have hsumle : (v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2) ≤ (((412 : ℝ)/625))^3 * (((371 : ℝ)/500)) + (((6037 : ℝ)/5000))^3 * (((229 : ℝ)/200)) := by linarith have hprod : u * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) ≤ ((177 : ℝ)/500) * ((((412 : ℝ)/625))^3 * (((371 : ℝ)/500)) + (((6037 : ℝ)/5000))^3 * (((229 : ℝ)/200))) := mul_le_mul hb hsumle hs0 hb0 have hprod2 := mul_le_mul_of_nonneg_left hprod (by rw [hth1]; norm_num : (0:ℝ) ≤ (thrTheta - 1) / 2) have hfin : u * (thrTheta - 1) * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) / 2 ≤ (((1189 : ℝ)/1000))^4 * (((283 : ℝ)/250)) + (((6101 : ℝ)/10000))^4 * (((701 : ℝ)/1000)) := by calc u * (thrTheta - 1) * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) / 2 = (thrTheta - 1) / 2 * (u * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2))) := by ring _ ≤ (thrTheta - 1) / 2 * (((177 : ℝ)/500) * ((((412 : ℝ)/625))^3 * (((371 : ℝ)/500)) + (((6037 : ℝ)/5000))^3 * (((229 : ℝ)/200)))) := hprod2 _ = ((177 : ℝ)/500) * ((1269:ℝ)/269) * ((((412 : ℝ)/625))^3 * (((371 : ℝ)/500)) + (((6037 : ℝ)/5000))^3 * (((229 : ℝ)/200))) / 2 := by rw [hth1]; ring _ ≤ (((1189 : ℝ)/1000))^4 * (((283 : ℝ)/250)) + (((6101 : ℝ)/10000))^4 * (((701 : ℝ)/1000)) := hkey calc u * (thrTheta - 1) * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) / 2 ≤ (((1189 : ℝ)/1000))^4 * (((283 : ℝ)/250)) + (((6101 : ℝ)/10000))^4 * (((701 : ℝ)/1000)) := hfin _ ≤ (u + v) ^ (thrTheta - 1) + (u + Real.sqrt thrT) ^ (thrTheta - 1) := by linarith [hL1, hL2] theorem edgeS_1 {u v : ℝ} (hu : 0 < u) (huv : u ≤ v) (hs : u ^ 2 + v ^ 2 = thrS) (ha : ((177 : ℝ)/500) ≤ u) (hb : u ≤ ((101 : ℝ)/250)) : u * (thrTheta - 1) * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) / 2 ≤ (u + v) ^ (thrTheta - 1) + (u + Real.sqrt thrT) ^ (thrTheta - 1) := by have hv0 : 0 < v := lt_of_lt_of_le hu huv have hth1 : thrTheta - 1 = ((1269:ℝ)/269) := by unfold thrTheta; norm_num have hwlo : ((3051 : ℝ)/10000) ≤ Real.sqrt thrT := by rw [Real.le_sqrt' (by norm_num)]; norm_num [thrT] have hwhi : Real.sqrt thrT ≤ ((763 : ℝ)/2500) := by rw [Real.sqrt_le_iff]; constructor <;> norm_num [thrT] have hv2 : v ^ 2 = thrS - u ^ 2 := by linarith have hvlo : ((8623 : ℝ)/10000) ≤ v := by have h1 : (((8623 : ℝ)/10000)) ^ 2 ≤ v ^ 2 := by rw [hv2]; unfold thrS; nlinarith nlinarith [h1, hv0, (by norm_num : (0:ℝ) ≤ ((8623 : ℝ)/10000))] have hvhi : v ≤ ((8841 : ℝ)/10000) := by have h1 : v ^ 2 ≤ (((8841 : ℝ)/10000)) ^ 2 := by rw [hv2]; unfold thrS; nlinarith nlinarith [h1, hv0, (by norm_num : (0:ℝ) ≤ ((8841 : ℝ)/10000))] have hL1 : (((12163 : ℝ)/10000))^4 * (((23 : ℝ)/20)) ≤ (u + v) ^ (thrTheta - 1) := by rw [rpow_p_eq (by positivity)] have hce : ((23 : ℝ)/20) ≤ (((12163 : ℝ)/10000)) ^ ((193:ℝ)/269) := by simpa using rpow_frac_le (c := ((23 : ℚ)/20)) (z := ((12163 : ℚ)/10000)) (by norm_num) (by norm_num) (by decide +kernel) have hA : ((12163 : ℝ)/10000) ≤ u + v := by linarith have hZ : (((12163 : ℝ)/10000))^4 ≤ (u + v)^4 := pow_le_pow_left₀ (by norm_num) hA 4 have he : (((12163 : ℝ)/10000)) ^ ((193:ℝ)/269) ≤ (u + v) ^ ((193:ℝ)/269) := Real.rpow_le_rpow (by norm_num) hA (by norm_num) exact le_trans (mul_le_mul_of_nonneg_left hce (by positivity)) (mul_le_mul hZ he (by positivity) (by positivity)) have hL2 : (((6591 : ℝ)/10000))^4 * (((741 : ℝ)/1000)) ≤ (u + Real.sqrt thrT) ^ (thrTheta - 1) := by rw [rpow_p_eq (by positivity)] have hce : ((741 : ℝ)/1000) ≤ (((6591 : ℝ)/10000)) ^ ((193:ℝ)/269) := by simpa using rpow_frac_le (c := ((741 : ℚ)/1000)) (z := ((6591 : ℚ)/10000)) (by norm_num) (by norm_num) (by decide +kernel) have hA : ((6591 : ℝ)/10000) ≤ u + Real.sqrt thrT := by linarith have hZ : (((6591 : ℝ)/10000))^4 ≤ (u + Real.sqrt thrT)^4 := pow_le_pow_left₀ (by norm_num) hA 4 have he : (((6591 : ℝ)/10000)) ^ ((193:ℝ)/269) ≤ (u + Real.sqrt thrT) ^ ((193:ℝ)/269) := Real.rpow_le_rpow (by norm_num) hA (by norm_num) exact le_trans (mul_le_mul_of_nonneg_left hce (by positivity)) (mul_le_mul hZ he (by positivity) (by positivity)) have hQ1 : (u + Real.sqrt thrT) ^ (thrTheta - 2) ≤ (((1773 : ℝ)/2500))^3 * (((391 : ℝ)/500)) := by rw [rpow_q_eq (by positivity)] have hce : (((1773 : ℝ)/2500)) ^ ((193:ℝ)/269) ≤ ((391 : ℝ)/500) := by simpa using le_rpow_frac (c := ((391 : ℚ)/500)) (z := ((1773 : ℚ)/2500)) (by norm_num) (by norm_num) (by decide +kernel) have hA : u + Real.sqrt thrT ≤ ((1773 : ℝ)/2500) := by linarith have hZ : (u + Real.sqrt thrT)^3 ≤ (((1773 : ℝ)/2500))^3 := pow_le_pow_left₀ (by positivity) hA 3 have he : (u + Real.sqrt thrT) ^ ((193:ℝ)/269) ≤ (((1773 : ℝ)/2500)) ^ ((193:ℝ)/269) := Real.rpow_le_rpow (by positivity) hA (by norm_num) calc (u + Real.sqrt thrT)^3 * (u + Real.sqrt thrT)^((193:ℝ)/269) ≤ (((1773 : ℝ)/2500))^3 * (((1773 : ℝ)/2500))^((193:ℝ)/269) := mul_le_mul hZ he (by positivity) (by positivity) _ ≤ (((1773 : ℝ)/2500))^3 * (((391 : ℝ)/500)) := mul_le_mul_of_nonneg_left hce (by positivity) have hQ2 : (v + Real.sqrt thrT) ^ (thrTheta - 2) ≤ (((11893 : ℝ)/10000))^3 * (((1133 : ℝ)/1000)) := by rw [rpow_q_eq (by positivity)] have hce : (((11893 : ℝ)/10000)) ^ ((193:ℝ)/269) ≤ ((1133 : ℝ)/1000) := by simpa using le_rpow_frac (c := ((1133 : ℚ)/1000)) (z := ((11893 : ℚ)/10000)) (by norm_num) (by norm_num) (by decide +kernel) have hA : v + Real.sqrt thrT ≤ ((11893 : ℝ)/10000) := by linarith have hZ : (v + Real.sqrt thrT)^3 ≤ (((11893 : ℝ)/10000))^3 := pow_le_pow_left₀ (by positivity) hA 3 have he : (v + Real.sqrt thrT) ^ ((193:ℝ)/269) ≤ (((11893 : ℝ)/10000)) ^ ((193:ℝ)/269) := Real.rpow_le_rpow (by positivity) hA (by norm_num) calc (v + Real.sqrt thrT)^3 * (v + Real.sqrt thrT)^((193:ℝ)/269) ≤ (((11893 : ℝ)/10000))^3 * (((11893 : ℝ)/10000))^((193:ℝ)/269) := mul_le_mul hZ he (by positivity) (by positivity) _ ≤ (((11893 : ℝ)/10000))^3 * (((1133 : ℝ)/1000)) := mul_le_mul_of_nonneg_left hce (by positivity) have hkey : ((101 : ℝ)/250) * ((1269:ℝ)/269) * ((((1773 : ℝ)/2500))^3 * (((391 : ℝ)/500)) + (((11893 : ℝ)/10000))^3 * (((1133 : ℝ)/1000))) / 2 ≤ (((12163 : ℝ)/10000))^4 * (((23 : ℝ)/20)) + (((6591 : ℝ)/10000))^4 * (((741 : ℝ)/1000)) := by norm_num have hs0 : (0:ℝ) ≤ (v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2) := by have h1 : (0:ℝ) ≤ (v + Real.sqrt thrT) ^ (thrTheta - 2) := Real.rpow_nonneg (by linarith) _ have h2 : (0:ℝ) ≤ (u + Real.sqrt thrT) ^ (thrTheta - 2) := Real.rpow_nonneg (by linarith) _ linarith have hZ0 : (0:ℝ) ≤ (((1773 : ℝ)/2500))^3 * (((391 : ℝ)/500)) + (((11893 : ℝ)/10000))^3 * (((1133 : ℝ)/1000)) := by positivity have hb0 : (0:ℝ) ≤ ((101 : ℝ)/250) := by norm_num have hsumle : (v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2) ≤ (((1773 : ℝ)/2500))^3 * (((391 : ℝ)/500)) + (((11893 : ℝ)/10000))^3 * (((1133 : ℝ)/1000)) := by linarith have hprod : u * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) ≤ ((101 : ℝ)/250) * ((((1773 : ℝ)/2500))^3 * (((391 : ℝ)/500)) + (((11893 : ℝ)/10000))^3 * (((1133 : ℝ)/1000))) := mul_le_mul hb hsumle hs0 hb0 have hprod2 := mul_le_mul_of_nonneg_left hprod (by rw [hth1]; norm_num : (0:ℝ) ≤ (thrTheta - 1) / 2) have hfin : u * (thrTheta - 1) * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) / 2 ≤ (((12163 : ℝ)/10000))^4 * (((23 : ℝ)/20)) + (((6591 : ℝ)/10000))^4 * (((741 : ℝ)/1000)) := by calc u * (thrTheta - 1) * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) / 2 = (thrTheta - 1) / 2 * (u * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2))) := by ring _ ≤ (thrTheta - 1) / 2 * (((101 : ℝ)/250) * ((((1773 : ℝ)/2500))^3 * (((391 : ℝ)/500)) + (((11893 : ℝ)/10000))^3 * (((1133 : ℝ)/1000)))) := hprod2 _ = ((101 : ℝ)/250) * ((1269:ℝ)/269) * ((((1773 : ℝ)/2500))^3 * (((391 : ℝ)/500)) + (((11893 : ℝ)/10000))^3 * (((1133 : ℝ)/1000))) / 2 := by rw [hth1]; ring _ ≤ (((12163 : ℝ)/10000))^4 * (((23 : ℝ)/20)) + (((6591 : ℝ)/10000))^4 * (((741 : ℝ)/1000)) := hkey calc u * (thrTheta - 1) * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) / 2 ≤ (((12163 : ℝ)/10000))^4 * (((23 : ℝ)/20)) + (((6591 : ℝ)/10000))^4 * (((741 : ℝ)/1000)) := hfin _ ≤ (u + v) ^ (thrTheta - 1) + (u + Real.sqrt thrT) ^ (thrTheta - 1) := by linarith [hL1, hL2] theorem edgeS_2 {u v : ℝ} (hu : 0 < u) (huv : u ≤ v) (hs : u ^ 2 + v ^ 2 = thrS) (ha : ((101 : ℝ)/250) ≤ u) (hb : u ≤ ((453 : ℝ)/1000)) : u * (thrTheta - 1) * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) / 2 ≤ (u + v) ^ (thrTheta - 1) + (u + Real.sqrt thrT) ^ (thrTheta - 1) := by have hv0 : 0 < v := lt_of_lt_of_le hu huv have hth1 : thrTheta - 1 = ((1269:ℝ)/269) := by unfold thrTheta; norm_num have hwlo : ((3051 : ℝ)/10000) ≤ Real.sqrt thrT := by rw [Real.le_sqrt' (by norm_num)]; norm_num [thrT] have hwhi : Real.sqrt thrT ≤ ((763 : ℝ)/2500) := by rw [Real.sqrt_le_iff]; constructor <;> norm_num [thrT] have hv2 : v ^ 2 = thrS - u ^ 2 := by linarith have hvlo : ((1047 : ℝ)/1250) ≤ v := by have h1 : (((1047 : ℝ)/1250)) ^ 2 ≤ v ^ 2 := by rw [hv2]; unfold thrS; nlinarith nlinarith [h1, hv0, (by norm_num : (0:ℝ) ≤ ((1047 : ℝ)/1250))] have hvhi : v ≤ ((539 : ℝ)/625) := by have h1 : v ^ 2 ≤ (((539 : ℝ)/625)) ^ 2 := by rw [hv2]; unfold thrS; nlinarith nlinarith [h1, hv0, (by norm_num : (0:ℝ) ≤ ((539 : ℝ)/625))] have hL1 : (((776 : ℝ)/625))^4 * (((1167 : ℝ)/1000)) ≤ (u + v) ^ (thrTheta - 1) := by rw [rpow_p_eq (by positivity)] have hce : ((1167 : ℝ)/1000) ≤ (((776 : ℝ)/625)) ^ ((193:ℝ)/269) := by simpa using rpow_frac_le (c := ((1167 : ℚ)/1000)) (z := ((776 : ℚ)/625)) (by norm_num) (by norm_num) (by decide +kernel) have hA : ((776 : ℝ)/625) ≤ u + v := by linarith have hZ : (((776 : ℝ)/625))^4 ≤ (u + v)^4 := pow_le_pow_left₀ (by norm_num) hA 4 have he : (((776 : ℝ)/625)) ^ ((193:ℝ)/269) ≤ (u + v) ^ ((193:ℝ)/269) := Real.rpow_le_rpow (by norm_num) hA (by norm_num) exact le_trans (mul_le_mul_of_nonneg_left hce (by positivity)) (mul_le_mul hZ he (by positivity) (by positivity)) have hL2 : (((7091 : ℝ)/10000))^4 * (((781 : ℝ)/1000)) ≤ (u + Real.sqrt thrT) ^ (thrTheta - 1) := by rw [rpow_p_eq (by positivity)] have hce : ((781 : ℝ)/1000) ≤ (((7091 : ℝ)/10000)) ^ ((193:ℝ)/269) := by simpa using rpow_frac_le (c := ((781 : ℚ)/1000)) (z := ((7091 : ℚ)/10000)) (by norm_num) (by norm_num) (by decide +kernel) have hA : ((7091 : ℝ)/10000) ≤ u + Real.sqrt thrT := by linarith have hZ : (((7091 : ℝ)/10000))^4 ≤ (u + Real.sqrt thrT)^4 := pow_le_pow_left₀ (by norm_num) hA 4 have he : (((7091 : ℝ)/10000)) ^ ((193:ℝ)/269) ≤ (u + Real.sqrt thrT) ^ ((193:ℝ)/269) := Real.rpow_le_rpow (by norm_num) hA (by norm_num) exact le_trans (mul_le_mul_of_nonneg_left hce (by positivity)) (mul_le_mul hZ he (by positivity) (by positivity)) have hQ1 : (u + Real.sqrt thrT) ^ (thrTheta - 2) ≤ (((3791 : ℝ)/5000))^3 * (((41 : ℝ)/50)) := by rw [rpow_q_eq (by positivity)] have hce : (((3791 : ℝ)/5000)) ^ ((193:ℝ)/269) ≤ ((41 : ℝ)/50) := by simpa using le_rpow_frac (c := ((41 : ℚ)/50)) (z := ((3791 : ℚ)/5000)) (by norm_num) (by norm_num) (by decide +kernel) have hA : u + Real.sqrt thrT ≤ ((3791 : ℝ)/5000) := by linarith have hZ : (u + Real.sqrt thrT)^3 ≤ (((3791 : ℝ)/5000))^3 := pow_le_pow_left₀ (by positivity) hA 3 have he : (u + Real.sqrt thrT) ^ ((193:ℝ)/269) ≤ (((3791 : ℝ)/5000)) ^ ((193:ℝ)/269) := Real.rpow_le_rpow (by positivity) hA (by norm_num) calc (u + Real.sqrt thrT)^3 * (u + Real.sqrt thrT)^((193:ℝ)/269) ≤ (((3791 : ℝ)/5000))^3 * (((3791 : ℝ)/5000))^((193:ℝ)/269) := mul_le_mul hZ he (by positivity) (by positivity) _ ≤ (((3791 : ℝ)/5000))^3 * (((41 : ℝ)/50)) := mul_le_mul_of_nonneg_left hce (by positivity) have hQ2 : (v + Real.sqrt thrT) ^ (thrTheta - 2) ≤ (((2919 : ℝ)/2500))^3 * (((559 : ℝ)/500)) := by rw [rpow_q_eq (by positivity)] have hce : (((2919 : ℝ)/2500)) ^ ((193:ℝ)/269) ≤ ((559 : ℝ)/500) := by simpa using le_rpow_frac (c := ((559 : ℚ)/500)) (z := ((2919 : ℚ)/2500)) (by norm_num) (by norm_num) (by decide +kernel) have hA : v + Real.sqrt thrT ≤ ((2919 : ℝ)/2500) := by linarith have hZ : (v + Real.sqrt thrT)^3 ≤ (((2919 : ℝ)/2500))^3 := pow_le_pow_left₀ (by positivity) hA 3 