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leandojo-lean4-formal-informal-strings-split

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StieltjesFunction.measure_singleton f : StieltjesFunction a : ℝ ⊢ ↑↑(StieltjesFunction.measure f) {a} = ofReal (↑f a - leftLim (↑f) a) obtain ⟨u, u_mono, u_lt_a, u_lim⟩ : ∃ u : ℕ → ℝ, StrictMono u ∧ (∀ n : ℕ, u n < a) ∧ Tendsto u atTop (𝓝 a) := exists_seq_strictMono_tendsto a case intro.intro.intro f : StieltjesFunction a : ℝ u : ℕ → ℝ u_mono : StrictMono u u_lt_a : ∀ (n : ℕ), u n < a u_lim : Tendsto u atTop (𝓝 a) ⊢ ↑↑(StieltjesFunction.measure f) {a} = ofReal (↑f a - leftLim (↑f) a) have A : {a} = ⋂ n, Ioc (u n) a := by refine' Subset.antisymm (fun x hx => by simp [mem_singleton_iff.1 hx, u_lt_a]) fun x hx => _ simp at hx have : a ≤ x := le_of_tendsto' u_lim fun n => (hx n).1.le simp [le_antisymm this (hx 0).2] case intro.intro.intro f : StieltjesFunction a : ℝ u : ℕ → ℝ u_mono : StrictMono u u_lt_a : ∀ (n : ℕ), u n < a u_lim : Tendsto u atTop (𝓝 a) A : {a} = ⋂ n, Ioc (u n) a L1 : Tendsto (fun n => ↑↑(StieltjesFunction.measure f) (Ioc (u n) a)) atTop (𝓝 (↑↑(StieltjesFunction.measure f) {a})) ⊢ ↑↑(StieltjesFunction.measure f) {a} = ofReal (↑f a - leftLim (↑f) a) have L2 : Tendsto (fun n => f.measure (Ioc (u n) a)) atTop (𝓝 (ofReal (f a - leftLim f a))) := by simp only [measure_Ioc] have : Tendsto (fun n => f (u n)) atTop (𝓝 (leftLim f a)) := by apply (f.mono.tendsto_leftLim a).comp exact tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _ u_lim (eventually_of_forall fun n => u_lt_a n) exact ENNReal.continuous_ofReal.continuousAt.tendsto.comp (tendsto_const_nhds.sub this) case intro.intro.intro f : StieltjesFunction a : ℝ u : ℕ → ℝ u_mono : StrictMono u u_lt_a : ∀ (n : ℕ), u n < a u_lim : Tendsto u atTop (𝓝 a) A : {a} = ⋂ n, Ioc (u n) a L1 : Tendsto (fun n => ↑↑(StieltjesFunction.measure f) (Ioc (u n) a)) atTop (𝓝 (↑↑(StieltjesFunction.measure f) {a})) L2 : Tendsto (fun n => ↑↑(StieltjesFunction.measure f) (Ioc (u n) a)) atTop (𝓝 (ofReal (↑f a - leftLim (↑f) a))) ⊢ ↑↑(StieltjesFunction.measure f) {a} = ofReal (↑f a - leftLim (↑f) a) exact tendsto_nhds_unique L1 L2 f : StieltjesFunction a : ℝ u : ℕ → ℝ u_mono : StrictMono u u_lt_a : ∀ (n : ℕ), u n < a u_lim : Tendsto u atTop (𝓝 a) ⊢ {a} = ⋂ n, Ioc (u n) a refine' Subset.antisymm (fun x hx => by simp [mem_singleton_iff.1 hx, u_lt_a]) fun x hx => _ f : StieltjesFunction a : ℝ u : ℕ → ℝ u_mono : StrictMono u u_lt_a : ∀ (n : ℕ), u n < a u_lim : Tendsto u atTop (𝓝 a) x : ℝ hx : x ∈ ⋂ n, Ioc (u n) a ⊢ x ∈ {a} simp at hx f : StieltjesFunction a : ℝ u : ℕ → ℝ u_mono : StrictMono u u_lt_a : ∀ (n : ℕ), u n < a u_lim : Tendsto u atTop (𝓝 a) x : ℝ hx : ∀ (i : ℕ), u i < x ∧ x ≤ a ⊢ x ∈ {a} have : a ≤ x := le_of_tendsto' u_lim fun n => (hx n).1.le f : StieltjesFunction a : ℝ u : ℕ → ℝ u_mono : StrictMono u u_lt_a : ∀ (n : ℕ), u n < a u_lim : Tendsto u atTop (𝓝 a) x : ℝ hx : ∀ (i : ℕ), u i < x ∧ x ≤ a this : a ≤ x ⊢ x ∈ {a} simp [le_antisymm this (hx 0).2] f : StieltjesFunction a : ℝ u : ℕ → ℝ u_mono : StrictMono u u_lt_a : ∀ (n : ℕ), u n < a u_lim : Tendsto u atTop (𝓝 a) x : ℝ hx : x ∈ {a} ⊢ x ∈ ⋂ n, Ioc (u n) a simp [mem_singleton_iff.1 hx, u_lt_a] f : StieltjesFunction a : ℝ u : ℕ → ℝ u_mono : StrictMono u u_lt_a : ∀ (n : ℕ), u n < a u_lim : Tendsto u atTop (𝓝 a) A : {a} = ⋂ n, Ioc (u n) a ⊢ Tendsto (fun n => ↑↑(StieltjesFunction.measure f) (Ioc (u n) a)) atTop (𝓝 (↑↑(StieltjesFunction.measure f) {a})) rw [A] f : StieltjesFunction a : ℝ u : ℕ → ℝ u_mono : StrictMono u u_lt_a : ∀ (n : ℕ), u n < a u_lim : Tendsto u atTop (𝓝 a) A : {a} = ⋂ n, Ioc (u n) a ⊢ Tendsto (fun n => ↑↑(StieltjesFunction.measure f) (Ioc (u n) a)) atTop (𝓝 (↑↑(StieltjesFunction.measure f) (⋂ n, Ioc (u n) a))) refine' tendsto_measure_iInter (fun n => measurableSet_Ioc) (fun m n hmn => _) _ case refine'_1 f : StieltjesFunction a : ℝ u : ℕ → ℝ u_mono : StrictMono u u_lt_a : ∀ (n : ℕ), u n < a u_lim : Tendsto u atTop (𝓝 a) A : {a} = ⋂ n, Ioc (u n) a m n : ℕ hmn : m ≤ n ⊢ Ioc (u n) a ≤ Ioc (u m) a exact Ioc_subset_Ioc (u_mono.monotone hmn) le_rfl case refine'_2 f : StieltjesFunction a : ℝ u : ℕ → ℝ u_mono : StrictMono u u_lt_a : ∀ (n : ℕ), u n < a u_lim : Tendsto u atTop (𝓝 a) A : {a} = ⋂ n, Ioc (u n) a ⊢ ∃ i, ↑↑(StieltjesFunction.measure f) (Ioc (u i) a) ≠ ⊤ exact ⟨0, by simpa only [measure_Ioc] using ENNReal.ofReal_ne_top⟩ f : StieltjesFunction a : ℝ u : ℕ → ℝ u_mono : StrictMono u u_lt_a : ∀ (n : ℕ), u n < a u_lim : Tendsto u atTop (𝓝 a) A : {a} = ⋂ n, Ioc (u n) a ⊢ ↑↑(StieltjesFunction.measure f) (Ioc (u 0) a) ≠ ⊤ simpa only [measure_Ioc] using ENNReal.ofReal_ne_top f : StieltjesFunction a : ℝ u : ℕ → ℝ u_mono : StrictMono u u_lt_a : ∀ (n : ℕ), u n < a u_lim : Tendsto u atTop (𝓝 a) A : {a} = ⋂ n, Ioc (u n) a L1 : Tendsto (fun n => ↑↑(StieltjesFunction.measure f) (Ioc (u n) a)) atTop (𝓝 (↑↑(StieltjesFunction.measure f) {a})) ⊢ Tendsto (fun n => ↑↑(StieltjesFunction.measure f) (Ioc (u n) a)) atTop (𝓝 (ofReal (↑f a - leftLim (↑f) a))) simp only [measure_Ioc] f : StieltjesFunction a : ℝ u : ℕ → ℝ u_mono : StrictMono u u_lt_a : ∀ (n : ℕ), u n < a u_lim : Tendsto u atTop (𝓝 a) A : {a} = ⋂ n, Ioc (u n) a L1 : Tendsto (fun n => ↑↑(StieltjesFunction.measure f) (Ioc (u n) a)) atTop (𝓝 (↑↑(StieltjesFunction.measure f) {a})) ⊢ Tendsto (fun n => ofReal (↑f a - ↑f (u n))) atTop (𝓝 (ofReal (↑f a - leftLim (↑f) a))) have : Tendsto (fun n => f (u n)) atTop (𝓝 (leftLim f a)) := by apply (f.mono.tendsto_leftLim a).comp exact tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _ u_lim (eventually_of_forall fun n => u_lt_a n) f : StieltjesFunction a : ℝ u : ℕ → ℝ u_mono : StrictMono u u_lt_a : ∀ (n : ℕ), u n < a u_lim : Tendsto u atTop (𝓝 a) A : {a} = ⋂ n, Ioc (u n) a L1 : Tendsto (fun n => ↑↑(StieltjesFunction.measure f) (Ioc (u n) a)) atTop (𝓝 (↑↑(StieltjesFunction.measure f) {a})) this : Tendsto (fun n => ↑f (u n)) atTop (𝓝 (leftLim (↑f) a)) ⊢ Tendsto (fun n => ofReal (↑f a - ↑f (u n))) atTop (𝓝 (ofReal (↑f a - leftLim (↑f) a))) exact ENNReal.continuous_ofReal.continuousAt.tendsto.comp (tendsto_const_nhds.sub this) f : StieltjesFunction a : ℝ u : ℕ → ℝ u_mono : StrictMono u u_lt_a : ∀ (n : ℕ), u n < a u_lim : Tendsto u atTop (𝓝 a) A : {a} = ⋂ n, Ioc (u n) a L1 : Tendsto (fun n => ↑↑(StieltjesFunction.measure f) (Ioc (u n) a)) atTop (𝓝 (↑↑(StieltjesFunction.measure f) {a})) ⊢ Tendsto (fun n => ↑f (u n)) atTop (𝓝 (leftLim (↑f) a)) apply (f.mono.tendsto_leftLim a).comp f : StieltjesFunction a : ℝ u : ℕ → ℝ u_mono : StrictMono u u_lt_a : ∀ (n : ℕ), u n < a u_lim : Tendsto u atTop (𝓝 a) A : {a} = ⋂ n, Ioc (u n) a L1 : Tendsto (fun n => ↑↑(StieltjesFunction.measure f) (Ioc (u n) a)) atTop (𝓝 (↑↑(StieltjesFunction.measure f) {a})) ⊢ Tendsto (fun n => u n) atTop (𝓝[Iio a] a) exact tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _ u_lim (eventually_of_forall fun n => u_lt_a n) ** Qed
StieltjesFunction.measure_Ico f : StieltjesFunction a b : ℝ ⊢ ↑↑(StieltjesFunction.measure f) (Ico a b) = ofReal (leftLim (↑f) b - leftLim (↑f) a) rcases le_or_lt b a with (hab \| hab) case inl f : StieltjesFunction a b : ℝ hab : b ≤ a ⊢ ↑↑(StieltjesFunction.measure f) (Ico a b) = ofReal (leftLim (↑f) b - leftLim (↑f) a) simp only [hab, measure_empty, Ico_eq_empty, not_lt] case inl f : StieltjesFunction a b : ℝ hab : b ≤ a ⊢ 0 = ofReal (leftLim (↑f) b - leftLim (↑f) a) symm case inl f : StieltjesFunction a b : ℝ hab : b ≤ a ⊢ ofReal (leftLim (↑f) b - leftLim (↑f) a) = 0 simp [ENNReal.ofReal_eq_zero, f.mono.leftLim hab] case inr f : StieltjesFunction a b : ℝ hab : a < b ⊢ ↑↑(StieltjesFunction.measure f) (Ico a b) = ofReal (leftLim (↑f) b - leftLim (↑f) a) have A : Disjoint {a} (Ioo a b) := by simp case inr f : StieltjesFunction a b : ℝ hab : a < b A : Disjoint {a} (Ioo a b) ⊢ ↑↑(StieltjesFunction.measure f) (Ico a b) = ofReal (leftLim (↑f) b - leftLim (↑f) a) simp [← Icc_union_Ioo_eq_Ico le_rfl hab, -singleton_union, hab.ne, f.mono.leftLim_le, measure_union A measurableSet_Ioo, f.mono.le_leftLim hab, ← ENNReal.ofReal_add] f : StieltjesFunction a b : ℝ hab : a < b ⊢ Disjoint {a} (Ioo a b) simp ** Qed
