import OAI.Probability.GaussianReplacement.RowOrder namespace OAI open MeasureTheory ProbabilityTheory Set Filter open scoped ENNReal NNReal RealInnerProductSpace Topology noncomputable section namespace CurrentProjection open InnerProductSpace open InformationTheory def arrayReference {d k : ℕ} {W : Type*} [MeasurableSpace W] (P : Measure ((Sphere d×W)×Rows k d)) : ℝ≥0∞ := conditionalReferenceKL (observationSignalLaw P).fst (gaussianObservations k d) lemma observationSignalLaw_fst_as_observed {d k : ℕ} {W : Type*} [MeasurableSpace W] (P : Measure ((Sphere d×W)×Rows k d)) : (observationSignalLaw P).fst = (observedMessage P).map (MeasurableEquiv.prodComm.trans ((MeasurableEquiv.refl W).prodCongr (observationEquiv k d))) := by rw [observationSignalLaw,Measure.fst,Measure.map_map measurable_fst measurable_observationSignalMap, observedMessage,Measure.map_map (MeasurableEquiv.measurable _) measurable_observedMessageMap] rfl lemma arrayReference_eq {d k : ℕ} {W : Type*} [MeasurableSpace W] (P : Measure ((Sphere d×W)×Rows k d)) [IsProbabilityMeasure P] : arrayReference P = observationReferenceKL P := by let e := MeasurableEquiv.prodComm.trans ((MeasurableEquiv.refl W).prodCongr (observationEquiv k d)) have href : (((gaussianRows k d).prod (gaussianLabels k)).prod P.fst.snd).map e = P.fst.snd.prod (gaussianObservations k d) := by change ((((gaussianRows k d).prod (gaussianLabels k)).prod P.fst.snd)).map ((Prod.map id (observationEquiv k d)) ∘ Prod.swap) = _ rw [←Measure.map_map (measurable_id.prodMap (observationEquiv k d).measurable) measurable_swap, Measure.prod_swap,←Measure.map_prod_map _ _ measurable_id (observationEquiv k d).measurable, Measure.map_id,←gaussianObservations_eq] rw [arrayReference,conditionalReferenceKL,observationSignalLaw_context, observationSignalLaw_fst_as_observed,←href] exact klDiv_map_equiv _ _ e lemma freshArrayLaw_assoc_first {d k r : ℕ} {W : Type*} [MeasurableSpace W] (P : Measure ((Sphere d×W)×Rows k d)) [IsProbabilityMeasure P] : ((freshArrayLaw (observationSignalLaw P) r).map MeasurableEquiv.prodAssoc).map (Prod.map id Prod.fst) = (observationSignalLaw P).fst := by rw [Measure.map_map (measurable_id.prodMap measurable_fst) MeasurableEquiv.prodAssoc.measurable] change (freshArrayLaw (observationSignalLaw P) r).fst = _ rw [freshArrayLaw_fst] lemma freshArrayLaw_aligned {d k r : ℕ} {W : Type*} [MeasurableSpace W] (P : Measure ((Sphere d×W)×Rows k d)) [IsProbabilityMeasure P] : ((freshArrayLaw (observationSignalLaw P) r).map MeasurableEquiv.prodAssoc).map (Prod.map id (appendEquiv (Vector d×ℝ) k r)) = (observationSignalLaw (alignedExperiment (r:=r) P)).fst := by rw [freshArrayLaw_eq,observationSignalLaw, show (P.map observationSignalMap).prod (gaussianRows r d) = (P.prod (gaussianRows r d)).map (Prod.map observationSignalMap id) by rw [←Measure.map_prod_map _ _ measurable_observationSignalMap