import Mathlib import OAI.Computability.DirectedFeedback.Probability.A13Reduction namespace OAI noncomputable section namespace DFVSGames.Appendix.A5FrequencyTransport open DFVSGames.Integration.BinaryLinear open DFVSGames.Appendix.LinearIdentities open DFVSGames.Appendix.NaturalTransport variable {V W : Type*} [AddCommGroup V] [Module F2 V] [AddCommGroup W] [Module F2 W] theorem quotient_relativeFrequency_apply (A A' : Submodule F2 V) (B' B : Submodule F2 W) (hA : A ≤ A') (hB : B' ≤ B) (Y : B' →ₗ[F2] (V ⧸ A')) (b : B'.comap B.subtype) : Submodule.quotientQuotientEquivQuotient A A' hA (A5ReverseIndex.relativeFrequency A A' B' B hA hB Y b) = Y ((SubmoduleInterval.lowerIntervalSpaceEquiv B ⟨B', hB⟩).symm b) := by change Submodule.quotientQuotientEquivQuotient A A' hA ((Submodule.quotientQuotientEquivQuotient A A' hA).symm (Y ((SubmoduleInterval.lowerIntervalSpaceEquiv B ⟨B', hB⟩).symm b))) = _ exact (Submodule.quotientQuotientEquivQuotient A A' hA).apply_symm_apply _ theorem quotient_compress_apply (A A' : Submodule F2 V) (B' B : Submodule F2 W) (hA : A ≤ A') (X : B →ₗ[F2] (V ⧸ A)) (b : B'.comap B.subtype) : Submodule.quotientQuotientEquivQuotient A A' hA (compress X (A'.map A.mkQ) (B'.comap B.subtype) b) = Submodule.mapQ A A' LinearMap.id hA (X b.val) := rfl theorem compression_eq_relativeFrequency_of_pointwise (A A' : Submodule F2 V) (B' B : Submodule F2 W) (hA : A ≤ A') (hB : B' ≤ B) (X : B →ₗ[F2] (V ⧸ A)) (Y : B' →ₗ[F2] (V ⧸ A')) (hY : ∀ b : B', Submodule.mapQ A A' LinearMap.id hA (X ⟨b.val, hB b.property⟩) = Y b) : compress X (A'.map A.mkQ) (B'.comap B.subtype) = A5ReverseIndex.relativeFrequency A A' B' B hA hB Y := by ext b apply (Submodule.quotientQuotientEquivQuotient A A' hA).injective rw [quotient_compress_apply, quotient_relativeFrequency_apply] exact hY ((SubmoduleInterval.lowerIntervalSpaceEquiv B ⟨B', hB⟩).symm b) variable [FiniteDimensional F2 V] [FiniteDimensional F2 W] [Finite V] [Finite W] omit [FiniteDimensional F2 V] [Finite V] in theorem quotientFactor_mapQ (A : Submodule F2 V) (C : Submodule F2 (V ⧸ A)) (hA : A ≤ C.comap A.mkQ) (u : V ⧸ A) : quotientFactor A C (Submodule.mapQ A (C.comap A.mkQ) LinearMap.id hA u) = C.mkQ u := by obtain ⟨x, rfl⟩ := A.mkQ_surjective u change quotientFactor A C ((C.comap A.mkQ).mkQ x) = C.mkQ (A.mkQ x) exact quotientFactor_mkQ A C x omit [FiniteDimensional F2 V] [FiniteDimensional F2 W] [Finite V] [Finite W] in theorem dualFactor_quotient (A : Submodule F2 V) (B : Submodule F2 W) (C : Submodule F2 (V ⧸ A)) (D : Submodule F2 B) (Z : D →ₗ[F2] ((V ⧸ A) ⧸ C)) (b : D.map B.subtype) : quotientFactor A C (dualFactor A B C D Z b) = Z ((outputFactor B D).symm b) := by change quotientFactor A C ((quotientFactor A C).symm (Z ((outputFactor B D).symm b))) = _ exact (quotientFactor A C).apply_symm_apply _ omit [FiniteDimensional F2 W] [Finite W] in theorem outputFactor_symm_as_inclusion (B : Submodule F2 W) (D : Submodule F2 B) (hB : D.map B.subtype ≤ B) (b : D.map B.subtype) : (((outputFactor B D).symm b : D) : B) = ⟨b.val, hB b.property⟩ := by apply Subtype.ext have hh := outputFactor_val B D ((outputFactor B D).symm b) rw [LinearEquiv.apply_symm_apply] at hh exact hh.symm omit [FiniteDimensional F2 V] [FiniteDimensional F2 W] [Finite V] [Finite W] in theorem compression_eq_relativeFrequency_dualFactor (A : Submodule F2 V) (B : Submodule F2 W) (C : Submodule F2 (V ⧸ A)) (D : Submodule F2 B) (hA : A ≤ C.comap A.mkQ) (hB : D.map B.subtype ≤ B) (X : B →ₗ[F2] (V ⧸ A)) : compress X ((C.comap A.mkQ).map A.mkQ) ((D.map B.subtype).comap B.subtype) = A5ReverseIndex.relativeFrequency A (C.comap A.mkQ) (D.map B.subtype) B hA hB (dualFactor A B C D (compress X C D)) := by apply compression_eq_relativeFrequency_of_pointwise intro b apply (quotientFactor A C).injective rw [quotientFactor_mapQ, dualFactor_quotient] change C.mkQ (X ⟨b.val, hB b.property⟩) = C.mkQ (X (((outputFactor B D).symm b : D) : B)) rw [outputFactor_symm_as_inclusion B D hB b] omit [FiniteDimensional F2 V] [FiniteDimensional F2 W] [Finite V] [Finite W] in theorem isAntecedent_relativeFrequency_dualFactor (A : Submodule F2 V) (B : Submodule F2 W) (C : Submodule F2 (V ⧸ A)) (D : Submodule F2 B) (hA : A ≤ C.comap A.mkQ) (hB : D.map B.subtype ≤ B) (X : B →ₗ[F2] (V ⧸ A)) (hdis : Disjoint X.range C) (hcover : D ⊔ X.ker = ⊤) : AntecedentCount.IsAntecedent ((D.map B.subtype).comap B.subtype) ((C.comap A.mkQ).map A.mkQ) (A5ReverseIndex.relativeFrequency A (C.comap A.mkQ) (D.map B.subtype) B hA hB (dualFactor A B C D (compress X C D))) X := by refine ⟨compression_eq_relativeFrequency_dualFactor A B C D hA hB X, ?_, ?