StieltjesFunction.measure_univ f : StieltjesFunction l u : ℝ hfl : Tendsto (↑f) atBot (𝓝 l) hfu : Tendsto (↑f) atTop (𝓝 u) ⊢ ↑↑(StieltjesFunction.measure f) univ = ofReal (u - l) refine' tendsto_nhds_unique (tendsto_measure_Iic_atTop _) _ f : StieltjesFunction l u : ℝ hfl : Tendsto (↑f) atBot (𝓝 l) hfu : Tendsto (↑f) atTop (𝓝 u) ⊢ Tendsto (fun x => ↑↑(StieltjesFunction.measure f) (Iic x)) atTop (𝓝 (ofReal (u - l))) simp_rw [measure_Iic f hfl] f : StieltjesFunction l u : ℝ hfl : Tendsto (↑f) atBot (𝓝 l) hfu : Tendsto (↑f) atTop (𝓝 u) ⊢ Tendsto (fun x => ofReal (↑f x - l)) atTop (𝓝 (ofReal (u - l))) exact ENNReal.tendsto_ofReal (Tendsto.sub_const hfu _) ** Qed
MeasureTheory.NullMeasurableSet.sUnion ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 m0 : MeasurableSpace α μ : Measure α s✝ t : Set α s : Set (Set α) hs : Set.Countable s h : ∀ (t : Set α), t ∈ s → NullMeasurableSet t ⊢ NullMeasurableSet (⋃₀ s) rw [sUnion_eq_biUnion] ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 m0 : MeasurableSpace α μ : Measure α s✝ t : Set α s : Set (Set α) hs : Set.Countable s h : ∀ (t : Set α), t ∈ s → NullMeasurableSet t ⊢ NullMeasurableSet (⋃ i ∈ s, i) exact MeasurableSet.biUnion hs h ** Qed
MeasureTheory.NullMeasurableSet.exists_measurable_superset_ae_eq ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 m0 : MeasurableSpace α μ : Measure α s t : Set α h : NullMeasurableSet s ⊢ ∃ t, t ⊇ s ∧ MeasurableSet t ∧ t =ᵐ[μ] s rcases h with ⟨t, htm, hst⟩ case intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 m0 : MeasurableSpace α μ : Measure α s t✝ t : Set α htm : MeasurableSet t hst : s =ᵐ[μ] t ⊢ ∃ t, t ⊇ s ∧ MeasurableSet t ∧ t =ᵐ[μ] s refine' ⟨t ∪ toMeasurable μ (s \ t), _, htm.union (measurableSet_toMeasurable _ _), _⟩ case intro.intro.refine'_1 ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 m0 : MeasurableSpace α μ : Measure α s t✝ t : Set α htm : MeasurableSet t hst : s =ᵐ[μ] t ⊢ t ∪ toMeasurable μ (s \ t) ⊇ s exact diff_subset_iff.1 (subset_toMeasurable _ _) case intro.intro.refine'_2 ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 m0 : MeasurableSpace α μ : Measure α s t✝ t : Set α htm : MeasurableSet t hst : s =ᵐ[μ] t ⊢ t ∪ toMeasurable μ (s \ t) =ᵐ[μ] s have : toMeasurable μ (s \ t) =ᵐ[μ] (∅ : Set α) := by simp [ae_le_set.1 hst.le] case intro.intro.refine'_2 ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 m0 : MeasurableSpace α μ : Measure α s t✝ t : Set α htm : MeasurableSet t hst : s =ᵐ[μ] t this : toMeasurable μ (s \ t) =ᵐ[μ] ∅ ⊢ t ∪ toMeasurable μ (s \ t) =ᵐ[μ] s simpa only [union_empty] using hst.symm.union this ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 m0 : MeasurableSpace α μ : Measure α s t✝ t : Set α htm : MeasurableSet t hst : s =ᵐ[μ] t ⊢ toMeasurable μ (s \ t) =ᵐ[μ] ∅ simp [ae_le_set.1 hst.le] ** Qed
MeasureTheory.measure_iUnion ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 m0✝ : MeasurableSpace α μ✝ : Measure α s t : Set α m0 : MeasurableSpace α μ : Measure α inst✝ : Countable ι f : ι → Set α hn : Pairwise (Disjoint on f) h : ∀ (i : ι), MeasurableSet (f i) ⊢ ↑↑μ (⋃ i, f i) = ∑' (i : ι), ↑↑μ (f i) rw [measure_eq_extend (MeasurableSet.iUnion h), extend_iUnion MeasurableSet.empty _ MeasurableSet.iUnion _ hn h] ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 m0✝ : MeasurableSpace α μ✝ : Measure α s t : Set α m0 : MeasurableSpace α μ : Measure α inst✝ : Countable ι f : ι → Set α hn : Pairwise (Disjoint on f) h : ∀ (i : ι), MeasurableSet (f i) ⊢ ∑' (i : ι), extend (fun t _ht => ↑↑μ t) (f i) = ∑' (i : ι), ↑↑μ (f i) simp [measure_eq_extend, h] ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 m0✝ : MeasurableSpace α μ✝ : Measure α s t : Set α m0 : MeasurableSpace α μ : Measure α inst✝ : Countable ι f : ι → Set α hn : Pairwise (Disjoint on f) h : ∀ (i : ι), MeasurableSet (f i) ⊢ ↑↑μ ∅ = 0 exact μ.empty ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 m0✝ : MeasurableSpace α μ✝ : Measure α s t : Set α m0 : MeasurableSpace α μ : Measure α inst✝ : Countable ι f : ι → Set α hn : Pairwise (Disjoint on f) h : ∀ (i : ι), MeasurableSet (f i) ⊢ ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), MeasurableSet (f i)) → Pairwise (Disjoint on f) → ↑↑μ (⋃ i, f i) = ∑' (i : ℕ), ↑↑μ (f i) exact μ.m_iUnion ** Qed
MeasureTheory.measure_union_add_inter₀ ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 m0 : MeasurableSpace α μ : Measure α s✝ t s : Set α ht : NullMeasurableSet t ⊢ ↑↑μ (s ∪ t) + ↑↑μ (s ∩ t) = ↑↑μ s + ↑↑μ t rw [← measure_inter_add_diff₀ (s ∪ t) ht, union_inter_cancel_right, union_diff_right, ← measure_inter_add_diff₀ s ht, add_comm, ← add_assoc, add_right_comm] ** Qed
MeasureTheory.measure_union_add_inter₀' ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 m0 : MeasurableSpace α μ : Measure α s t✝ : Set α hs : NullMeasurableSet s t : Set α ⊢ ↑↑μ (s ∪ t) + ↑↑μ (s ∩ t) = ↑↑μ s + ↑↑μ t rw [union_comm, inter_comm, measure_union_add_inter₀ t hs, add_comm] ** Qed
MeasureTheory.measure_union₀ ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 m0 : MeasurableSpace α μ : Measure α s t : Set α ht : NullMeasurableSet t hd : AEDisjoint μ s t ⊢ ↑↑μ (s ∪ t) = ↑↑μ s + ↑↑μ t rw [← measure_union_add_inter₀ s ht, hd, add_zero] ** Qed
MeasureTheory.measure_union₀' ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 m0 : MeasurableSpace α μ : Measure α s t : Set α hs : NullMeasurableSet s hd : AEDisjoint μ s t ⊢ ↑↑μ (s ∪ t) = ↑↑μ s + ↑↑μ t rw [union_comm, measure_union₀ hs (AEDisjoint.symm hd), add_comm] ** Qed
MeasureTheory.measure_add_measure_compl₀ ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 m0 : MeasurableSpace α μ : Measure α s✝ t s : Set α hs : NullMeasurableSet s ⊢ ↑↑μ s + ↑↑μ sᶜ = ↑↑μ univ rw [← measure_union₀' hs aedisjoint_compl_right, union_compl_self] ** Qed
MeasureTheory.trim_eq_self α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α ⊢ Measure.trim μ (_ : inst✝ ≤ inst✝) = μ simp [Measure.trim] ** Qed
MeasureTheory.toOuterMeasure_trim_eq_trim_toOuterMeasure α : Type u_1 m m0 : MeasurableSpace α μ✝ : Measure α s : Set α μ : Measure α hm : m ≤ m0 ⊢ ↑(Measure.trim μ hm) = OuterMeasure.trim ↑μ rw [Measure.trim, toMeasure_toOuterMeasure (ms := m)] ** Qed
MeasureTheory.trim_measurableSet_eq α : Type u_1 m m0 : MeasurableSpace α μ : Measure α s : Set α hm : m ≤ m0 hs : MeasurableSet s ⊢ ↑↑(Measure.trim μ hm) s = ↑↑μ s rw [Measure.trim, toMeasure_apply (ms := m) _ _ hs] ** Qed
MeasureTheory.le_trim α : Type u_1 m m0 : MeasurableSpace α μ : Measure α s : Set α hm : m ≤ m0 ⊢ ↑↑μ s ≤ ↑↑(Measure.trim μ hm) s simp_rw [Measure.trim] α : Type u_1 m m0 : MeasurableSpace α μ : Measure α s : Set α hm : m ≤ m0 ⊢ ↑↑μ s ≤ ↑↑(OuterMeasure.toMeasure ↑μ (_ : m ≤ OuterMeasure.caratheodory ↑μ)) s exact @le_toMeasure_apply _ m _ _ _ ** Qed
MeasureTheory.measure_trim_toMeasurable_eq_zero α : Type u_1 m m0 : MeasurableSpace α μ : Measure α s : Set α hm : m ≤ m0 hs : ↑↑(Measure.trim μ hm) s = 0 ⊢ ↑↑(Measure.trim μ hm) (toMeasurable (Measure.trim μ hm) s) = 0 rwa [@measure_toMeasurable _ m] ** Qed