measurable_id,Measure.map_id], Measure.map_map (by fun_prop) (measurable_observationSignalMap.prodMap measurable_id), Measure.map_map MeasurableEquiv.prodAssoc.measurable (by fun_prop), Measure.map_map (measurable_id.prodMap (appendEquiv _ _ _).measurable) (by fun_prop), observationSignalLaw,alignedExperiment,Measure.fst, Measure.map_map measurable_fst measurable_observationSignalMap, Measure.map_map (measurable_fst.comp measurable_observationSignalMap) measurable_alignedMap] congr 1 funext z dsimp [actualExperiment,observationSignalMap] rw [ownObservations_append] rfl lemma arrayReference_aligned {d k r : ℕ} {W : Type*} [MeasurableSpace W] (P : Measure ((Sphere d×W)×Rows k d)) [IsProbabilityMeasure P] : arrayReference (alignedExperiment (r:=r) P) = arrayReference P + freshReference (observationSignalLaw P) r := by let Q := (freshArrayLaw (observationSignalLaw P) r).map MeasurableEquiv.prodAssoc let : IsProbabilityMeasure Q := inferInstance have he := conditionalReferenceKL_outcome_equiv Q ((gaussianObservations k d).prod (gaussianObservations r d)) (appendEquiv (Vector d×ℝ) k r) rw [gaussianObservations_append] at he dsimp only [Q] at he rw [freshArrayLaw_aligned,conditionalReferenceKL_chain,freshArrayLaw_assoc_first, MeasurableEquiv.map_symm_map] at he exact he lemma arrayReference_prefix_step {d m : ℕ} {W : Type*} [MeasurableSpace W] (P : Measure ((Sphere d×W)×Rows m d)) [IsProbabilityMeasure P] (i : ℕ) (hi : i < m) : arrayReference (prefixLaw P (i+1) (Nat.succ_le_of_lt hi)) = arrayReference (prefixLaw P i (Nat.le_of_lt hi)) + singleReference (nextActualRowLaw P i hi) := by let Q := (singleObservationLaw (nextActualRowLaw P i hi)).map MeasurableEquiv.prodAssoc let : IsProbabilityMeasure Q := inferInstance have hfirst : Q.map (Prod.map id Prod.fst) = (observationSignalLaw (prefixLaw P i (Nat.le_of_lt hi))).fst := by rw [Measure.map_map (measurable_id.prodMap measurable_fst) MeasurableEquiv.prodAssoc.measurable] change (singleObservationLaw (nextActualRowLaw P i hi)).fst = _ rw [singleObservationLaw_fst,nextActualRowLaw_fst] have htotal : Q.map (Prod.map id (snocEquiv (Vector d×ℝ) i)) = (observationSignalLaw (prefixLaw P (i+1) (Nat.succ_le_of_lt hi))).fst := by have he := congrArg (fun μ : Measure ((W×Observations (i+1) d)×Sphere d) => μ.fst) (nextActualRowLaw_single P i hi) rw [Measure.fst,Measure.map_map measurable_fst ((MeasurableEquiv.measurable _).prodMap measurable_id)] at he dsimp only [Q,singleObservationLaw] rw [Measure.map_map (measurable_id.prodMap (snocEquiv _ _).measurable) MeasurableEquiv.prodAssoc.measurable, Measure.fst,Measure.map_map (by fun_prop) measurable_fst] exact he have he := conditionalReferenceKL_outcome_equiv Q ((gaussianObservations i d).prod (gaussianObservation d)) (snocEquiv (Vector d×ℝ) i) rw [show (((gaussianObservations i d).prod (gaussianObservation d)).map (snocEquiv (Vector d×ℝ) i)) = gaussianObservations (i+1) d