have he : (v + Real.sqrt thrT) ^ ((193:ℝ)/269) ≤ (((2919 : ℝ)/2500)) ^ ((193:ℝ)/269) := Real.rpow_le_rpow (by positivity) hA (by norm_num) calc (v + Real.sqrt thrT)^3 * (v + Real.sqrt thrT)^((193:ℝ)/269) ≤ (((2919 : ℝ)/2500))^3 * (((2919 : ℝ)/2500))^((193:ℝ)/269) := mul_le_mul hZ he (by positivity) (by positivity) _ ≤ (((2919 : ℝ)/2500))^3 * (((559 : ℝ)/500)) := mul_le_mul_of_nonneg_left hce (by positivity) have hkey : ((453 : ℝ)/1000) * ((1269:ℝ)/269) * ((((3791 : ℝ)/5000))^3 * (((41 : ℝ)/50)) + (((2919 : ℝ)/2500))^3 * (((559 : ℝ)/500))) / 2 ≤ (((776 : ℝ)/625))^4 * (((1167 : ℝ)/1000)) + (((7091 : ℝ)/10000))^4 * (((781 : ℝ)/1000)) := by norm_num have hs0 : (0:ℝ) ≤ (v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2) := by have h1 : (0:ℝ) ≤ (v + Real.sqrt thrT) ^ (thrTheta - 2) := Real.rpow_nonneg (by linarith) _ have h2 : (0:ℝ) ≤ (u + Real.sqrt thrT) ^ (thrTheta - 2) := Real.rpow_nonneg (by linarith) _ linarith have hZ0 : (0:ℝ) ≤ (((3791 : ℝ)/5000))^3 * (((41 : ℝ)/50)) + (((2919 : ℝ)/2500))^3 * (((559 : ℝ)/500)) := by positivity have hb0 : (0:ℝ) ≤ ((453 : ℝ)/1000) := by norm_num have hsumle : (v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2) ≤ (((3791 : ℝ)/5000))^3 * (((41 : ℝ)/50)) + (((2919 : ℝ)/2500))^3 * (((559 : ℝ)/500)) := by linarith have hprod : u * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) ≤ ((453 : ℝ)/1000) * ((((3791 : ℝ)/5000))^3 * (((41 : ℝ)/50)) + (((2919 : ℝ)/2500))^3 * (((559 : ℝ)/500))) := mul_le_mul hb hsumle hs0 hb0 have hprod2 := mul_le_mul_of_nonneg_left hprod (by rw [hth1]; norm_num : (0:ℝ) ≤ (thrTheta - 1) / 2) have hfin : u * (thrTheta - 1) * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) / 2 ≤ (((776 : ℝ)/625))^4 * (((1167 : ℝ)/1000)) + (((7091 : ℝ)/10000))^4 * (((781 : ℝ)/1000)) := by calc u * (thrTheta - 1) * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) / 2 = (thrTheta - 1) / 2 * (u * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2))) := by ring _ ≤ (thrTheta - 1) / 2 * (((453 : ℝ)/1000) * ((((3791 : ℝ)/5000))^3 * (((41 : ℝ)/50)) + (((2919 : ℝ)/2500))^3 * (((559 : ℝ)/500)))) := hprod2 _ = ((453 : ℝ)/1000) * ((1269:ℝ)/269) * ((((3791 : ℝ)/5000))^3 * (((41 : ℝ)/50)) + (((2919 : ℝ)/2500))^3 * (((559 : ℝ)/500))) / 2 := by rw [hth1]; ring _ ≤ (((776 : ℝ)/625))^4 * (((1167 : ℝ)/1000)) + (((7091 : ℝ)/10000))^4 * (((781 : ℝ)/1000)) := hkey calc u * (thrTheta - 1) * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) / 2 ≤ (((776 : ℝ)/625))^4 * (((1167 : ℝ)/1000)) + (((7091 : ℝ)/10000))^4 * (((781 : ℝ)/1000)) := hfin _ ≤ (u + v) ^ (thrTheta - 1) + (u + Real.sqrt thrT) ^ (thrTheta - 1) := by linarith [hL1, hL2] theorem edgeS_3 {u v : ℝ} (hu : 0 < u) (huv : u ≤ v) (hs : u ^ 2 + v ^ 2 = thrS) (ha : ((453 : ℝ)/1000) ≤ u) (hb : u ≤ ((503 : ℝ)/1000)) : u * (thrTheta - 1) * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) / 2 ≤ (u + v) ^ (thrTheta - 1) + (u + Real.sqrt thrT) ^ (thrTheta - 1) := by have hv0 : 0 < v := lt_of_lt_of_le hu huv have hth1 : thrTheta - 1 = ((1269:ℝ)/269) := by unfold thrTheta; norm_num have hwlo : ((3051 : ℝ)/10000) ≤ Real.sqrt thrT := by rw [Real.le_sqrt' (by norm_num)]; norm_num [thrT] have hwhi : Real.sqrt thrT ≤ ((763 : ℝ)/2500) := by rw [Real.sqrt_le_iff]; constructor <;> norm_num [thrT] have hv2 : v ^ 2 = thrS - u ^ 2 := by linarith have hvlo : ((4043 : ℝ)/5000) ≤ v := by have h1 : (((4043 : ℝ)/5000)) ^ 2 ≤ v ^ 2 := by rw [hv2]; unfold thrS; nlinarith nlinarith [h1, hv0, (by norm_num : (0:ℝ) ≤ ((4043 : ℝ)/5000))] have hvhi : v ≤ ((8377 : ℝ)/10000) := by have h1 : v ^ 2 ≤ (((8377 : ℝ)/10000)) ^ 2 := by rw [hv2]; unfold thrS; nlinarith nlinarith [h1, hv0, (by norm_num : (0:ℝ) ≤ ((8377 : ℝ)/10000))] have hL1 : (((1577 : ℝ)/1250))^4 * (((1181 : ℝ)/1000)) ≤ (u + v) ^ (thrTheta - 1) := by rw [rpow_p_eq (by positivity)] have hce : ((1181 : ℝ)/1000) ≤ (((1577 : ℝ)/1250)) ^ ((193:ℝ)/269) := by simpa using rpow_frac_le (c := ((1181 : ℚ)/1000)) (z := ((1577 : ℚ)/1250)) (by norm_num) (by norm_num) (by decide +kernel) have hA : ((1577 : ℝ)/1250) ≤ u + v := by linarith have hZ : (((1577 : ℝ)/1250))^4 ≤ (u + v)^4 := pow_le_pow_left₀ (by norm_num) hA 4 have he : (((1577 : ℝ)/1250)) ^ ((193:ℝ)/269) ≤ (u + v) ^ ((193:ℝ)/269) := Real.rpow_le_rpow (by norm_num) hA (by norm_num) exact le_trans (mul_le_mul_of_nonneg_left hce (by positivity)) (mul_le_mul hZ he (by positivity) (by positivity)) have hL2 : (((7581 : ℝ)/10000))^4 * (((819 : ℝ)/1000)) ≤ (u + Real.sqrt thrT) ^ (thrTheta - 1) := by rw [rpow_p_eq (by positivity)] have hce : ((819 : ℝ)/1000) ≤ (((7581 : ℝ)/10000)) ^ ((193:ℝ)/269) := by simpa using rpow_frac_le (c := ((819 : ℚ)/1000)) (z := ((7581 : ℚ)/10000)) (by norm_num) (by norm_num) (by decide +kernel) have hA : ((7581 : ℝ)/10000) ≤ u + Real.sqrt thrT := by linarith have hZ : (((7581 : ℝ)/10000))^4 ≤ (u + Real.sqrt thrT)^4 := pow_le_pow_left₀ (by norm_num) hA 4 have he : (((7581 : ℝ)/10000)) ^ ((193:ℝ)/269) ≤ (u + Real.sqrt thrT) ^ ((193:ℝ)/269) := Real.rpow_le_rpow (by norm_num) hA (by norm_num) exact le_trans (mul_le_mul_of_nonneg_left hce (by positivity)) (mul_le_mul hZ he (by positivity) (by positivity)) have hQ1 : (u + Real.sqrt thrT) ^ (thrTheta - 2) ≤ (((4041 : ℝ)/5000))^3 * (((859 : ℝ)/1000)) := by rw [rpow_q_eq (by positivity)] have hce : (((4041 : ℝ)/5000)) ^ ((193:ℝ)/269) ≤ ((859 : ℝ)/1000) := by simpa using le_rpow_frac (c := ((859 : ℚ)/1000)) (z := ((4041 : ℚ)/5000)) (by norm_num) (by norm_num) (by decide +kernel) have hA : u + Real.sqrt thrT ≤ ((4041 : ℝ)/5000) := by linarith have hZ : (u + Real.sqrt thrT)^3 ≤ (((4041 : ℝ)/5000))^3 := pow_le_pow_left₀ (by positivity) hA 3 have he : (u + Real.sqrt thrT) ^ ((193:ℝ)/269) ≤ (((4041 : ℝ)/5000)) ^ ((193:ℝ)/269) := Real.rpow_le_rpow (by positivity) hA (by norm_num) calc (u + Real.sqrt thrT)^3 * (u + Real.sqrt thrT)^((193:ℝ)/269) ≤ (((4041 : ℝ)/5000))^3 * (((4041 : ℝ)/5000))^((193:ℝ)/269) := mul_le_mul hZ he (by positivity) (by positivity) _ ≤ (((4041 : ℝ)/5000))^3 * (((859 : ℝ)/1000)) := mul_le_mul_of_nonneg_left hce (by positivity) have hQ2 : (v + Real.sqrt thrT) ^ (thrTheta - 2) ≤ (((11429 : ℝ)/10000))^3 * (((1101 : ℝ)/1000)) := by rw [rpow_q_eq (by positivity)] have hce : (((11429 : ℝ)/10000)) ^ ((193:ℝ)/269) ≤ ((1101 : ℝ)/1000) := by simpa using le_rpow_frac (c := ((1101 : ℚ)/1000)) (z := ((11429 : ℚ)/10000)) (by norm_num) (by norm_num) (by decide +kernel) have hA : v + Real.sqrt thrT ≤ ((11429 : ℝ)/10000) := by linarith have hZ : (v + Real.sqrt thrT)^3 ≤ (((11429 : ℝ)/10000))^3 := pow_le_pow_left₀ (by positivity) hA 3 have he : (v + Real.sqrt thrT) ^ ((193:ℝ)/269) ≤ (((11429 : ℝ)/10000)) ^ ((193:ℝ)/269) := Real.rpow_le_rpow (by positivity) hA (by norm_num) calc (v + Real.sqrt thrT)^3 * (v + Real.sqrt thrT)^((193:ℝ)/269) ≤ (((11429 : ℝ)/10000))^3 * (((11429 : ℝ)/10000))^((193:ℝ)/269) := mul_le_mul hZ he (by positivity) (by positivity) _ ≤ (((11429 : ℝ)/10000))^3 * (((1101 : ℝ)/1000)) := mul_le_mul_of_nonneg_left hce (by positivity) have hkey : ((503 : ℝ)/1000) * ((1269:ℝ)/269) * ((((4041 : ℝ)/5000))^3 * (((859 : ℝ)/1000)) + (((11429 : ℝ)/10000))^3 * (((1101 : ℝ)/1000))) / 2 ≤ (((1577 : ℝ)/1250))^4 * (((1181 : ℝ)/1000)) + (((7581 : ℝ)/10000))^4 * (((819 : ℝ)/1000)) := by norm_num have hs0 : (0:ℝ) ≤ (v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2) := by have h1 : (0:ℝ) ≤ (v + Real.sqrt thrT) ^ (thrTheta - 2) := Real.rpow_nonneg (by linarith) _ have h2 : (0:ℝ) ≤ (u + Real.sqrt thrT) ^ (thrTheta - 2) := Real.rpow_nonneg (by linarith) _ linarith have hZ0 : (0:ℝ) ≤ (((4041 : ℝ)/5000))^3 * (((859 : ℝ)/1000)) + (((11429 : ℝ)/10000))^3 * (((1101 : ℝ)/1000)) := by positivity have hb0 : (0:ℝ) ≤ ((503 : ℝ)/1000) := by norm_num have hsumle : (v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2) ≤ (((4041 : ℝ)/5000))^3 * (((859 : ℝ)/1000)) + (((11429 : ℝ)/10000))^3 * (((1101 : ℝ)/1000)) := by linarith have hprod : u * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) ≤ ((503 : ℝ)/1000) * ((((4041 : ℝ)/5000))^3 * (((859 : ℝ)/1000)) + (((11429 : ℝ)/10000))^3 * (((1101 : ℝ)/1000))) := mul_le_mul hb hsumle hs0 hb0 have hprod2 := mul_le_mul_of_nonneg_left hprod (by rw [hth1]; norm_num : (0:ℝ) ≤ (thrTheta - 1) / 2) have hfin : u * (thrTheta - 1) * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) / 2 ≤ (((1577 : ℝ)/1250))^4 * (((1181 : ℝ)/1000)) + (((7581 : ℝ)/10000))^4 * (((819 : ℝ)/1000)) := by calc u * (thrTheta - 1) * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) / 2 = (thrTheta - 1) / 2 * (u * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2))) := by ring _ ≤ (thrTheta - 1) / 2 * (((503 : ℝ)/1000) * ((((4041 : ℝ)/5000))^3 * (((859 : ℝ)/1000)) + (((11429 : ℝ)/10000))^3 * (((1101 : ℝ)/1000)))) := hprod2 _ = ((503 : ℝ)/1000) * ((1269:ℝ)/269) * ((((4041 : ℝ)/5000))^3 * (((859 : ℝ)/1000)) + (((11429 : ℝ)/10000))^3 * (((1101 : ℝ)/1000))) / 2 := by rw [hth1]; ring _ ≤ (((1577 : ℝ)/1250))^4 * (((1181 : ℝ)/1000)) + (((7581 : ℝ)/10000))^4 * (((819 : ℝ)/1000)) := hkey calc u * (thrTheta - 1) * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) / 2 ≤ (((1577 : ℝ)/1250))^4 * (((1181 : ℝ)/1000)) + (((7581 : ℝ)/10000))^4 * (((819 : ℝ)/1000)) := hfin _ ≤ (u + v) ^ (thrTheta - 1) + (u + Real.sqrt thrT) ^ (thrTheta - 1) := by linarith [hL1, hL2] theorem edgeS_4 {u v : ℝ} (hu : 0 < u) (huv : u ≤ v) (hs : u ^ 2 + v ^ 2 = thrS) (ha : ((503 : ℝ)/1000) ≤ u) (hb : u ≤ ((69 : ℝ)/125)) : u * (thrTheta - 1) * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) / 2 ≤ (u + v) ^ (thrTheta - 1) + (u + Real.sqrt thrT) ^ (thrTheta - 1) := by have hv0 : 0 < v := lt_of_lt_of_le hu huv have hth1 : thrTheta - 1 = ((1269:ℝ)/269) := by unfold thrTheta; norm_num have hwlo : ((3051 : ℝ)/10000) ≤ Real.sqrt thrT := by rw [Real.le_sqrt' (by norm_num)]; norm_num [thrT] have hwhi : Real.sqrt thrT ≤ ((763 : ℝ)/2500) := by rw [Real.sqrt_le_iff]; constructor <;> norm_num [thrT] have hv2 : v ^ 2 = thrS - u ^ 2 := by linarith have hvlo : ((7759 : ℝ)/10000) ≤ v := by have h1 : (((7759 : ℝ)/10000)) ^ 2 ≤ v ^ 2 := by rw [hv2]; unfold thrS; nlinarith nlinarith [h1, hv0, (by norm_num : (0:ℝ) ≤ ((7759 : ℝ)/10000))] have hvhi : v ≤ ((8087 : ℝ)/10000) := by have h1 : v ^ 2 ≤ (((8087 : ℝ)/10000)) ^ 2 := by rw [hv2]; unfold thrS; nlinarith nlinarith [h1, hv0, (by norm_num : (0:ℝ) ≤ ((8087 : ℝ)/10000))] have hL1 : (((12789 : ℝ)/10000))^4 * (((1193 : ℝ)/1000)) ≤ (u + v) ^ (thrTheta - 1) := by rw [rpow_p_eq (by positivity)] have hce : ((1193 : ℝ)/1000) ≤ (((12789 : ℝ)/10000)) ^ ((193:ℝ)/269) := by simpa using rpow_frac_le (c := ((1193 : ℚ)/1000)) (z := ((12789 : ℚ)/10000)) (by norm_num) (by norm_num) (by decide +kernel) have hA : ((12789 : ℝ)/10000) ≤ u + v := by linarith have hZ : (((12789 : ℝ)/10000))^4 ≤ (u + v)^4 := pow_le_pow_left₀ (by norm_num) hA 4 have he : (((12789 : ℝ)/10000)) ^ ((193:ℝ)/269) ≤ (u + v) ^ ((193:ℝ)/269) := Real.rpow_le_rpow (by norm_num) hA (by norm_num) exact le_trans (mul_le_mul_of_nonneg_left hce (by positivity)) (mul_le_mul hZ he (by positivity) (by positivity)) have hL2 : (((8081 : ℝ)/10000))^4 * (((429 : ℝ)/500)) ≤ (u + Real.sqrt thrT) ^ (thrTheta - 1) := by rw [rpow_p_eq (by positivity)] have hce : ((429 : ℝ)/500) ≤ (((8081 : ℝ)/10000)) ^ ((193:ℝ)/269) := by simpa using rpow_frac_le (c := ((429 : ℚ)/500)) (z := ((8081 : ℚ)/10000)) (by norm_num) (by norm_num) (by decide +kernel) have hA : ((8081 : ℝ)/10000) ≤ u + Real.sqrt thrT := by linarith have hZ : (((8081 : ℝ)/10000))^4 ≤ (u + Real.sqrt thrT)^4 := pow_le_pow_left₀ (by norm_num) hA 4 have he : (((8081 : ℝ)/10000)) ^ ((193:ℝ)/269) ≤ (u + Real.sqrt thrT) ^ ((193:ℝ)/269) := Real.rpow_le_rpow (by norm_num) hA (by norm_num) exact le_trans (mul_le_mul_of_nonneg_left hce (by positivity)) (mul_le_mul hZ he (by positivity) (by positivity)) have hQ1 : (u + Real.sqrt thrT) ^ (thrTheta - 2) ≤ (((2143 : ℝ)/2500))^3 * (((112 : ℝ)/125)) := by rw [rpow_q_eq (by positivity)] have hce : (((2143 : ℝ)/2500)) ^ ((193:ℝ)/269) ≤ ((112 : ℝ)/125) := by simpa using