| import OAI.Combinatorics.ContingencyTables.Transport.ObservableBounds |
|
|
| namespace OAI |
|
|
| /-! |
| # The manuscript's original omission coordinates |
|
|
| Complementation reverses defect labels and their multipliers. We prove the |
| actual chain energy, conditional projection, and variance identities before |
| transferring the variance and observable inequalities back to the original |
| factorial lift f. The final theorems assume its omission property. |
| -/ |
|
|
| namespace ContingencyTables.ComplementChain |
|
|
| open scoped BigOperators Classical |
| open CommonBasesFPRAS BinaryContraction DegreeCompletion PairedComplement ObservableVariance ObservableBounds |
|
|
| variable {Γ : Type*} [Fintype Γ] [DecidableEq Γ] |
|
|
| omit [Fintype Γ] [DecidableEq Γ] in |
| lemma plain_complement (gs : List (Γ × Γ)) (S : Set Γ) : |
| PlainState gs Sᶜ ↔ PlainState gs S := by |
| unfold PlainState PairSingle |
| simp only [Set.mem_compl_iff] |
| constructor <;> intro h p hp <;> have hh := h p hp <;> tauto |
|
|
| omit [Fintype Γ] [DecidableEq Γ] in |
| lemma defect_complement (gs : List (Γ × Γ)) (p r : Γ × Γ) (S : Set Γ) : |
| DefectState gs p r Sᶜ ↔ DefectState gs r p S := by |
| unfold DefectState PairEmpty PairFull PairSingle |
| simp only [Set.mem_compl_iff, not_not] |
| constructor |
| · rintro ⟨hp,hr,hpr,hep,hfr,hs⟩ |
| refine ⟨hr,hp,hpr.symm,hfr,hep,?_⟩ |
| intro t ht htr htp |
| have h := hs t ht htp htr |
| tauto |
| · rintro ⟨hr,hp,hrp,her,hfp,hs⟩ |
| refine ⟨hp,hr,hrp.symm,hfp,her,?_⟩ |
| intro t ht htp htr |
| have h := hs t ht htr htp |
| tauto |
|
|
| omit [Fintype Γ] [DecidableEq Γ] in |
| lemma state_complement (gs : List (Γ × Γ)) (S : Set Γ) : |
| ChainState gs Sᶜ ↔ ChainState gs S := by |
| unfold ChainState |
| rw [plain_complement] |
| constructor |
| · rintro (h | ⟨p,r,h⟩) |
| · exact Or.inl h |
| · exact Or.inr ⟨r,p,(defect_complement gs p r S).mp h⟩ |
| · rintro (h | ⟨p,r,h⟩) |
| · exact Or.inl h |
| · exact Or.inr ⟨r,p,(defect_complement gs r p S).mpr h⟩ |
|
|
| omit [Fintype Γ] [DecidableEq Γ] in |
| lemma adjacent_complement (S T : Set Γ) : ExchangeAdjacent Sᶜ Tᶜ ↔ ExchangeAdjacent S T := by |
| have hf (S T : Set Γ) (h : ExchangeAdjacent S T) : ExchangeAdjacent Sᶜ Tᶜ := by |
| obtain ⟨A,a,b,hab,ha,hb,rfl,rfl⟩ := h |
| let B := (insert a (insert b A))ᶜ |
| refine ⟨B,b,a,hab.symm,?_,?_,?_,?_⟩ |
| · simp [B] |
| · simp [B] |
| · ext x |
| simp only [B, Set.mem_compl_iff, Set.mem_insert_iff] |
| constructor |
| · intro hx |
| by_cases hxb : x = b |
| · exact Or.inl hxb |
| · exact Or.inr (by tauto) |
| · intro hx |
| rcases hx with rfl | hx |
| · exact not_or.mpr ⟨hab.symm, hb⟩ |
| · tauto |
| · ext x |
| simp only [B, Set.mem_compl_iff, Set.mem_insert_iff] |
| constructor |
| · intro hx |
| by_cases hxa : x = a |
| · exact Or.inl hxa |
| · exact Or.inr (by tauto) |
| · intro hx |
| rcases hx with rfl | hx |