from snocEquiv_map (gaussianObservation d) i,htotal, conditionalReferenceKL_chain,hfirst] at he dsimp only [Q] at he rw [MeasurableEquiv.map_symm_map] at he exact he def baselineCost (d k : ℕ) : ℝ≥0∞ := klDiv (projectionJoint (uniformSphere d) (gaussianRows k d)) ((gaussianRows k d).prod (gaussianLabels k)) def hybridBaselineConstant : ℝ := 20 + |Real.log (normalizer 1)+1/2| lemma hybridBaselineConstant_pos : 0 < hybridBaselineConstant := by unfold hybridBaselineConstant positivity lemma baselineCost_finite {d k : ℕ} (hk : 2 ≤ k) (hkd : k < d-1) : baselineCost d k ≠ ∞ ∧ (baselineCost d k).toReal ≤ hybridBaselineConstant*d := by have hd : 2 ≤ d := by omega have hdim : AdmissibleDimensions d 1 (k-1) 1 := by unfold AdmissibleDimensions omega have he : 1+(k-1) = k := by omega have h := sphere_projection_reference_finite hdim hd rw [he] at h refine ⟨h.1,h.2.trans ?_⟩ have hcast : (k:ℝ) ≤ d := by exact_mod_cast (show k ≤ d by omega) have hg := le_abs_self (Real.log (normalizer 1)+1/2) have hg0 := abs_nonneg (Real.log (normalizer 1)+1/2) dsimp [hybridBaselineConstant] nlinarith lemma arrayReference_finite_bound {d k : ℕ} {W : Type*} [MeasurableSpace W] [StandardBorelSpace W] [Nonempty W] [Countable W] [MeasurableSingletonClass W] (P : Measure ((Sphere d×W)×Rows k d)) [IsProbabilityMeasure P] (hd : 1 ≤ d) (hInd : (P.map swapLast).fst = (uniformSphere d).prod (gaussianRows k d)) {B : ℝ} (hB : 0 ≤ B) (hH : shannonEntropy P.fst.snd ≤ ENNReal.ofReal B) (hk : 2 ≤ k) (hkd : k < d-1) : arrayReference P ≠ ∞ ∧ (arrayReference P).toReal ≤ hybridBaselineConstant*d+B := by let : IsProbabilityMeasure (uniformSphere d) := uniformSphere_probability_of_pos hd have hHf : shannonEntropy (observedMessage P).snd ≠ ∞ := by rw [observedMessage_snd] exact ne_top_of_le_ne_top ENNReal.ofReal_ne_top hH have hI := countable_information_le_entropy (observedMessage P) hHf have hbase := baselineCost_finite hk hkd rw [arrayReference_eq,observationReferenceKL_decomposition P (uniformSphere d) hInd] change baselineCost d k + mutualInformation (observedMessage P) ≠ ∞ ∧ (baselineCost d k + mutualInformation (observedMessage P)).toReal ≤ hybridBaselineConstant*d+B have hIf : mutualInformation (observedMessage P) ≠ ∞ := hI.1 refine ⟨ENNReal.add_ne_top.mpr ⟨hbase.1,hI.1⟩,?_⟩ rw [ENNReal.toReal_add hbase.1 hIf] have hHb := ENNReal.toReal_mono ENNReal.ofReal_ne_top hH rw [ENNReal.toReal_ofReal hB] at hHb apply add_le_add hbase.2 apply hI.2.trans simpa only [observedMessage_snd] using hHb lemma arrayReference_exposed_chain {d k : ℕ} {W : Type*} [MeasurableSpace W] [StandardBorelSpace W] [Nonempty W] [Nonempty (Sphere d)] (P : Measure ((Sphere d×W)×Rows k d)) [IsProbabilityMeasure P] (hd : 1 ≤ d) (hInd : (P.map swapLast).fst = (uniformSphere d).prod (gaussianRows k d)) : arrayReference P + exposedInformation P = baselineCost d k + mutualInformation P.fst + rowInformation P (gaussianRows k d) := by let : IsProbabilityMeasure (uniformSphere d) := uniformSphere_probability_of_pos hd rw [arrayReference_eq] exact observationReferenceKL_chain_exposed P (uniformSphere d) hInd lemma exposedInformation_finite {d k : ℕ} {W : Type*} [MeasurableSpace W] [StandardBorelSpace W] [Nonempty W] [Nonempty (Sphere d)] [Countable W] [MeasurableSingletonClass W] (P : Measure ((Sphere d×W)×Rows k d)) [IsProbabilityMeasure P] (hH : shannonEntropy P.fst.snd ≠ ∞) : exposedInformation P ≠ ∞ := by have he := exposedInformation_chain P let : IsProbabilityMeasure (P.map swapLast) := inferInstance have hi := countable_information_le_entropy (P.map swapLast) (by rwa [swapLast_snd]) exact ne_top_of_le_ne_top hi.1 (by change exposedInformation P ≤ mutualInformation (P.map swapLast); rw [he]; exact le_add_left le_rfl) lemma rowInformation_finite_bound {d k : ℕ} {W : Type*} [MeasurableSpace W] [Countable W] [MeasurableSingletonClass W] (P : Measure ((Sphere d×W)×Rows k d)) [IsProbabilityMeasure P] (hd : 1 ≤ d) (hInd : (P.map swapLast).fst = (uniformSphere d).prod (gaussianRows k d)) {B : ℝ} (hB : 0 ≤ B) (hH : shannonEntropy P.fst.snd ≤ ENNReal.ofReal B) : rowInformation P (gaussianRows k d) ≠ ∞ ∧ (rowInformation P (gaussianRows k d)).toReal ≤ B := by let : IsProbabilityMeasure (uniformSphere d) := uniformSphere_probability_of_pos hd have h := rowInformation_le_entropy P (uniformSphere d) (gaussianRows k d) hInd (by rw [swapLast_snd]; exact ne_top_of_le_ne_top ENNReal.ofReal_ne_top hH) rw [swapLast_snd] at h have hHb := ENNReal.toReal_mono ENNReal.ofReal_ne_top hH rw [ENNReal.toReal_ofReal hB] at hHb exact ⟨h.1,h.2.trans hHb⟩ lemma freshReference_from_array_bound {d i r : ℕ} {W : Type*} [MeasurableSpace W] [StandardBorelSpace W] [Nonempty W] [Countable W] [MeasurableSingletonClass W] (P : Measure ((Sphere d×W)×Rows i d)) [IsProbabilityMeasure P] (hd : 1 ≤ d) (hInd : (P.map swapLast).fst = (uniformSphere d).prod (gaussianRows i d)) {B : ℝ} (hB : 0 ≤ B) (hH : shannonEntropy P.fst.snd ≤ ENNReal.ofReal B) (hk : 2 ≤ i+r) (hkd : i+r < d-1) : freshReference (observationSignalLaw P) r ≠ ∞ ∧ (freshReference (observationSignalLaw P) r).toReal ≤ hybridBaselineConstant*d+B := by let : IsProbabilityMeasure (uniformSphere d) := uniformSphere_probability_of_pos hd have h := arrayReference_finite_bound (alignedExperiment (r:=r) P) hd (alignedExperiment_independence P (uniformSphere d) hInd) hB (by rwa [alignedExperiment_fst]) hk hkd have hl : freshReference (observationSignalLaw P) r ≤ arrayReference (alignedExperiment (r:=r) P) := by rw [arrayReference_aligned] exact le_add_left le_rfl exact ⟨ne_top_of_le_ne_top h.1 hl,(ENNReal.toReal_mono h.1 hl).trans h.2⟩ lemma freshSlope_from_array_bound {d i r : ℕ} {W : Type*} [MeasurableSpace W] [StandardBorelSpace W] [Nonempty W] [Countable W] [MeasurableSingletonClass W] (P : Measure ((Sphere d×W)×Rows i d)) [IsProbabilityMeasure