MeasureTheory.trim_trim α : Type u_1 m m0 : MeasurableSpace α μ : Measure α s : Set α m₁ m₂ : MeasurableSpace α hm₁₂ : m₁ ≤ m₂ hm₂ : m₂ ≤ m0 ⊢ Measure.trim (Measure.trim μ hm₂) hm₁₂ = Measure.trim μ (_ : m₁ ≤ m0) refine @Measure.ext _ m₁ _ _ (fun t ht => ?_) α : Type u_1 m m0 : MeasurableSpace α μ : Measure α s : Set α m₁ m₂ : MeasurableSpace α hm₁₂ : m₁ ≤ m₂ hm₂ : m₂ ≤ m0 t : Set α ht : MeasurableSet t ⊢ ↑↑(Measure.trim (Measure.trim μ hm₂) hm₁₂) t = ↑↑(Measure.trim μ (_ : m₁ ≤ m0)) t rw [trim_measurableSet_eq hm₁₂ ht, trim_measurableSet_eq (hm₁₂.trans hm₂) ht, trim_measurableSet_eq hm₂ (hm₁₂ t ht)] ** Qed
MeasureTheory.restrict_trim α : Type u_1 m m0 : MeasurableSpace α μ✝ : Measure α s : Set α hm : m ≤ m0 μ : Measure α hs : MeasurableSet s ⊢ Measure.restrict (Measure.trim μ hm) s = Measure.trim (Measure.restrict μ s) hm refine @Measure.ext _ m _ _ (fun t ht => ?_) α : Type u_1 m m0 : MeasurableSpace α μ✝ : Measure α s : Set α hm : m ≤ m0 μ : Measure α hs : MeasurableSet s t : Set α ht : MeasurableSet t ⊢ ↑↑(Measure.restrict (Measure.trim μ hm) s) t = ↑↑(Measure.trim (Measure.restrict μ s) hm) t rw [@Measure.restrict_apply α m _ _ _ ht, trim_measurableSet_eq hm ht, Measure.restrict_apply (hm t ht), trim_measurableSet_eq hm (@MeasurableSet.inter α m t s ht hs)] ** Qed
MeasureTheory.exists_null_pairwise_disjoint_diff ι : Type u_1 α : Type u_2 m : MeasurableSpace α μ : Measure α s✝ t u v : Set α inst✝ : Countable ι s : ι → Set α hd : Pairwise (AEDisjoint μ on s) ⊢ ∃ t, (∀ (i : ι), MeasurableSet (t i)) ∧ (∀ (i : ι), ↑↑μ (t i) = 0) ∧ Pairwise (Disjoint on fun i => s i \ t i) refine' ⟨fun i => toMeasurable μ (s i ∩ ⋃ j ∈ ({i}ᶜ : Set ι), s j), fun i => measurableSet_toMeasurable _ _, fun i => _, _⟩ case refine'_1 ι : Type u_1 α : Type u_2 m : MeasurableSpace α μ : Measure α s✝ t u v : Set α inst✝ : Countable ι s : ι → Set α hd : Pairwise (AEDisjoint μ on s) i : ι ⊢ ↑↑μ ((fun i => toMeasurable μ (s i ∩ ⋃ j ∈ {i}ᶜ, s j)) i) = 0 simp only [measure_toMeasurable, inter_iUnion] case refine'_1 ι : Type u_1 α : Type u_2 m : MeasurableSpace α μ : Measure α s✝ t u v : Set α inst✝ : Countable ι s : ι → Set α hd : Pairwise (AEDisjoint μ on s) i : ι ⊢ ↑↑μ (⋃ i_1 ∈ {i}ᶜ, s i ∩ s i_1) = 0 exact (measure_biUnion_null_iff <\| to_countable _).2 fun j hj => hd (Ne.symm hj) case refine'_2 ι : Type u_1 α : Type u_2 m : MeasurableSpace α μ : Measure α s✝ t u v : Set α inst✝ : Countable ι s : ι → Set α hd : Pairwise (AEDisjoint μ on s) ⊢ Pairwise (Disjoint on fun i => s i \ (fun i => toMeasurable μ (s i ∩ ⋃ j ∈ {i}ᶜ, s j)) i) simp only [Pairwise, disjoint_left, onFun, mem_diff, not_and, and_imp, Classical.not_not] case refine'_2 ι : Type u_1 α : Type u_2 m : MeasurableSpace α μ : Measure α s✝ t u v : Set α inst✝ : Countable ι s : ι → Set α hd : Pairwise (AEDisjoint μ on s) ⊢ ∀ ⦃i j : ι⦄, i ≠ j → ∀ ⦃a : α⦄, a ∈ s i → ¬a ∈ toMeasurable μ (s i ∩ ⋃ j ∈ {i}ᶜ, s j) → a ∈ s j → a ∈ toMeasurable μ (s j ∩ ⋃ j_1 ∈ {j}ᶜ, s j_1) intro i j hne x hi hU hj case refine'_2 ι : Type u_1 α : Type u_2 m : MeasurableSpace α μ : Measure α s✝ t u v : Set α inst✝ : Countable ι s : ι → Set α hd : Pairwise (AEDisjoint μ on s) i j : ι hne : i ≠ j x : α hi : x ∈ s i hU : ¬x ∈ toMeasurable μ (s i ∩ ⋃ j ∈ {i}ᶜ, s j) hj : x ∈ s j ⊢ x ∈ toMeasurable μ (s j ∩ ⋃ j_1 ∈ {j}ᶜ, s j_1) replace hU : x ∉ s i ∩ iUnion λ j => iUnion λ _ => s j := λ h => hU (subset_toMeasurable _ _ h) case refine'_2 ι : Type u_1 α : Type u_2 m : MeasurableSpace α μ : Measure α s✝ t u v : Set α inst✝ : Countable ι s : ι → Set α hd : Pairwise (AEDisjoint μ on s) i j : ι hne : i ≠ j x : α hi : x ∈ s i hj : x ∈ s j hU : ¬x ∈ s i ∩ ⋃ j ∈ {i}ᶜ, s j ⊢ x ∈ toMeasurable μ (s j ∩ ⋃ j_1 ∈ {j}ᶜ, s j_1) simp only [mem_inter_iff, mem_iUnion, not_and, not_exists] at hU case refine'_2 ι : Type u_1 α : Type u_2 m : MeasurableSpace α μ : Measure α s✝ t u v : Set α inst✝ : Countable ι s : ι → Set α hd : Pairwise (AEDisjoint μ on s) i j : ι hne : i ≠ j x : α hi : x ∈ s i hj : x ∈ s j hU : x ∈ s i → ∀ (x_1 : ι), x_1 ∈ {i}ᶜ → ¬x ∈ s x_1 ⊢ x ∈ toMeasurable μ (s j ∩ ⋃ j_1 ∈ {j}ᶜ, s j_1) exact (hU hi j hne.symm hj).elim ** Qed
MeasureTheory.AEDisjoint.symm ι : Type u_1 α : Type u_2 m : MeasurableSpace α μ : Measure α s t u v : Set α h : AEDisjoint μ s t ⊢ AEDisjoint μ t s rwa [AEDisjoint, inter_comm] ** Qed
Disjoint.aedisjoint ι : Type u_1 α : Type u_2 m : MeasurableSpace α μ : Measure α s t u v : Set α h : Disjoint s t ⊢ AEDisjoint μ s t rw [AEDisjoint, disjoint_iff_inter_eq_empty.1 h, measure_empty] ** Qed
MeasureTheory.AEDisjoint.iUnion_left_iff ι : Type u_1 α : Type u_2 m : MeasurableSpace α μ : Measure α s✝ t u v : Set α inst✝ : Countable ι s : ι → Set α ⊢ AEDisjoint μ (⋃ i, s i) t ↔ ∀ (i : ι), AEDisjoint μ (s i) t simp only [AEDisjoint, iUnion_inter, measure_iUnion_null_iff] ** Qed
MeasureTheory.AEDisjoint.iUnion_right_iff ι : Type u_1 α : Type u_2 m : MeasurableSpace α μ : Measure α s t✝ u v : Set α inst✝ : Countable ι t : ι → Set α ⊢ AEDisjoint μ s (⋃ i, t i) ↔ ∀ (i : ι), AEDisjoint μ s (t i) simp only [AEDisjoint, inter_iUnion, measure_iUnion_null_iff] ** Qed
MeasureTheory.AEDisjoint.union_left_iff ι : Type u_1 α : Type u_2 m : MeasurableSpace α μ : Measure α s t u v : Set α ⊢ AEDisjoint μ (s ∪ t) u ↔ AEDisjoint μ s u ∧ AEDisjoint μ t u simp [union_eq_iUnion, and_comm] ** Qed
MeasureTheory.AEDisjoint.union_right_iff ι : Type u_1 α : Type u_2 m : MeasurableSpace α μ : Measure α s t u v : Set α ⊢ AEDisjoint μ s (t ∪ u) ↔ AEDisjoint μ s t ∧ AEDisjoint μ s u simp [union_eq_iUnion, and_comm] ** Qed
MeasureTheory.AEDisjoint.exists_disjoint_diff ι : Type u_1 α : Type u_2 m : MeasurableSpace α μ : Measure α s t u v : Set α h : AEDisjoint μ s t x : α hx : x ∈ s \ toMeasurable μ (s ∩ t) ⊢ x ∈ ?m.9910 h \ t simp ι : Type u_1 α : Type u_2 m : MeasurableSpace α μ : Measure α s t u v : Set α h : AEDisjoint μ s t x : α hx : x ∈ s \ toMeasurable μ (s ∩ t) ⊢ x ∈ ?m.9910 h ∧ ¬x ∈ t exact ⟨hx.1, fun hxt => hx.2 <\| subset_toMeasurable _ _ ⟨hx.1, hxt⟩⟩ ** Qed
LinearMap.exists_map_addHaar_eq_smul_addHaar 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : CompleteSpace 𝕜 inst✝¹⁰ : NormedAddCommGroup E inst✝⁹ : MeasurableSpace E inst✝⁸ : BorelSpace E inst✝⁷ : NormedSpace 𝕜 E inst✝⁶ : NormedAddCommGroup F inst✝⁵ : MeasurableSpace F inst✝⁴ : BorelSpace F inst✝³ : NormedSpace 𝕜 F L : E →ₗ[𝕜] F μ : Measure E ν : Measure F inst✝² : IsAddHaarMeasure μ inst✝¹ : IsAddHaarMeasure ν inst✝ : LocallyCompactSpace E h : Function.Surjective ↑L ⊢ ∃ c, 0 < c ∧ map (↑L) μ = c • ν rcases L.exists_map_addHaar_eq_smul_addHaar' μ ν h with ⟨c, c_pos, -, hc⟩ ** case intro.intro.intro 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : CompleteSpace 𝕜 inst✝¹⁰ : NormedAddCommGroup E inst✝⁹ : MeasurableSpace E inst✝⁸ : BorelSpace E inst✝⁷ : NormedSpace 𝕜 E inst✝⁶ : NormedAddCommGroup F inst✝⁵ : MeasurableSpace F inst✝⁴ : BorelSpace F inst✝³ : NormedSpace 𝕜 F L : E →ₗ[𝕜] F μ : Measure E ν : Measure F inst✝² : IsAddHaarMeasure μ inst✝¹ : IsAddHaarMeasure ν inst✝ : LocallyCompactSpace E h : Function.Surjective ↑L c : ℝ≥0∞ c_pos : 0 < c hc : map (↑L) μ = (c * ↑↑addHaar univ) • ν ⊢ ∃ c, 0 < c ∧ map (↑L) μ = c • ν exact ⟨_, by simp [c_pos, NeZero.ne addHaar], hc⟩ 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : CompleteSpace 𝕜 inst✝¹⁰ : NormedAddCommGroup E inst✝⁹ : MeasurableSpace E inst✝⁸ : BorelSpace E inst✝⁷ : NormedSpace 𝕜 E inst✝⁶ : NormedAddCommGroup F inst✝⁵ : MeasurableSpace F inst✝⁴ : BorelSpace F inst✝³ : NormedSpace 𝕜 F L : E →ₗ[𝕜] F μ : Measure E ν : Measure F inst✝² : IsAddHaarMeasure μ inst✝¹ : IsAddHaarMeasure ν inst✝ : LocallyCompactSpace E h : Function.Surjective ↑L c : ℝ≥0∞ c_pos : 0 < c hc : map (↑L) μ = (c * ↑↑addHaar univ) • ν ⊢ 0 < c * ↑↑addHaar univ simp [c_pos, NeZero.ne addHaar] Qed