_⟩ · rw [Submodule.map_comap_eq_of_surjective A.mkQ_surjective C] exact hdis · rw [Submodule.comap_map_eq_of_injective (f := B.subtype) Subtype.val_injective D] exact hcover end DFVSGames.Appendix.A5FrequencyTransport end noncomputable section namespace DFVSGames.Appendix.A5ForwardIndex open DFVSGames.Integration.BinaryLinear open DFVSGames.Fourier.MatrixRestrictions (order) open scoped BigOperators attribute [local instance] Classical.propDecidable attribute [local instance] OperatorNorm.quotientFinite OperatorNorm.compressedDualFintype variable {E F : Type*} [AddCommGroup E] [Module F2 E] [AddCommGroup F] [Module F2 F] [FiniteDimensional F2 E] [FiniteDimensional F2 F] [Finite E] [Finite F] [Fintype (E →ₗ[F2] F)] [Fintype (F →ₗ[F2] E)] abbrev Index (d : ℕ) := Σ r : A13Index.BoundedPositiveTriple (E := E) (F := F) d, A5Passage.Passage r.val.2.2 (d - A13Index.size r.val) def descendant {d : ℕ} (a : Index (E := E) (F := F) d) : A13Index.Triple (E := E) (F := F) := ⟨a.2.2.val.1.comap a.1.val.1.mkQ, a.2.2.val.2.map a.1.val.2.1.subtype, NaturalTransport.dualFactor a.1.val.1 a.1.val.2.1 a.2.2.val.1 a.2.2.val.2 (LinearIdentities.compress a.1.val.2.2 a.2.2.val.1 a.2.2.val.2)⟩ def sourceI {d : ℕ} (a : Index (E := E) (F := F) d) : ℕ := Module.finrank F2 a.1.val.1 def sourceJ {d : ℕ} (a : Index (E := E) (F := F) d) : ℕ := Module.finrank F2 (F ⧸ a.1.val.2.1) def targetRank (r : A13Index.Triple (E := E) (F := F)) : ℕ := Module.finrank F2 r.2.2.range omit [FiniteDimensional F2 E] [Finite E] [Finite F] [Fintype (E →ₗ[F2] F)] [Fintype (F →ₗ[F2] E)] in theorem descendant_rank {d : ℕ} (a : Index (E := E) (F := F) d) : targetRank (descendant a) = Module.finrank F2 a.1.val.2.2.range := A5Passage.descendant_rank a.1.val.1 a.1.val.2.1 a.1.val.2.2 a.2 omit [FiniteDimensional F2 E] [Finite E] [Finite F] [Fintype (E →ₗ[F2] F)] [Fintype (F →ₗ[F2] E)] in theorem original_size {d : ℕ} (a : Index (E := E) (F := F) d) : sourceI a + sourceJ a + targetRank (descendant a) = A13Index.size a.1.val := by rw [descendant_rank] rfl omit [FiniteDimensional F2 E] [Finite E] [Finite F] [Fintype (E →ₗ[F2] F)] [Fintype (F →ₗ[F2] E)] in theorem original_order_bounds {d : ℕ} (a : Index (E := E) (F := F) d) : 1 ≤ sourceI a + sourceJ a + targetRank (descendant a) ∧ sourceI a + sourceJ a + targetRank (descendant a) ≤ d := by rw [original_size] exact a.1.property omit [Finite E] [Finite F] [Fintype (E →ₗ[F2] F)] [Fintype (F →ₗ[F2] E)] in theorem descendant_size_le {d : ℕ} (a : Index (E := E) (F := F) d) : A13Index.size (descendant a) ≤ d := A5Passage.effective_order_rank_le a.1.val.1 a.1.val.2.1 a.1.val.2.2 d a.1.property.2 a.2 omit [FiniteDimensional F2 E] [FiniteDimensional F2 F] [Finite E] [Finite F] [Fintype (E →ₗ[F2] F)] [Fintype (F →ₗ[F2] E)] in theorem earlierA_le {d : ℕ} (a : Index (E := E) (F := F) d) : a.1.val.1 ≤ (descendant a).1 := by intro x hx change a.1.val.1.mkQ x ∈ a.2.2.val.1 rw [show a.1.val.1.mkQ x = 0 from (Submodule.Quotient.mk_eq_zero _).mpr hx] exact a.2.2.val.1.zero_mem omit [FiniteDimensional F2 E] [FiniteDimensional F2 F] [Finite E] [Finite F] [Fintype (E →ₗ[F2] F)] [Fintype (F →ₗ[F2] E)] in theorem earlierB_le {d : ℕ} (a : Index (E := E) (F := F) d) : (descendant a).2.1 ≤ a.1.val.2.1 := by rintro x ⟨b, hb, rfl⟩ exact b.property abbrev Fiber (d : ℕ) (r : A13Index.Triple (E := E) (F := F)) (i j : ℕ) := {a : Index (E := E) (F := F) d // descendant a = r ∧ sourceI a = i ∧ sourceJ a = j} def encode (d : ℕ) (r : A13Index.Triple (E := E) (F := F)) (i j : ℕ) (a : Fiber d r i j) : A5ReverseIndex.Ancestor r.1 r.2.1 r.2.2 i j := by rcases a with ⟨a, hr, hi, hj⟩ cases hr refine ⟨⟨a.1.val.1, earlierA_le a, hi⟩, ⟨a.1.val.2.1, earlierB_le a, hj⟩, ⟨a.1.val.2.2, ?_⟩⟩ exact A5FrequencyTransport.isAntecedent_relativeFrequency_dualFactor a.1.val.1 a.1.val.2.1 a.2.2.val.1 a.2.2.val.2 (earlierA_le a) (earlierB_le a) a.1.val.2.2 a.2.2.property.1.symm a.2.2.property.2.2.1 def earlierTriple {A' : Submodule F2 E} {B' : Submodule F2 F} {Y : B' →ₗ[F2] (E ⧸ A')} {i j : ℕ} (a : A5ReverseIndex.Ancestor A' B' Y i j) : A13Index.Triple (E := E) (F := F) := ⟨a.1.val, a.2.1.val, a.2.2.val⟩ omit [Fintype (E →ₗ[F2] F)] [Fintype (F →ₗ[F2] E)] in omit [FiniteDimensional F2 E] [FiniteDimensional F2 F] [Finite E] [Finite F] in theorem earlierTriple_encode (d : ℕ) (r : A13Index.Triple (E := E) (F := F)) (i j : ℕ) (a : Fiber d r i j) : earlierTriple (encode d r i j a) = a.val.1.val := by rcases a with ⟨a, hr, hi, hj⟩ cases hr rfl omit [Fintype (E →ₗ[F2] F)] [Fintype (F →ₗ[F2] E)] in omit [FiniteDimensional F2 E] [FiniteDimensional F2 F] [Finite E] [Finite F] in theorem encode_injective (d : ℕ) (r : A13Index.Triple (E := E) (F := F)) (i j : ℕ) : Function.Injective (encode d r i j) := by intro a b hab have hraw : a.val.1.val = b.val.1.val := by simpa only [earlierTriple_encode] using congrArg earlierTriple hab have hbase : a.val.1 = b.val.1 := Subtype.ext hraw have hresult : descendant a.val = descendant b.val := a.property.1.trans b.property.1.symm rcases a with ⟨⟨r₁, p⟩, ha⟩ rcases b with ⟨⟨r₂, q⟩, hb⟩ change r₁ = r₂ at hbase cases hbase have hA := congrArg (fun z : A13Index.Triple (E := E) (F := F) => z.1) hresult have hB := congrArg (fun z : A13Index.Triple (E := E) (F := F) => z.2.1) hresult change p.2.val.1.comap r₁.val.1.mkQ = q.2.val.1.comap r₁.val.1.mkQ at hA change p.2.val.2.map r₁.val.2.1.subtype = q.2.val.2.map r₁.val.2.1.subtype at hB have hpq : p = q := A5Passage.forget_injective r₁.val.2.2 _ (Prod.ext (A5Passage.comap_mkQ_injective r₁.val.1 hA) (A5Passage.map_subtype_injective r₁.val.2.1 hB)) cases hpq rfl local instance indexFintype (d : ℕ) : Fintype (Index (E := E) (F := F) d) := by letI := A13Index.boundedPositiveTripleFintype (E := E) (F := F) d exact inferInstance variable [Fintype (Submodule F2 E)] [Fintype (Submodule F2 F)] omit [Fintype (E →ₗ[F2] F)] [Fintype (F →ₗ[F2] E)] in