P] (hd : 16 ≤ d) (hInd : (P.map swapLast).fst = (uniformSphere d).prod (gaussianRows i d)) {B : ℝ} (hB : 0 ≤ B) (hH : shannonEntropy P.fst.snd ≤ ENNReal.ofReal B) (hk : 2 ≤ i+r+d/8) (hkd : i+r+d/8 < d-1) : freshSlope (observationSignalLaw P) r ≠ ∞ ∧ (freshSlope (observationSignalLaw P) r).toReal ≤ 16*hybridBaselineConstant+B/(d/8:ℕ) := by have h := freshReference_from_array_bound (r:=r+d/8) P (by omega) hInd hB hH (by omega) (by omega) have ha := freshSlope_average (observationSignalLaw P) r (d/8) (by omega) h.1 refine ⟨ha.1,ha.2.trans ?_⟩ have hp : (0:ℝ)<(d/8:ℕ) := by exact_mod_cast (show 0 (z.1.2,z.2)) = ν.prod (stdGaussian (Vector d)) := by have he := congrArg (fun μ : Measure (Sphere d×Rows m d) => μ.map (Prod.map id (fun X => X ⟨i,hi⟩))) hInd rw [←Measure.map_prod_map _ _ measurable_id (measurable_pi_apply _),Measure.map_id] at he have hev : (gaussianRows m d).map (fun X => X ⟨i,hi⟩) = stdGaussian (Vector d) := (measurePreserving_eval (fun _ : Fin m => stdGaussian (Vector d)) ⟨i,hi⟩).map_eq rw [hev] at he rw [nextActualRowLaw,Measure.map_map (measurable_fst.snd.prodMk measurable_snd) (measurable_nextActualRow hi)] rw [Measure.fst,Measure.map_map measurable_fst swapLast.measurable, Measure.map_map (measurable_id.prodMap (measurable_pi_apply _)) (measurable_fst.comp swapLast.measurable)] at he exact he lemma gaussian_sphere_inner_mixture {d : ℕ} (ν : Measure (Sphere d)) [IsProbabilityMeasure ν] : (ν.prod (stdGaussian (Vector d))).map (fun z => ⟪(z.1:Vector d),z.2⟫) = gaussianReal 0 1 := by ext t ht rw [Measure.map_apply (by fun_prop) ht,Measure.prod_apply (ht.preimage (by fun_prop))] have he (s : Sphere d) : stdGaussian (Vector d) {x | ⟪(s:Vector d),x⟫ ∈ t} = gaussianReal 0 1 t := by have h := congrArg (fun μ : Measure ℝ => μ t) (gaussian_inner_map_sphere s) rwa [Measure.map_apply (by fun_prop) ht] at h change (∫⁻ s : Sphere d, stdGaussian (Vector d) {x | ⟪(s:Vector d),x⟫ ∈ t} ∂ν) = _ simp_rw [he] simp lemma nextActualRowLaw_label {d m : ℕ} {W : Type*} [MeasurableSpace W] (P : Measure ((Sphere d×W)×Rows m d)) [IsProbabilityMeasure P] (ν : Measure (Sphere d)) [IsProbabilityMeasure ν] (hInd : (P.map swapLast).fst = ν.prod (gaussianRows m d)) (i : ℕ) (hi : i < m) (r : ℕ) : (residualLabelLaw (nextActualRowLaw P i hi) r).snd = gaussianReal 0 1 := by rw [residualLabelLaw,Measure.snd,Measure.map_map measurable_snd measurable_residualLabelObservation] have hel : Measurable (fun z : ((W×Observations i d)×Sphere d)×Vector d => ⟪(z.1.2:Vector d),z.2⟫) := by fun_prop change ((nextActualRowLaw P i hi).prod (gaussianRows r d)).map ((fun z => ⟪(z.1.2:Vector d),z.2⟫) ∘ Prod.fst) = _ rw [←Measure.map_map hel measurable_fst,←Measure.fst,Measure.fst_prod] have he := congrArg (fun μ : Measure (Sphere d×Vector d) => μ.map (fun z => ⟪(z.1:Vector d),z.2⟫)) (nextActualRowLaw_signal_row P ν hInd i hi) rw [Measure.map_map (by