| · exact not_or.mpr ⟨hab, ha⟩ |
| · tauto |
| exact ⟨fun h => by simpa using hf Sᶜ Tᶜ h, hf S T⟩ |
|
|
| omit [Fintype Γ] [DecidableEq Γ] in |
| def complementEquiv : Set Γ ≃ Set Γ where |
| toFun := compl |
| invFun := compl |
| left_inv := compl_compl |
| right_inv := compl_compl |
|
|
| omit [Fintype Γ] [DecidableEq Γ] in |
| def reverseMultiplier (w : (Γ × Γ) → (Γ × Γ) → ℝ) : (Γ × Γ) → (Γ × Γ) → ℝ := |
| fun p r => w r p |
|
|
| lemma multiplier_complement (gs : List (Γ × Γ)) (w : (Γ × Γ) → (Γ × Γ) → ℝ) (S : Set Γ) : |
| chainMultiplier gs (reverseMultiplier w) Sᶜ = chainMultiplier gs w S := by |
| by_cases he : ∃ p r, DefectState gs p r S |
| · obtain ⟨p,r,h⟩ := he |
| rw [chainMultiplier_defect gs _ Sᶜ ((defect_complement gs r p S).mpr h), |
| chainMultiplier_defect gs w S h] |
| rfl |
| · have hec : ¬ ∃ p r, DefectState gs p r Sᶜ := by |
| rintro ⟨p,r,h⟩ |
| exact he ⟨r,p,(defect_complement gs p r S).mp h⟩ |
| simp only [chainMultiplier, dite_eq_right he, dite_eq_right hec] |
|
|
| lemma weight_complement (gs : List (Γ × Γ)) (w : (Γ × Γ) → (Γ × Γ) → ℝ) |
| (f : Set Γ → ℝ) (S : Set Γ) : |
| chainWeight gs (reverseMultiplier w) (complementWeight f) Sᶜ = chainWeight gs w f S := by |
| simp only [chainWeight, multiplier_complement, complementWeight, compl_compl] |
|
|
| lemma chain_sum_complement (gs : List (Γ × Γ)) (w : (Γ × Γ) → (Γ × Γ) → ℝ) |
| (f J : Set Γ → ℝ) : |
| (∑ S ∈ chainStates gs, chainWeight gs (reverseMultiplier w) (complementWeight f) S * J S) = |
| ∑ S ∈ chainStates gs, chainWeight gs w f S * J Sᶜ := by |
| simp only [chainStates, Finset.sum_filter] |
| apply Fintype.sum_equiv complementEquiv |
| intro S |
| change (if ChainState gs S then _ else 0) = (if ChainState gs Sᶜ then _ else 0) |
| rw [state_complement] |
| by_cases h : ChainState gs S |
| · simp only [ite_eq_left h] |
| have hw := weight_complement gs w f Sᶜ |
| simp only [compl_compl] at hw |
| rw [hw] |
| simp only [complementEquiv, Equiv.coe_fn_mk, compl_compl] |
| · simp only [ite_eq_right h] |
|
|
| lemma total_complement (gs : List (Γ × Γ)) (w : (Γ × Γ) → (Γ × Γ) → ℝ) (f : Set Γ → ℝ) : |
| chainTotal gs (reverseMultiplier w) (complementWeight f) = chainTotal gs w f := by |
| simpa only [chainTotal, mul_one] using chain_sum_complement gs w f (fun _ => 1) |
|
|
| lemma mean_complement (gs : List (Γ × Γ)) (w : (Γ × Γ) → (Γ × Γ) → ℝ) (f H : Set Γ → ℝ) : |
| chainMean gs (reverseMultiplier w) (complementWeight f) (complementWeight H) = chainMean gs w f H := by |
| unfold chainMean |
| rw [total_complement, chain_sum_complement] |
| simp only [complementWeight, compl_compl] |
|
|
| lemma variance_complement (gs : List (Γ × Γ)) (w : (Γ × Γ) → (Γ × Γ) → ℝ) (f H : Set Γ → ℝ) : |
| chainVariance gs (reverseMultiplier w) (complementWeight f) (complementWeight H) = |
| chainVariance gs w f H := by |
| unfold chainVariance |
| rw [total_complement, mean_complement, chain_sum_complement] |