theorem card_fiber_le (d : ℕ) (r : A13Index.Triple (E := E) (F := F)) (i j : ℕ) : Nat.card (Fiber d r i j) ≤ 2 ^ (3 * (d * (i + j + targetRank r))) := by by_cases hn : Nonempty (Fiber d r i j) · obtain ⟨a⟩ := hn have hsize : A13Index.size r ≤ d := by rw [← a.property.1] exact descendant_size_le a.val have hA : Module.finrank F2 r.1 ≤ d := by unfold A13Index.size order at hsize omega have hB : Module.finrank F2 (F ⧸ r.2.1) ≤ d := by unfold A13Index.size order at hsize omega have ht : i + j + targetRank r ≤ d := by have h := (original_order_bounds a.val).2 rwa [a.property.1, a.property.2.1, a.property.2.2] at h have hcount := A5ReverseIndex.card_ancestor_le_source r.1 r.2.1 r.2.2 d i j (targetRank r) (le_refl _) hA hB ht exact (Nat.card_le_card_of_injective (encode d r i j) (encode_injective d r i j)).trans hcount · have : IsEmpty (Fiber d r i j) := ⟨fun a => hn ⟨a⟩⟩ simp end DFVSGames.Appendix.A5ForwardIndex end noncomputable section open scoped BigOperators namespace DFVSGames.Appendix.A5WeightedReindex attribute [local instance] Classical.propDecidable theorem exponent_weight_le (d t k : ℕ) (ht : t ≤ d) (hk : k ≤ t) : (2 : ℝ) ^ (100 * (d - t) ^ 2 + 27 * d * t + 6 * d * k) ≤ (2 : ℝ) ^ (100 * d * d) * ((2 : ℝ)⁻¹) ^ (63 * d * t + 4 * d * k) := by have he : (100 * (d - t) ^ 2 + 27 * d * t + 6 * d * k) + (63 * d * t + 4 * d * k) ≤ 100 * d * d := by simpa only [pow_two, Nat.mul_assoc, Nat.add_assoc] using Induction.induction_exponent_budget d t k ht hk have hp := pow_le_pow_right₀ (by norm_num : (1 : ℝ) ≤ 2) he rw [pow_add] at hp have hn : (2 : ℝ) ^ (63 * d * t + 4 * d * k) ≠ 0 := pow_ne_zero _ two_ne_zero have hm := mul_le_mul_of_nonneg_right hp (inv_nonneg.mpr (pow_nonneg (by norm_num : (0 : ℝ) ≤ 2) (63 * d * t + 4 * d * k))) rw [mul_assoc, mul_inv_cancel₀ hn, mul_one] at hm simpa only [inv_pow] using hm theorem counted_weight_le (d t k c : ℕ) (ht : t ≤ d) (hk : k ≤ t) (hc : c ≤ 2 ^ (3 * d * t)) : (c : ℝ) * (2 : ℝ) ^ (100 * (d - t) ^ 2 + 24 * d * t + 6 * d * k) ≤ (2 : ℝ) ^ (100 * d * d) * ((2 : ℝ)⁻¹) ^ (63 * d * t + 4 * d * k) := by have hc' : (c : ℝ) ≤ (2 : ℝ) ^ (3 * d * t) := by exact_mod_cast hc calc _ ≤ (2 : ℝ) ^ (3 * d * t) * (2 : ℝ) ^ (100 * (d - t) ^ 2 + 24 * d * t + 6 * d * k) := mul_le_mul_of_nonneg_right hc' (pow_nonneg (by norm_num) _) _ = (2 : ℝ) ^ (100 * (d - t) ^ 2 + 27 * d * t + 6 * d * k) := by rw [← pow_add] congr 1 ring _ ≤ _ := exponent_weight_le d t k ht hk variable {α β : Type*} [Fintype α] [Fintype β] def fiber (result : α → β) (i j : α → ℕ) (b : β) (r s : ℕ) : Finset α := Finset.univ.filter (fun a => result a = b ∧ i a = r ∧ j a = s) theorem sum_grouped (d : ℕ) (result : α → β) (i j : α → ℕ) (hbound : ∀ a, i a ≤ d ∧ j a ≤ d) (C : β → ℕ → ℕ → ℝ) : (∑ a, C (result a) (i a) (j a)) = ∑ b, ∑ r ∈ Finset.range (d + 1), ∑ s ∈ Finset.range (d + 1), ((fiber result i j b r s).card : ℝ) * C b r s := by classical let key : α → β × ℕ × ℕ := fun a => (result a, i a, j a) let target : Finset (β × ℕ × ℕ) := Finset.univ ×ˢ (Finset.range (d + 1) ×ˢ Finset.range (d + 1)) let G : β × ℕ × ℕ → ℝ := fun p => C p.1 p.2.1 p.2.2 have hmap : ∀ a ∈ (Finset.univ : Finset α), key a ∈ target := by intro a _ simp only [target, key, Finset.mem_product, Finset.mem_univ, Finset.mem_range, true_and] exact ⟨Nat.lt_succ_iff.mpr (hbound a).1, Nat.lt_succ_iff.mpr (hbound a).2⟩ have h := Finset.sum_fiberwise_of_maps_to' (s := Finset.univ) (t := target) (g := key) hmap G simpa only [target, Finset.sum_product, G, key, Prod.mk.injEq, Finset.sum_const, nsmul_eq_mul, fiber] using h.symm theorem weighted_reindex (d : ℕ) (hd : 1 ≤ d) (result : α → β) (i j : α → ℕ) (k : β → ℕ) (w : β → ℝ) (hw : ∀ b, 0 ≤ w b) (horder : ∀ a, 1 ≤ i a + j a + k (result a) ∧ i a + j a + k (result a) ≤ d) (hcard : ∀ b r s, Nat.card {a : α // result a = b ∧ i a = r ∧ j a = s} ≤ 2 ^ (3 * d * (r + s + k b))) : (∑ a, (2 : ℝ) ^ (100 * (d - (i a + j a + k (result a))) ^ 2 + 24 * d * (i a + j a + k (result a)) + 6 * d * k (result a)) * w (result a)) ≤ (2 : ℝ) ^ (100 * d * d) * ∑ b, ((2 : ℝ)⁻¹) ^ (31 * d * (k b + 1) + 4 * d * k b) * w b := by classical let C : β → ℕ → ℕ → ℝ := fun b r s => (2 : ℝ) ^ (100 * (d - (r + s + k b)) ^ 2 + 24 * d * (r + s + k b) + 6 * d * k b) * w b let envelope : β → ℝ := fun b => (2 : ℝ) ^ (100 * d * d) * ((2 : ℝ)⁻¹) ^ (4 * d * k b) * w b have henvelope : ∀ b, 0 ≤ envelope b := by intro b exact mul_nonneg (mul_nonneg (pow_nonneg (by norm_num) _) (pow_nonneg (by norm_num) _)) (hw b) have hbound : ∀ a, i a ≤ d ∧ j a ≤ d := by intro a have h := (horder a).2 omega have hfiber : ∀ b r s, ((fiber result i j b r s).card : ℝ) * C b r s ≤ envelope b * (if 1 ≤ r + s + k b then ((2 : ℝ)⁻¹) ^ (63 * d * (r + s + k b)) else 0) := by intro b r s by_cases hne : (fiber result i j b r s).Nonempty · obtain ⟨a, ha⟩ := hne rcases Finset.mem_filter.mp ha with ⟨_, hr, hi, hj⟩ have ht := horder a rw [hr, hi, hj] at ht have hc : (fiber result i j b r s).card ≤ 2 ^ (3 * d * (r + s + k b)) := by simpa only [Nat.card_eq_fintype_card, Fintype.card_subtype, fiber] using hcard b r s have hweight := counted_weight_le d (r + s + k b) (k b) (fiber result i j b r s).card ht.2 (by omega) hc rw [ite_eq_left ht.1] calc _ = (((fiber result i j b r s).card : ℝ) * (2 : ℝ) ^ (100 * (d - (r + s + k b)) ^ 2 + 24 * d * (r + s + k b) + 6 * d * k b)) * w b := by dsimp only [C] ring _ ≤ ((2 : ℝ) ^ (100 * d * d) * ((2 : ℝ)⁻¹) ^ (63 * d * (r + s + k b) + 4 * d * k b)) * w b := mul_le_mul_of_nonneg_right hweight (hw b) _ = envelope b * ((2 : ℝ)⁻¹) ^ (63 * d * (r + s + k b)) := by dsimp only [envelope] rw [pow_add] ring · have hz : (fiber result i j b r s).card = 0 := by exact Finset.card_eq_zero.mpr (Finset.not_nonempty_iff_eq_empty.mp hne) rw [hz, Nat.cast_zero, zero_mul] apply mul_nonneg (henvelope b) split_ifs <;> positivity change (∑ a, C (result a) (i a) (j a)) ≤ _ rw [sum_grouped d result i j hbound C] calc _ ≤ ∑ b, ∑ r ∈ Finset.range (d + 1), ∑ s ∈ Finset.range (d + 1), envelope b * (if 1 ≤ r + s + k b then ((2 : ℝ)⁻¹) ^ (63 * d * (r + s + k b)) else 0) := by apply Finset.sum_le_sum intro b _ apply Finset.sum_le_sum intro r _ exact Finset.sum_le_sum (fun s _ => hfiber b r s) _ = ∑ b, envelope b * (∑ r ∈ Finset.range (d + 1), ∑ s ∈ Finset.range (d + 1), if 1 ≤ r + s + k b then ((2 : ℝ)⁻¹) ^ (63 * d * (r + s + k b)) else 0) := by simp only [Finset.mul_sum] _ ≤ ∑ b, envelope b * ((2 : ℝ)⁻¹) ^ (31 * d * (k b + 1)) := by apply Finset.sum_le_sum intro b _ exact mul_le_mul_of_nonneg_left (A5Scalar.a16_original_range d (k b) hd) (henvelope b) _ = _ := by rw [Finset.mul_sum] apply Finset.sum_congr rfl intro b _ dsimp only [envelope] rw [pow_add] ring end DFVSGames.Appendix.A5WeightedReindex end noncomputable section namespace DFVSGames.Appendix.A5ReverseSum open scoped BigOperators open DFVSGames.Integration.BinaryLinear open DFVSGames.Fourier.MatrixRestrictions (order) open Derivatives open A5ForwardIndex attribute [local instance] Classical.propDecidable attribute [local instance] OperatorNorm.quotientFinite OperatorNorm.compressedDualFintype attribute [local instance] SubspaceExtensions.subspaceFintype variable {E F : Type*} [AddCommGroup E] [Module F2 E] [AddCommGroup F] [Module F2 F] [FiniteDimensional F2 E] [FiniteDimensional F2 F] [Finite E] [Finite F] [Fintype (E →ₗ[F2] F)] [Fintype (F →ₗ[F2] E)] def actualMoment (f : (E →ₗ[F2] F) → ℝ) (r : A13Index.Triple (E := E) (F := F)) : ℝ := 𝔼 T, (𝔼 N, mapDerivative r.2.2 (hybridDerivative r.1 r.2.1 T f) N ^ 2) ^ 2 def targetWeight (d : ℕ) (f : (E →ₗ[F2] F) → ℝ) (r : A13Index.Triple (E := E) (F := F)) : ℝ := if order r.1 r.2.1 ≤ d then actualMoment f r else 0 omit [FiniteDimensional F2 E] [FiniteDimensional F2 F] in theorem targetWeight_nonneg (d : ℕ) (f : (E →ₗ[F2] F) → ℝ) (r : A13Index.Triple (E := E) (F := F)) : 0 ≤ targetWeight d f r := by unfold targetWeight split_ifs · exact Finset.expect_nonneg fun _ _ => sq_nonneg _ · exact le_refl _ theorem targetWeight_descendant (d : ℕ) (f : (E →ₗ[F2] F) → ℝ) (a : Index (E := E) (F := F) d) : targetWeight d f (descendant a) = actualMoment f (descendant a) := by have h := descendant_size_le a have ho : order (descendant a).1 (descendant a).2.1 ≤ d := by unfold A13Index.size at h omega exact ite_eq_left ho omit [FiniteDimensional F2 E] [FiniteDimensional F2 F] [Fintype (E →ₗ[F2] F)] [Fintype (F →ₗ[F2] E)] in theorem sum_triples_cutoff (d : ℕ) (H : ∀ (A : Submodule F2 E) (B : Submodule F2 F), (B →ₗ[F2] (E ⧸ A)) → ℝ) : (∑ r : A13Index.Triple (E := E) (F := F), if order r.1 r.2.1 ≤ d then H r.1 r.2.1 r.2.2 else 0) = ∑ s : HybridIndex (E := E) (F := F) d, ∑ Y : s.val.2 →ₗ[F2] (E ⧸ s.val.1), H s.val.1 s.val.2 Y := by classical let G : Submodule F2 E × Submodule F2 F → ℝ := fun p => ∑ Y : p.2 →ₗ[F2] (E ⧸ p.1), H p.1 p.2 Y have hh : (∑ s : HybridIndex (E := E) (F := F) d, G s.val) = ∑ p : Submodule F2 E × Submodule F2 F, if order p.1 p.2 ≤ d then G p else 0 := by calc _ = ∑ p ∈ Finset.univ.filter (fun p : Submodule F2 E × Submodule F2 F => order p.1 p.2 ≤ d), G p := by exact (Finset.sum_subtype _ (by simp) G).symm _ = _ := Finset.sum_filter _ _ change _ = ∑ s : HybridIndex (E := E) (F := F) d, G s.val rw [hh, Fintype.sum_prod_type] rw [Fintype.sum_sigma] apply Finset.sum_congr rfl intro A _ rw [Fintype.sum_sigma] apply Finset.sum_congr rfl intro B _ by_cases h : order A B ≤ d · simp only [h, ite_true, G] · simp only [h, ite_false, Finset.sum_const_zero] omit [FiniteDimensional F2 E] [FiniteDimensional F2 F] in theorem target_sum_eq (d : ℕ) (f : (E →ₗ[F2] F) → ℝ) : (∑ r : A13Index.Triple (E := E) (F := F), ((2 : ℝ)⁻¹) ^ (31 * d * (targetRank r + 1) + 4 * d * targetRank r) * targetWeight d f r) = ∑ s : HybridIndex (E := E) (F := F) d, 𝔼 T, ∑ Y : s.val.2 →ₗ[F2] (E ⧸ s.val.1), ((2 : ℝ)⁻¹) ^ (31 * d * (Module.finrank F2 Y.range + 1) + 4 * d * Module.finrank F2 Y.range) * (𝔼 N, mapDerivative Y (hybridDerivative s.val.1 s.val.2 T f) N ^ 2) ^ 2 := by let H : ∀ (A : Submodule F2 E) (B : Submodule F2 F), (B →ₗ[F2] (E ⧸ A)) → ℝ := fun A B Y => ((2 : ℝ)⁻¹) ^ (31 * d * (Module.finrank F2 Y.range + 1) + 4 * d * Module.finrank F2 Y.range) * actualMoment f ⟨A, B, Y⟩ calc _ = ∑ r : A13Index.Triple (E := E) (F := F), if order r.1 r.2.1 ≤ d then H r.1 r.2.1 r.2.2 else 0 := by apply Finset.sum_congr rfl rintro ⟨A, B, Y⟩ _ simp only [targetWeight, targetRank, H, mul_ite, mul_zero] _ = ∑ s : HybridIndex (E := E) (F := F) d, ∑ Y : s.val.2 →ₗ[F2] (E ⧸ s.val.1), H s.val.1 s.val.2 Y := sum_triples_cutoff d H _ = _ := ?