mem_parallelepiped_iff ι : Type u_1 ι' : Type u_2 E : Type u_3 F : Type u_4 inst✝⁵ : Fintype ι inst✝⁴ : Fintype ι' inst✝³ : AddCommGroup E inst✝² : Module ℝ E inst✝¹ : AddCommGroup F inst✝ : Module ℝ F v : ι → E x : E ⊢ x ∈ parallelepiped v ↔ ∃ t _ht, x = ∑ i : ι, t i • v i simp [parallelepiped, eq_comm] ** Qed
image_parallelepiped ι : Type u_1 ι' : Type u_2 E : Type u_3 F : Type u_4 inst✝⁵ : Fintype ι inst✝⁴ : Fintype ι' inst✝³ : AddCommGroup E inst✝² : Module ℝ E inst✝¹ : AddCommGroup F inst✝ : Module ℝ F f : E →ₗ[ℝ] F v : ι → E ⊢ ↑f '' parallelepiped v = parallelepiped (↑f ∘ v) simp only [parallelepiped, ← image_comp] ι : Type u_1 ι' : Type u_2 E : Type u_3 F : Type u_4 inst✝⁵ : Fintype ι inst✝⁴ : Fintype ι' inst✝³ : AddCommGroup E inst✝² : Module ℝ E inst✝¹ : AddCommGroup F inst✝ : Module ℝ F f : E →ₗ[ℝ] F v : ι → E ⊢ ((fun a => ↑f a) ∘ fun t => ∑ i : ι, t i • v i) '' Icc 0 1 = (fun a => ∑ x : ι, a x • (↑f ∘ v) x) '' Icc 0 1 congr 1 with t case h ι : Type u_1 ι' : Type u_2 E : Type u_3 F : Type u_4 inst✝⁵ : Fintype ι inst✝⁴ : Fintype ι' inst✝³ : AddCommGroup E inst✝² : Module ℝ E inst✝¹ : AddCommGroup F inst✝ : Module ℝ F f : E →ₗ[ℝ] F v : ι → E t : F ⊢ t ∈ ((fun a => ↑f a) ∘ fun t => ∑ i : ι, t i • v i) '' Icc 0 1 ↔ t ∈ (fun a => ∑ x : ι, a x • (↑f ∘ v) x) '' Icc 0 1 simp only [Function.comp_apply, map_sum, LinearMap.map_smulₛₗ, RingHom.id_apply] ** Qed
Basis.addHaar_reindex ι : Type u_1 ι' : Type u_2 E : Type u_3 F : Type u_4 inst✝⁷ : Fintype ι inst✝⁶ : Fintype ι' inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ E inst✝² : NormedSpace ℝ F inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E b : Basis ι ℝ E e : ι ≃ ι' ⊢ addHaar (reindex b e) = addHaar b rw [Basis.addHaar, b.parallelepiped_reindex e, ← Basis.addHaar] ** Qed
Basis.addHaar_self ι : Type u_1 ι' : Type u_2 E : Type u_3 F : Type u_4 inst✝⁷ : Fintype ι inst✝⁶ : Fintype ι' inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ E inst✝² : NormedSpace ℝ F inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E b : Basis ι ℝ E ⊢ ↑↑(addHaar b) (_root_.parallelepiped ↑b) = 1 rw [Basis.addHaar] ι : Type u_1 ι' : Type u_2 E : Type u_3 F : Type u_4 inst✝⁷ : Fintype ι inst✝⁶ : Fintype ι' inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ E inst✝² : NormedSpace ℝ F inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E b : Basis ι ℝ E ⊢ ↑↑(addHaarMeasure (parallelepiped b)) (_root_.parallelepiped ↑b) = 1 exact addHaarMeasure_self ** Qed
MeasureTheory.Measure.dirac_apply_of_mem α : Type u_1 β : Type ?u.2369 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β s : Set α a : α h : a ∈ s ⊢ ↑↑(dirac a) s = 1 have : ∀ t : Set α, a ∈ t → t.indicator (1 : α → ℝ≥0∞) a = 1 := fun t ht => indicator_of_mem ht 1 α : Type u_1 β : Type ?u.2369 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β s : Set α a : α h : a ∈ s this : ∀ (t : Set α), a ∈ t → indicator t 1 a = 1 ⊢ ↑↑(dirac a) s = 1 refine' le_antisymm (this univ trivial ▸ _) (this s h ▸ le_dirac_apply) α : Type u_1 β : Type ?u.2369 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β s : Set α a : α h : a ∈ s this : ∀ (t : Set α), a ∈ t → indicator t 1 a = 1 ⊢ ↑↑(dirac a) s ≤ indicator univ 1 a rw [← dirac_apply' a MeasurableSet.univ] α : Type u_1 β : Type ?u.2369 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β s : Set α a : α h : a ∈ s this : ∀ (t : Set α), a ∈ t → indicator t 1 a = 1 ⊢ ↑↑(dirac a) s ≤ ↑↑(dirac a) univ exact measure_mono (subset_univ s) ** Qed
MeasureTheory.Measure.dirac_apply α : Type u_1 β : Type ?u.3265 inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β s✝ : Set α inst✝ : MeasurableSingletonClass α a : α s : Set α ⊢ ↑↑(dirac a) s = indicator s 1 a by_cases h : a ∈ s case neg α : Type u_1 β : Type ?u.3265 inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β s✝ : Set α inst✝ : MeasurableSingletonClass α a : α s : Set α h : ¬a ∈ s ⊢ ↑↑(dirac a) s = indicator s 1 a rw [indicator_of_not_mem h, ← nonpos_iff_eq_zero] case neg α : Type u_1 β : Type ?u.3265 inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β s✝ : Set α inst✝ : MeasurableSingletonClass α a : α s : Set α h : ¬a ∈ s ⊢ ↑↑(dirac a) s ≤ 0 calc dirac a s ≤ dirac a {a}ᶜ := measure_mono (subset_compl_comm.1 <\| singleton_subset_iff.2 h) _ = 0 := by simp [dirac_apply' _ (measurableSet_singleton _).compl] case pos α : Type u_1 β : Type ?u.3265 inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β s✝ : Set α inst✝ : MeasurableSingletonClass α a : α s : Set α h : a ∈ s ⊢ ↑↑(dirac a) s = indicator s 1 a rw [dirac_apply_of_mem h, indicator_of_mem h, Pi.one_apply] α : Type u_1 β : Type ?u.3265 inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β s✝ : Set α inst✝ : MeasurableSingletonClass α a : α s : Set α h : ¬a ∈ s ⊢ ↑↑(dirac a) {a}ᶜ = 0 simp [dirac_apply' _ (measurableSet_singleton _).compl] ** Qed
MeasureTheory.Measure.map_dirac α : Type u_2 β : Type u_1 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β s✝ : Set α f : α → β hf : Measurable f a : α s : Set β hs : MeasurableSet s ⊢ ↑↑(map f (dirac a)) s = ↑↑(dirac (f a)) s simp [hs, map_apply hf hs, hf hs, indicator_apply] ** Qed
MeasureTheory.Measure.tsum_indicator_apply_singleton α : Type u_1 β : Type ?u.19731 inst✝³ : MeasurableSpace α inst✝² : MeasurableSpace β s✝ : Set α inst✝¹ : Countable α inst✝ : MeasurableSingletonClass α μ : Measure α s : Set α hs : MeasurableSet s ⊢ ∑' (x : α), indicator s (fun x => ↑↑μ {x}) x = ↑↑(sum fun a => ↑↑μ {a} • dirac a) s simp only [Measure.sum_apply _ hs, Measure.smul_apply, smul_eq_mul, Measure.dirac_apply, Set.indicator_apply, mul_ite, Pi.one_apply, mul_one, mul_zero] α : Type u_1 β : Type ?u.19731 inst✝³ : MeasurableSpace α inst✝² : MeasurableSpace β s✝ : Set α inst✝¹ : Countable α inst✝ : MeasurableSingletonClass α μ : Measure α s : Set α hs : MeasurableSet s ⊢ ↑↑(sum fun a => ↑↑μ {a} • dirac a) s = ↑↑μ s rw [μ.sum_smul_dirac] ** Qed
MeasureTheory.mem_ae_dirac_iff α : Type u_1 β : Type ?u.22669 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β s : Set α a : α hs : MeasurableSet s ⊢ s ∈ ae (dirac a) ↔ a ∈ s by_cases a ∈ s <;> simp [mem_ae_iff, dirac_apply', hs.compl, indicator_apply, ] * Qed
MeasureTheory.ae_dirac_eq α : Type u_1 β : Type ?u.24458 inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β s : Set α inst✝ : MeasurableSingletonClass α a : α ⊢ ae (dirac a) = pure a ext s case a α : Type u_1 β : Type ?u.24458 inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β s✝ : Set α inst✝ : MeasurableSingletonClass α a : α s : Set α ⊢ s ∈ ae (dirac a) ↔ s ∈ pure a simp [mem_ae_iff, imp_false] ** Qed
MeasureTheory.ae_eq_dirac α : Type u_2 β : Type ?u.26627 inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β s : Set α δ : Type u_1 inst✝ : MeasurableSingletonClass α a : α f : α → δ ⊢ f =ᶠ[ae (dirac a)] const α (f a) simp [Filter.EventuallyEq] ** Qed