| simp only [complementWeight, compl_compl] |
|
|
| lemma conductance_complement (gs : List (Γ × Γ)) (w : (Γ × Γ) → (Γ × Γ) → ℝ) |
| (f : Set Γ → ℝ) (ST : Set Γ × Set Γ) : |
| chainConductance gs (reverseMultiplier w) (complementWeight f) (ST.1ᶜ,ST.2ᶜ) = |
| chainConductance gs w f ST := by |
| simp only [chainConductance, multiplier_complement, complementWeight, compl_compl] |
|
|
| lemma energy_complement (gs : List (Γ × Γ)) (w : (Γ × Γ) → (Γ × Γ) → ℝ) (f H : Set Γ → ℝ) : |
| restrictedEnergy (chainEdges gs) (chainConductance gs (reverseMultiplier w) (complementWeight f)) |
| (complementWeight H) ∅ ∅ = |
| restrictedEnergy (chainEdges gs) (chainConductance gs w f) H ∅ ∅ := by |
| have hr (ST : Set Γ × Set Γ) : EdgeRespects ∅ ∅ ST := by simp [EdgeRespects, Respects] |
| unfold restrictedEnergy |
| simp only [hr, ite_true, chainEdges, Finset.sum_filter] |
| congr 1 |
| let E : (Set Γ × Set Γ) ≃ (Set Γ × Set Γ) := Equiv.prodCongr complementEquiv complementEquiv |
| apply Fintype.sum_equiv E |
| intro ST |
| change (if ChainState gs ST.1 ∧ ChainState gs ST.2 ∧ ExchangeAdjacent ST.1 ST.2 then _ else 0) = |
| (if ChainState gs ST.1ᶜ ∧ ChainState gs ST.2ᶜ ∧ ExchangeAdjacent ST.1ᶜ ST.2ᶜ then _ else 0) |
| rw [state_complement, state_complement, adjacent_complement] |
| by_cases h : ChainState gs ST.1 ∧ ChainState gs ST.2 ∧ ExchangeAdjacent ST.1 ST.2 |
| · simp only [ite_eq_left h] |
| have hc := conductance_complement gs w f (ST.1ᶜ,ST.2ᶜ) |
| simp only [compl_compl] at hc |
| rw [hc] |
| rfl |
| · simp only [ite_eq_right h] |
|
|
| lemma dirichlet_complement (gs : List (Γ × Γ)) (w : (Γ × Γ) → (Γ × Γ) → ℝ) (f H : Set Γ → ℝ) : |
| chainDirichlet gs (reverseMultiplier w) (complementWeight f) (complementWeight H) = |
| chainDirichlet gs w f H := by |
| unfold chainDirichlet |
| rw [total_complement, energy_complement] |
|
|
| lemma plain_mass_complement (gs : List (Γ × Γ)) (hgs : SlotsFresh gs ∅) |
| (hcover : pairSupport gs = Set.univ) (f : Set Γ → ℝ) : |
| slotTotal gs (complementWeight f) ∅ = slotTotal gs f ∅ := by |
| exact slotTotal_complement gs f ∅ ∅ (by simp) (by simpa using hgs) (by simpa using hcover) |
|
|
| lemma defect_mass_complement (gs : List (Γ × Γ)) (hgs : SlotsFresh gs ∅) |
| (hcover : pairSupport gs = Set.univ) (f : Set Γ → ℝ) |
| (p r : Γ × Γ) (hp : p ∈ gs) (hr : r ∈ gs) (hpr : p ≠ r) : |
| slotDefect gs (complementWeight f) ∅ p r = slotDefect gs f ∅ r p := by |
| exact slotDefect_complement gs f ∅ ∅ (by simp) (by simpa using hgs) (by simpa using hcover) p r hp hr hpr |
|
|
| lemma projection_complement (gs : List (Γ × Γ)) (hgs : SlotsFresh gs ∅) |
| (hcover : pairSupport gs = Set.univ) (f H : Set Γ → ℝ) : |
| observableProjection gs (complementWeight f) (complementWeight H) = |
| complementWeight (observableProjection gs f H) := by |
| have hmul : (fun S => complementWeight f S * complementWeight H S) = |