_ apply Finset.sum_congr rfl intro s _ rw [Finset.expect_sum_comm] apply Finset.sum_congr rfl intro Y _ rw [← Finset.mul_expect] rfl theorem source_summand_eq (d : ℕ) (f : (E →ₗ[F2] F) → ℝ) (a : Index (E := E) (F := F) d) : (2 : ℝ) ^ (100 * (d - (sourceI a + sourceJ a + targetRank (descendant a))) ^ 2 + 24 * d * (sourceI a + sourceJ a + targetRank (descendant a)) + 6 * d * targetRank (descendant a)) * targetWeight d f (descendant a) = (2 : ℝ) ^ (100 * (d - A13Index.size a.1.val) ^ 2 + 24 * d * A13Index.size a.1.val + 6 * d * Module.finrank F2 a.1.val.2.2.range) * actualMoment f (descendant a) := by rw [original_size, descendant_rank, targetWeight_descendant] theorem source_sum_eq (d : ℕ) (f : (E →ₗ[F2] F) → ℝ) : (∑ a : Index (E := E) (F := F) d, (2 : ℝ) ^ (100 * (d - (sourceI a + sourceJ a + targetRank (descendant a))) ^ 2 + 24 * d * (sourceI a + sourceJ a + targetRank (descendant a)) + 6 * d * targetRank (descendant a)) * targetWeight d f (descendant a)) = ∑ r : A13Index.BoundedPositiveTriple (E := E) (F := F) d, (2 : ℝ) ^ (100 * (d - A13Index.size r.val) ^ 2 + 24 * d * A13Index.size r.val + 6 * d * Module.finrank F2 r.val.2.2.range) * A5Passage.descendantSum r.val.1 r.val.2.1 r.val.2.2 (d - A13Index.size r.val) f := by simp_rw [source_summand_eq] rw [Fintype.sum_sigma] apply Finset.sum_congr rfl intro r _ rw [A5Passage.descendantSum, Finset.mul_sum] apply Finset.sum_congr rfl intro q _ rfl theorem weighted_descendantSum_le (d : ℕ) (hd : 1 ≤ d) (f : (E →ₗ[F2] F) → ℝ) : (∑ r : A13Index.BoundedPositiveTriple (E := E) (F := F) d, (2 : ℝ) ^ (100 * (d - A13Index.size r.val) ^ 2 + 24 * d * A13Index.size r.val + 6 * d * Module.finrank F2 r.val.2.2.range) * A5Passage.descendantSum r.val.1 r.val.2.1 r.val.2.2 (d - A13Index.size r.val) f) ≤ (2 : ℝ) ^ (100 * d * d) * ∑ s : HybridIndex (E := E) (F := F) d, 𝔼 T, ∑ Y : s.val.2 →ₗ[F2] (E ⧸ s.val.1), ((2 : ℝ)⁻¹) ^ (31 * d * (Module.finrank F2 Y.range + 1) + 4 * d * Module.finrank F2 Y.range) * (𝔼 N, mapDerivative Y (hybridDerivative s.val.1 s.val.2 T f) N ^ 2) ^ 2 := by have h := A5WeightedReindex.weighted_reindex d hd (descendant (E := E) (F := F) (d := d)) sourceI sourceJ targetRank (targetWeight d f) (targetWeight_nonneg d f) original_order_bounds (fun r i j => by simpa only [Nat.mul_assoc] using card_fiber_le d r i j) rwa [source_sum_eq, target_sum_eq] at h end DFVSGames.Appendix.A5ReverseSum end noncomputable section namespace DFVSGames.Appendix.A5Induction open scoped BigOperators open DFVSGames.Fourier.MatrixCharacters DFVSGames.Fourier.MatrixFourier open DFVSGames.Fourier.MatrixRestrictions open DFVSGames.Appendix.Derivatives DFVSGames.Appendix.DerivativeDegree open DFVSGames.Appendix.LevelInequality (frequencyRank) attribute [local instance] Classical.propDecidable attribute [local instance] OperatorNorm.quotientFinite OperatorNorm.compressedDualFintype attribute [local instance] SubspaceExtensions.subspaceFintype universe u v def BoundAtDegree (d : ℕ) : Prop := ∀ {E : Type u} {F : Type v} [AddCommGroup E] [Module F2 E] [AddCommGroup F] [Module F2 F] [FiniteDimensional F2 E] [FiniteDimensional F2 F] [Finite E] [Finite F] [Fintype (E →ₗ[F2] F)] [Fintype (F →ₗ[F2] E)] (f : (E →ₗ[F2] F) → ℝ), DegreeAtMost d f → (𝔼 M, f M ^ 4) ≤ (2 : ℝ) ^ (100 * d * d) * hybridMomentSum d f theorem damping_le (d k : ℕ) : (1 / 2 : ℝ) ^ (31 * d * (k + 1) + 4 * d * k) ≤ (1 / 2 : ℝ) ^ (31 * d) * (1 / 2 : ℝ) ^ (8 * d * k) := by have hpow (n : ℕ) : (1 / 2 : ℝ) ^ n ≤ 1 := by induction n with | zero => norm_num | succ n ih => rw [pow_succ] have hn := pow_nonneg (by norm_num : (0 : ℝ) ≤ 1 / 2) n nlinarith have he : 31 * d * (k + 1) + 4 * d * k = (31 * d + 8 * d * k) + 27 * d * k := by ring rw [he, pow_add, pow_add] have hh := mul_le_mul_of_nonneg_left (hpow (27 * d * k)) (mul_nonneg (pow_nonneg (by norm_num : (0 : ℝ) ≤ 1 / 2) (31 * d)) (pow_nonneg (by norm_num : (0 : ℝ) ≤ 1 / 2) (8 * d * k))) simpa only [mul_one] using hh theorem coefficient_absorption (d : ℕ) (hd : 1 ≤ d) : (162 : ℝ) * ((2 : ℝ) ^ (6 * d * d) + (2 : ℝ) ^ (100 * d * d) * (2 * (1 / 2 : ℝ) ^ (31 * d))) ≤ (2 : ℝ) ^ (100 * d * d) := by have hfirst : (2 : ℝ) ^ (100 * d * d) * ((2 : ℝ) ^ (94 * d * d))⁻¹ = (2 : ℝ) ^ (6 * d * d) := by have he : 100 * d * d = 6 * d * d + 94 * d * d := by ring rw [he, pow_add, mul_assoc, mul_inv_cancel₀ (pow_ne_zero _ (by norm_num : (2 : ℝ) ≠ 0)), mul_one] have he : 31 * d = (31 * d - 1) + 1 := by omega have hsecond : ((2 : ℝ) ^ (31 * d - 1))⁻¹ = 2 * (1 / 2 : ℝ) ^ (31 * d) := by calc ((2 : ℝ) ^ (31 * d - 1))⁻¹ = (1 / 2 : ℝ) ^ (31 * d - 1) := by rw [one_div, inv_pow] _ = 2 * ((1 / 2 : ℝ) ^ (31 * d - 1) * (1 / 2 : ℝ)) := by ring _ = _ := by rw [← pow_succ, ← he] have hscaled := mul_le_mul_of_nonneg_right (A5Scalar.absorption hd) (pow_nonneg (by norm_num : (0 : ℝ) ≤ 2) (100 * d * d)) calc (162 : ℝ) * ((2 : ℝ) ^ (6 * d * d) + (2 : ℝ) ^ (100 * d * d) * (2 * (1 / 2 : ℝ) ^ (31 * d))) = 162 * (((2 : ℝ) ^ (94 * d * d))⁻¹ + ((2 : ℝ) ^ (31 * d - 1))⁻¹) * (2 : ℝ) ^ (100 * d * d) := by symm calc _ = 162 * ((2 : ℝ) ^ (100 * d * d) * ((2 : ℝ) ^ (94 * d * d))⁻¹ + (2 : ℝ) ^ (100 * d * d) * ((2 : ℝ) ^ (31 * d - 1))⁻¹) := by ring _ = _ := by rw [hfirst, hsecond] _ ≤ _ := by simpa only [one_mul] using hscaled variable {E : Type u} {F : Type v} variable [AddCommGroup