MeasureTheory.restrict_dirac' α : Type u_1 β : Type ?u.27642 inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β s : Set α a : α hs : MeasurableSet s inst✝ : Decidable (a ∈ s) ⊢ Measure.restrict (dirac a) s = if a ∈ s then dirac a else 0 split_ifs with has case pos α : Type u_1 β : Type ?u.27642 inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β s : Set α a : α hs : MeasurableSet s inst✝ : Decidable (a ∈ s) has : a ∈ s ⊢ Measure.restrict (dirac a) s = dirac a apply restrict_eq_self_of_ae_mem case pos.hs α : Type u_1 β : Type ?u.27642 inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β s : Set α a : α hs : MeasurableSet s inst✝ : Decidable (a ∈ s) has : a ∈ s ⊢ ∀ᵐ (x : α) ∂dirac a, x ∈ s rw [ae_dirac_iff] <;> assumption case neg α : Type u_1 β : Type ?u.27642 inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β s : Set α a : α hs : MeasurableSet s inst✝ : Decidable (a ∈ s) has : ¬a ∈ s ⊢ Measure.restrict (dirac a) s = 0 rw [restrict_eq_zero, dirac_apply' _ hs, indicator_of_not_mem has] ** Qed
MeasureTheory.restrict_dirac α : Type u_1 β : Type ?u.28539 inst✝³ : MeasurableSpace α inst✝² : MeasurableSpace β s : Set α a : α inst✝¹ : MeasurableSingletonClass α inst✝ : Decidable (a ∈ s) ⊢ Measure.restrict (dirac a) s = if a ∈ s then dirac a else 0 split_ifs with has case pos α : Type u_1 β : Type ?u.28539 inst✝³ : MeasurableSpace α inst✝² : MeasurableSpace β s : Set α a : α inst✝¹ : MeasurableSingletonClass α inst✝ : Decidable (a ∈ s) has : a ∈ s ⊢ Measure.restrict (dirac a) s = dirac a apply restrict_eq_self_of_ae_mem case pos.hs α : Type u_1 β : Type ?u.28539 inst✝³ : MeasurableSpace α inst✝² : MeasurableSpace β s : Set α a : α inst✝¹ : MeasurableSingletonClass α inst✝ : Decidable (a ∈ s) has : a ∈ s ⊢ ∀ᵐ (x : α) ∂dirac a, x ∈ s rwa [ae_dirac_eq] case neg α : Type u_1 β : Type ?u.28539 inst✝³ : MeasurableSpace α inst✝² : MeasurableSpace β s : Set α a : α inst✝¹ : MeasurableSingletonClass α inst✝ : Decidable (a ∈ s) has : ¬a ∈ s ⊢ Measure.restrict (dirac a) s = 0 rw [restrict_eq_zero, dirac_apply, indicator_of_not_mem has] ** Qed
MeasureTheory.FiniteMeasure.apply_mono Ω : Type u_1 inst✝ : MeasurableSpace Ω μ : FiniteMeasure Ω s₁ s₂ : Set Ω h : s₁ ⊆ s₂ ⊢ (fun s => ENNReal.toNNReal (↑↑↑μ s)) s₁ ≤ (fun s => ENNReal.toNNReal (↑↑↑μ s)) s₂ change ((μ : Measure Ω) s₁).toNNReal ≤ ((μ : Measure Ω) s₂).toNNReal Ω : Type u_1 inst✝ : MeasurableSpace Ω μ : FiniteMeasure Ω s₁ s₂ : Set Ω h : s₁ ⊆ s₂ ⊢ ENNReal.toNNReal (↑↑↑μ s₁) ≤ ENNReal.toNNReal (↑↑↑μ s₂) have key : (μ : Measure Ω) s₁ ≤ (μ : Measure Ω) s₂ := (μ : Measure Ω).mono h Ω : Type u_1 inst✝ : MeasurableSpace Ω μ : FiniteMeasure Ω s₁ s₂ : Set Ω h : s₁ ⊆ s₂ key : ↑↑↑μ s₁ ≤ ↑↑↑μ s₂ ⊢ ENNReal.toNNReal (↑↑↑μ s₁) ≤ ENNReal.toNNReal (↑↑↑μ s₂) apply (ENNReal.toNNReal_le_toNNReal (measure_ne_top _ s₁) (measure_ne_top _ s₂)).mpr key ** Qed
MeasureTheory.FiniteMeasure.apply_le_mass Ω : Type u_1 inst✝ : MeasurableSpace Ω μ : FiniteMeasure Ω s : Set Ω ⊢ (fun s => ENNReal.toNNReal (↑↑↑μ s)) s ≤ mass μ simpa using apply_mono μ (subset_univ s) ** Qed
MeasureTheory.FiniteMeasure.mass_nonzero_iff Ω : Type u_1 inst✝ : MeasurableSpace Ω μ : FiniteMeasure Ω ⊢ mass μ ≠ 0 ↔ μ ≠ 0 rw [not_iff_not] Ω : Type u_1 inst✝ : MeasurableSpace Ω μ : FiniteMeasure Ω ⊢ mass μ = 0 ↔ μ = 0 exact FiniteMeasure.mass_zero_iff μ ** Qed
MeasureTheory.FiniteMeasure.eq_of_forall_toMeasure_apply_eq Ω : Type u_1 inst✝ : MeasurableSpace Ω μ ν : FiniteMeasure Ω h : ∀ (s : Set Ω), MeasurableSet s → ↑↑↑μ s = ↑↑↑ν s ⊢ μ = ν apply Subtype.ext case a Ω : Type u_1 inst✝ : MeasurableSpace Ω μ ν : FiniteMeasure Ω h : ∀ (s : Set Ω), MeasurableSet s → ↑↑↑μ s = ↑↑↑ν s ⊢ ↑μ = ↑ν ext1 s s_mble case a.h Ω : Type u_1 inst✝ : MeasurableSpace Ω μ ν : FiniteMeasure Ω h : ∀ (s : Set Ω), MeasurableSet s → ↑↑↑μ s = ↑↑↑ν s s : Set Ω s_mble : MeasurableSet s ⊢ ↑↑↑μ s = ↑↑↑ν s exact h s s_mble ** Qed
MeasureTheory.FiniteMeasure.eq_of_forall_apply_eq Ω : Type u_1 inst✝ : MeasurableSpace Ω μ ν : FiniteMeasure Ω h : ∀ (s : Set Ω), MeasurableSet s → (fun s => ENNReal.toNNReal (↑↑↑μ s)) s = (fun s => ENNReal.toNNReal (↑↑↑ν s)) s ⊢ μ = ν ext1 s s_mble case h Ω : Type u_1 inst✝ : MeasurableSpace Ω μ ν : FiniteMeasure Ω h : ∀ (s : Set Ω), MeasurableSet s → (fun s => ENNReal.toNNReal (↑↑↑μ s)) s = (fun s => ENNReal.toNNReal (↑↑↑ν s)) s s : Set Ω s_mble : MeasurableSet s ⊢ ↑↑↑μ s = ↑↑↑ν s simpa [ennreal_coeFn_eq_coeFn_toMeasure] using congr_arg ((↑) : ℝ≥0 → ℝ≥0∞) (h s s_mble) ** Qed
MeasureTheory.FiniteMeasure.coeFn_zero Ω : Type u_1 inst✝⁴ : MeasurableSpace Ω R : Type u_2 inst✝³ : SMul R ℝ≥0 inst✝² : SMul R ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ ⊢ (fun s => ENNReal.toNNReal (↑↑↑0 s)) = 0 funext case h Ω : Type u_1 inst✝⁴ : MeasurableSpace Ω R : Type u_2 inst✝³ : SMul R ℝ≥0 inst✝² : SMul R ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ x✝ : Set Ω ⊢ ENNReal.toNNReal (↑↑↑0 x✝) = OfNat.ofNat 0 x✝ rfl ** Qed
MeasureTheory.FiniteMeasure.coeFn_add Ω : Type u_1 inst✝⁴ : MeasurableSpace Ω R : Type u_2 inst✝³ : SMul R ℝ≥0 inst✝² : SMul R ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ μ ν : FiniteMeasure Ω ⊢ (fun s => ENNReal.toNNReal (↑↑↑(μ + ν) s)) = (fun s => ENNReal.toNNReal (↑↑↑μ s)) + fun s => ENNReal.toNNReal (↑↑↑ν s) funext case h Ω : Type u_1 inst✝⁴ : MeasurableSpace Ω R : Type u_2 inst✝³ : SMul R ℝ≥0 inst✝² : SMul R ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ μ ν : FiniteMeasure Ω x✝ : Set Ω ⊢ ENNReal.toNNReal (↑↑↑(μ + ν) x✝) = ((fun s => ENNReal.toNNReal (↑↑↑μ s)) + fun s => ENNReal.toNNReal (↑↑↑ν s)) x✝ simp only [Pi.add_apply, ← ENNReal.coe_eq_coe, ne_eq, ennreal_coeFn_eq_coeFn_toMeasure, ENNReal.coe_add] case h Ω : Type u_1 inst✝⁴ : MeasurableSpace Ω R : Type u_2 inst✝³ : SMul R ℝ≥0 inst✝² : SMul R ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ μ ν : FiniteMeasure Ω x✝ : Set Ω ⊢ ↑↑↑(μ + ν) x✝ = ↑↑↑μ x✝ + ↑↑↑ν x✝ norm_cast ** Qed
MeasureTheory.FiniteMeasure.coeFn_smul Ω : Type u_1 inst✝⁵ : MeasurableSpace Ω R : Type u_2 inst✝⁴ : SMul R ℝ≥0 inst✝³ : SMul R ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝ : IsScalarTower R ℝ≥0 ℝ≥0 c : R μ : FiniteMeasure Ω ⊢ (fun s => ENNReal.toNNReal (↑↑↑(c • μ) s)) = c • fun s => ENNReal.toNNReal (↑↑↑μ s) funext case h Ω : Type u_1 inst✝⁵ : MeasurableSpace Ω R : Type u_2 inst✝⁴ : SMul R ℝ≥0 inst✝³ : SMul R ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝ : IsScalarTower R ℝ≥0 ℝ≥0 c : R μ : FiniteMeasure Ω x✝ : Set Ω ⊢ ENNReal.toNNReal (↑↑↑(c • μ) x✝) = (c • fun s => ENNReal.toNNReal (↑↑↑μ s)) x✝ simp only [Pi.smul_apply, ← ENNReal.coe_eq_coe, ne_eq, ennreal_coeFn_eq_coeFn_toMeasure, ENNReal.coe_smul] case h Ω : Type u_1 inst✝⁵ : MeasurableSpace Ω R : Type u_2 inst✝⁴ : SMul R ℝ≥0 inst✝³ : SMul R ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝ : IsScalarTower R ℝ≥0 ℝ≥0 c : R μ : FiniteMeasure Ω x✝ : Set Ω ⊢ ↑↑↑(c • μ) x✝ = c • ↑↑↑μ x✝ norm_cast ** Qed
MeasureTheory.FiniteMeasure.restrict_apply Ω : Type u_1 inst✝⁴ : MeasurableSpace Ω R : Type u_2 inst✝³ : SMul R ℝ≥0 inst✝² : SMul R ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ μ : FiniteMeasure Ω A s : Set Ω s_mble : MeasurableSet s ⊢ (fun s => ENNReal.toNNReal (↑↑↑(restrict μ A) s)) s = (fun s => ENNReal.toNNReal (↑↑↑μ s)) (s ∩ A) apply congr_arg ENNReal.toNNReal Ω : Type u_1 inst✝⁴ : MeasurableSpace Ω R : Type u_2 inst✝³ : SMul R ℝ≥0 inst✝² : SMul R ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ μ : FiniteMeasure Ω A s : Set Ω s_mble : MeasurableSet s ⊢ ↑↑↑(restrict μ A) s = ↑↑↑μ (s ∩ A) exact Measure.restrict_apply s_mble ** Qed
MeasureTheory.FiniteMeasure.restrict_eq_zero_iff Ω : Type u_1 inst✝⁴ : MeasurableSpace Ω R : Type u_2 inst✝³ : SMul R ℝ≥0 inst✝² : SMul R ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ μ : FiniteMeasure Ω A : Set Ω ⊢ restrict μ A = 0 ↔ (fun s => ENNReal.toNNReal (↑↑↑μ s)) A = 0 rw [← mass_zero_iff, restrict_mass] ** Qed