| complementWeight (fun S => f S * H S) := rfl |
| funext S |
| change observableProjection gs (complementWeight f) (complementWeight H) S = |
| observableProjection gs f H Sᶜ |
| by_cases he : ∃ p r, DefectState gs p r S |
| · obtain ⟨p,r,h⟩ := he |
| have hc : DefectState gs r p Sᶜ := (defect_complement gs r p S).mpr h |
| rw [observableProjection_defect gs _ _ S h, observableProjection_defect gs f H Sᶜ hc] |
| unfold defectMean |
| rw [hmul, defect_mass_complement gs hgs hcover _ p r h.1 h.2.1 h.2.2.1, |
| defect_mass_complement gs hgs hcover f p r h.1 h.2.1 h.2.2.1] |
| · have hec : ¬ ∃ p r, DefectState gs p r Sᶜ := by |
| rintro ⟨p,r,h⟩ |
| exact he ⟨r,p,(defect_complement gs p r S).mp h⟩ |
| simp only [observableProjection, dite_eq_right he, dite_eq_right hec, complementWeight] |
|
|
| lemma slotVariance_complement (gs : List (Γ × Γ)) (hgs : SlotsFresh gs ∅) |
| (hcover : pairSupport gs = Set.univ) (f H : Set Γ → ℝ) : |
| slotVariance gs (complementWeight f) (complementWeight H) ∅ = slotVariance gs f H ∅ := by |
| unfold slotVariance |
| change slotTotal gs (complementWeight (fun S => f S*(H S)^2)) ∅ - |
| (slotTotal gs (complementWeight (fun S => f S*H S)) ∅)^2/slotTotal gs (complementWeight f) ∅ = _ |
| rw [plain_mass_complement gs hgs hcover, plain_mass_complement gs hgs hcover, |
| plain_mass_complement gs hgs hcover] |
|
|
| /-- The actual manuscript variance theorem in original omission coordinates. -/ |
| theorem omission_twoLift_transversal_variance (gs : List (Γ × Γ)) (f g H : Set Γ → ℝ) |
| (w : (Γ × Γ) → (Γ × Γ) → ℝ) (q : ℝ) (hq : 0 ≤ q) |
| (hf : DegreePositive f gs.length) (hg : DegreePositive g gs.length) |
| (hsigf : OmissionProperty f gs.length) (hsigg : AdditionProperty g gs.length) |
| (hgs : SlotsFresh gs ∅) (hcover : pairSupport gs = Set.univ) |
| (hw : ∀ p r, 0 < w p r) |
| (hgood : ∀ p ∈ gs, ∀ r ∈ gs, p ≠ r → |
| slotTotal gs f ∅/(4*slotDefect gs f ∅ p r) ≤ w p r) |
| (hagree : ∀ S, PlainState gs S → f S = g S) |
| (hcompare : ∀ S, ChainState gs S → f S ≤ q*g S) : |
| slotVariance gs f H ∅ ≤ (gs.length : ℝ)*(1+8*(gs.length : ℝ)*q^2)* |
| restrictedEnergy (chainEdges gs) (chainConductance gs w f) H ∅ ∅ := by |
| have hcard := slots_ground_card gs hgs hcover |
| have hhalf : Fintype.card Γ - gs.length = gs.length := by omega |
| have hfc : DegreePositive (complementWeight f) gs.length := by |
| simpa only [hhalf] using complement_degreePositive f gs.length (by omega) hf |
| have hgc : DegreePositive (complementWeight g) gs.length := by |
| simpa only [hhalf] using complement_degreePositive g gs.length (by omega) hg |
| have hsf : AdditionProperty (complementWeight f) gs.length := by |
| simpa only [OmissionProperty, hhalf] using hsigf |
| have hsg : OmissionProperty (complementWeight g) gs.length := by |
| have hcc : complementWeight (complementWeight g) = g := by funext S; simp [complementWeight] |
| simpa only [OmissionProperty, hhalf, hcc] using hsigg |