E] [Module F2 E] [AddCommGroup F] [Module F2 F] variable [FiniteDimensional F2 E] [FiniteDimensional F2 F] variable [Finite E] [Finite F] variable [Fintype (E →ₗ[F2] F)] [Fintype (F →ₗ[F2] E)] def dampedMapMoment (d : ℕ) (f : (E →ₗ[F2] F) → ℝ) : ℝ := ∑ X : F →ₗ[F2] E, (1 / 2 : ℝ) ^ (31 * d * (frequencyRank X + 1) + 4 * d * frequencyRank X) * (𝔼 N, mapDerivative X f N ^ 2) ^ 2 theorem dampedMapMoment_le (d : ℕ) (f : (E →ₗ[F2] F) → ℝ) (hf : DegreeAtMost d f) : dampedMapMoment d f ≤ (2 * (1 / 2 : ℝ) ^ (31 * d)) * (𝔼 M, f M ^ 2) ^ 2 := by have hterms : dampedMapMoment d f ≤ (1 / 2 : ℝ) ^ (31 * d) * ∑ X : F →ₗ[F2] E, (1 / 2 : ℝ) ^ (8 * d * frequencyRank X) * (𝔼 N, mapDerivative X f N ^ 2) ^ 2 := by unfold dampedMapMoment rw [Finset.mul_sum] apply Finset.sum_le_sum intro X _ have hh := mul_le_mul_of_nonneg_right (damping_le d (frequencyRank X)) (sq_nonneg (𝔼 N, mapDerivative X f N ^ 2)) simpa only [mul_assoc] using hh have hbase := weighted_fourth_derivative_energy f d (fun Y hY => hf Y hY) have hbase' : (∑ X : F →ₗ[F2] E, (1 / 2 : ℝ) ^ (8 * d * frequencyRank X) * (𝔼 N, mapDerivative X f N ^ 2)^2) ≤ 2 * (𝔼 M, f M ^ 2)^2 := by simpa only [one_div, frequencyRank] using hbase calc dampedMapMoment d f ≤ _ := hterms _ ≤ (1 / 2 : ℝ) ^ (31 * d) * (2 * (𝔼 M, f M ^ 2) ^ 2) := mul_le_mul_of_nonneg_left hbase' (pow_nonneg (by norm_num : (0 : ℝ) ≤ 1 / 2) _) _ = _ := by ring def dampedHybridMoment (d : ℕ) (f : (E →ₗ[F2] F) → ℝ) : ℝ := ∑ s : HybridIndex (E := E) (F := F) d, 𝔼 T, dampedMapMoment d (hybridDerivative s.val.1 s.val.2 T f) theorem dampedHybridMoment_le (d : ℕ) (f : (E →ₗ[F2] F) → ℝ) (hf : DegreeAtMost d f) : dampedHybridMoment d f ≤ (2 * (1 / 2 : ℝ) ^ (31 * d)) * hybridMomentSum d f := by unfold dampedHybridMoment hybridMomentSum rw [Finset.mul_sum] apply Finset.sum_le_sum intro s _ calc (𝔼 T, dampedMapMoment d (hybridDerivative s.val.1 s.val.2 T f)) ≤ 𝔼 T, (2 * (1 / 2 : ℝ) ^ (31 * d)) * (𝔼 N, hybridDerivative s.val.1 s.val.2 T f N ^ 2) ^ 2 := by apply Finset.expect_le_expect intro T _ exact dampedMapMoment_le d _ ((degree_hybridDerivative s.val.1 s.val.2 T f hf).mono (Nat.sub_le _ _)) _ = _ := by rw [← Finset.mul_expect] omit [Finite E] [Finite F] in theorem hybridProjector_zero_pair (f : (E →ₗ[F2] F) → ℝ) : hybridProjector (⊥ : Submodule F2 E) (⊤ : Submodule F2 F) f = f := by apply function_eq_of_linearCoeff_eq intro Y simp [hybridProjector, linearCoeff_spectralProjector, LinearIdentities.Hybrid] theorem energy_sq_le_hybridMomentSum (d : ℕ) (f : (E →ₗ[F2] F) → ℝ) : (𝔼 M, f M ^ 2) ^ 2 ≤ hybridMomentSum d f := by have hzero : Module.finrank F2 (F ⧸ (⊤ : Submodule F2 F)) = 0 := by have hq := (⊤ : Submodule F2 F).finrank_quotient_add_finrank have he : Module.finrank F2 (F ⧸ (⊤ : Submodule F2 F)) + Module.finrank F2 F = Module.finrank F2 F := by simpa using hq omega let s0 : HybridIndex (E := E) (F := F) d := ⟨(⊥, ⊤), by change Module.finrank F2 (⊥ : Submodule F2 E) + Module.finrank F2 (F ⧸ (⊤ : Submodule F2 F)) ≤ d rw [hzero] simp⟩ have havg := hybridDerivative_energy_average (⊥ : Submodule F2 E) (⊤ : Submodule F2 F) f rw [hybridProjector_zero_pair] at havg have hcs : (𝔼 T, 𝔼 N, hybridDerivative (⊥ : Submodule F2 E) (⊤ : Submodule F2 F) T f N ^ 2) ^ 2 ≤ 𝔼 T, (𝔼 N, hybridDerivative (⊥ : Submodule F2 E) (⊤ : Submodule F2 F) T f N ^ 2) ^ 2 := by simpa using Finset.expect_mul_sq_le_sq_mul_sq Finset.univ (fun T => 𝔼 N, hybridDerivative (⊥ : Submodule F2 E) (⊤ : Submodule F2 F) T f N ^ 2) (fun _ => (1 : ℝ)) rw [havg] at hcs apply hcs.trans change (𝔼 T, (𝔼 N, hybridDerivative s0.val.1 s0.val.2 T f N ^ 2) ^ 2) ≤ _ unfold hybridMomentSum let H : HybridIndex (E := E) (F := F) d → ℝ := fun s => 𝔼 T, (𝔼 N, hybridDerivative s.val.1 s.val.2 T f N ^ 2)^2 change H s0 ≤ ∑ s, H s exact Finset.single_le_sum (f := H) (s := Finset.univ) (fun s _ => Finset.expect_nonneg (fun T _ => sq_nonneg (𝔼 N, hybridDerivative s.val.1 s.val.2 T f N ^ 2))) (Finset.mem_univ s0) theorem degree_zero_bound (f : (E →ₗ[F2] F) → ℝ) (hf : DegreeAtMost 0 f) : (𝔼 M, f M ^ 4) ≤ (2 : ℝ) ^ (100 * 0 * 0) * hybridMomentSum 0 f := by rw [DegreeZero.fourth_moment_eq_energy_sq f (fun Y hY => hf Y hY)] simpa using energy_sq_le_hybridMomentSum 0 f theorem recurrence_closure (d : ℕ) (hd : 1 ≤ d) (f : (E →ₗ[F2] F) → ℝ) (hf : DegreeAtMost d f) (hrec : (𝔼 M, f M ^ 4) / 162 ≤ (2 : ℝ) ^ (6 * d * d) * (𝔼 M, f M ^ 2)^2 + (2 : ℝ) ^ (100 * d * d) * dampedHybridMoment d f) : (𝔼 M, f M ^ 4) ≤ (2 : ℝ) ^ (100 * d * d) * hybridMomentSum d f := by have hS : 0 ≤ hybridMomentSum d f := Finset.sum_nonneg (fun s _ => Finset.expect_nonneg (fun T _ => sq_nonneg _)) have herror := mul_le_mul_of_nonneg_left (energy_sq_le_hybridMomentSum d f) (pow_nonneg (by norm_num : (0 : ℝ) ≤ 2) (6 * d * d)) have hdamp := mul_le_mul_of_nonneg_left (dampedHybridMoment_le d f hf) (pow_nonneg (by norm_num : (0 : ℝ) ≤ 2) (100 * d * d)) have hrec' : (𝔼 M, f M ^ 4) / 162 ≤ ((2 : ℝ) ^ (6 * d * d) + (2 : ℝ) ^ (100 * d * d) * (2 * (1 / 2 : ℝ) ^ (31 * d))) * hybridMomentSum d f := by calc _ ≤ _ := hrec _ ≤ (2 : ℝ) ^ (6 * d * d) * hybridMomentSum d f + (2 : ℝ) ^ (100 * d * d) * ((2 * (1 / 2 : ℝ) ^ (31 * d)) * hybridMomentSum d f) := add_le_add herror hdamp _ = _ := by ring calc (𝔼 M, f M ^ 4) = 162 * ((𝔼 M, f M ^ 4) / 162) := by ring _ ≤ 162 * (((2 : ℝ) ^ (6 * d * d) + (2 : ℝ) ^ (100 * d * d) * (2 * (1 / 2 : ℝ) ^ (31 * d))) * hybridMomentSum d f) := mul_le_mul_of_nonneg_left hrec' (by norm_num) _ = (162 * ((2 : ℝ) ^ (6 * d * d) + (2 : ℝ) ^ (100 * d * d) * (2 * (1 / 2 : ℝ) ^ (31 * d)))) * hybridMomentSum d f := by ring _ ≤ _ := mul_le_mul_of_nonneg_right (coefficient_absorption d hd) hS omit [Finite E] [Finite F] [Fintype (E →ₗ[F2] F)] [Fintype (F →ₗ[F2] E)] in theorem lifted_outer_dimensions (X : F →ₗ[F2] E) (C : Submodule F2 (E ⧸ X.range)) (D : Submodule F2 X.ker) (d : ℕ) (hcut : Module.finrank F2 X.range + order C D ≤ d) : Module.finrank F2 (C.comap X.range.mkQ) ≤ d ∧ Module.finrank F2 (F ⧸ D.map X.ker.subtype) ≤ d := by have hC := HybridComposition.finrank_comap_mkQ X.range C have hD := Submodule.finrank_map_subtype_eq X.ker D have hKD := D.finrank_quotient_add_finrank have hFD := (D.map X.ker.subtype).finrank_quotient_add_finrank have hX := X.finrank_range_add_finrank_ker change Module.finrank F2 X.range + (Module.finrank F2 C + Module.finrank F2 (X.ker ⧸ D)) ≤ d at hcut constructor <;> omega theorem a5_energySquare_le_power (X : F →ₗ[F2] E) (C : Submodule F2 (E ⧸ X.range)) (D : Submodule F2 X.ker) [Fintype (OperatorPartitions.A5GeometricIndex X (C.comap X.range.mkQ) (D.map X.ker.subtype))] (d : ℕ) (hcut : Module.finrank F2 X.range + order C D ≤ d) (f : (E →ₗ[F2] F) → ℝ) (S : Parameter X.range X.ker) : (𝔼 N, hybridDerivative C D S (mapDerivative X f) N ^ 2) ^ 2 ≤ (2 : ℝ) ^ (6 * d * Module.finrank F2 X.range) * ∑ p : OperatorPartitions.A5GeometricIndex X (C.comap X.range.mkQ) (D.map X.ker.subtype), (𝔼 N, mapDerivative (LinearIdentities.compress X p.val.1 p.val.2) (hybridDerivative p.val.1 p.val.2 (embed X.range X.ker S) f) N ^ 2)^2 := by have hI : X.range ≤ C.comap X.range.mkQ := by intro x hx change X.range.mkQ x ∈ C have hz : X.range.mkQ x = 0 := by change x ∈ LinearMap.ker X.range.mkQ simpa only [Submodule.ker_mkQ] using hx rw [hz] exact C.zero_mem have hB : D.map X.ker.subtype ≤ X.ker := by intro x hx obtain ⟨y, hy, rfl⟩ := Submodule.mem_map.mp hx exact y.property have hCmap : (C.comap X.range.mkQ).map X.range.mkQ = C := by ext x constructor · intro hx obtain ⟨y, hy, rfl⟩ := Submodule.mem_map.mp hx exact hy · intro hx obtain ⟨y, rfl⟩ := X.range.mkQ_surjective x exact Submodule.mem_map.mpr ⟨y, hx, rfl⟩ have hDcomap : (D.map X.ker.subtype).comap X.ker.subtype = D := by ext x constructor · intro hx obtain ⟨y, hy, heq⟩ := Submodule.mem_map.mp hx have hxy : y = x := Subtype.ext heq simpa only [hxy] using hy · intro hx exact Submodule.mem_map.mpr ⟨x, hx, rfl⟩ have hdims := lifted_outer_dimensions X C D d hcut have hcount := A5IndexCount.card_geometricIndex_cube_le X (C.comap X.range.mkQ) (D.map X.ker.subtype) hI d hdims.1 hdims.2 rw [Nat.card_eq_fintype_card] at hcount have hcountR : (Fintype.card (OperatorPartitions.A5GeometricIndex X (C.comap X.range.mkQ) (D.map X.ker.subtype)) : ℝ)^3 ≤ (2 : ℝ) ^ (6 * d * Module.finrank F2 X.range) := by exact_mod_cast hcount have hn := OperatorNorm.a5_energySquare_le X (C.comap X.range.mkQ) (D.map X.ker.subtype) hI hB f S have hfinal := hn.trans (mul_le_mul_of_nonneg_right hcountR (Finset.sum_nonneg (fun p _ => sq_nonneg _))) let energy (C' : Submodule F2 (E ⧸ X.range)) (D' : Submodule F2 X.ker) : ℝ := (𝔼 N, hybridDerivative C' D' S (mapDerivative X f) N ^ 2)^2 have he : energy ((C.comap X.range.mkQ).map X.range.mkQ) ((D.map X.ker.subtype).comap X.ker.subtype) = energy C D := congrArg₂ energy hCmap hDcomap change energy ((C.comap X.range.mkQ).map X.range.mkQ) ((D.map X.ker.subtype).comap X.ker.subtype) ≤ _ at hfinal rw [he] at hfinal exact hfinal def postA5Sum (X : F →ₗ[F2] E) (e : ℕ) (f : (E →ₗ[F2] F) → ℝ) : ℝ := ∑ q : A5Passage.Passage X e, 𝔼 S : Parameter X.range X.ker, (𝔼 N, mapDerivative (LinearIdentities.compress X q.2.val.1 q.2.val.2) (hybridDerivative q.2.val.1 q.2.val.2 (embed X.range X.ker S) f) N ^ 2)^2 theorem mapDerivative_hybridMomentSum_le (d e : ℕ) (X : F →ₗ[F2] E) (f : (E →ₗ[F2] F) → ℝ) (hcut : Module.finrank F2 X.range + e ≤ d) : hybridMomentSum e (mapDerivative X f) ≤ (2 : ℝ) ^ (6 * d * Module.finrank F2 X.range) * postA5Sum X e f := by unfold hybridMomentSum postA5Sum rw [Fintype.sum_sigma, Finset.mul_sum] apply Finset.sum_le_sum intro s _ let : Fintype (OperatorPartitions.A5GeometricIndex X (s.val.1.comap X.range.mkQ) (s.val.2.map X.ker.subtype)) := A5Passage.pairFintype X s have hs : Module.finrank F2 X.range + order s.val.1 s.val.2 ≤ d := by have ho := s.property omega calc (𝔼 S, (𝔼 N, hybridDerivative s.val.1 s.val.2 S (mapDerivative X f) N ^ 2)^2) ≤ 𝔼 S, (2 : ℝ) ^ (6 * d * Module.finrank F2 X.range) * ∑ p : A5Passage.Pair X s, (𝔼 N, mapDerivative (LinearIdentities.compress X p.val.1 p.val.2) (hybridDerivative p.val.1 p.val.2 (embed X.range X.ker S) f) N ^ 2)^2 := by apply Finset.expect_le_expect intro S _ exact a5_energySquare_le_power X s.val.1 s.val.2 d hs f S _ = _ := by rw [← Finset.mul_expect, Finset.expect_sum_comm] theorem postA5Sum_hybrid_average (A : Submodule F2 E) (B : Submodule F2 F) (X : B →ₗ[F2] (E ⧸ A)) (e : ℕ) (f : (E →ₗ[F2] F) → ℝ) : (𝔼 T, postA5Sum X e (hybridDerivative A B T f)) = A5Passage.descendantSum A B X e f := by unfold postA5Sum A5Passage.descendantSum rw [Finset.expect_sum_comm] apply Finset.sum_congr rfl intro q _ simp_rw [NaturalTransport.nested_mapDerivative_energy A B q.2.val.1 q.2.val.2 f] exact Restriction.real_translation_average (fun S : Parameter X.range X.ker => embed A B (embed X.range X.ker S)) (fun U => (𝔼 N, mapDerivative (NaturalTransport.dualFactor A B q.2.val.1 q.2.val.2 (LinearIdentities.compress X q.2.val.1 q.2.val.2)) (hybridDerivative (q.2.val.1.comap A.mkQ) (q.2.val.2.map B.subtype) U f) N ^ 2)^2) theorem a15_of_strong_induction (d t : ℕ) (A : Submodule F2 E) (B : Submodule F2 F) (X : B →ₗ[F2] (E ⧸ A)) (ht : t = order A B + Module.finrank F2 X.range) (htpos : 0 < t) (htd : t ≤ d) (ih : ∀ e, e < d → BoundAtDegree.