MeasureTheory.FiniteMeasure.restrict_nonzero_iff Ω : Type u_1 inst✝⁴ : MeasurableSpace Ω R : Type u_2 inst✝³ : SMul R ℝ≥0 inst✝² : SMul R ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ μ : FiniteMeasure Ω A : Set Ω ⊢ restrict μ A ≠ 0 ↔ (fun s => ENNReal.toNNReal (↑↑↑μ s)) A ≠ 0 rw [← mass_nonzero_iff, restrict_mass] ** Qed
MeasureTheory.FiniteMeasure.testAgainstNN_const ** Ω : Type u_1 inst✝⁵ : MeasurableSpace Ω R : Type u_2 inst✝⁴ : SMul R ℝ≥0 inst✝³ : SMul R ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝ : TopologicalSpace Ω μ : FiniteMeasure Ω c : ℝ≥0 ⊢ testAgainstNN μ (const Ω c) = c * mass μ simp [← ENNReal.coe_eq_coe] Qed
MeasureTheory.FiniteMeasure.testAgainstNN_mono Ω : Type u_1 inst✝⁵ : MeasurableSpace Ω R : Type u_2 inst✝⁴ : SMul R ℝ≥0 inst✝³ : SMul R ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝ : TopologicalSpace Ω μ : FiniteMeasure Ω f g : Ω →ᵇ ℝ≥0 f_le_g : ↑f ≤ ↑g ⊢ testAgainstNN μ f ≤ testAgainstNN μ g simp only [← ENNReal.coe_le_coe, testAgainstNN_coe_eq] Ω : Type u_1 inst✝⁵ : MeasurableSpace Ω R : Type u_2 inst✝⁴ : SMul R ℝ≥0 inst✝³ : SMul R ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝ : TopologicalSpace Ω μ : FiniteMeasure Ω f g : Ω →ᵇ ℝ≥0 f_le_g : ↑f ≤ ↑g ⊢ ∫⁻ (ω : Ω), ↑(↑f ω) ∂↑μ ≤ ∫⁻ (ω : Ω), ↑(↑g ω) ∂↑μ exact lintegral_mono fun ω => ENNReal.coe_mono (f_le_g ω) ** Qed
MeasureTheory.FiniteMeasure.testAgainstNN_zero Ω : Type u_1 inst✝⁵ : MeasurableSpace Ω R : Type u_2 inst✝⁴ : SMul R ℝ≥0 inst✝³ : SMul R ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝ : TopologicalSpace Ω μ : FiniteMeasure Ω ⊢ testAgainstNN μ 0 = 0 simpa only [zero_mul] using μ.testAgainstNN_const 0 ** Qed
MeasureTheory.FiniteMeasure.testAgainstNN_one Ω : Type u_1 inst✝⁵ : MeasurableSpace Ω R : Type u_2 inst✝⁴ : SMul R ℝ≥0 inst✝³ : SMul R ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝ : TopologicalSpace Ω μ : FiniteMeasure Ω ⊢ testAgainstNN μ 1 = mass μ simp only [testAgainstNN, coe_one, Pi.one_apply, ENNReal.coe_one, lintegral_one] Ω : Type u_1 inst✝⁵ : MeasurableSpace Ω R : Type u_2 inst✝⁴ : SMul R ℝ≥0 inst✝³ : SMul R ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝ : TopologicalSpace Ω μ : FiniteMeasure Ω ⊢ ENNReal.toNNReal (↑↑↑μ univ) = mass μ rfl ** Qed
MeasureTheory.FiniteMeasure.zero_testAgainstNN Ω : Type u_1 inst✝⁵ : MeasurableSpace Ω R : Type u_2 inst✝⁴ : SMul R ℝ≥0 inst✝³ : SMul R ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝ : TopologicalSpace Ω ⊢ testAgainstNN 0 = 0 funext case h Ω : Type u_1 inst✝⁵ : MeasurableSpace Ω R : Type u_2 inst✝⁴ : SMul R ℝ≥0 inst✝³ : SMul R ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝ : TopologicalSpace Ω x✝ : Ω →ᵇ ℝ≥0 ⊢ testAgainstNN 0 x✝ = OfNat.ofNat 0 x✝ simp only [zero_testAgainstNN_apply, Pi.zero_apply] ** Qed
MeasureTheory.FiniteMeasure.testAgainstNN_lipschitz Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω μ : FiniteMeasure Ω ⊢ LipschitzWith (mass μ) fun f => testAgainstNN μ f rw [lipschitzWith_iff_dist_le_mul] ** Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω μ : FiniteMeasure Ω ⊢ ∀ (x y : Ω →ᵇ ℝ≥0), dist (testAgainstNN μ x) (testAgainstNN μ y) ≤ ↑(mass μ) * dist x y intro f₁ f₂ Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω μ : FiniteMeasure Ω f₁ f₂ : Ω →ᵇ ℝ≥0 ⊢ dist (testAgainstNN μ f₁) (testAgainstNN μ f₂) ≤ ↑(mass μ) * dist f₁ f₂ ** suffices abs (μ.testAgainstNN f₁ - μ.testAgainstNN f₂ : ℝ) ≤ μ.mass * dist f₁ f₂ by rwa [NNReal.dist_eq] ** Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω μ : FiniteMeasure Ω f₁ f₂ : Ω →ᵇ ℝ≥0 ⊢ \|↑(testAgainstNN μ f₁) - ↑(testAgainstNN μ f₂)\| ≤ ↑(mass μ) * dist f₁ f₂ apply abs_le.mpr Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω μ : FiniteMeasure Ω f₁ f₂ : Ω →ᵇ ℝ≥0 ⊢ -(↑(mass μ) * dist f₁ f₂) ≤ ↑(testAgainstNN μ f₁) - ↑(testAgainstNN μ f₂) ∧ ↑(testAgainstNN μ f₁) - ↑(testAgainstNN μ f₂) ≤ ↑(mass μ) * dist f₁ f₂ constructor Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω μ : FiniteMeasure Ω f₁ f₂ : Ω →ᵇ ℝ≥0 this : \|↑(testAgainstNN μ f₁) - ↑(testAgainstNN μ f₂)\| ≤ ↑(mass μ) * dist f₁ f₂ ⊢ dist (testAgainstNN μ f₁) (testAgainstNN μ f₂) ≤ ↑(mass μ) * dist f₁ f₂ rwa [NNReal.dist_eq] case left Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω μ : FiniteMeasure Ω f₁ f₂ : Ω →ᵇ ℝ≥0 ⊢ -(↑(mass μ) * dist f₁ f₂) ≤ ↑(testAgainstNN μ f₁) - ↑(testAgainstNN μ f₂) have key' := μ.testAgainstNN_lipschitz_estimate f₂ f₁ case left Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω μ : FiniteMeasure Ω f₁ f₂ : Ω →ᵇ ℝ≥0 key' : testAgainstNN μ f₂ ≤ testAgainstNN μ f₁ + nndist f₂ f₁ * mass μ ⊢ -(↑(mass μ) * dist f₁ f₂) ≤ ↑(testAgainstNN μ f₁) - ↑(testAgainstNN μ f₂) rw [mul_comm] at key' case left Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω μ : FiniteMeasure Ω f₁ f₂ : Ω →ᵇ ℝ≥0 key' : testAgainstNN μ f₂ ≤ testAgainstNN μ f₁ + mass μ * nndist f₂ f₁ ⊢ -(↑(mass μ) * dist f₁ f₂) ≤ ↑(testAgainstNN μ f₁) - ↑(testAgainstNN μ f₂) ** suffices ↑(μ.testAgainstNN f₂) ≤ ↑(μ.testAgainstNN f₁) + ↑μ.mass * dist f₁ f₂ by linarith ** case left Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω μ : FiniteMeasure Ω f₁ f₂ : Ω →ᵇ ℝ≥0 key' : testAgainstNN μ f₂ ≤ testAgainstNN μ f₁ + mass μ * nndist f₂ f₁ ⊢ ↑(testAgainstNN μ f₂) ≤ ↑(testAgainstNN μ f₁) + ↑(mass μ) * dist f₁ f₂ have key := NNReal.coe_mono key' case left Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω μ : FiniteMeasure Ω f₁ f₂ : Ω →ᵇ ℝ≥0 key' : testAgainstNN μ f₂ ≤ testAgainstNN μ f₁ + mass μ * nndist f₂ f₁ key : ↑(testAgainstNN μ f₂) ≤ ↑(testAgainstNN μ f₁ + mass μ * nndist f₂ f₁) ⊢ ↑(testAgainstNN μ f₂) ≤ ↑(testAgainstNN μ f₁) + ↑(mass μ) * dist f₁ f₂ rwa [NNReal.coe_add, NNReal.coe_mul, nndist_comm] at key Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω μ : FiniteMeasure Ω f₁ f₂ : Ω →ᵇ ℝ≥0 key' : testAgainstNN μ f₂ ≤ testAgainstNN μ f₁ + mass μ * nndist f₂ f₁ this : ↑(testAgainstNN μ f₂) ≤ ↑(testAgainstNN μ f₁) + ↑(mass μ) * dist f₁ f₂ ⊢ -(↑(mass μ) * dist f₁ f₂) ≤ ↑(testAgainstNN μ f₁) - ↑(testAgainstNN μ f₂) linarith case right Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω μ : FiniteMeasure Ω f₁ f₂ : Ω →ᵇ ℝ≥0 ⊢ ↑(testAgainstNN μ f₁) - ↑(testAgainstNN μ f₂) ≤ ↑(mass μ) * dist f₁ f₂ have key' := μ.testAgainstNN_lipschitz_estimate f₁ f₂ case right Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω μ : FiniteMeasure Ω f₁ f₂ : Ω →ᵇ ℝ≥0 key' : testAgainstNN μ f₁ ≤ testAgainstNN μ f₂ + nndist f₁ f₂ * mass μ ⊢ ↑(testAgainstNN μ f₁) - ↑(testAgainstNN μ f₂) ≤ ↑(mass μ) * dist f₁ f₂ rw [mul_comm] at key' case right Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω μ : FiniteMeasure Ω f₁ f₂ : Ω →ᵇ ℝ≥0 key' : testAgainstNN μ f₁ ≤ testAgainstNN μ f₂ + mass μ * nndist f₁ f₂ ⊢ ↑(testAgainstNN μ f₁) - ↑(testAgainstNN μ f₂) ≤ ↑(mass μ) * dist f₁ f₂ ** suffices ↑(μ.testAgainstNN f₁) ≤ ↑(μ.testAgainstNN f₂) + ↑μ.mass * dist f₁ f₂ by linarith ** case right Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω μ : FiniteMeasure Ω f₁ f₂ : Ω →ᵇ ℝ≥0 key' : testAgainstNN μ f₁ ≤ testAgainstNN μ f₂ + mass μ * nndist f₁ f₂ ⊢ ↑(testAgainstNN μ f₁) ≤ ↑(testAgainstNN μ f₂) + ↑(mass μ) * dist f₁ f₂ have key := NNReal.coe_mono key' case right Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω μ : FiniteMeasure Ω f₁ f₂ : Ω →ᵇ ℝ≥0 key' : testAgainstNN μ f₁ ≤ testAgainstNN μ f₂ + mass μ * nndist f₁ f₂ key : ↑(testAgainstNN μ f₁) ≤ ↑(testAgainstNN μ f₂ + mass μ * nndist f₁ f₂) ⊢ ↑(testAgainstNN μ f₁) ≤ ↑(testAgainstNN μ f₂) + ↑(mass μ) * dist f₁ f₂ rwa [NNReal.coe_add, NNReal.coe_mul] at key Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω μ : FiniteMeasure Ω f₁ f₂ : Ω →ᵇ ℝ≥0 key' : testAgainstNN μ f₁ ≤ testAgainstNN μ f₂ + mass μ * nndist f₁ f₂ this : ↑(testAgainstNN μ f₁) ≤ ↑(testAgainstNN μ f₂) + ↑(mass μ) * dist f₁ f₂ ⊢ ↑(testAgainstNN μ f₁) - ↑(testAgainstNN μ f₂) ≤ ↑(mass μ) * dist f₁ f₂ linarith Qed