le_rpow_frac (c := ((112 : ℚ)/125)) (z := ((2143 : ℚ)/2500)) (by norm_num) (by norm_num) (by decide +kernel) have hA : u + Real.sqrt thrT ≤ ((2143 : ℝ)/2500) := by linarith have hZ : (u + Real.sqrt thrT)^3 ≤ (((2143 : ℝ)/2500))^3 := pow_le_pow_left₀ (by positivity) hA 3 have he : (u + Real.sqrt thrT) ^ ((193:ℝ)/269) ≤ (((2143 : ℝ)/2500)) ^ ((193:ℝ)/269) := Real.rpow_le_rpow (by positivity) hA (by norm_num) calc (u + Real.sqrt thrT)^3 * (u + Real.sqrt thrT)^((193:ℝ)/269) ≤ (((2143 : ℝ)/2500))^3 * (((2143 : ℝ)/2500))^((193:ℝ)/269) := mul_le_mul hZ he (by positivity) (by positivity) _ ≤ (((2143 : ℝ)/2500))^3 * (((112 : ℝ)/125)) := mul_le_mul_of_nonneg_left hce (by positivity) have hQ2 : (v + Real.sqrt thrT) ^ (thrTheta - 2) ≤ (((11139 : ℝ)/10000))^3 * (((1081 : ℝ)/1000)) := by rw [rpow_q_eq (by positivity)] have hce : (((11139 : ℝ)/10000)) ^ ((193:ℝ)/269) ≤ ((1081 : ℝ)/1000) := by simpa using le_rpow_frac (c := ((1081 : ℚ)/1000)) (z := ((11139 : ℚ)/10000)) (by norm_num) (by norm_num) (by decide +kernel) have hA : v + Real.sqrt thrT ≤ ((11139 : ℝ)/10000) := by linarith have hZ : (v + Real.sqrt thrT)^3 ≤ (((11139 : ℝ)/10000))^3 := pow_le_pow_left₀ (by positivity) hA 3 have he : (v + Real.sqrt thrT) ^ ((193:ℝ)/269) ≤ (((11139 : ℝ)/10000)) ^ ((193:ℝ)/269) := Real.rpow_le_rpow (by positivity) hA (by norm_num) calc (v + Real.sqrt thrT)^3 * (v + Real.sqrt thrT)^((193:ℝ)/269) ≤ (((11139 : ℝ)/10000))^3 * (((11139 : ℝ)/10000))^((193:ℝ)/269) := mul_le_mul hZ he (by positivity) (by positivity) _ ≤ (((11139 : ℝ)/10000))^3 * (((1081 : ℝ)/1000)) := mul_le_mul_of_nonneg_left hce (by positivity) have hkey : ((69 : ℝ)/125) * ((1269:ℝ)/269) * ((((2143 : ℝ)/2500))^3 * (((112 : ℝ)/125)) + (((11139 : ℝ)/10000))^3 * (((1081 : ℝ)/1000))) / 2 ≤ (((12789 : ℝ)/10000))^4 * (((1193 : ℝ)/1000)) + (((8081 : ℝ)/10000))^4 * (((429 : ℝ)/500)) := by norm_num have hs0 : (0:ℝ) ≤ (v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2) := by have h1 : (0:ℝ) ≤ (v + Real.sqrt thrT) ^ (thrTheta - 2) := Real.rpow_nonneg (by linarith) _ have h2 : (0:ℝ) ≤ (u + Real.sqrt thrT) ^ (thrTheta - 2) := Real.rpow_nonneg (by linarith) _ linarith have hZ0 : (0:ℝ) ≤ (((2143 : ℝ)/2500))^3 * (((112 : ℝ)/125)) + (((11139 : ℝ)/10000))^3 * (((1081 : ℝ)/1000)) := by positivity have hb0 : (0:ℝ) ≤ ((69 : ℝ)/125) := by norm_num have hsumle : (v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2) ≤ (((2143 : ℝ)/2500))^3 * (((112 : ℝ)/125)) + (((11139 : ℝ)/10000))^3 * (((1081 : ℝ)/1000)) := by linarith have hprod : u * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) ≤ ((69 : ℝ)/125) * ((((2143 : ℝ)/2500))^3 * (((112 : ℝ)/125)) + (((11139 : ℝ)/10000))^3 * (((1081 : ℝ)/1000))) := mul_le_mul hb hsumle hs0 hb0 have hprod2 := mul_le_mul_of_nonneg_left hprod (by rw [hth1]; norm_num : (0:ℝ) ≤ (thrTheta - 1) / 2) have hfin : u * (thrTheta - 1) * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) / 2 ≤ (((12789 : ℝ)/10000))^4 * (((1193 : ℝ)/1000)) + (((8081 : ℝ)/10000))^4 * (((429 : ℝ)/500)) := by calc u * (thrTheta - 1) * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) / 2 = (thrTheta - 1) / 2 * (u * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2))) := by ring _ ≤ (thrTheta - 1) / 2 * (((69 : ℝ)/125) * ((((2143 : ℝ)/2500))^3 * (((112 : ℝ)/125)) + (((11139 : ℝ)/10000))^3 * (((1081 : ℝ)/1000)))) := hprod2 _ = ((69 : ℝ)/125) * ((1269:ℝ)/269) * ((((2143 : ℝ)/2500))^3 * (((112 : ℝ)/125)) + (((11139 : ℝ)/10000))^3 * (((1081 : ℝ)/1000))) / 2 := by rw [hth1]; ring _ ≤ (((12789 : ℝ)/10000))^4 * (((1193 : ℝ)/1000)) + (((8081 : ℝ)/10000))^4 * (((429 : ℝ)/500)) := hkey calc u * (thrTheta - 1) * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) / 2 ≤ (((12789 : ℝ)/10000))^4 * (((1193 : ℝ)/1000)) + (((8081 : ℝ)/10000))^4 * (((429 : ℝ)/500)) := hfin _ ≤ (u + v) ^ (thrTheta - 1) + (u + Real.sqrt thrT) ^ (thrTheta - 1) := by linarith [hL1, hL2] theorem edgeS_5 {u v : ℝ} (hu : 0 < u) (huv : u ≤ v) (hs : u ^ 2 + v ^ 2 = thrS) (ha : ((69 : ℝ)/125) ≤ u) (hb : u ≤ ((601 : ℝ)/1000)) : u * (thrTheta - 1) * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) / 2 ≤ (u + v) ^ (thrTheta - 1) + (u + Real.sqrt thrT) ^ (thrTheta - 1) := by have hv0 : 0 < v := lt_of_lt_of_le hu huv have hth1 : thrTheta - 1 = ((1269:ℝ)/269) := by unfold thrTheta; norm_num have hwlo : ((3051 : ℝ)/10000) ≤ Real.sqrt thrT := by rw [Real.le_sqrt' (by norm_num)]; norm_num [thrT] have hwhi : Real.sqrt thrT ≤ ((763 : ℝ)/2500) := by rw [Real.sqrt_le_iff]; constructor <;> norm_num [thrT] have hv2 : v ^ 2 = thrS - u ^ 2 := by linarith have hvlo : ((3693 : ℝ)/5000) ≤ v := by have h1 : (((3693 : ℝ)/5000)) ^ 2 ≤ v ^ 2 := by rw [hv2]; unfold thrS; nlinarith nlinarith [h1, hv0, (by norm_num : (0:ℝ) ≤ ((3693 : ℝ)/5000))] have hvhi : v ≤ ((97 : ℝ)/125) := by have h1 : v ^ 2 ≤ (((97 : ℝ)/125)) ^ 2 := by rw [hv2]; unfold thrS; nlinarith nlinarith [h1, hv0, (by norm_num : (0:ℝ) ≤ ((97 : ℝ)/125))] have hL1 : (((6453 : ℝ)/5000))^4 * (((6 : ℝ)/5)) ≤ (u + v) ^ (thrTheta - 1) := by rw [rpow_p_eq (by positivity)] have hce : ((6 : ℝ)/5) ≤ (((6453 : ℝ)/5000)) ^ ((193:ℝ)/269) := by simpa using rpow_frac_le (c := ((6 : ℚ)/5)) (z := ((6453 : ℚ)/5000)) (by norm_num) (by norm_num) (by decide +kernel) have hA : ((6453 : ℝ)/5000) ≤ u + v := by linarith have hZ : (((6453 : ℝ)/5000))^4 ≤ (u + v)^4 := pow_le_pow_left₀ (by norm_num) hA 4 have he : (((6453 : ℝ)/5000)) ^ ((193:ℝ)/269) ≤ (u + v) ^ ((193:ℝ)/269) := Real.rpow_le_rpow (by norm_num) hA (by norm_num) exact le_trans (mul_le_mul_of_nonneg_left hce (by positivity)) (mul_le_mul hZ he (by positivity) (by positivity)) have hL2 : (((8571 : ℝ)/10000))^4 * (((179 : ℝ)/200)) ≤ (u + Real.sqrt thrT) ^ (thrTheta - 1) := by rw [rpow_p_eq (by positivity)] have hce : ((179 : ℝ)/200) ≤ (((8571 : ℝ)/10000)) ^ ((193:ℝ)/269) := by simpa using rpow_frac_le (c := ((179 : ℚ)/200)) (z := ((8571 : ℚ)/10000)) (by norm_num) (by norm_num) (by decide +kernel) have hA : ((8571 : ℝ)/10000) ≤ u + Real.sqrt thrT := by linarith have hZ : (((8571 : ℝ)/10000))^4 ≤ (u + Real.sqrt thrT)^4 := pow_le_pow_left₀ (by norm_num) hA 4 have he : (((8571 : ℝ)/10000)) ^ ((193:ℝ)/269) ≤ (u + Real.sqrt thrT) ^ ((193:ℝ)/269) := Real.rpow_le_rpow (by norm_num) hA (by norm_num) exact le_trans (mul_le_mul_of_nonneg_left hce (by positivity)) (mul_le_mul hZ he (by positivity) (by positivity)) have hQ1 : (u + Real.sqrt thrT) ^ (thrTheta - 2) ≤ (((4531 : ℝ)/5000))^3 * (((233 : ℝ)/250)) := by rw [rpow_q_eq (by positivity)] have hce : (((4531 : ℝ)/5000)) ^ ((193:ℝ)/269) ≤ ((233 : ℝ)/250) := by simpa using le_rpow_frac (c := ((233 : ℚ)/250)) (z := ((4531 : ℚ)/5000)) (by norm_num) (by norm_num) (by decide +kernel) have hA : u + Real.sqrt thrT ≤ ((4531 : ℝ)/5000) := by linarith have hZ : (u + Real.sqrt thrT)^3 ≤ (((4531 : ℝ)/5000))^3 := pow_le_pow_left₀ (by positivity) hA 3 have he : (u + Real.sqrt thrT) ^ ((193:ℝ)/269) ≤ (((4531 : ℝ)/5000)) ^ ((193:ℝ)/269) := Real.rpow_le_rpow (by positivity) hA (by norm_num) calc (u + Real.sqrt thrT)^3 * (u + Real.sqrt thrT)^((193:ℝ)/269) ≤ (((4531 : ℝ)/5000))^3 * (((4531 : ℝ)/5000))^((193:ℝ)/269) := mul_le_mul hZ he (by positivity) (by positivity) _ ≤ (((4531 : ℝ)/5000))^3 * (((233 : ℝ)/250)) := mul_le_mul_of_nonneg_left hce (by positivity) have hQ2 : (v + Real.sqrt thrT) ^ (thrTheta - 2) ≤ (((2703 : ℝ)/2500))^3 * (((529 : ℝ)/500)) := by rw [rpow_q_eq (by positivity)] have hce : (((2703 : ℝ)/2500)) ^ ((193:ℝ)/269) ≤ ((529 : ℝ)/500) := by simpa using le_rpow_frac (c := ((529 : ℚ)/500)) (z := ((2703 : ℚ)/2500)) (by norm_num) (by norm_num) (by decide +kernel) have hA : v + Real.sqrt thrT ≤ ((2703 : ℝ)/2500) := by linarith have hZ : (v + Real.sqrt thrT)^3 ≤ (((2703 : ℝ)/2500))^3 := pow_le_pow_left₀ (by positivity) hA 3 have he : (v + Real.sqrt thrT) ^ ((193:ℝ)/269) ≤ (((2703 : ℝ)/2500)) ^ ((193:ℝ)/269) := Real.rpow_le_rpow (by positivity) hA (by norm_num) calc (v + Real.sqrt thrT)^3 * (v + Real.sqrt thrT)^((193:ℝ)/269) ≤ (((2703 : ℝ)/2500))^3 * (((2703 : ℝ)/2500))^((193:ℝ)/269) := mul_le_mul hZ he (by positivity) (by positivity) _ ≤ (((2703 : ℝ)/2500))^3 * (((529 : ℝ)/500)) := mul_le_mul_of_nonneg_left hce (by positivity) have hkey : ((601 : ℝ)/1000) * ((1269:ℝ)/269) * ((((4531 : ℝ)/5000))^3 * (((233 : ℝ)/250)) + (((2703 : ℝ)/2500))^3 * (((529 : ℝ)/500))) / 2 ≤ (((6453 : ℝ)/5000))^4 * (((6 : ℝ)/5)) + (((8571 : ℝ)/10000))^4 * (((179 : ℝ)/200)) := by norm_num have hs0 : (0:ℝ) ≤ (v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2) := by have h1 : (0:ℝ) ≤ (v + Real.sqrt thrT) ^ (thrTheta - 2) := Real.rpow_nonneg (by linarith) _ have h2 : (0:ℝ) ≤ (u + Real.sqrt thrT) ^ (thrTheta - 2) := Real.rpow_nonneg (by linarith) _ linarith have hZ0 : (0:ℝ) ≤ (((4531 : ℝ)/5000))^3 * (((233 : ℝ)/250)) + (((2703 : ℝ)/2500))^3 * (((529 : ℝ)/500)) := by positivity have hb0 : (0:ℝ) ≤ ((601 : ℝ)/1000) := by norm_num have hsumle : (v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2) ≤ (((4531 : ℝ)/5000))^3 * (((233 : ℝ)/250)) + (((2703 : ℝ)/2500))^3 * (((529 : ℝ)/500)) := by linarith have hprod : u * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) ≤ ((601 : ℝ)/1000) * ((((4531 : ℝ)/5000))^3 * (((233 : ℝ)/250)) + (((2703 : ℝ)/2500))^3 * (((529 : ℝ)/500))) := mul_le_mul hb hsumle hs0 hb0 have hprod2 := mul_le_mul_of_nonneg_left hprod (by rw [hth1]; norm_num : (0:ℝ) ≤ (thrTheta - 1) / 2) have hfin : u * (thrTheta - 1) * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) / 2 ≤ (((6453 : ℝ)/5000))^4 * (((6 : ℝ)/5)) + (((8571 : ℝ)/10000))^4 * (((179 : ℝ)/200)) := by calc u * (thrTheta - 1) * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) / 2 = (thrTheta - 1) / 2 * (u * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2))) := by ring _ ≤ (thrTheta - 1) / 2 * (((601 : ℝ)/1000) * ((((4531 : ℝ)/5000))^3 * (((233 : ℝ)/250)) + (((2703 : ℝ)/2500))^3 * (((529 : ℝ)/500)))) := hprod2 _ = ((601 : ℝ)/1000) * ((1269:ℝ)/269) * ((((4531 : ℝ)/5000))^3 * (((233 : ℝ)/250)) + (((2703 : ℝ)/2500))^3 * (((529 : ℝ)/500))) / 2 := by rw [hth1]; ring _ ≤ (((6453 : ℝ)/5000))^4 * (((6 : ℝ)/5)) + (((8571 : ℝ)/10000))^4 * (((179 : ℝ)/200)) := hkey calc u * (thrTheta - 1) * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) / 2 ≤ (((6453 : ℝ)/5000))^4 * (((6 : ℝ)/5)) + (((8571 : ℝ)/10000))^4 * (((179 : ℝ)/200)) := hfin _ ≤ (u + v) ^ (thrTheta - 1) + (u + Real.sqrt thrT) ^ (thrTheta - 1) := by linarith [hL1, hL2] theorem edgeS_6 {u v : ℝ} (hu : 0 < u) (huv : u ≤ v) (hs : u ^ 2 + v ^ 2 = thrS) (ha : ((601 : ℝ)/1000) ≤ u) (hb : u ≤ ((651 : ℝ)/1000)) : u * (thrTheta - 1) * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) / 2 ≤ (u + v) ^ (thrTheta - 1) + (u + Real.sqrt thrT) ^ (thrTheta - 1) := by have hv0 : 0 < v := lt_of_lt_of_le hu huv have hth1 : thrTheta - 1 = ((1269:ℝ)/269) := by unfold thrTheta; norm_num have hwlo : ((3051 : ℝ)/10000) ≤ Real.sqrt thrT := by rw [Real.le_sqrt' (by norm_num)]; norm_num [thrT] have hwhi : Real.sqrt thrT ≤ ((763 : ℝ)/2500) := by rw [Real.sqrt_le_iff]; constructor <;> norm_num [thrT] have hv2 : v ^ 2 = thrS - u ^ 2 := by linarith have hvlo : ((139 : ℝ)/200) ≤ v := by have h1 : (((139 : ℝ)/200)) ^ 2 ≤ v ^ 2 := by rw [hv2]; unfold thrS; nlinarith nlinarith [h1, hv0, (by norm_num : (0:ℝ) ≤ ((139 : ℝ)/200))] have hvhi : v ≤ ((7387 : ℝ)/10000) := by have h1 : v ^ 2 ≤ (((7387 : ℝ)/10000)) ^ 2 := by rw [hv2]; unfold thrS; nlinarith nlinarith [h1, hv0, (by norm_num : (0:ℝ) ≤ ((7387 : ℝ)/10000))] have hL1 : (((162 : ℝ)/125))^4 * (((301 : ℝ)/250)) ≤ (u + v) ^ (thrTheta - 1) := by rw [rpow_p_eq (by positivity)] have hce : ((301 : ℝ)/250) ≤ (((162 : ℝ)/125)) ^ ((193:ℝ)/269) := by simpa using rpow_frac_le (c := ((301 : ℚ)/250)) (z := ((162 : ℚ)/125)) (by norm_num) (by norm_num) (by decide +kernel) have hA : ((162 : ℝ)/125) ≤ u + v := by linarith have hZ : (((162 : ℝ)/125))^4 ≤ (u + v)^4 := pow_le_pow_left₀ (by norm_num) hA 4 have he : (((162 : ℝ)/125)) ^ ((193:ℝ)/269) ≤ (u + v) ^ ((193:ℝ)/269) := Real.rpow_le_rpow (by norm_num) hA (by norm_num) exact