fun_prop) (measurable_fst.snd.prodMk measurable_snd),gaussian_sphere_inner_mixture] at he exact he lemma hybrid_one_step {d m i r q : ℕ} {W : Type*} [MeasurableSpace W] [StandardBorelSpace W] [Nonempty W] [Countable W] [MeasurableSingletonClass W] (P : Measure ((Sphere d×W)×Rows m d)) [IsProbabilityMeasure P] (hd : 16 ≤ d) (hInd : (P.map swapLast).fst = (uniformSphere d).prod (gaussianRows m d)) {B : ℝ} (hB : 0 ≤ B) (hH : shannonEntropy P.fst.snd ≤ ENNReal.ofReal B) (hi : i < m) (hr : r ≤ d) (hd8 : d ≤ 8*r) (hq : 1 ≤ q) (hqr : 2*q ≤ r) (hmargin : 2 ≤ r-q) (htotal : i+r+2 < d-1) : (exposedInformation (alignedExperiment (r:=r+1) (prefixLaw P i (Nat.le_of_lt hi)))).toReal - (exposedInformation (alignedExperiment (r:=r) (prefixLaw P (i+1) (Nat.succ_le_of_lt hi)))).toReal ≤ hybridRowConstant + (klDiv (nextActualRowLaw P i hi) ((nextActualRowLaw P i hi).fst.prod (stdGaussian (Vector d)))).toReal/q + (freshSlope (observationSignalLaw (prefixLaw P i (Nat.le_of_lt hi))) (r+1)).toReal - (freshSlope (observationSignalLaw (prefixLaw P i (Nat.le_of_lt hi))) r).toReal := by let : IsProbabilityMeasure (uniformSphere d) := uniformSphere_probability_of_pos (by omega) let : Nonempty (Sphere d) := nonempty_of_isProbabilityMeasure (uniformSphere d) let P0 := prefixLaw P i (Nat.le_of_lt hi) let P1 := prefixLaw P (i+1) (Nat.succ_le_of_lt hi) have hI0 : (P0.map swapLast).fst = (uniformSphere d).prod (gaussianRows i d) := prefixLaw_independence P (uniformSphere d) hInd _ _ have hI1 : (P1.map swapLast).fst = (uniformSphere d).prod (gaussianRows (i+1) d) := prefixLaw_independence P (uniformSphere d) hInd _ _ have hH0 : shannonEntropy P0.fst.snd ≤ ENNReal.ofReal B := by rwa [prefixLaw_fst] have hH1 : shannonEntropy P1.fst.snd ≤ ENNReal.ofReal B := by rwa [prefixLaw_fst] have hR0 := rowInformation_finite_bound P0 (by omega) hI0 hB hH0 have hR1 := rowInformation_finite_bound P1 (by omega) hI1 hB hH1 have hKchain := prefix_rowInformation_step P i hi change rowInformation P1 _ = rowInformation P0 _ + _ at hKchain have hK : klDiv (nextActualRowLaw P i hi) ((nextActualRowLaw P i hi).fst.prod (stdGaussian (Vector d))) ≠ ∞ := ne_top_of_le_ne_top hR1.1 (by rw [hKchain]; exact le_add_left le_rfl) have hF0 := freshReference_from_array_bound (r:=r+1) P0 (by omega) hI0 hB hH0 (by omega) (by omega) have hF1 := freshReference_from_array_bound (r:=r+1+1) P0 (by omega) hI0 hB hH0 (by omega) (by omega) have hT0 := (freshSlope_average (observationSignalLaw P0) r 1 (by omega) hF0.1).1 have hT1 := (freshSlope_average (observationSignalLaw P0) (r+1) 1 (by omega) hF1.1).1 have hcross := actual_fresh_row_bound (nextActualRowLaw P i hi) hK hr hd8 hq hqr hmargin (by rwa [nextActualRowLaw_fst]) (by rwa [nextActualRowLaw_fst]) (nextActualRowLaw_label P (uniformSphere d) hInd i hi r) rw [nextActualRowLaw_fst] at hcross have hA0 := arrayReference_finite_bound (alignedExperiment (r:=r) P0) (by