| have hgood' : ∀ p ∈ gs, ∀ r ∈ gs, p ≠ r → |
| slotTotal gs (complementWeight f) ∅/(4*slotDefect gs (complementWeight f) ∅ p r) ≤ reverseMultiplier w p r := by |
| intro p hp r hr hpr |
| rw [plain_mass_complement gs hgs hcover, defect_mass_complement gs hgs hcover f p r hp hr hpr] |
| exact hgood r hr p hp hpr.symm |
| have ht := validDegree_twoLift_transversal_variance gs (complementWeight f) (complementWeight g) |
| (complementWeight H) (reverseMultiplier w) q hq hfc hgc hsf hsg hgs hcover |
| (fun p r => hw r p) hgood' |
| (fun S hS => hagree Sᶜ ((plain_complement gs S).mpr hS)) |
| (fun S hS => hcompare Sᶜ ((state_complement gs S).mpr hS)) |
| rwa [slotVariance_complement gs hgs hcover f H, energy_complement gs w f H] at ht |
|
|
| /-- The paper's observable Poincaré theorem for the literal first factorial |
| lift, assuming its omission property only at the actual subset degree. -/ |
| theorem omission_observable_poincare (gs : List (Γ × Γ)) (hgs : SlotsFresh gs ∅) (hne : gs ≠ []) |
| (hcover : pairSupport gs = Set.univ) (w : (Γ × Γ) → (Γ × Γ) → ℝ) |
| (hw : ∀ p r, 0 < w p r) (f H : Set Γ → ℝ) |
| (hf : DegreePositive f gs.length) (hsig : OmissionProperty f gs.length) |
| (q : ℝ) (hq : 1 ≤ q) |
| (hlower : ∀ p ∈ gs, ∀ r ∈ gs, p ≠ r → slotTotal gs f ∅/(4*slotDefect gs f ∅ p r) ≤ w p r) |
| (hupper : ∀ p ∈ gs, ∀ r ∈ gs, p ≠ r → w p r ≤ 4*slotTotal gs f ∅/slotDefect gs f ∅ p r) : |
| chainVariance gs w f (observableProjection gs f H) ≤ |
| observableConstant gs.length q * chainDirichlet gs w f H := by |
| have hcard := slots_ground_card gs hgs hcover |
| have hhalf : Fintype.card Γ - gs.length = gs.length := by omega |
| have hfc : DegreePositive (complementWeight f) gs.length := by |
| simpa only [hhalf] using complement_degreePositive f gs.length (by omega) hf |
| have hsf : AdditionProperty (complementWeight f) gs.length := by |
| simpa only [OmissionProperty, hhalf] using hsig |
| have hl : ∀ p ∈ gs, ∀ r ∈ gs, p ≠ r → |
| slotTotal gs (complementWeight f) ∅/(4*slotDefect gs (complementWeight f) ∅ p r) ≤ reverseMultiplier w p r := by |
| intro p hp r hr hpr |
| rw [plain_mass_complement gs hgs hcover, defect_mass_complement gs hgs hcover f p r hp hr hpr] |
| exact hlower r hr p hp hpr.symm |
| have hu : ∀ p ∈ gs, ∀ r ∈ gs, p ≠ r → |
| reverseMultiplier w p r ≤ 4*slotTotal gs (complementWeight f) ∅/slotDefect gs (complementWeight f) ∅ p r := by |
| intro p hp r hr hpr |
| rw [plain_mass_complement gs hgs hcover, defect_mass_complement gs hgs hcover f p r hp hr hpr] |
| exact hupper r hr p hp hpr.symm |
| have ht := validDegree_observable_poincare gs hgs hne hcover (reverseMultiplier w) |
| (fun p r => hw r p) (complementWeight f) (complementWeight H) hfc hsf q hq hl hu |
| rwa [projection_complement gs hgs hcover, variance_complement, dirichlet_complement] at ht |
|
|
| end ContingencyTables.ComplementChain |
|
|
| end OAI |
|
|