le_trans (mul_le_mul_of_nonneg_left hce (by positivity)) (mul_le_mul hZ he (by positivity) (by positivity)) have hL2 : (((9061 : ℝ)/10000))^4 * (((931 : ℝ)/1000)) ≤ (u + Real.sqrt thrT) ^ (thrTheta - 1) := by rw [rpow_p_eq (by positivity)] have hce : ((931 : ℝ)/1000) ≤ (((9061 : ℝ)/10000)) ^ ((193:ℝ)/269) := by simpa using rpow_frac_le (c := ((931 : ℚ)/1000)) (z := ((9061 : ℚ)/10000)) (by norm_num) (by norm_num) (by decide +kernel) have hA : ((9061 : ℝ)/10000) ≤ u + Real.sqrt thrT := by linarith have hZ : (((9061 : ℝ)/10000))^4 ≤ (u + Real.sqrt thrT)^4 := pow_le_pow_left₀ (by norm_num) hA 4 have he : (((9061 : ℝ)/10000)) ^ ((193:ℝ)/269) ≤ (u + Real.sqrt thrT) ^ ((193:ℝ)/269) := Real.rpow_le_rpow (by norm_num) hA (by norm_num) exact le_trans (mul_le_mul_of_nonneg_left hce (by positivity)) (mul_le_mul hZ he (by positivity) (by positivity)) have hQ1 : (u + Real.sqrt thrT) ^ (thrTheta - 2) ≤ (((4781 : ℝ)/5000))^3 * (((969 : ℝ)/1000)) := by rw [rpow_q_eq (by positivity)] have hce : (((4781 : ℝ)/5000)) ^ ((193:ℝ)/269) ≤ ((969 : ℝ)/1000) := by simpa using le_rpow_frac (c := ((969 : ℚ)/1000)) (z := ((4781 : ℚ)/5000)) (by norm_num) (by norm_num) (by decide +kernel) have hA : u + Real.sqrt thrT ≤ ((4781 : ℝ)/5000) := by linarith have hZ : (u + Real.sqrt thrT)^3 ≤ (((4781 : ℝ)/5000))^3 := pow_le_pow_left₀ (by positivity) hA 3 have he : (u + Real.sqrt thrT) ^ ((193:ℝ)/269) ≤ (((4781 : ℝ)/5000)) ^ ((193:ℝ)/269) := Real.rpow_le_rpow (by positivity) hA (by norm_num) calc (u + Real.sqrt thrT)^3 * (u + Real.sqrt thrT)^((193:ℝ)/269) ≤ (((4781 : ℝ)/5000))^3 * (((4781 : ℝ)/5000))^((193:ℝ)/269) := mul_le_mul hZ he (by positivity) (by positivity) _ ≤ (((4781 : ℝ)/5000))^3 * (((969 : ℝ)/1000)) := mul_le_mul_of_nonneg_left hce (by positivity) have hQ2 : (v + Real.sqrt thrT) ^ (thrTheta - 2) ≤ (((10439 : ℝ)/10000))^3 * (((129 : ℝ)/125)) := by rw [rpow_q_eq (by positivity)] have hce : (((10439 : ℝ)/10000)) ^ ((193:ℝ)/269) ≤ ((129 : ℝ)/125) := by simpa using le_rpow_frac (c := ((129 : ℚ)/125)) (z := ((10439 : ℚ)/10000)) (by norm_num) (by norm_num) (by decide +kernel) have hA : v + Real.sqrt thrT ≤ ((10439 : ℝ)/10000) := by linarith have hZ : (v + Real.sqrt thrT)^3 ≤ (((10439 : ℝ)/10000))^3 := pow_le_pow_left₀ (by positivity) hA 3 have he : (v + Real.sqrt thrT) ^ ((193:ℝ)/269) ≤ (((10439 : ℝ)/10000)) ^ ((193:ℝ)/269) := Real.rpow_le_rpow (by positivity) hA (by norm_num) calc (v + Real.sqrt thrT)^3 * (v + Real.sqrt thrT)^((193:ℝ)/269) ≤ (((10439 : ℝ)/10000))^3 * (((10439 : ℝ)/10000))^((193:ℝ)/269) := mul_le_mul hZ he (by positivity) (by positivity) _ ≤ (((10439 : ℝ)/10000))^3 * (((129 : ℝ)/125)) := mul_le_mul_of_nonneg_left hce (by positivity) have hkey : ((651 : ℝ)/1000) * ((1269:ℝ)/269) * ((((4781 : ℝ)/5000))^3 * (((969 : ℝ)/1000)) + (((10439 : ℝ)/10000))^3 * (((129 : ℝ)/125))) / 2 ≤ (((162 : ℝ)/125))^4 * (((301 : ℝ)/250)) + (((9061 : ℝ)/10000))^4 * (((931 : ℝ)/1000)) := by norm_num have hs0 : (0:ℝ) ≤ (v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2) := by have h1 : (0:ℝ) ≤ (v + Real.sqrt thrT) ^ (thrTheta - 2) := Real.rpow_nonneg (by linarith) _ have h2 : (0:ℝ) ≤ (u + Real.sqrt thrT) ^ (thrTheta - 2) := Real.rpow_nonneg (by linarith) _ linarith have hZ0 : (0:ℝ) ≤ (((4781 : ℝ)/5000))^3 * (((969 : ℝ)/1000)) + (((10439 : ℝ)/10000))^3 * (((129 : ℝ)/125)) := by positivity have hb0 : (0:ℝ) ≤ ((651 : ℝ)/1000) := by norm_num have hsumle : (v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2) ≤ (((4781 : ℝ)/5000))^3 * (((969 : ℝ)/1000)) + (((10439 : ℝ)/10000))^3 * (((129 : ℝ)/125)) := by linarith have hprod : u * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) ≤ ((651 : ℝ)/1000) * ((((4781 : ℝ)/5000))^3 * (((969 : ℝ)/1000)) + (((10439 : ℝ)/10000))^3 * (((129 : ℝ)/125))) := mul_le_mul hb hsumle hs0 hb0 have hprod2 := mul_le_mul_of_nonneg_left hprod (by rw [hth1]; norm_num : (0:ℝ) ≤ (thrTheta - 1) / 2) have hfin : u * (thrTheta - 1) * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) / 2 ≤ (((162 : ℝ)/125))^4 * (((301 : ℝ)/250)) + (((9061 : ℝ)/10000))^4 * (((931 : ℝ)/1000)) := by calc u * (thrTheta - 1) * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) / 2 = (thrTheta - 1) / 2 * (u * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2))) := by ring _ ≤ (thrTheta - 1) / 2 * (((651 : ℝ)/1000) * ((((4781 : ℝ)/5000))^3 * (((969 : ℝ)/1000)) + (((10439 : ℝ)/10000))^3 * (((129 : ℝ)/125)))) := hprod2 _ = ((651 : ℝ)/1000) * ((1269:ℝ)/269) * ((((4781 : ℝ)/5000))^3 * (((969 : ℝ)/1000)) + (((10439 : ℝ)/10000))^3 * (((129 : ℝ)/125))) / 2 := by rw [hth1]; ring _ ≤ (((162 : ℝ)/125))^4 * (((301 : ℝ)/250)) + (((9061 : ℝ)/10000))^4 * (((931 : ℝ)/1000)) := hkey calc u * (thrTheta - 1) * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) / 2 ≤ (((162 : ℝ)/125))^4 * (((301 : ℝ)/250)) + (((9061 : ℝ)/10000))^4 * (((931 : ℝ)/1000)) := hfin _ ≤ (u + v) ^ (thrTheta - 1) + (u + Real.sqrt thrT) ^ (thrTheta - 1) := by linarith [hL1, hL2] theorem edgeS_7 {u v : ℝ} (hu : 0 < u) (huv : u ≤ v) (hs : u ^ 2 + v ^ 2 = thrS) (ha : ((651 : ℝ)/1000) ≤ u) (hb : u ≤ ((7 : ℝ)/10)) : u * (thrTheta - 1) * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) / 2 ≤ (u + v) ^ (thrTheta - 1) + (u + Real.sqrt thrT) ^ (thrTheta - 1) := by have hv0 : 0 < v := lt_of_lt_of_le hu huv have hth1 : thrTheta - 1 = ((1269:ℝ)/269) := by unfold thrTheta; norm_num have hwlo : ((3051 : ℝ)/10000) ≤ Real.sqrt thrT := by rw [Real.le_sqrt' (by norm_num)]; norm_num [thrT] have hwhi : Real.sqrt thrT ≤ ((763 : ℝ)/2500) := by rw [Real.sqrt_le_iff]; constructor <;> norm_num [thrT] have hv2 : v ^ 2 = thrS - u ^ 2 := by linarith have hvlo : ((807 : ℝ)/1250) ≤ v := by have h1 : (((807 : ℝ)/1250)) ^ 2 ≤ v ^ 2 := by rw [hv2]; unfold thrS; nlinarith nlinarith [h1, hv0, (by norm_num : (0:ℝ) ≤ ((807 : ℝ)/1250))] have hvhi : v ≤ ((6951 : ℝ)/10000) := by have h1 : v ^ 2 ≤ (((6951 : ℝ)/10000)) ^ 2 := by rw [hv2]; unfold thrS; nlinarith nlinarith [h1, hv0, (by norm_num : (0:ℝ) ≤ ((6951 : ℝ)/10000))] have hL1 : (((6483 : ℝ)/5000))^4 * (((301 : ℝ)/250)) ≤ (u + v) ^ (thrTheta - 1) := by rw [rpow_p_eq (by positivity)] have hce : ((301 : ℝ)/250) ≤ (((6483 : ℝ)/5000)) ^ ((193:ℝ)/269) := by simpa using rpow_frac_le (c := ((301 : ℚ)/250)) (z := ((6483 : ℚ)/5000)) (by norm_num) (by norm_num) (by decide +kernel) have hA : ((6483 : ℝ)/5000) ≤ u + v := by linarith have hZ : (((6483 : ℝ)/5000))^4 ≤ (u + v)^4 := pow_le_pow_left₀ (by norm_num) hA 4 have he : (((6483 : ℝ)/5000)) ^ ((193:ℝ)/269) ≤ (u + v) ^ ((193:ℝ)/269) := Real.rpow_le_rpow (by norm_num) hA (by norm_num) exact le_trans (mul_le_mul_of_nonneg_left hce (by positivity)) (mul_le_mul hZ he (by positivity) (by positivity)) have hL2 : (((9561 : ℝ)/10000))^4 * (((121 : ℝ)/125)) ≤ (u + Real.sqrt thrT) ^ (thrTheta - 1) := by rw [rpow_p_eq (by positivity)] have hce : ((121 : ℝ)/125) ≤ (((9561 : ℝ)/10000)) ^ ((193:ℝ)/269) := by simpa using rpow_frac_le (c := ((121 : ℚ)/125)) (z := ((9561 : ℚ)/10000)) (by norm_num) (by norm_num) (by decide +kernel) have hA : ((9561 : ℝ)/10000) ≤ u + Real.sqrt thrT := by linarith have hZ : (((9561 : ℝ)/10000))^4 ≤ (u + Real.sqrt thrT)^4 := pow_le_pow_left₀ (by norm_num) hA 4 have he : (((9561 : ℝ)/10000)) ^ ((193:ℝ)/269) ≤ (u + Real.sqrt thrT) ^ ((193:ℝ)/269) := Real.rpow_le_rpow (by norm_num) hA (by norm_num) exact le_trans (mul_le_mul_of_nonneg_left hce (by positivity)) (mul_le_mul hZ he (by positivity) (by positivity)) have hQ1 : (u + Real.sqrt thrT) ^ (thrTheta - 2) ≤ (((2513 : ℝ)/2500))^3 * (((251 : ℝ)/250)) := by rw [rpow_q_eq (by positivity)] have hce : (((2513 : ℝ)/2500)) ^ ((193:ℝ)/269) ≤ ((251 : ℝ)/250) := by simpa using le_rpow_frac (c := ((251 : ℚ)/250)) (z := ((2513 : ℚ)/2500)) (by norm_num) (by norm_num) (by decide +kernel) have hA : u + Real.sqrt thrT ≤ ((2513 : ℝ)/2500) := by linarith have hZ : (u + Real.sqrt thrT)^3 ≤ (((2513 : ℝ)/2500))^3 := pow_le_pow_left₀ (by positivity) hA 3 have he : (u + Real.sqrt thrT) ^ ((193:ℝ)/269) ≤ (((2513 : ℝ)/2500)) ^ ((193:ℝ)/269) := Real.rpow_le_rpow (by positivity) hA (by norm_num) calc (u + Real.sqrt thrT)^3 * (u + Real.sqrt thrT)^((193:ℝ)/269) ≤ (((2513 : ℝ)/2500))^3 * (((2513 : ℝ)/2500))^((193:ℝ)/269) := mul_le_mul hZ he (by positivity) (by positivity) _ ≤ (((2513 : ℝ)/2500))^3 * (((251 : ℝ)/250)) := mul_le_mul_of_nonneg_left hce (by positivity) have hQ2 : (v + Real.sqrt thrT) ^ (thrTheta - 2) ≤ (((10003 : ℝ)/10000))^3 * (((1001 : ℝ)/1000)) := by rw [rpow_q_eq (by positivity)] have hce : (((10003 : ℝ)/10000)) ^ ((193:ℝ)/269) ≤ ((1001 : ℝ)/1000) := by simpa using le_rpow_frac (c := ((1001 : ℚ)/1000)) (z := ((10003 : ℚ)/10000)) (by norm_num) (by norm_num) (by decide +kernel) have hA : v + Real.sqrt thrT ≤ ((10003 : ℝ)/10000) := by linarith have hZ : (v + Real.sqrt thrT)^3 ≤ (((10003 : ℝ)/10000))^3 := pow_le_pow_left₀ (by positivity) hA 3 have he : (v + Real.sqrt thrT) ^ ((193:ℝ)/269) ≤ (((10003 : ℝ)/10000)) ^ ((193:ℝ)/269) := Real.rpow_le_rpow (by positivity) hA (by norm_num) calc (v + Real.sqrt thrT)^3 * (v + Real.sqrt thrT)^((193:ℝ)/269) ≤ (((10003 : ℝ)/10000))^3 * (((10003 : ℝ)/10000))^((193:ℝ)/269) := mul_le_mul hZ he (by positivity) (by positivity) _ ≤ (((10003 : ℝ)/10000))^3 * (((1001 : ℝ)/1000)) := mul_le_mul_of_nonneg_left hce (by positivity) have hkey : ((7 : ℝ)/10) * ((1269:ℝ)/269) * ((((2513 : ℝ)/2500))^3 * (((251 : ℝ)/250)) + (((10003 : ℝ)/10000))^3 * (((1001 : ℝ)/1000))) / 2 ≤ (((6483 : ℝ)/5000))^4 * (((301 : ℝ)/250)) + (((9561 : ℝ)/10000))^4 * (((121 : ℝ)/125)) := by norm_num have hs0 : (0:ℝ) ≤ (v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2) := by have h1 : (0:ℝ) ≤ (v + Real.sqrt thrT) ^ (thrTheta - 2) := Real.rpow_nonneg (by linarith) _ have h2 : (0:ℝ) ≤ (u + Real.sqrt thrT) ^ (thrTheta - 2) := Real.rpow_nonneg (by linarith) _ linarith have hZ0 : (0:ℝ) ≤ (((2513 : ℝ)/2500))^3 * (((251 : ℝ)/250)) + (((10003 : ℝ)/10000))^3 * (((1001 : ℝ)/1000)) := by positivity have hb0 : (0:ℝ) ≤ ((7 : ℝ)/10) := by norm_num have hsumle : (v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2) ≤ (((2513 : ℝ)/2500))^3 * (((251 : ℝ)/250)) + (((10003 : ℝ)/10000))^3 * (((1001 : ℝ)/1000)) := by linarith have hprod : u * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) ≤ ((7 : ℝ)/10) * ((((2513 : ℝ)/2500))^3 * (((251 : ℝ)/250)) + (((10003 : ℝ)/10000))^3 * (((1001 : ℝ)/1000))) := mul_le_mul hb hsumle hs0 hb0 have hprod2 := mul_le_mul_of_nonneg_left hprod (by rw [hth1]; norm_num : (0:ℝ) ≤ (thrTheta - 1) / 2) have hfin : u * (thrTheta - 1) * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) / 2 ≤ (((6483 : ℝ)/5000))^4 * (((301 : ℝ)/250)) + (((9561 : ℝ)/10000))^4 * (((121 : ℝ)/125)) := by calc u * (thrTheta - 1) * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) / 2 = (thrTheta - 1) / 2 * (u * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2))) := by ring _ ≤ (thrTheta - 1) / 2 * (((7 : ℝ)/10) * ((((2513 : ℝ)/2500))^3 * (((251 : ℝ)/250)) + (((10003 : ℝ)/10000))^3 * (((1001 : ℝ)/1000)))) := hprod2 _ = ((7 : ℝ)/10) * ((1269:ℝ)/269) * ((((2513 : ℝ)/2500))^3 * (((251 : ℝ)/250)) + (((10003 : ℝ)/10000))^3 * (((1001 : ℝ)/1000))) / 2 := by rw [hth1]; ring _ ≤ (((6483 : ℝ)/5000))^4 * (((301 : ℝ)/250)) + (((9561 : ℝ)/10000))^4 * (((121 : ℝ)/125)) := hkey calc u * (thrTheta - 1) * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) / 2 ≤ (((6483 : ℝ)/5000))^4 * (((301 : ℝ)/250)) + (((9561 : ℝ)/10000))^4 * (((121 : ℝ)/125)) := hfin _ ≤ (u + v) ^ (thrTheta - 1) + (u + Real.sqrt thrT) ^ (thrTheta - 1) := by linarith [hL1, hL2] theorem edgeS {u v : ℝ} (hu : 0 < u) (huv : u ≤ v) (hs : u ^ 2 + v ^ 2 = thrS) (hwu : Real.sqrt thrT ≤ u) : u * (thrTheta - 1) * ((v + Real.sqrt thrT) ^ (thrTheta - 2) + (u + Real.sqrt thrT) ^ (thrTheta - 2)) / 2 ≤ (u + v) ^ (thrTheta - 1) + (u + Real.sqrt thrT) ^ (thrTheta - 1) := by have hwlo : ((3051 : ℝ)/10000) ≤ Real.sqrt thrT := by rw [Real.le_sqrt' (by norm_num)]; norm_num [thrT] have hw305 : ((61 : ℝ)/200) ≤ Real.sqrt thrT := le_trans (by norm_num) hwlo have