omega) (alignedExperiment_independence P0 (uniformSphere d) hI0) hB (by rwa [alignedExperiment_fst]) (by omega) (by omega) have hAl := arrayReference_finite_bound (alignedExperiment (r:=r+1) P0) (by omega) (alignedExperiment_independence P0 (uniformSphere d) hI0) hB (by rwa [alignedExperiment_fst]) (by omega) (by omega) have hAr := arrayReference_finite_bound (alignedExperiment (r:=r) P1) (by omega) (alignedExperiment_independence P1 (uniformSphere d) hI1) hB (by rwa [alignedExperiment_fst]) (by omega) (by omega) have hJ0 := exposedInformation_finite (alignedExperiment (r:=r+1) P0) (by rw [alignedExperiment_fst]; exact ne_top_of_le_ne_top ENNReal.ofReal_ne_top hH0) have hJ1 := exposedInformation_finite (alignedExperiment (r:=r) P1) (by rw [alignedExperiment_fst]; exact ne_top_of_le_ne_top ENNReal.ofReal_ne_top hH1) have hMI : mutualInformation P.fst ≠ ∞ := (countable_information_le_entropy P.fst (ne_top_of_le_ne_top ENNReal.ofReal_ne_top hH)).1 have hbase := (baselineCost_finite (d:=d) (k:=i+(r+1)) (by omega) (by omega)).1 have hc0 := arrayReference_exposed_chain (alignedExperiment (r:=r+1) P0) (by omega) (alignedExperiment_independence P0 (uniformSphere d) hI0) have hc1 := arrayReference_exposed_chain (alignedExperiment (r:=r) P1) (by omega) (alignedExperiment_independence P1 (uniformSphere d) hI1) rw [alignedExperiment_fst,rowInformation_alignedExperiment] at hc0 hc1 change _ = baselineCost d (i+(r+1)) + mutualInformation P0.fst + _ at hc0 change _ = baselineCost d (i+1+r) + mutualInformation P1.fst + _ at hc1 have hPf0 : P0.fst = P.fst := prefixLaw_fst _ _ _ have hPf1 : P1.fst = P.fst := prefixLaw_fst _ _ _ rw [hPf0] at hc0 have hbe : baselineCost d (i+1+r) = baselineCost d (i+(r+1)) := congrArg (baselineCost d) (by omega) rw [hPf1,hbe] at hc1 have hc0r := congrArg ENNReal.toReal hc0 have hc1r := congrArg ENNReal.toReal hc1 rw [ENNReal.toReal_add hAl.1 hJ0,ENNReal.toReal_add (ENNReal.add_ne_top.mpr ⟨hbase,hMI⟩) hR0.1, ENNReal.toReal_add hbase hMI] at hc0r rw [ENNReal.toReal_add hAr.1 hJ1,ENNReal.toReal_add (ENNReal.add_ne_top.mpr ⟨hbase,hMI⟩) hR1.1, ENNReal.toReal_add hbase hMI] at hc1r have hKreal := congrArg ENNReal.toReal hKchain rw [ENNReal.toReal_add hR0.1 hK] at hKreal have har := arrayReference_actual_step P i hi r have harr := congrArg ENNReal.toReal har rw [ENNReal.toReal_add hA0.1 hcross.1] at harr have hal : arrayReference (alignedExperiment (r:=r+1) P0) = arrayReference (alignedExperiment (r:=r) P0) + freshSlope (observationSignalLaw P0) r := by rw [arrayReference_aligned,arrayReference_aligned,freshReference_succ,add_assoc] have halr := congrArg ENNReal.toReal hal rw [ENNReal.toReal_add hA0.1 hT0] at halr change _ ≤ hybridRowConstant + _ + (freshSlope (observationSignalLaw P0) (r+1)).toReal - (freshSlope (observationSignalLaw P0) r).toReal rw [nextActualRowLaw_fst] at hKreal ⊢ linarith [hcross.2] end CurrentProjection end end OAI