MeasureTheory.FiniteMeasure.tendsto_iff_forall_testAgainstNN_tendsto Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_3 F : Filter γ μs : γ → FiniteMeasure Ω μ : FiniteMeasure Ω ⊢ Tendsto μs F (𝓝 μ) ↔ ∀ (f : Ω →ᵇ ℝ≥0), Tendsto (fun i => testAgainstNN (μs i) f) F (𝓝 (testAgainstNN μ f)) rw [FiniteMeasure.tendsto_iff_forall_toWeakDualBCNN_tendsto] Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_3 F : Filter γ μs : γ → FiniteMeasure Ω μ : FiniteMeasure Ω ⊢ (∀ (f : Ω →ᵇ ℝ≥0), Tendsto (fun i => ↑(toWeakDualBCNN (μs i)) f) F (𝓝 (↑(toWeakDualBCNN μ) f))) ↔ ∀ (f : Ω →ᵇ ℝ≥0), Tendsto (fun i => testAgainstNN (μs i) f) F (𝓝 (testAgainstNN μ f)) rfl ** Qed
MeasureTheory.FiniteMeasure.tendsto_zero_of_tendsto_zero_mass Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_3 F : Filter γ μs : γ → FiniteMeasure Ω mass_lim : Tendsto (fun i => mass (μs i)) F (𝓝 0) ⊢ Tendsto μs F (𝓝 0) rw [tendsto_iff_forall_testAgainstNN_tendsto] Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_3 F : Filter γ μs : γ → FiniteMeasure Ω mass_lim : Tendsto (fun i => mass (μs i)) F (𝓝 0) ⊢ ∀ (f : Ω →ᵇ ℝ≥0), Tendsto (fun i => testAgainstNN (μs i) f) F (𝓝 (testAgainstNN 0 f)) intro f Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_3 F : Filter γ μs : γ → FiniteMeasure Ω mass_lim : Tendsto (fun i => mass (μs i)) F (𝓝 0) f : Ω →ᵇ ℝ≥0 ⊢ Tendsto (fun i => testAgainstNN (μs i) f) F (𝓝 (testAgainstNN 0 f)) convert tendsto_zero_testAgainstNN_of_tendsto_zero_mass mass_lim f case h.e'_5.h.e'_3 Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_3 F : Filter γ μs : γ → FiniteMeasure Ω mass_lim : Tendsto (fun i => mass (μs i)) F (𝓝 0) f : Ω →ᵇ ℝ≥0 ⊢ testAgainstNN 0 f = 0 rw [zero_testAgainstNN_apply] ** Qed
MeasureTheory.FiniteMeasure.tendsto_testAgainstNN_filter_of_le_const Ω : Type u_1 inst✝³ : MeasurableSpace Ω inst✝² : TopologicalSpace Ω inst✝¹ : OpensMeasurableSpace Ω ι : Type u_2 L : Filter ι inst✝ : IsCountablyGenerated L μ : FiniteMeasure Ω fs : ι → Ω →ᵇ ℝ≥0 c : ℝ≥0 fs_le_const : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂↑μ, ↑(fs i) ω ≤ c f : Ω →ᵇ ℝ≥0 fs_lim : ∀ᵐ (ω : Ω) ∂↑μ, Tendsto (fun i => ↑(fs i) ω) L (𝓝 (↑f ω)) ⊢ Tendsto (fun i => testAgainstNN μ (fs i)) L (𝓝 (testAgainstNN μ f)) apply (ENNReal.tendsto_toNNReal (f.lintegral_lt_top_of_nnreal (μ : Measure Ω)).ne).comp Ω : Type u_1 inst✝³ : MeasurableSpace Ω inst✝² : TopologicalSpace Ω inst✝¹ : OpensMeasurableSpace Ω ι : Type u_2 L : Filter ι inst✝ : IsCountablyGenerated L μ : FiniteMeasure Ω fs : ι → Ω →ᵇ ℝ≥0 c : ℝ≥0 fs_le_const : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂↑μ, ↑(fs i) ω ≤ c f : Ω →ᵇ ℝ≥0 fs_lim : ∀ᵐ (ω : Ω) ∂↑μ, Tendsto (fun i => ↑(fs i) ω) L (𝓝 (↑f ω)) ⊢ Tendsto (fun i => ∫⁻ (ω : Ω), ↑(↑(fs i) ω) ∂↑μ) L (𝓝 (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ)) exact tendsto_lintegral_nn_filter_of_le_const μ fs_le_const fs_lim ** Qed
MeasureTheory.FiniteMeasure.tendsto_of_forall_integral_tendsto Ω : Type u_1 inst✝² : MeasurableSpace Ω inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_2 F : Filter γ μs : γ → FiniteMeasure Ω μ : FiniteMeasure Ω h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), ↑f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), ↑f x ∂↑μ)) ⊢ Tendsto μs F (𝓝 μ) apply (@tendsto_iff_forall_lintegral_tendsto Ω _ _ _ γ F μs μ).mpr Ω : Type u_1 inst✝² : MeasurableSpace Ω inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_2 F : Filter γ μs : γ → FiniteMeasure Ω μ : FiniteMeasure Ω h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), ↑f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), ↑f x ∂↑μ)) ⊢ ∀ (f : Ω →ᵇ ℝ≥0), Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ)) intro f Ω : Type u_1 inst✝² : MeasurableSpace Ω inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_2 F : Filter γ μs : γ → FiniteMeasure Ω μ : FiniteMeasure Ω h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), ↑f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), ↑f x ∂↑μ)) f : Ω →ᵇ ℝ≥0 ⊢ Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ)) have key := @ENNReal.tendsto_toReal_iff _ F _ (fun i => (f.lintegral_lt_top_of_nnreal (μs i)).ne) _ (f.lintegral_lt_top_of_nnreal μ).ne Ω : Type u_1 inst✝² : MeasurableSpace Ω inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_2 F : Filter γ μs : γ → FiniteMeasure Ω μ : FiniteMeasure Ω h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), ↑f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), ↑f x ∂↑μ)) f : Ω →ᵇ ℝ≥0 key : Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) ↔ Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ)) ⊢ Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ)) apply key.mp Ω : Type u_1 inst✝² : MeasurableSpace Ω inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_2 F : Filter γ μs : γ → FiniteMeasure Ω μ : FiniteMeasure Ω h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), ↑f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), ↑f x ∂↑μ)) f : Ω →ᵇ ℝ≥0 key : Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) ↔ Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ)) ⊢ Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) have lip : LipschitzWith 1 ((↑) : ℝ≥0 → ℝ) := isometry_subtype_coe.lipschitz Ω : Type u_1 inst✝² : MeasurableSpace Ω inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_2 F : Filter γ μs : γ → FiniteMeasure Ω μ : FiniteMeasure Ω h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), ↑f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), ↑f x ∂↑μ)) f : Ω →ᵇ ℝ≥0 key : Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) ↔ Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ)) lip : LipschitzWith 1 NNReal.toReal ⊢ Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) set f₀ := BoundedContinuousFunction.comp _ lip f with _def_f₀ Ω : Type u_1 inst✝² : MeasurableSpace Ω inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_2 F : Filter γ μs : γ → FiniteMeasure Ω μ : FiniteMeasure Ω h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), ↑f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), ↑f x ∂↑μ)) f : Ω →ᵇ ℝ≥0 key : Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) ↔ Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ)) lip : LipschitzWith 1 NNReal.toReal f₀ : Ω →ᵇ ℝ := comp NNReal.toReal lip f _def_f₀ : f₀ = comp NNReal.toReal lip f ⊢ Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) have f₀_eq : ⇑f₀ = ((↑) : ℝ≥0 → ℝ) ∘ ⇑f := rfl Ω : Type u_1 inst✝² : MeasurableSpace Ω inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_2 F : Filter γ μs : γ → FiniteMeasure Ω μ : FiniteMeasure Ω h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), ↑f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), ↑f x ∂↑μ)) f : Ω →ᵇ ℝ≥0 key : Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) ↔ Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ)) lip : LipschitzWith 1 NNReal.toReal f₀ : Ω →ᵇ ℝ := comp NNReal.toReal lip f _def_f₀ : f₀ = comp NNReal.toReal lip f f₀_eq : ↑f₀ = NNReal.toReal ∘ ↑f ⊢ Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) have f₀_nn : 0 ≤ ⇑f₀ := fun _ => by simp only [f₀_eq, Pi.zero_apply, Function.comp_apply, NNReal.zero_le_coe] Ω : Type u_1 inst✝² : MeasurableSpace Ω inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_2 F : Filter γ μs : γ → FiniteMeasure Ω μ : FiniteMeasure Ω h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), ↑f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), ↑f