hu2 : u ^ 2 ≤ thrS / 2 := by nlinarith [hs, sq_nonneg (v - u)] have hs7 : thrS / 2 ≤ ((7:ℝ)/10) * ((7:ℝ)/10) := by unfold thrS; norm_num have hu7 : u ≤ (7:ℝ)/10 := by nlinarith [hu2, hs7, hu] rcases lt_or_ge u ((177 : ℝ)/500) with h1 | h1 · exact edgeS_0 hu huv hs (le_trans hw305 hwu) h1.le rcases lt_or_ge u ((101 : ℝ)/250) with h2 | h2 · exact edgeS_1 hu huv hs (h1) h2.le rcases lt_or_ge u ((453 : ℝ)/1000) with h3 | h3 · exact edgeS_2 hu huv hs (h2) h3.le rcases lt_or_ge u ((503 : ℝ)/1000) with h4 | h4 · exact edgeS_3 hu huv hs (h3) h4.le rcases lt_or_ge u ((69 : ℝ)/125) with h5 | h5 · exact edgeS_4 hu huv hs (h4) h5.le rcases lt_or_ge u ((601 : ℝ)/1000) with h6 | h6 · exact edgeS_5 hu huv hs (h5) h6.le rcases lt_or_ge u ((651 : ℝ)/1000) with h7 | h7 · exact edgeS_6 hu huv hs (h6) h7.le · exact edgeS_7 hu huv hs (h7) hu7 theorem edgePsi : EdgePsi := by intro x hx1 hx2 have hx0 : 0 < x := lt_trans thrT_pos hx1 have hy0 : 0 < thrS - x := by linarith [hx2, thrS_pos] have hu0 : 0 < Real.sqrt x := Real.sqrt_pos_of_pos hx0 have hv0 : 0 < Real.sqrt (thrS - x) := Real.sqrt_pos_of_pos hy0 have hu2 : Real.sqrt x ^ 2 = x := Real.sq_sqrt hx0.le have hv2 : Real.sqrt (thrS - x) ^ 2 = thrS - x := Real.sq_sqrt hy0.le have hs : Real.sqrt x ^ 2 + Real.sqrt (thrS - x) ^ 2 = thrS := by rw [hu2, hv2]; ring have huv : Real.sqrt x ≤ Real.sqrt (thrS - x) := Real.sqrt_le_sqrt (by linarith) have hrw : Real.sqrt thrT ≤ Real.sqrt x := Real.sqrt_le_sqrt hx1.le have hS := edgeS hu0 huv hs hrw have hw0 : (0:ℝ) < Real.sqrt thrT := Real.sqrt_pos_of_pos thrT_pos have hC0 : (0:ℝ) < Real.sqrt x + Real.sqrt thrT := by linarith have hCB : Real.sqrt x + Real.sqrt thrT ≤ Real.sqrt (thrS - x) + Real.sqrt thrT := by linarith have hq : (2:ℝ) ≤ thrTheta - 1 := by unfold thrTheta; norm_num have htrap := rpow_sec_le_trap (q := thrTheta - 1) hq hC0 hCB have htrap' : (Real.sqrt (thrS - x) + Real.sqrt thrT) ^ (thrTheta - 1) - (Real.sqrt x + Real.sqrt thrT) ^ (thrTheta - 1) ≤ (thrTheta - 1) * (Real.sqrt (thrS - x) - Real.sqrt x) * ((Real.sqrt (thrS - x) + Real.sqrt thrT) ^ (thrTheta - 2) + (Real.sqrt x + Real.sqrt thrT) ^ (thrTheta - 2)) / 2 := by rw [show thrTheta - 1 - 1 = thrTheta - 2 by ring] at htrap rw [show (Real.sqrt (thrS - x) + Real.sqrt thrT) - (Real.sqrt x + Real.sqrt thrT) = Real.sqrt (thrS - x) - Real.sqrt x by ring] at htrap exact htrap have hstep := mul_le_mul_of_nonneg_left htrap' hu0.le have hvu : 0 ≤ Real.sqrt (thrS - x) - Real.sqrt x := by linarith rw [phi_eS_sub] unfold phi have hbr : 0 ≤ (Real.sqrt x + Real.sqrt (thrS - x)) ^ (thrTheta - 1) + (Real.sqrt x + Real.sqrt thrT) ^ (thrTheta - 1) - Real.sqrt x * ((thrTheta - 1) * ((Real.sqrt (thrS - x) + Real.sqrt thrT) ^ (thrTheta - 2) + (Real.sqrt x + Real.sqrt thrT) ^ (thrTheta - 2))) / 2 := by linarith [hS] have hkey2 : 0 ≤ (Real.sqrt (thrS - x) - Real.sqrt x) * ((Real.sqrt x + Real.sqrt (thrS - x)) ^ (thrTheta - 1) + (Real.sqrt x + Real.sqrt thrT) ^ (thrTheta - 1) - Real.sqrt x * ((thrTheta - 1) * ((Real.sqrt (thrS - x) + Real.sqrt thrT) ^ (thrTheta - 2) + (Real.sqrt x + Real.sqrt thrT) ^ (thrTheta - 2))) / 2) := mul_nonneg hvu hbr nlinarith [hstep, hkey2] /-! ### Section VI: the interior critical point, unconditionally The boundary of the corner domain is handled by `edge_one_dim` (Section VII, using the unconditional `EdgePsi`). At an interior maximum of the branch sum the two directional derivatives along the simplex vanish; the resulting relations force two of the three branch bases `a = sqrt x`, `b = sqrt y`, `c = sqrt z` to coincide, because the three numbers `a + b - c`, `b + c - a`, `a + c - b` are roots of one strictly convex equation. The value is then the one-variable profile `xiProf` on the collision families, which is bounded by the interval certificate `xiSeg_all`. -/ open scoped Topology noncomputable def bProf (v : ℝ) : ℝ := v + Real.sqrt ((1 - v ^ 2) / 2) lemma hasDerivAt_bProf {v : ℝ} (hv : 0 < 1 - v ^ 2) : HasDerivAt bProf (1 - v / (2 * Real.sqrt ((1 - v ^ 2) / 2))) v := by have h1 : HasDerivAt (fun v : ℝ => 1 - v ^ 2) (-2 * v) v := by simpa using ((hasDerivAt_pow 2 v).const_sub 1) have h2 : HasDerivAt (fun v : ℝ => (1 - v ^ 2) / 2) ((-2 * v) / 2) v := h1.div_const 2 have hpos : 0 < (1 - v ^ 2) / 2 := by linarith have h3 := (Real.hasDerivAt_sqrt hpos.ne').comp v h2 have h4 : HasDerivAt (fun v : ℝ => v + Real.sqrt ((1 - v ^ 2) / 2)) (1 + (1 / (2 * Real.sqrt ((1 - v ^ 2) / 2)) * ((-2 * v) / 2))) v := (hasDerivAt_id v).add h3 have h5 : bProf = fun v : ℝ => v + Real.sqrt ((1 - v ^ 2) / 2) := rfl rw [h5] convert h4 using 1 have hs : 0 < Real.sqrt ((1 - v ^ 2) / 2) := Real.sqrt_pos_of_pos hpos field_simp ring lemma deriv_bProf_eq {v : ℝ} (hv : 0 < 1 - v ^ 2) : deriv bProf v = 1 - v / (2 * Real.sqrt ((1 - v ^ 2) / 2)) := (hasDerivAt_bProf hv).deriv lemma bProf_monoOn : MonotoneOn bProf (Set.Icc 0 (408/500)) := by refine monotoneOn_of_deriv_nonneg (convex_Icc _ _) ?_ ?_ ?_ · exact (by unfold bProf; fun_prop : Continuous bProf).continuousOn · intro x hx rw [interior_Icc] at hx obtain ⟨h1, h2⟩ := hx have hp : 0 < 1 - x ^ 2 := by nlinarith [mul_pos (by linarith : (0:ℝ) < x) (by linarith : (0:ℝ) < 408/500 - x)] exact (hasDerivAt_bProf hp).differentiableAt.differentiableWithinAt · intro x hx rw [interior_Icc] at hx obtain ⟨h1, h2⟩ := hx have hp : 0 < 1 - x ^ 2 := by nlinarith [mul_pos (by linarith : (0:ℝ) < x) (by linarith : (0:ℝ) < 408/500 - x)] rw [deriv_bProf_eq hp] have h3 : x ^ 2 ≤ 2/3 := by nlinarith [mul_pos (by linarith : (0:ℝ) < x) (by linarith : (0:ℝ) < 408/500 - x)] have hs : 0 < Real.sqrt ((1 - x ^ 2) / 2) := Real.sqrt_pos_of_pos (by linarith) rw [sub_nonneg, div_le_one (by linarith)] have hle : x / 2 ≤ Real.sqrt ((1 - x ^ 2) / 2) := by rw [Real.le_sqrt' (by linarith : (0:ℝ) < x / 2)] nlinarith linarith lemma bProf_antiOn : AntitoneOn bProf (Set.Icc (817/1000) 1) := by refine antitoneOn_of_deriv_nonpos (convex_Icc _ _) ?_ ?_ ?_ · exact (by unfold bProf; fun_prop : Continuous bProf).continuousOn · intro x hx rw [interior_Icc] at hx obtain ⟨h1, h2⟩ := hx have hp : 0 < 1 - x ^ 2 := by nlinarith [mul_nonneg (by linarith : (0:ℝ) ≤ x) (by linarith : (0:ℝ) ≤ 1 - x)] exact (hasDerivAt_bProf hp).differentiableAt.differentiableWithinAt · intro x hx rw [interior_Icc] at hx obtain ⟨h1, h2⟩ := hx have hp : 0 < 1 - x ^ 2 := by nlinarith [mul_nonneg (by linarith : (0:ℝ) ≤ x) (by linarith : (0:ℝ) ≤ 1 - x)] rw [deriv_bProf_eq hp] have h3 : 2/3 ≤ x ^ 2 := by nlinarith [sq_nonneg (x - 817/1000)] have hs : 0 < Real.sqrt ((1 - x ^ 2) / 2) := Real.sqrt_pos_of_pos (by linarith) rw [sub_nonpos, le_div_iff₀ (by linarith : (0:ℝ) < 2 * Real.sqrt ((1 - x ^ 2) / 2))] have hle : Real.sqrt ((1 - x ^ 2) / 2) ≤ x / 2 := by rw [Real.sqrt_le_iff] exact ⟨by linarith, by nlinarith⟩ linarith lemma bProf_le {v : ℝ} (h0 : 0 ≤ v) (h1 : v ≤ 1) : bProf v ≤ Real.sqrt (3/2) := by have hb : 0 ≤ bProf v := by unfold bProf; positivity rw [Real.le_sqrt hb (by norm_num)] set s := Real.sqrt ((1 - v ^ 2) / 2) with hsdef have hs2 : s ^ 2 = (1 - v ^ 2) / 2 := Real.sq_sqrt (by nlinarith) have hs0 : 0 ≤ s := Real.sqrt_nonneg _ have hbs : bProf v = v + s := by unfold bProf; rw [hsdef] have key : 4 * v * s ≤ 2 - v ^ 2 := by have h : (4 * v * s) ^ 2 ≤ (2 - v ^ 2) ^ 2 := by nlinarith [sq_nonneg (3 * v ^ 2 - 2), hs2] nlinarith [h, hs0, h0, h1, sq_nonneg (4 * v * s + (2 - v ^ 2)), sq_nonneg (4 * v * s - (2 - v ^ 2))] rw [hbs] nlinarith [key, hs2, hs0] lemma xiSeg_core {v M N : ℝ} (hv0 : 0 < v) (hB : bProf v ≤ M) (hM : 0 < M) (hN : M ^ ((193:ℝ)/269) ≤ N) (hnum : 2 * M^4 * N ≤ (29/4) * v) : 2 * bProf v ^ (thrTheta - 1) ≤ (29/4) * v := by have hb : 0 < bProf v := by unfold bProf; nlinarith [Real.sqrt_nonneg ((1 - v^2)/2)] have h1 : bProf v ^ (thrTheta - 1) ≤ M ^ (thrTheta - 1) := Real.rpow_le_rpow hb.le hB (by unfold thrTheta; norm_num) have h2 : M ^ (thrTheta - 1) = M^4 * M ^ ((193:ℝ)/269) := rpow_p_eq hM have h3 : M^4 * M ^ ((193:ℝ)/269) ≤ M^4 * N := mul_le_mul_of_nonneg_left hN (by positivity) linarith -- segment 0: L on [3051/10000, 697/2000] lemma xiSeg_0 {v : ℝ} (hlo : (3051/10000 : ℝ) ≤ v) (hhi : v ≤ (697/2000 : ℝ)) : 2 * bProf v ^ (thrTheta - 1) ≤ (29/4 : ℝ) * v := by refine xiSeg_core (M := (505639/500000 : ℝ)) (N := (1008079/1000000 : ℝ)) (by linarith) ?_ (by norm_num) ?_ ?_ · have hm : bProf v ≤ bProf (697/2000 : ℝ) := bProf_monoOn (by constructor <;> linarith) (by constructor <;> norm_num) hhi have hc : bProf (697/2000 : ℝ) ≤ (505639/500000 : ℝ) := by unfold bProf have hs : Real.sqrt ((1 - ((697/2000 : ℝ))^2)/2) ≤ (331389/500000 : ℝ) := by rw [Real.sqrt_le_iff]; exact ⟨by norm_num, by norm_num⟩ linarith linarith · simpa using le_rpow_frac (c := ((1008079:ℚ)/1000000)) (z := ((505639:ℚ)/500000)) (by norm_num) (by norm_num) (by decide +kernel) · have hnum : 2 * (505639/500000 : ℝ)^4 * (1008079/1000000 : ℝ) ≤ (29/4) * (3051/10000 : ℝ) := by norm_num linarith -- segment 1: L on [697/2000, 973/2500] lemma xiSeg_1 {v : ℝ} (hlo : (697/2000 : ℝ) ≤ v) (hhi : v ≤ (973/2500 : ℝ)) : 2 * bProf v ^ (thrTheta - 1) ≤ (29/4 : ℝ) * v := by refine xiSeg_core (M := (520277/500000 : ℝ)) (N := (1028933/1000000 : ℝ)) (by linarith) ?_ (by norm_num) ?_ ?_ · have hm : bProf v ≤ bProf (973/2500 : ℝ) := bProf_monoOn (by constructor <;> linarith) (by constructor <;> norm_num) hhi have hc : bProf (973/2500 : ℝ) ≤ (520277/500000 : ℝ) := by unfold bProf have hs : Real.sqrt ((1 - ((973/2500 : ℝ))^2)/2) ≤ (325677/500000 : ℝ) := by rw [Real.sqrt_le_iff]; exact ⟨by norm_num, by norm_num⟩ linarith linarith · simpa using le_rpow_frac (c := ((1028933:ℚ)/1000000)) (z := ((520277:ℚ)/500000)) (by norm_num) (by norm_num) (by decide +kernel) · have hnum : 2 * (520277/500000 : ℝ)^4 * (1028933/1000000 : ℝ) ≤ (29/4) * (697/2000 : ℝ) := by norm_num linarith -- segment 2: L on [973/2500, 4273/10000] lemma xiSeg_2 {v : ℝ} (hlo : (973/2500 : ℝ) ≤ v) (hhi : v ≤ (4273/10000 : ℝ)) : 2 * bProf v ^ (thrTheta - 1) ≤ (29/4 : ℝ) * v := by refine xiSeg_core (M := (1066603/1000000 : ℝ)) (N := (1047349/1000000 : ℝ)) (by linarith) ?_ (by norm_num) ?_ ?_ · have hm : bProf v ≤ bProf (4273/10000 : ℝ) := bProf_monoOn (by constructor <;> linarith) (by constructor <;> norm_num) hhi have hc : bProf (4273/10000 : ℝ) ≤ (1066603/1000000 : ℝ) := by unfold bProf have hs : Real.sqrt ((1 - ((4273/10000 : ℝ))^2)/2) ≤ (639303/1000000 : ℝ) := by rw [Real.sqrt_le_iff]; exact ⟨by norm_num, by norm_num⟩ linarith linarith · simpa using le_rpow_frac (c := ((1047349:ℚ)/1000000)) (z := ((1066603:ℚ)/1000000)) (by norm_num) (by norm_num) (by decide +kernel) · have hnum : 2 * (1066603/1000000 : ℝ)^4 * (1047349/1000000 : ℝ) ≤ (29/4) * (973/2500 : ℝ) := by norm_num linarith -- segment 3: L on [4273/10000, 463/1000] lemma xiSeg_3 {v : ℝ} (hlo : (4273/10000 : ℝ) ≤ v) (hhi : v ≤ (463/1000 : ℝ)) : 2 * bProf v ^ (thrTheta - 1) ≤ (29/4 : ℝ) * v := by refine xiSeg_core (M := (4359/4000 : ℝ)) (N := (1063607/1000000 : ℝ)) (by linarith) ?_ (by norm_num) ?_ ?_ · have hm : bProf v ≤ bProf (463/1000 : ℝ) := bProf_monoOn (by constructor <;> linarith) (by constructor <;> norm_num) hhi have hc : bProf (463/1000 : ℝ) ≤ (4359/4000 : ℝ) := by unfold bProf have hs : Real.sqrt ((1 - ((463/1000 : ℝ))^2)/2) ≤ (2507/4000 : ℝ) := by rw [Real.sqrt_le_iff]; exact ⟨by norm_num, by norm_num⟩ linarith linarith · simpa using le_rpow_frac (c := ((1063607:ℚ)/1000000)) (z := ((4359:ℚ)/4000)) (by norm_num) (by norm_num) (by decide +kernel) · have hnum : 2 * (4359/4000 : ℝ)^4 * (1063607/1000000 : ℝ) ≤ (29/4) * (4273/10000 : ℝ) := by norm_num linarith -- segment 4: L on [463/1000, 993/2000] lemma xiSeg_4 {v : ℝ} (hlo : (463/1000 : ℝ) ≤ v) (hhi : v ≤ (993/2000 : ℝ)) : 2 * bProf v ^ (thrTheta - 1) ≤ (29/4 : ℝ) * v := by refine xiSeg_core (M := (222059/200000 : ℝ)) (N := (269489/250000 : ℝ)) (by linarith) ?_ (by norm_num) ?_ ?