{u, v} e) (f : (E →ₗ[F2] F) → ℝ) (hf : DegreeAtMost d f) : (𝔼 T, 𝔼 N, mapDerivative X (hybridDerivative A B T f) N ^ 4) ≤ (2 : ℝ) ^ (100 * (d - t) * (d - t) + 6 * d * Module.finrank F2 X.range) * A5Passage.descendantSum A B X (d - t) f := by have hlt : d - t < d := by omega have hcut : Module.finrank F2 X.range + (d - t) ≤ d := by omega have hpoint (T : E →ₗ[F2] F) : (𝔼 N, mapDerivative X (hybridDerivative A B T f) N ^ 4) ≤ (2 : ℝ) ^ (100 * (d - t) * (d - t) + 6 * d * Module.finrank F2 X.range) * postA5Sum X (d - t) (hybridDerivative A B T f) := by have hdeg : DegreeAtMost (d - t) (mapDerivative X (hybridDerivative A B T f)) := by simpa only [ht, frequencyRank] using degree_map_hybridDerivative A B X T f hf have hIH := ih (d - t) hlt (mapDerivative X (hybridDerivative A B T f)) hdeg have hnorm := mapDerivative_hybridMomentSum_le d (d - t) X (hybridDerivative A B T f) hcut calc (𝔼 N, mapDerivative X (hybridDerivative A B T f) N ^ 4) ≤ (2 : ℝ) ^ (100 * (d - t) * (d - t)) * hybridMomentSum (d - t) (mapDerivative X (hybridDerivative A B T f)) := hIH _ ≤ (2 : ℝ) ^ (100 * (d - t) * (d - t)) * ((2 : ℝ) ^ (6 * d * Module.finrank F2 X.range) * postA5Sum X (d - t) (hybridDerivative A B T f)) := mul_le_mul_of_nonneg_left hnorm (pow_nonneg (by norm_num : (0 : ℝ) ≤ 2) _) _ = _ := by rw [pow_add]; ring calc (𝔼 T, 𝔼 N, mapDerivative X (hybridDerivative A B T f) N ^ 4) ≤ 𝔼 T, (2 : ℝ) ^ (100 * (d - t) * (d - t) + 6 * d * Module.finrank F2 X.range) * postA5Sum X (d - t) (hybridDerivative A B T f) := Finset.expect_le_expect (fun T _ => hpoint T) _ = _ := by rw [← Finset.mul_expect, postA5Sum_hybrid_average] theorem boundedTripleSum_le_descendants (d : ℕ) (ih : ∀ e, e < d → BoundAtDegree.{u, v} e) (f : (E →ₗ[F2] F) → ℝ) (hf : DegreeAtMost d f) : A13Reduction.boundedPositiveTripleSum d f ≤ ∑ p : A13Index.BoundedPositiveTriple (E := E) (F := F) d, (2 : ℝ) ^ (100 * (d - A13Index.size p.val)^2 + 24 * d * A13Index.size p.val + 6 * d * Module.finrank F2 p.val.2.2.range) * A5Passage.descendantSum p.val.1 p.val.2.1 p.val.2.2 (d - A13Index.size p.val) f := by unfold A13Reduction.boundedPositiveTripleSum apply Finset.sum_le_sum intro p _ have h15 := a15_of_strong_induction d (A13Index.size p.val) p.val.1 p.val.2.1 p.val.2.2 rfl p.property.1 p.property.2 ih f hf have hw := mul_le_mul_of_nonneg_left h15 (pow_nonneg (by norm_num : (0 : ℝ) ≤ 2) (24 * d * A13Index.size p.val)) have he : 24 * d * A13Index.size p.val + (100 * (d - A13Index.size p.val) * (d - A13Index.size p.val) + 6 * d * Module.finrank F2 p.val.2.2.range) = 100 * (d - A13Index.size p.val)^2 + 24 * d * A13Index.size p.val + 6 * d * Module.finrank F2 p.val.2.2.range := by ring calc A13Reduction.tripleTerm d p.val f ≤ (2 : ℝ) ^ (24 * d * A13Index.size p.val) * ((2 : ℝ) ^ (100 * (d - A13Index.size p.val) * (d - A13Index.size p.val) + 6 * d * Module.finrank F2 p.val.2.2.range) * A5Passage.descendantSum p.val.1 p.val.2.1 p.val.2.2 (d - A13Index.size p.val) f) := hw _ = ((2 : ℝ) ^ (24 * d * A13Index.size p.val) * (2 : ℝ) ^ (100 * (d - A13Index.size p.val) * (d - A13Index.size p.val) + 6 * d * Module.finrank F2 p.val.2.2.range)) * A5Passage.descendantSum p.val.1 p.val.2.1 p.val.2.2 (d - A13Index.size p.val) f := by ring _ = _ := by rw [← pow_add, he] theorem boundAtDegree (d : ℕ) : BoundAtDegree.{u, v} d := by induction d using Nat.strong_induction_on with | h d ih => intro E F instE instME instF instMF instDE instDF instFiniteE instFiniteF instHom instDual f hf by_cases hz : d = 0 · subst d exact degree_zero_bound f hf · have hd : 1 ≤ d := by omega have hreverse : (∑ p : A13Index.BoundedPositiveTriple (E := E) (F := F) d, (2 : ℝ) ^ (100 * (d - A13Index.size p.val)^2 + 24 * d * A13Index.size p.val + 6 * d * Module.finrank F2 p.val.2.2.range) * A5Passage.descendantSum p.val.1 p.val.2.1 p.val.2.2 (d - A13Index.size p.val) f) ≤ (2 : ℝ) ^ (100 * d * d) * dampedHybridMoment d f := by simpa only [dampedHybridMoment, dampedMapMoment, frequencyRank, one_div] using A5ReverseSum.weighted_descendantSum_le d hd f have hbounded := (boundedTripleSum_le_descendants d ih f hf).trans hreverse have hA13 := A13Reduction.degree_reduction_truncated f d hf apply recurrence_closure d hd f hf simpa only [pow_two, Nat.mul_assoc] using hA13.trans (add_le_add le_rfl hbounded) theorem derivative_inequality (d : ℕ) (f : (E →ₗ[F2] F) → ℝ) (hf : DegreeAtMost d f) : (𝔼 M, f M ^ 4) ≤ (2 : ℝ) ^ (100 * d ^ 2) * hybridMomentSum d f := by simpa only [pow_two, Nat.mul_assoc] using boundAtDegree.{u, v} d f hf end DFVSGames.Appendix.A5Induction end end OAI