_ · have hm : bProf v ≤ bProf (993/2000 : ℝ) := bProf_monoOn (by constructor <;> linarith) (by constructor <;> norm_num) hhi have hc : bProf (993/2000 : ℝ) ≤ (222059/200000 : ℝ) := by unfold bProf have hs : Real.sqrt ((1 - ((993/2000 : ℝ))^2)/2) ≤ (122759/200000 : ℝ) := by rw [Real.sqrt_le_iff]; exact ⟨by norm_num, by norm_num⟩ linarith linarith · simpa using le_rpow_frac (c := ((269489:ℚ)/250000)) (z := ((222059:ℚ)/200000)) (by norm_num) (by norm_num) (by decide +kernel) · have hnum : 2 * (222059/200000 : ℝ)^4 * (269489/250000 : ℝ) ≤ (29/4) * (463/1000 : ℝ) := by norm_num linarith -- segment 5: L on [993/2000, 5279/10000] lemma xiSeg_5 {v : ℝ} (hlo : (993/2000 : ℝ) ≤ v) (hhi : v ≤ (5279/10000 : ℝ)) : 2 * bProf v ^ (thrTheta - 1) ≤ (29/4 : ℝ) * v := by refine xiSeg_core (M := (1128451/1000000 : ℝ)) (N := (545287/500000 : ℝ)) (by linarith) ?_ (by norm_num) ?_ ?_ · have hm : bProf v ≤ bProf (5279/10000 : ℝ) := bProf_monoOn (by constructor <;> linarith) (by constructor <;> norm_num) hhi have hc : bProf (5279/10000 : ℝ) ≤ (1128451/1000000 : ℝ) := by unfold bProf have hs : Real.sqrt ((1 - ((5279/10000 : ℝ))^2)/2) ≤ (600551/1000000 : ℝ) := by rw [Real.sqrt_le_iff]; exact ⟨by norm_num, by norm_num⟩ linarith linarith · simpa using le_rpow_frac (c := ((545287:ℚ)/500000)) (z := ((1128451:ℚ)/1000000)) (by norm_num) (by norm_num) (by decide +kernel) · have hnum : 2 * (1128451/1000000 : ℝ)^4 * (545287/500000 : ℝ) ≤ (29/4) * (993/2000 : ℝ) := by norm_num linarith -- segment 6: L on [5279/10000, 2787/5000] lemma xiSeg_6 {v : ℝ} (hlo : (5279/10000 : ℝ) ≤ v) (hhi : v ≤ (2787/5000 : ℝ)) : 2 * bProf v ^ (thrTheta - 1) ≤ (29/4 : ℝ) * v := by refine xiSeg_core (M := (143059/125000 : ℝ)) (N := (55083/50000 : ℝ)) (by linarith) ?_ (by norm_num) ?_ ?_ · have hm : bProf v ≤ bProf (2787/5000 : ℝ) := bProf_monoOn (by constructor <;> linarith) (by constructor <;> norm_num) hhi have hc : bProf (2787/5000 : ℝ) ≤ (143059/125000 : ℝ) := by unfold bProf have hs : Real.sqrt ((1 - ((2787/5000 : ℝ))^2)/2) ≤ (9173/15625 : ℝ) := by rw [Real.sqrt_le_iff]; exact ⟨by norm_num, by norm_num⟩ linarith linarith · simpa using le_rpow_frac (c := ((55083:ℚ)/50000)) (z := ((143059:ℚ)/125000)) (by norm_num) (by norm_num) (by decide +kernel) · have hnum : 2 * (143059/125000 : ℝ)^4 * (55083/50000 : ℝ) ≤ (29/4) * (5279/10000 : ℝ) := by norm_num linarith -- segment 7: L on [2787/5000, 117/200] lemma xiSeg_7 {v : ℝ} (hlo : (2787/5000 : ℝ) ≤ v) (hhi : v ≤ (117/200 : ℝ)) : 2 * bProf v ^ (thrTheta - 1) ≤ (29/4 : ℝ) * v := by refine xiSeg_core (M := (144811/125000 : ℝ)) (N := (277831/250000 : ℝ)) (by linarith) ?_ (by norm_num) ?_ ?_ · have hm : bProf v ≤ bProf (117/200 : ℝ) := bProf_monoOn (by constructor <;> linarith) (by constructor <;> norm_num) hhi have hc : bProf (117/200 : ℝ) ≤ (144811/125000 : ℝ) := by unfold bProf have hs : Real.sqrt ((1 - ((117/200 : ℝ))^2)/2) ≤ (35843/62500 : ℝ) := by rw [Real.sqrt_le_iff]; exact ⟨by norm_num, by norm_num⟩ linarith linarith · simpa using le_rpow_frac (c := ((277831:ℚ)/250000)) (z := ((144811:ℚ)/125000)) (by norm_num) (by norm_num) (by decide +kernel) · have hnum : 2 * (144811/125000 : ℝ)^4 * (277831/250000 : ℝ) ≤ (29/4) * (2787/5000 : ℝ) := by norm_num linarith -- segment 8: L on [117/200, 6109/10000] lemma xiSeg_8 {v : ℝ} (hlo : (117/200 : ℝ) ≤ v) (hhi : v ≤ (6109/10000 : ℝ)) : 2 * bProf v ^ (thrTheta - 1) ≤ (29/4 : ℝ) * v := by refine xiSeg_core (M := (585361/500000 : ℝ)) (N := (1119731/1000000 : ℝ)) (by linarith) ?_ (by norm_num) ?_ ?_ · have hm : bProf v ≤ bProf (6109/10000 : ℝ) := bProf_monoOn (by constructor <;> linarith) (by constructor <;> norm_num) hhi have hc : bProf (6109/10000 : ℝ) ≤ (585361/500000 : ℝ) := by unfold bProf have hs : Real.sqrt ((1 - ((6109/10000 : ℝ))^2)/2) ≤ (279911/500000 : ℝ) := by rw [Real.sqrt_le_iff]; exact ⟨by norm_num, by norm_num⟩ linarith linarith · simpa using le_rpow_frac (c := ((1119731:ℚ)/1000000)) (z := ((585361:ℚ)/500000)) (by norm_num) (by norm_num) (by decide +kernel) · have hnum : 2 * (585361/500000 : ℝ)^4 * (1119731/1000000 : ℝ) ≤ (29/4) * (117/200 : ℝ) := by norm_num linarith -- segment 9: L on [6109/10000, 397/625] lemma xiSeg_9 {v : ℝ} (hlo : (6109/10000 : ℝ) ≤ v) (hhi : v ≤ (397/625 : ℝ)) : 2 * bProf v ^ (thrTheta - 1) ≤ (29/4 : ℝ) * v := by refine xiSeg_core (M := (1181333/1000000 : ℝ)) (N := (281751/250000 : ℝ)) (by linarith) ?_ (by norm_num) ?_ ?_ · have hm : bProf v ≤ bProf (397/625 : ℝ) := bProf_monoOn (by constructor <;> linarith) (by constructor <;> norm_num) hhi have hc : bProf (397/625 : ℝ) ≤ (1181333/1000000 : ℝ) := by unfold bProf have hs : Real.sqrt ((1 - ((397/625 : ℝ))^2)/2) ≤ (546133/1000000 : ℝ) := by rw [Real.sqrt_le_iff]; exact ⟨by norm_num, by norm_num⟩ linarith linarith · simpa using le_rpow_frac (c := ((281751:ℚ)/250000)) (z := ((1181333:ℚ)/1000000)) (by norm_num) (by norm_num) (by decide +kernel) · have hnum : 2 * (1181333/1000000 : ℝ)^4 * (281751/250000 : ℝ) ≤ (29/4) * (6109/10000 : ℝ) := by norm_num linarith -- segment 10: L on [397/625, 6579/10000] lemma xiSeg_10 {v : ℝ} (hlo : (397/625 : ℝ) ≤ v) (hhi : v ≤ (6579/10000 : ℝ)) : 2 * bProf v ^ (thrTheta - 1) ≤ (29/4 : ℝ) * v := by refine xiSeg_core (M := (595213/500000 : ℝ)) (N := (1133221/1000000 : ℝ)) (by linarith) ?_ (by norm_num) ?_ ?_ · have hm : bProf v ≤ bProf (6579/10000 : ℝ) := bProf_monoOn (by constructor <;> linarith) (by constructor <;> norm_num) hhi have hc : bProf (6579/10000 : ℝ) ≤ (595213/500000 : ℝ) := by unfold bProf have hs : Real.sqrt ((1 - ((6579/10000 : ℝ))^2)/2) ≤ (266263/500000 : ℝ) := by rw [Real.sqrt_le_iff]; exact ⟨by norm_num, by norm_num⟩ linarith linarith · simpa using le_rpow_frac (c := ((1133221:ℚ)/1000000)) (z := ((595213:ℚ)/500000)) (by norm_num) (by norm_num) (by decide +kernel) · have hnum : 2 * (595213/500000 : ℝ)^4 * (1133221/1000000 : ℝ) ≤ (29/4) * (397/625 : ℝ) := by norm_num linarith -- segment 11: L on [6579/10000, 6777/10000] lemma xiSeg_11 {v : ℝ} (hlo : (6579/10000 : ℝ) ≤ v) (hhi : v ≤ (6777/10000 : ℝ)) : 2 * bProf v ^ (thrTheta - 1) ≤ (29/4 : ℝ) * v := by refine xiSeg_core (M := (1197663/1000000 : ℝ)) (N := (1138159/1000000 : ℝ)) (by linarith) ?_ (by norm_num) ?_ ?_ · have hm : bProf v ≤ bProf (6777/10000 : ℝ) := bProf_monoOn (by constructor <;> linarith) (by constructor <;> norm_num) hhi have hc : bProf (6777/10000 : ℝ) ≤ (1197663/1000000 : ℝ) := by unfold bProf have hs : Real.sqrt ((1 - ((6777/10000 : ℝ))^2)/2) ≤ (519963/1000000 : ℝ) := by rw [Real.sqrt_le_iff]; exact ⟨by norm_num, by norm_num⟩ linarith linarith · simpa using le_rpow_frac (c := ((1138159:ℚ)/1000000)) (z := ((1197663:ℚ)/1000000)) (by norm_num) (by norm_num) (by decide +kernel) · have hnum : 2 * (1197663/1000000 : ℝ)^4 * (1138159/1000000 : ℝ) ≤ (29/4) * (6579/10000 : ℝ) := by norm_num linarith -- segment 12: L on [6777/10000, 57/80] lemma xiSeg_12 {v : ℝ} (hlo : (6777/10000 : ℝ) ≤ v) (hhi : v ≤ (57/80 : ℝ)) : 2 * bProf v ^ (thrTheta - 1) ≤ (29/4 : ℝ) * v := by refine xiSeg_core (M := (604329/500000 : ℝ)) (N := (572823/500000 : ℝ)) (by linarith) ?_ (by norm_num) ?_ ?_ · have hm : bProf v ≤ bProf (57/80 : ℝ) := bProf_monoOn (by constructor <;> linarith) (by constructor <;> norm_num) hhi have hc : bProf (57/80 : ℝ) ≤ (604329/500000 : ℝ) := by unfold bProf have hs : Real.sqrt ((1 - ((57/80 : ℝ))^2)/2) ≤ (248079/500000 : ℝ) := by rw [Real.sqrt_le_iff]; exact ⟨by norm_num, by norm_num⟩ linarith linarith · simpa using le_rpow_frac (c := ((572823:ℚ)/500000)) (z := ((604329:ℚ)/500000)) (by norm_num) (by norm_num) (by decide +kernel) · have hnum : 2 * (604329/500000 : ℝ)^4 * (572823/500000 : ℝ) ≤ (29/4) * (6777/10000 : ℝ) := by norm_num linarith -- segment 13: L on [57/80, 7647/10000] lemma xiSeg_13 {v : ℝ} (hlo : (57/80 : ℝ) ≤ v) (hhi : v ≤ (7647/10000 : ℝ)) : 2 * bProf v ^ (thrTheta - 1) ≤ (29/4 : ℝ) * v := by refine xiSeg_core (M := (1220351/1000000 : ℝ)) (N := (288397/250000 : ℝ)) (by linarith) ?_ (by norm_num) ?_ ?_ · have hm : bProf v ≤ bProf (7647/10000 : ℝ) := bProf_monoOn (by constructor <;> linarith) (by constructor <;> norm_num) hhi have hc : bProf (7647/10000 : ℝ) ≤ (1220351/1000000 : ℝ) := by unfold bProf have hs : Real.sqrt ((1 - ((7647/10000 : ℝ))^2)/2) ≤ (455651/1000000 : ℝ) := by rw [Real.sqrt_le_iff]; exact ⟨by norm_num, by norm_num⟩ linarith linarith · simpa using le_rpow_frac (c := ((288397:ℚ)/250000)) (z := ((1220351:ℚ)/1000000)) (by norm_num) (by norm_num) (by decide +kernel) · have hnum : 2 * (1220351/1000000 : ℝ)^4 * (288397/250000 : ℝ) ≤ (29/4) * (57/80 : ℝ) := by norm_num linarith -- segment 14: X on [7647/10000, 817/1000] lemma xiSeg_14 {v : ℝ} (hlo : (7647/10000 : ℝ) ≤ v) (hhi : v ≤ (817/1000 : ℝ)) : 2 * bProf v ^ (thrTheta - 1) ≤ (29/4 : ℝ) * v := by refine xiSeg_core (M := (244949/200000 : ℝ)) (N := (578283/500000 : ℝ)) (by linarith) ?_ (by norm_num) ?_ ?_ · have hm : bProf v ≤ Real.sqrt (3/2) := bProf_le (by linarith) (by linarith) have hc : Real.sqrt (3/2) ≤ (244949/200000 : ℝ) := by rw [Real.sqrt_le_iff]; exact ⟨by norm_num, by norm_num⟩ linarith · simpa using le_rpow_frac (c := ((578283:ℚ)/500000)) (z := ((244949:ℚ)/200000)) (by norm_num) (by norm_num) (by decide +kernel) · have hnum : 2 * (244949/200000 : ℝ)^4 * (578283/500000 : ℝ) ≤ (29/4) * (7647/10000 : ℝ) := by norm_num linarith -- segment 15: R on [817/1000, 1] lemma xiSeg_15 {v : ℝ} (hlo : (817/1000 : ℝ) ≤ v) (hhi : v ≤ (1 : ℝ)) : 2 * bProf v ^ (thrTheta - 1) ≤ (29/4 : ℝ) * v := by refine xiSeg_core (M := (244949/200000 : ℝ)) (N := (578283/500000 : ℝ)) (by linarith) ?_ (by norm_num) ?_ ?_ · have hm : bProf v ≤ bProf (817/1000 : ℝ) := bProf_antiOn (by constructor <;> norm_num) (by constructor <;> linarith) hlo have hc : bProf (817/1000 : ℝ) ≤ (244949/200000 : ℝ) := by unfold bProf have hs : Real.sqrt ((1 - ((817/1000 : ℝ))^2)/2) ≤ (81549/200000 : ℝ) := by rw [Real.sqrt_le_iff]; exact ⟨by norm_num, by norm_num⟩ linarith linarith · simpa using le_rpow_frac (c := ((578283:ℚ)/500000)) (z := ((244949:ℚ)/200000)) (by norm_num) (by norm_num) (by decide +kernel) · have hnum : 2 * (244949/200000 : ℝ)^4 * (578283/500000 : ℝ) ≤ (29/4) * (817/1000 : ℝ) := by norm_num linarith lemma xiSeg_all {v : ℝ} (h0 : (3051/10000 : ℝ) ≤ v) (h1 : v ≤ 1) : 2 * bProf v ^ (thrTheta - 1) ≤ (29/4 : ℝ) * v := by rcases le_or_gt v (697/2000 : ℝ) with h | h · exact xiSeg_0 h0 h rcases le_or_gt v (973/2500 : ℝ) with h_ | h_ · exact xiSeg_1 (le_of_lt h) h_ rcases le_or_gt v (4273/10000 : ℝ) with h__ | h__ · exact xiSeg_2 (le_of_lt h_) h__ rcases le_or_gt v (463/1000 : ℝ) with h___ | h___ · exact xiSeg_3 (le_of_lt h__) h___ rcases le_or_gt v (993/2000 : ℝ) with h____ | h____ · exact xiSeg_4 (le_of_lt h___) h____ rcases le_or_gt v (5279/10000 : ℝ) with h_____ | h_____ · exact xiSeg_5 (le_of_lt h____) h_____ rcases le_or_gt v (2787/5000 : ℝ) with h______ | h______ · exact xiSeg_6 (le_of_lt h_____) h______ rcases le_or_gt v (117/200 : ℝ) with h_______ | h_______ · exact xiSeg_7 (le_of_lt h______) h_______ rcases le_or_gt v (6109/10000 : ℝ) with h________ | h________ · exact xiSeg_8 (le_of_lt h_______) h________ rcases le_or_gt v (397/625 : ℝ) with h_________ | h_________ · exact xiSeg_9 (le_of_lt h________) h_________ rcases le_or_gt v (6579/10000 : ℝ) with h__________ | h__________ · exact xiSeg_10 (le_of_lt h_________) h__________ rcases le_or_gt v (6777/10000 : ℝ) with h___________ | h___________ · exact xiSeg_11 (le_of_lt h__________) h___________ rcases le_or_gt v (57/80 : ℝ) with h____________ | h____________ · exact xiSeg_12 (le_of_lt h___________) h____________ rcases le_or_gt v (7647/10000 : ℝ) with h_____________ | h_____________ · exact xiSeg_13 (le_of_lt h____________) h_____________ rcases le_or_gt v (817/1000 : ℝ) with h______________ | h______________ · exact xiSeg_14 (le_of_lt h_____________) h______________ rcases le_or_gt v (1 : ℝ) with h_______________ | h_______________ · exact xiSeg_15 (le_of_lt h______________) h_______________ · exact absurd h_______________ (by linarith) lemma seven_le_two : (29/4 : ℝ) ≤ (2:ℝ) ^ ((769:ℝ)/269) := by refine le_rpow_div (z := (2:ℝ)) (M := (29/4:ℝ)) (m := 769) (n := 269) (by norm_num) (by norm_num) (by norm_num) ?_ have hnat : ((29:ℚ)/4)^269 ≤ (2:ℚ)^769 := by decide +kernel have hcast : ((((29:ℚ)/4)^269 : ℚ) : ℝ) ≤ ((((2:ℚ)^769 : ℚ)) : ℝ) := by exact_mod_cast hnat push_cast at hcast simpa using hcast noncomputable def xiProf (u : ℝ) : ℝ := 2 * (Real.sqrt u + Real.sqrt ((1 - u) / 2)) ^ (thrTheta - 1) / Real.sqrt u theorem xiProf_le (u : ℝ) (hu : thrT ≤ u) (hu1 : u ≤ 1) : xiProf u ≤ (2:ℝ) ^ ((769:ℝ)/269) := by have hu0 : 0 < u := lt_of_lt_of_le thrT_pos hu have hv0 : 0 < Real.sqrt u := Real.sqrt_pos_of_pos hu0 have hsq : Real.sqrt u ^ 2 = u := Real.sq_sqrt hu0.le have hw3051 : ((3051 : ℝ)/10000) ≤ Real.sqrt thrT := by rw [Real.le_sqrt' (by norm_num)]; norm_num [thrT] have hlo : (3051/10000 : ℝ) ≤ Real.sqrt u := le_trans hw3051 (Real.sqrt_le_sqrt hu) have hhi : Real.sqrt u ≤ 1 := by rw [show (1:ℝ) = Real.sqrt 1 by simp] exact Real.sqrt_le_sqrt hu1 have hkey := xiSeg_all hlo hhi have hB : bProf (Real.sqrt u) = Real.sqrt u + Real.sqrt ((1-u)/2) := by unfold bProf; rw [hsq] have hseven := seven_le_two unfold xiProf rw [← hB, div_le_iff₀ hv0] nlinarith [hkey, hseven, hv0, Real.sqrt_nonneg u] /-! ### The critical point algebra -/ noncomputable def Axy (x y : ℝ) : ℝ := (Real.sqrt x + Real.sqrt y) ^ (thrTheta - 1) noncomputable def G3 (x y z : ℝ) : ℝ := (Real.sqrt x + Real.sqrt y) ^ thrTheta + (Real.sqrt x + Real.sqrt z) ^ thrTheta + (Real.sqrt y + Real.sqrt z) ^ thrTheta /-- The normalised sum is `2^(-769/269)` times the branch sum `G3`. -/ theorem corner_sum_eq (p : Fin 3 → ℝ) : (nC p 1 2) ^ ((769:ℝ)/269) + (nC p 0 2) ^ ((769:ℝ)/269) + (nC p 0 1) ^ ((769:ℝ)/269) = (2:ℝ) ^ (-((769:ℝ)/269)) * G3 (p 0) (p 1) (p 2) := by unfold nC G3 thrTheta rw [sq_half_rpow (by positivity), sq_half_rpow (by positivity), sq_half_rpow (by positivity)] ring lemma rpow_theta_succ {A : ℝ} (hA : 0 < A) : A ^ thrTheta = A ^ (thrTheta - 1) * A := by conv_lhs => rw [show thrTheta = (thrTheta - 1) + 1 by ring] rw [Real.rpow_add hA, Real.rpow_one] lemma G3_eq_lin {x y z : ℝ} (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : G3 x y z = Real.sqrt x * (Axy x y + Axy x z) + Real.sqrt y * (Axy x y + Axy y z) + Real.sqrt z * (Axy x z + Axy y z) := by have h1 : (0:ℝ) < Real.sqrt x + Real.sqrt y := by positivity have h2 : (0:ℝ) < Real.sqrt x + Real.sqrt z := by positivity have h3 : (0:ℝ) < Real.sqrt y + Real.sqrt z := by positivity unfold G3 Axy rw [rpow_theta_succ h1, rpow_theta_succ h2, rpow_theta_succ h3] ring lemma G3_eq_mu {x y z mu : ℝ} (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) (hsum : x + y + z = 1) (h1 : Axy x y + Axy x z = mu * Real.sqrt x) (h2 : Axy x y + Axy y z = mu * Real.sqrt y) (h3 : Axy x z + Axy y z = mu * Real.sqrt z) : G3 x y z = mu := by rw [G3_eq_lin hx hy hz, h1, h2, h3] have hx2 : Real.sqrt x * Real.sqrt x = x := by rw [← sq, Real.sq_sqrt hx.le] have hy2 : Real.sqrt y * Real.sqrt y = y := by rw [← sq, Real.sq_sqrt hy.le] have hz2 : Real.sqrt z * Real.sqrt z = z := by rw [← sq, Real.sq_sqrt hz.le] have e1 : Real.sqrt x * (mu * Real.sqrt x) = mu * x := by rw [show Real.sqrt x * (mu * Real.sqrt x) = mu * (Real.sqrt x * Real.sqrt x) by ring, hx2] have e2 : Real.sqrt y * (mu * Real.sqrt y) = mu * y := by rw [show Real.sqrt y * (mu * Real.sqrt y) = mu * (Real.sqrt y * Real.sqrt y) by ring, hy2] have e3 : Real.sqrt z * (mu * Real.sqrt z) = mu * z := by rw [show Real.sqrt z * (mu * Real.sqrt z) = mu * (Real.sqrt z * Real.sqrt z) by ring, hz2] rw [e1, e2, e3] have hm : mu * x + mu * y + mu * z = mu * (x + y + z) := by ring rw [hm, hsum, mul_one] lemma no_three_roots {S mu p q r : ℝ} (hpq : p < q) (hqr : q < r) (hu : 0 < (S + p)/2) (hw : 0 < (S + r)/2) (hp : ((S+p)/2) ^ (thrTheta-1) = mu * p / 2) (hq : ((S+q)/2) ^ (thrTheta-1) = mu * q / 2) (hr : ((S+r)/2) ^ (thrTheta-1) = mu * r / 2) : False := by have hθ : 1 < thrTheta - 1 := by unfold thrTheta; norm_num have huv : (S+p)/2 < (S+q)/2 := by linarith have hvw : (S+q)/2 < (S+r)/2 := by linarith have hslope := (strictConvexOn_rpow hθ).slope_strict_mono_adjacent (x := (S+p)/2) (y := (S+q)/2) (z := (S+r)/2) (Set.mem_Ici.mpr hu.le) (Set.mem_Ici.mpr hw.le) huv hvw have hs1 : (((S+q)/2) ^ (thrTheta-1) - ((S+p)/2) ^ (thrTheta-1)) / ((S+q)/2 - (S+p)/2) = mu := by rw [hp, hq, div_eq_iff (by linarith : (S+q)/2 - (S+p)/2 ≠ 0)] ring have hs2 : (((S+r)/2) ^ (thrTheta-1) - ((S+q)/2) ^ (thrTheta-1)) / ((S+r)/2 - (S+q)/2) = mu := by rw [hq, hr, div_eq_iff (by linarith : (S+r)/2 - (S+q)/2 ≠ 0)] ring rw [hs1, hs2] at hslope exact lt_irrefl mu hslope lemma two_equal {a b c P Q R mu : ℝ} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hP : P = (a + b) ^ (thrTheta - 1)) (hQ : Q = (a + c) ^ (thrTheta - 1)) (hR : R = (b + c) ^ (thrTheta - 1)) (h1 : P + Q = mu * a) (h2 : Q + R = mu * c) (h3 : P + R = mu * b) : a = c ∨ b = c ∨ a = b := by by_contra hcon push Not at hcon obtain ⟨hac, hbc, hab⟩ := hcon have hP2 : 2*P = mu*(a+b-c) := by linarith [h1, h2, h3] have hQ2 : 2*Q = mu*(a+c-b) := by linarith [h1, h2, h3] have hR2 : 2*R = mu*(b+c-a) := by linarith [h1, h2, h3] have eα : ((a+b+c + (a+b-c))/2) ^ (thrTheta-1) = mu * (a+b-c)/2 := by rw [show (a+b+c + (a+b-c))/2 = a+b by ring, ← hP]; linarith have eβ : ((a+b+c + (b+c-a))/2) ^ (thrTheta-1) = mu * (b+c-a)/2 := by rw [show (a+b+c + (b+c-a))/2 = b+c by ring, ← hR]; linarith have eγ : ((a+b+c + (a+c-b))/2) ^ (thrTheta-1) = mu * (a+c-b)/2 := by rw [show (a+b+c + (a+c-b))/2 = a+c by ring, ← hQ]; linarith have hposα : (0:ℝ) < (a+b+c + (a+b-c))/2 := by rw [show (a+b+c + (a+b-c))/2 = a+b by ring]; linarith have hposβ : (0:ℝ) < (a+b+c + (b+c-a))/2 := by rw [show (a+b+c + (b+c-a))/2 = b+c by ring]; linarith have hposγ : (0:ℝ) < (a+b+c + (a+c-b))/2 := by rw [show (a+b+c + (a+c-b))/2 = a+c by ring]; linarith have hαβ : a+b-c ≠ b+c-a := by intro h; exact hac (by linarith) have hαγ : a+b-c ≠ a+c-b := by intro h; exact hbc (by linarith) have hβγ : b+c-a ≠ a+c-b := by intro h; exact hab (by linarith) rcases lt_or_gt_of_ne hαβ with hab1 | hab1 · rcases lt_or_gt_of_ne hαγ with hac1 | hac1 · rcases lt_or_gt_of_ne hβγ with hbc1 | hbc1 · exact no_three_roots hab1 hbc1 hposα hposγ eα eβ eγ · exact no_three_roots hac1 hbc1 hposα hposβ eα eγ eβ · exact no_three_roots hac1 hab1 hposγ hposβ eγ eα eβ · rcases lt_or_gt_of_ne hαγ with hac1 | hac1 · exact no_three_roots hab1 hac1 hposβ hposγ eβ eα eγ · rcases lt_or_gt_of_ne hβγ with hbc1 | hbc1 · exact no_three_roots hbc1 hac1 hposβ hposα eβ eγ eα · exact no_three_roots hbc1 hab1 hposγ hposα eγ eβ eα /-- At an interior critical point the value is bounded by `2^(769/269)`. -/ lemma critical_le {x y z mu : ℝ} (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) (hsum : x + y + z = 1) (h1 : Axy x y + Axy x z = mu * Real.sqrt x) (h2 : Axy x y + Axy y z = mu * Real.sqrt y) (h3 : Axy x z + Axy y z = mu * Real.sqrt z) (htx : thrT < x) (hty : thrT < y) (htz : thrT < z) : G3 x y z ≤ (2:ℝ)^((769:ℝ)/269) := by have hG : G3 x y z = mu := G3_eq_mu hx hy hz hsum h1 h2 h3 have htwo := two_equal (Real.sqrt_pos_of_pos hx) (Real.sqrt_pos_of_pos hy) (Real.sqrt_pos_of_pos hz) (rfl : Axy x y = (Real.sqrt x + Real.sqrt y) ^ (thrTheta - 1)) (rfl : Axy x z = (Real.sqrt x + Real.sqrt z) ^ (thrTheta - 1)) (rfl : Axy y z = (Real.sqrt y + Real.sqrt z) ^ (thrTheta - 1)) h1 h3 h2 rw [hG] rcases htwo with hxz | hyz | hxy · have hxz' : x = z := (Real.sqrt_inj hx.le hz.le).mp hxz have hPR : (Real.sqrt x + Real.sqrt y)^(thrTheta-1) = (Real.sqrt y + Real.sqrt z)^(thrTheta-1) := by rw [← hxz]; ring_nf have hmu2 : mu * Real.sqrt y = 2 * (Real.sqrt x + Real.sqrt y)^(thrTheta-1) := by unfold Axy at h2; linarith [h2, hPR] have hy1 : (1 - y)/2 = x := by linarith have hxi : mu = xiProf y := by unfold xiProf rw [show Real.sqrt ((1 - y)/2) = Real.sqrt x by rw [hy1], show Real.sqrt y + Real.sqrt x = Real.sqrt x + Real.sqrt y by ring] rw [eq_div_iff (Real.sqrt_pos_of_pos hy).ne', mul_comm mu (Real.sqrt y)] linarith [hmu2] rw [hxi] exact xiProf_le y (le_of_lt hty) (by linarith) · have hyz' : y = z := (Real.sqrt_inj hy.le hz.le).mp hyz have hPQ : (Real.sqrt x + Real.sqrt y)^(thrTheta-1) = (Real.sqrt x + Real.sqrt z)^(thrTheta-1) := by rw [← hyz] have hmu2 : mu * Real.sqrt x = 2 * (Real.sqrt x + Real.sqrt y)^(thrTheta-1) := by unfold Axy at h1; linarith [h1, hPQ] have hx1 : (1 - x)/2 = y := by linarith have hxi : mu = xiProf x := by unfold xiProf rw [show Real.sqrt ((1 - x)/2) = Real.sqrt y by rw [hx1]] rw [eq_div_iff (Real.sqrt_pos_of_pos hx).ne', mul_comm mu (Real.sqrt x)] linarith [hmu2] rw [hxi] exact xiProf_le x (le_of_lt htx) (by linarith) · have hxy' : x = y := (Real.sqrt_inj hx.le hy.le).mp hxy have hQR : (Real.sqrt x + Real.sqrt z)^(thrTheta-1) = (Real.sqrt y + Real.sqrt z)^(thrTheta-1) := by rw [hxy'] have hmu2 : mu * Real.sqrt z = 2 * (Real.sqrt x + Real.sqrt z)^(thrTheta-1) := by unfold Axy at h3; linarith [h3, hQR] have hz1 : (1 - z)/2 = x := by linarith have hxi : mu = xiProf z := by unfold xiProf rw [show Real.sqrt ((1 - z)/2) = Real.sqrt x by rw [hz1], show Real.sqrt z + Real.sqrt x = Real.sqrt x + Real.sqrt z by ring] rw [eq_div_iff (Real.sqrt_pos_of_pos hz).ne', mul_comm mu (Real.sqrt z)] linarith [hmu2] rw [hxi] exact xiProf_le z (le_of_lt htz) (by linarith) /-! ### Directional derivatives of `G3` and the critical equations -/ lemma hasDerivAt_sqrt_add {x : ℝ} (hx : 0 < x) : HasDerivAt (fun t : ℝ => Real.sqrt (x + t)) (1 / (2 * Real.sqrt x)) 0 := by have h1 : HasDerivAt (fun t : ℝ => x + t) 1 0 := by simpa using (hasDerivAt_id (0:ℝ)).const_add x have h2 := (Real.hasDerivAt_sqrt (by simpa using hx.ne')).comp 0 h1 simpa [Function.comp_def] using h2 lemma hasDerivAt_sqrt_sub {z : ℝ} (hz : 0 < z) : HasDerivAt (fun t : ℝ => Real.sqrt (z - t)) (-(1 / (2 * Real.sqrt z))) 0 := by have h1 : HasDerivAt (fun t : ℝ => z - t) (-1) 0 := by simpa using (hasDerivAt_id (0:ℝ)).const_sub z have h2 := (Real.hasDerivAt_sqrt (by simpa using hz.ne')).comp 0 h1 simpa [Function.comp_def] using h2 lemma hasDerivAt_G3_dir1 {x y z : ℝ} (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : HasDerivAt (fun t : ℝ => G3 (x + t) y (z - t)) ((thrTheta/2) * ((Real.sqrt x + Real.sqrt y) ^ (thrTheta-1) / Real.sqrt x + (Real.sqrt x + Real.sqrt z) ^ (thrTheta-1) * (1/Real.sqrt x - 1/Real.sqrt z) - (Real.sqrt y + Real.sqrt z) ^ (thrTheta-1) / Real.sqrt z)) 0 := by have hUA := hasDerivAt_sqrt_add hx have hWC := hasDerivAt_sqrt_sub hz have hA : HasDerivAt (fun t : ℝ => Real.sqrt (x + t) + Real.sqrt y) (1/(2*Real.sqrt x)) 0 := by simpa using hUA.add_const (Real.sqrt y) have hW : HasDerivAt (fun t : ℝ => Real.sqrt (z - t) + Real.sqrt y) (-(1/(2*Real.sqrt z))) 0 := by simpa using hWC.add_const (Real.sqrt y) have hB : HasDerivAt (fun t : ℝ => Real.sqrt (x + t) + Real.sqrt (z - t)) (1/(2*Real.sqrt x) + -(1/(2*Real.sqrt z))) 0 := hUA.add hWC have hpA : (0:ℝ) < Real.sqrt (x + 0) + Real.sqrt y := by positivity have hpB : (0:ℝ) < Real.sqrt (x + 0) + Real.sqrt (z - 0) := by positivity have hpW : (0:ℝ) < Real.sqrt (z - 0) + Real.sqrt y := by positivity have h1 := (Real.hasDerivAt_rpow_const (p := thrTheta) (Or.inl hpA.ne')).comp 0 hA have h2 := (Real.hasDerivAt_rpow_const (p := thrTheta) (Or.inl hpB.ne')).comp 0 hB have h3 := (Real.hasDerivAt_rpow_const (p := thrTheta) (Or.inl hpW.ne')).comp 0 hW have hsum := (h1.add h2).add h3 have hsum' : HasDerivAt (fun t : ℝ => (Real.sqrt (x + t) + Real.sqrt y) ^ thrTheta + (Real.sqrt (x + t) + Real.sqrt (z - t)) ^ thrTheta + (Real.sqrt y + Real.sqrt (z - t)) ^ thrTheta) (thrTheta * (Real.sqrt (x + 0) + Real.sqrt y) ^ (thrTheta - 1) * (1 / (2 * Real.sqrt x)) + thrTheta * (Real.sqrt (x + 0) + Real.sqrt (z - 0)) ^ (thrTheta - 1) * (1 / (2 * Real.sqrt x) + -(1 / (2 * Real.sqrt z))) + thrTheta * (Real.sqrt (z - 0) + Real.sqrt y) ^ (thrTheta - 1) * -(1 / (2 * Real.sqrt z))) 0 := by refine HasDerivAt.congr_of_eventuallyEq hsum ?_ filter_upwards with t simp only [Function.comp_apply, Pi.add_apply] ring_nf have hfun : (fun t : ℝ => (Real.sqrt (x + t) + Real.sqrt y) ^ thrTheta + (Real.sqrt (x + t) + Real.sqrt (z - t)) ^ thrTheta + (Real.sqrt y + Real.sqrt (z - t)) ^ thrTheta) = (fun t : ℝ => G3 (x + t) y (z - t)) := by funext t; unfold G3; ring rw [← hfun] convert hsum' using 1 field_simp ring lemma hasDerivAt_G3_dir2 {x y z : ℝ} (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : HasDerivAt (fun t : ℝ => G3 x (y + t) (z - t)) ((thrTheta/2) * ((Real.sqrt x + Real.sqrt y) ^ (thrTheta-1) / Real.sqrt y - (Real.sqrt x + Real.sqrt z) ^ (thrTheta-1) / Real.sqrt z + (Real.sqrt y + Real.sqrt z) ^ (thrTheta-1) * (1/Real.sqrt y - 1/Real.sqrt z))) 0 := by have hUA := hasDerivAt_sqrt_add hy have hWC := hasDerivAt_sqrt_sub hz have hA : HasDerivAt (fun t : ℝ => Real.sqrt (y + t) + Real.sqrt x) (1/(2*Real.sqrt y)) 0 := by simpa using hUA.add_const (Real.sqrt x) have hW : HasDerivAt (fun t : ℝ => Real.sqrt (z - t) + Real.sqrt x) (-(1/(2*Real.sqrt z))) 0 := by simpa using hWC.add_const (Real.sqrt x) have hB : HasDerivAt (fun t : ℝ => Real.sqrt (y + t) + Real.sqrt (z - t)) (1/(2*Real.sqrt y) + -(1/(2*Real.sqrt z))) 0 := hUA.add hWC have hpA : (0:ℝ) < Real.sqrt (y + 0) + Real.sqrt x := by positivity have hpB : (0:ℝ) < Real.sqrt (y + 0) + Real.sqrt (z - 0) := by positivity have hpW : (0:ℝ) < Real.sqrt (z - 0) + Real.sqrt x := by positivity have h1 := (Real.hasDerivAt_rpow_const (p := thrTheta) (Or.inl hpA.ne')).comp 0 hA have h2 := (Real.hasDerivAt_rpow_const (p := thrTheta) (Or.inl hpB.ne')).comp 0 hB have h3 := (Real.hasDerivAt_rpow_const (p := thrTheta) (Or.inl hpW.ne')).comp 0 hW have hsum := (h1.add h2).add h3 have hsum' : HasDerivAt (fun t : ℝ => (Real.sqrt (y + t) + Real.sqrt x) ^ thrTheta + (Real.sqrt (z - t) + Real.sqrt x) ^ thrTheta + (Real.sqrt (y + t) + Real.sqrt (z - t)) ^ thrTheta) (thrTheta * (Real.sqrt (y + 0) + Real.sqrt x) ^ (thrTheta - 1) * (1 / (2 * Real.sqrt y)) + thrTheta * (Real.sqrt (z - 0) + Real.sqrt x) ^ (thrTheta - 1) * -(1 / (2 * Real.sqrt z)) + thrTheta * (Real.sqrt (y + 0) + Real.sqrt (z - 0)) ^ (thrTheta - 1) * (1 / (2 * Real.sqrt y) + -(1 / (2 * Real.sqrt z)))) 0 := by refine HasDerivAt.congr_of_eventuallyEq ((h1.add h3).add h2) ?_ filter_upwards with t simp only [Function.comp_apply, Pi.add_apply] have hfun : (fun t : ℝ => (Real.sqrt (y + t) + Real.sqrt x) ^ thrTheta + (Real.sqrt (z - t) + Real.sqrt x) ^ thrTheta + (Real.sqrt (y + t) + Real.sqrt (z - t)) ^ thrTheta) = (fun t : ℝ => G3 x (y + t) (z - t)) := by funext t; unfold G3; ring rw [← hfun] convert hsum' using 1 field_simp ring lemma critical_of_localmax {x y z : ℝ} (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) (hloc1 : IsLocalMax (fun t : ℝ => G3 (x + t) y (z - t)) 0) (hloc2 : IsLocalMax (fun t : ℝ => G3 x (y + t) (z - t)) 0) : ∃ mu, (Axy x y + Axy x z = mu * Real.sqrt x) ∧ (Axy x y + Axy y z = mu * Real.sqrt y) ∧ (Axy x z + Axy y z = mu * Real.sqrt z) := by have hd1 := hloc1.hasDerivAt_eq_zero (hasDerivAt_G3_dir1 hx hy hz) have hd2 := hloc2.hasDerivAt_eq_zero (hasDerivAt_G3_dir2 hx hy hz) have hth : (thrTheta/2) ≠ 0 := by unfold thrTheta; norm_num have e1 : (Real.sqrt x + Real.sqrt y) ^ (thrTheta-1) / Real.sqrt x + (Real.sqrt x + Real.sqrt z) ^ (thrTheta-1) * (1/Real.sqrt x - 1/Real.sqrt z) - (Real.sqrt y + Real.sqrt z) ^ (thrTheta-1) / Real.sqrt z = 0 := by rcases mul_eq_zero.mp hd1 with h | h · exact absurd h hth · exact h have e2 : (Real.sqrt x + Real.sqrt y) ^ (thrTheta-1) / Real.sqrt y - (Real.sqrt x + Real.sqrt z) ^ (thrTheta-1) / Real.sqrt z + (Real.sqrt y + Real.sqrt z) ^ (thrTheta-1) * (1/Real.sqrt y - 1/Real.sqrt z) = 0 := by rcases mul_eq_zero.mp hd2 with h | h · exact absurd h hth · exact h refine ⟨((Real.sqrt x + Real.sqrt z) ^ (thrTheta-1) + (Real.sqrt y + Real.sqrt z) ^ (thrTheta-1)) / Real.sqrt z, ?_, ?_, ?_⟩ · unfold Axy have hx0 := (Real.sqrt_pos_of_pos hx).ne' have hz0 := (Real.sqrt_pos_of_pos hz).ne' field_simp at e1 ⊢ ring_nf at e1 ⊢ linarith [e1] · unfold Axy have hy0 := (Real.sqrt_pos_of_pos hy).ne' have hz0 := (Real.sqrt_pos_of_pos hz).ne' field_simp at e2 ⊢ ring_nf at e2 ⊢ linarith [e2] · unfold Axy have hz0 := (Real.sqrt_pos_of_pos hz).ne' rw [div_mul_cancel₀ _ hz0] /-! ### The corner maximum principle, unconditionally -/ lemma edge_pair_le {a b : ℝ} (ha : thrT ≤ a) (hb : thrT ≤ b) (hab : a + b = thrS) : ((Real.sqrt a + Real.sqrt b) ^ 2 / 2) ^ ((769:ℝ)/269) + ((Real.sqrt a + Real.sqrt thrT) ^ 2 / 2) ^ ((769:ℝ)/269) + ((Real.sqrt b + Real.sqrt thrT) ^ 2 / 2) ^ ((769:ℝ)/269) ≤ 1 := le_trans (edge_one_dim edgePsi ha hb hab) cornerBound theorem cornerMaxOne (p : Fin 3 → ℝ) (hsum : ∑ i, p i = 1) (ht : ∀ i, thrT ≤ p i) : (nC p 1 2) ^ ((769:ℝ)/269) + (nC p 0 2) ^ ((769:ℝ)/269) + (nC p 0 1) ^ ((769:ℝ)/269) ≤ 1 := by classical set S : Set (Fin 3 → ℝ) := {q | (∀ i, thrT ≤ q i) ∧ ∑ i, q i = 1} with hS have hcont : ContinuousOn (fun q : Fin 3 → ℝ => (nC q 1 2) ^ ((769:ℝ)/269) + (nC q 0 2) ^ ((769:ℝ)/269) + (nC q 0 1) ^ ((769:ℝ)/269)) S := by refine ContinuousOn.add (ContinuousOn.add ?_ ?_) ?_ · exact (by unfold nC; fun_prop : Continuous (fun q : Fin 3 → ℝ => nC q 1 2)).continuousOn.rpow_const (fun q _ => Or.inr (by norm_num)) · exact (by unfold nC; fun_prop : Continuous (fun q : Fin 3 → ℝ => nC q 0 2)).continuousOn.rpow_const (fun q _ => Or.inr (by norm_num)) · exact (by unfold nC; fun_prop : Continuous (fun q : Fin 3 → ℝ => nC q 0 1)).continuousOn.rpow_const (fun q _ => Or.inr (by norm_num)) have hSclosed : IsClosed S := by rw [hS] have h1 : IsClosed {q : Fin 3 → ℝ | ∀ i, thrT ≤ q i} := by rw [show {q : Fin 3 → ℝ | ∀ i, thrT ≤ q i} = ⋂ i, {q | thrT ≤ q i} by ext q; simp] exact isClosed_iInter fun i => isClosed_le continuous_const (continuous_apply i) exact h1.inter (isClosed_eq (by fun_prop) continuous_const) have hK : IsCompact (Set.univ.pi (fun _ : Fin 3 => Set.Icc thrT 1)) := isCompact_univ_pi (fun _ => isCompact_Icc) have hsub : S ⊆ Set.univ.pi (fun _ : Fin 3 => Set.Icc thrT 1) := by rw [hS] intro q hq simp only [Set.mem_pi, Set.mem_univ, Set.mem_Icc] intro i _ refine ⟨hq.1 i, ?_⟩ have h1 : q i ≤ ∑ j, q j := Finset.single_le_sum (f := q) (fun j _ => le_trans thrT_pos.le (hq.1 j)) (Finset.mem_univ i) rw [hq.2] at h1; exact h1 have hSne : S.Nonempty := ⟨p, by rw [hS]; exact ⟨ht, hsum⟩⟩ obtain ⟨q, hqS, hmax⟩ := (hK.of_isClosed_subset hSclosed hsub).exists_isMaxOn hSne hcont have hpq : (nC p 1 2) ^ ((769:ℝ)/269) + (nC p 0 2) ^ ((769:ℝ)/269) + (nC p 0 1) ^ ((769:ℝ)/269) ≤ (nC q 1 2) ^ ((769:ℝ)/269) + (nC q 0 2) ^ ((769:ℝ)/269) + (nC q 0 1) ^ ((769:ℝ)/269) := hmax (by rw [hS]; exact ⟨ht, hsum⟩) have hq0 : 0 < q 0 := lt_of_lt_of_le thrT_pos (hqS.1 0) have hq1 : 0 < q 1 := lt_of_lt_of_le thrT_pos (hqS.1 1) have hq2 : 0 < q 2 := lt_of_lt_of_le thrT_pos (hqS.1 2) have hqsum : q 0 + q 1 + q 2 = 1 := by have := hqS.2; rw [Fin.sum_univ_three] at this; exact this by_cases hcase : ∃ i, q i = thrT · obtain ⟨i, hi⟩ := hcase have key : (nC q 1 2) ^ ((769:ℝ)/269) + (nC q 0 2) ^ ((769:ℝ)/269) + (nC q 0 1) ^ ((769:ℝ)/269) ≤ 1 := by fin_cases i · have hk : q 0 = thrT := hi have hpair : q 1 + q 2 = thrS := by rw [hk] at hqsum; linarith [thrS_eq] have h := edge_pair_le (hqS.1 1) (hqS.1 2) hpair have eB : ((Real.sqrt thrT + Real.sqrt (q 2))^2/2)^((769:ℝ)/269) = ((Real.sqrt (q 2) + Real.sqrt thrT)^2/2)^((769:ℝ)/269) := by rw [add_comm (Real.sqrt thrT)] have eC : ((Real.sqrt thrT + Real.sqrt (q 1))^2/2)^((769:ℝ)/269) = ((Real.sqrt (q 1) + Real.sqrt thrT)^2/2)^((769:ℝ)/269) := by rw [add_comm (Real.sqrt thrT)] simp only [nC, hk] rw [eB, eC] linarith [h] · have hk : q 1 = thrT := hi have hpair : q 2 + q 0 = thrS := by rw [hk] at hqsum; linarith [thrS_eq] have h := edge_pair_le (hqS.1 2) (hqS.1 0) hpair have eB : ((Real.sqrt thrT + Real.sqrt (q 2))^2/2)^((769:ℝ)/269) = ((Real.sqrt (q 2) + Real.sqrt thrT)^2/2)^((769:ℝ)/269) := by rw [add_comm (Real.sqrt thrT)] have eC : ((Real.sqrt (q 0) + Real.sqrt (q 2))^2/2)^((769:ℝ)/269) = ((Real.sqrt (q 2) + Real.sqrt (q 0))^2/2)^((769:ℝ)/269) := by rw [add_comm (Real.sqrt (q 0))] simp only [nC, hk] rw [eB, eC] linarith [h] · have hk : q 2 = thrT := hi have hpair : q 0 + q 1 = thrS := by rw [hk] at hqsum; linarith [thrS_eq] have h := edge_pair_le (hqS.1 0) (hqS.1 1) hpair simp only [nC, hk] linarith [h] linarith · push Not at hcase have hgt0 : thrT < q 0 := lt_of_le_of_ne (hqS.1 0) (Ne.symm (hcase 0)) have hgt1 : thrT < q 1 := lt_of_le_of_ne (hqS.1 1) (Ne.symm (hcase 1)) have hgt2 : thrT < q 2 := lt_of_le_of_ne (hqS.1 2) (Ne.symm (hcase 2)) have hqpos : (0:ℝ) < (2:ℝ) ^ (-((769:ℝ)/269)) := Real.rpow_pos_of_pos (by norm_num) _ have slice1 : ∀ t : ℝ, thrT ≤ q 0 + t → thrT ≤ q 2 - t → G3 (q 0 + t) (q 1) (q 2 - t) ≤ G3 (q 0) (q 1) (q 2) := by intro t h1 h2 have hmem : (![q 0 + t, q 1, q 2 - t] : Fin 3 → ℝ) ∈ S := by rw [hS] refine ⟨?_, ?_⟩ · intro j fin_cases j · simpa using h1 · simpa using hqS.1 1 · simpa using h2 · rw [Fin.sum_univ_three] simp only [Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.head_cons, Matrix.cons_val_two, Matrix.tail_cons] linarith [hqsum] have hle := hmax hmem simp only [Set.mem_ofPred_eq] at hle rw [corner_sum_eq, corner_sum_eq] at hle have e0 : (![q 0 + t, q 1, q 2 - t] : Fin 3 → ℝ) 0 = q 0 + t := rfl have e1 : (![q 0 + t, q 1, q 2 - t] : Fin 3 → ℝ) 1 = q 1 := rfl have e2 : (![q 0 + t, q 1, q 2 - t] : Fin 3 → ℝ) 2 = q 2 - t := rfl rw [e0, e1, e2] at hle exact le_of_mul_le_mul_left hle hqpos have slice2 : ∀ t : ℝ, thrT ≤ q 1 + t → thrT ≤ q 2 - t → G3 (q 0) (q 1 + t) (q 2 - t) ≤ G3 (q 0) (q 1) (q 2) := by intro t h1 h2 have hmem : (![q 0, q 1 + t, q 2 - t] : Fin 3 → ℝ) ∈ S := by rw [hS] refine ⟨?_, ?_⟩ · intro j fin_cases j · simpa using hqS.1 0 · simpa using h1 · simpa using h2 · rw [Fin.sum_univ_three] simp only [Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.head_cons, Matrix.cons_val_two, Matrix.tail_cons] linarith [hqsum] have hle := hmax hmem simp only [Set.mem_ofPred_eq] at hle rw [corner_sum_eq, corner_sum_eq] at hle have e0 : (![q 0, q 1 + t, q 2 - t] : Fin 3 → ℝ) 0 = q 0 := rfl have e1 : (![q 0, q 1 + t, q 2 - t] : Fin 3 → ℝ) 1 = q 1 + t := rfl have e2 : (![q 0, q 1 + t, q 2 - t] : Fin 3 → ℝ) 2 = q 2 - t := rfl rw [e0, e1, e2] at hle exact le_of_mul_le_mul_left hle hqpos have ev1 : ∀ᶠ t : ℝ in 𝓝 0, thrT ≤ q 0 + t := by have h := eventually_gt_nhds (a := (0:ℝ)) (b := thrT - q 0) (show thrT - q 0 < 0 by linarith) filter_upwards [h] with t ht'; linarith have ev2 : ∀ᶠ t : ℝ in 𝓝 0, thrT ≤ q 2 - t := by have h := eventually_lt_nhds (a := (0:ℝ)) (b := q 2 - thrT) (show (0:ℝ) < q 2 - thrT by linarith) filter_upwards [h] with t ht'; linarith have ev3 : ∀ᶠ t : ℝ in 𝓝 0, thrT ≤ q 1 + t := by have h := eventually_gt_nhds (a := (0:ℝ)) (b := thrT - q 1) (show thrT - q 1 < 0 by linarith) filter_upwards [h] with t ht'; linarith have hloc1 : IsLocalMax (fun t : ℝ => G3 (q 0 + t) (q 1) (q 2 - t)) 0 := by rw [IsLocalMax] filter_upwards [ev1, ev2] with t h1 h2 simpa using slice1 t h1 h2 have hloc2 : IsLocalMax (fun t : ℝ => G3 (q 0) (q 1 + t) (q 2 - t)) 0 := by rw [IsLocalMax] filter_upwards [ev3, ev2] with t h1 h2 simpa using slice2 t h1 h2 obtain ⟨mu, hm1, hm2, hm3⟩ := critical_of_localmax hq0 hq1 hq2 hloc1 hloc2 have hbound := critical_le hq0 hq1 hq2 hqsum hm1 hm2 hm3 hgt0 hgt1 hgt2 have hfin : (nC q 1 2) ^ ((769:ℝ)/269) + (nC q 0 2) ^ ((769:ℝ)/269) + (nC q 0 1) ^ ((769:ℝ)/269) ≤ 1 := by rw [corner_sum_eq] calc (2:ℝ) ^ (-((769:ℝ)/269)) * G3 (q 0) (q 1) (q 2) ≤ (2:ℝ) ^ (-((769:ℝ)/269)) * (2:ℝ) ^ ((769:ℝ)/269) := mul_le_mul_of_nonneg_left hbound hqpos.le _ = 1 := by rw [← Real.rpow_add (by norm_num : (0:ℝ) < 2)] norm_num linarith /-- The sharp two-variable inequality `CaseIIBound`, unconditionally. -/ theorem caseIIBound_direct : CaseIIBound := by intro p hp0 hp1 ht exact cornerMaxOne p hp1 ht /-- **Theorem 3** (the paper's `chi3`): the monotone complexity exponent of `U_k` is at least `0.769` for `k = 3`. Unconditional. -/ theorem theorem3 : IsMonotoneGraphExponent 3 ((769 : ℝ) / 1000) := theorem3_of_caseII caseIIBound_direct end end T3 end UnateLin