x ∂↑μ)) f : Ω →ᵇ ℝ≥0 key : Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) ↔ Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ)) lip : LipschitzWith 1 NNReal.toReal f₀ : Ω →ᵇ ℝ := comp NNReal.toReal lip f _def_f₀ : f₀ = comp NNReal.toReal lip f f₀_eq : ↑f₀ = NNReal.toReal ∘ ↑f f₀_nn : 0 ≤ ↑f₀ ⊢ Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) have f₀_ae_nn : 0 ≤ᵐ[(μ : Measure Ω)] ⇑f₀ := eventually_of_forall f₀_nn Ω : Type u_1 inst✝² : MeasurableSpace Ω inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_2 F : Filter γ μs : γ → FiniteMeasure Ω μ : FiniteMeasure Ω h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), ↑f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), ↑f x ∂↑μ)) f : Ω →ᵇ ℝ≥0 key : Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) ↔ Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ)) lip : LipschitzWith 1 NNReal.toReal f₀ : Ω →ᵇ ℝ := comp NNReal.toReal lip f _def_f₀ : f₀ = comp NNReal.toReal lip f f₀_eq : ↑f₀ = NNReal.toReal ∘ ↑f f₀_nn : 0 ≤ ↑f₀ f₀_ae_nn : 0 ≤ᵐ[↑μ] ↑f₀ ⊢ Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) have f₀_ae_nns : ∀ i, 0 ≤ᵐ[(μs i : Measure Ω)] ⇑f₀ := fun i => eventually_of_forall f₀_nn Ω : Type u_1 inst✝² : MeasurableSpace Ω inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_2 F : Filter γ μs : γ → FiniteMeasure Ω μ : FiniteMeasure Ω h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), ↑f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), ↑f x ∂↑μ)) f : Ω →ᵇ ℝ≥0 key : Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) ↔ Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ)) lip : LipschitzWith 1 NNReal.toReal f₀ : Ω →ᵇ ℝ := comp NNReal.toReal lip f _def_f₀ : f₀ = comp NNReal.toReal lip f f₀_eq : ↑f₀ = NNReal.toReal ∘ ↑f f₀_nn : 0 ≤ ↑f₀ f₀_ae_nn : 0 ≤ᵐ[↑μ] ↑f₀ f₀_ae_nns : ∀ (i : γ), 0 ≤ᵐ[↑(μs i)] ↑f₀ ⊢ Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) have aux := integral_eq_lintegral_of_nonneg_ae f₀_ae_nn f₀.continuous.measurable.aestronglyMeasurable Ω : Type u_1 inst✝² : MeasurableSpace Ω inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_2 F : Filter γ μs : γ → FiniteMeasure Ω μ : FiniteMeasure Ω h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), ↑f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), ↑f x ∂↑μ)) f : Ω →ᵇ ℝ≥0 key : Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) ↔ Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ)) lip : LipschitzWith 1 NNReal.toReal f₀ : Ω →ᵇ ℝ := comp NNReal.toReal lip f _def_f₀ : f₀ = comp NNReal.toReal lip f f₀_eq : ↑f₀ = NNReal.toReal ∘ ↑f f₀_nn : 0 ≤ ↑f₀ f₀_ae_nn : 0 ≤ᵐ[↑μ] ↑f₀ f₀_ae_nns : ∀ (i : γ), 0 ≤ᵐ[↑(μs i)] ↑f₀ aux : ∫ (a : Ω), ↑f₀ a ∂↑μ = ENNReal.toReal (∫⁻ (a : Ω), ENNReal.ofReal (↑f₀ a) ∂↑μ) ⊢ Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) have auxs := fun i => integral_eq_lintegral_of_nonneg_ae (f₀_ae_nns i) f₀.continuous.measurable.aestronglyMeasurable Ω : Type u_1 inst✝² : MeasurableSpace Ω inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_2 F : Filter γ μs : γ → FiniteMeasure Ω μ : FiniteMeasure Ω h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), ↑f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), ↑f x ∂↑μ)) f : Ω →ᵇ ℝ≥0 key : Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) ↔ Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ)) lip : LipschitzWith 1 NNReal.toReal f₀ : Ω →ᵇ ℝ := comp NNReal.toReal lip f _def_f₀ : f₀ = comp NNReal.toReal lip f f₀_eq : ↑f₀ = NNReal.toReal ∘ ↑f f₀_nn : 0 ≤ ↑f₀ f₀_ae_nn : 0 ≤ᵐ[↑μ] ↑f₀ f₀_ae_nns : ∀ (i : γ), 0 ≤ᵐ[↑(μs i)] ↑f₀ aux : ∫ (a : Ω), ↑f₀ a ∂↑μ = ENNReal.toReal (∫⁻ (a : Ω), ENNReal.ofReal (↑f₀ a) ∂↑μ) auxs : ∀ (i : γ), ∫ (a : Ω), ↑f₀ a ∂↑(μs i) = ENNReal.toReal (∫⁻ (a : Ω), ENNReal.ofReal (↑f₀ a) ∂↑(μs i)) ⊢ Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) simp_rw [f₀_eq, Function.comp_apply, ENNReal.ofReal_coe_nnreal] at aux auxs Ω : Type u_1 inst✝² : MeasurableSpace Ω inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_2 F : Filter γ μs : γ → FiniteMeasure Ω μ : FiniteMeasure Ω h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), ↑f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), ↑f x ∂↑μ)) f : Ω →ᵇ ℝ≥0 key : Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) ↔ Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ)) lip : LipschitzWith 1 NNReal.toReal f₀ : Ω →ᵇ ℝ := comp NNReal.toReal lip f _def_f₀ : f₀ = comp NNReal.toReal lip f f₀_eq : ↑f₀ = NNReal.toReal ∘ ↑f f₀_nn : 0 ≤ ↑f₀ f₀_ae_nn : 0 ≤ᵐ[↑μ] ↑f₀ f₀_ae_nns : ∀ (i : γ), 0 ≤ᵐ[↑(μs i)] ↑f₀ aux : ∫ (a : Ω), ↑(↑f a) ∂↑μ = ENNReal.toReal (∫⁻ (a : Ω), ↑(↑f a) ∂↑μ) auxs : ∀ (i : γ), ∫ (a : Ω), ↑(↑f a) ∂↑(μs i) = ENNReal.toReal (∫⁻ (a : Ω), ↑(↑f a) ∂↑(μs i)) ⊢ Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) simpa only [← aux, ← auxs] using h f₀ Ω : Type u_1 inst✝² : MeasurableSpace Ω inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_2 F : Filter γ μs : γ → FiniteMeasure Ω μ : FiniteMeasure Ω h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), ↑f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), ↑f x ∂↑μ)) f : Ω →ᵇ ℝ≥0 key : Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) ↔ Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ)) lip : LipschitzWith 1 NNReal.toReal f₀ : Ω →ᵇ ℝ := comp NNReal.toReal lip f _def_f₀ : f₀ = comp NNReal.toReal lip f f₀_eq : ↑f₀ = NNReal.toReal ∘ ↑f x✝ : Ω ⊢ OfNat.ofNat 0 x✝ ≤ ↑f₀ x✝ simp only [f₀_eq, Pi.zero_apply, Function.comp_apply, NNReal.zero_le_coe] ** Qed
MeasureTheory.Memℒp.integrable_sq α : Type u_1 F : Type u_2 m : MeasurableSpace α μ : Measure α inst✝ : NormedAddCommGroup F f : α → ℝ h : Memℒp f 2 ⊢ Integrable fun x => f x ^ 2 simpa [← memℒp_one_iff_integrable] using h.norm_rpow two_ne_zero ENNReal.two_ne_top ** Qed
MeasureTheory.memℒp_two_iff_integrable_sq_norm α : Type u_1 F : Type u_2 m : MeasurableSpace α μ : Measure α inst✝ : NormedAddCommGroup F f : α → F hf : AEStronglyMeasurable f μ ⊢ Memℒp f 2 ↔ Integrable fun x => ‖f x‖ ^ 2 rw [← memℒp_one_iff_integrable] α : Type u_1 F : Type u_2 m : MeasurableSpace α μ : Measure α inst✝ : NormedAddCommGroup F f : α → F hf : AEStronglyMeasurable f μ ⊢ Memℒp f 2 ↔ Memℒp (fun x => ‖f x‖ ^ 2) 1 convert (memℒp_norm_rpow_iff hf two_ne_zero ENNReal.two_ne_top).symm case h.e'_2.h.e'_5.h α : Type u_1 F : Type u_2 m : MeasurableSpace α μ : Measure α inst✝ : NormedAddCommGroup F f : α → F hf : AEStronglyMeasurable f μ x✝ : α ⊢ ‖f x✝‖ ^ 2 = ‖f x✝‖ ^ ENNReal.toReal 2 simp case h.e'_2.h.e'_6 α : Type u_1 F : Type u_2 m : MeasurableSpace α μ : Measure α inst✝ : NormedAddCommGroup F f : α → F hf : AEStronglyMeasurable f μ ⊢ 1 = 2 / 2 rw [div_eq_mul_inv, ENNReal.mul_inv_cancel two_ne_zero ENNReal.two_ne_top] ** Qed
MeasureTheory.Memℒp.inner_const ** α : Type u_1 m : MeasurableSpace α p : ℝ≥0∞ μ : Measure α E : Type u_2 𝕜 : Type u_3 inst✝² : IsROrC 𝕜 inst✝¹ : NormedAddCommGroup E inst✝ : InnerProductSpace 𝕜 E f : α → E hf : Memℒp f p c : E x : α ⊢ ‖inner (f x) c‖ ≤ ?m.29864 hf c * ‖f x‖ rw [mul_comm] α : Type u_1 m : MeasurableSpace α p : ℝ≥0∞ μ : Measure α E : Type u_2 𝕜 : Type u_3 inst✝² : IsROrC 𝕜 inst✝¹ : NormedAddCommGroup E inst✝ : InnerProductSpace 𝕜 E f : α → E hf : Memℒp f p c : E x : α ⊢ ‖inner (f x) c‖ ≤ ‖f x‖ * ?m.29864 hf c α : Type u_1 m : MeasurableSpace α p : ℝ≥0∞ μ : Measure α E : Type u_2 𝕜 : Type u_3 inst✝² : IsROrC 𝕜 inst✝¹ : NormedAddCommGroup E inst✝ : InnerProductSpace 𝕜 E ⊢ {f : α → E} → Memℒp f p → E → ℝ exact norm_inner_le_norm _ _ Qed
MeasureTheory.Integrable.const_inner α : Type u_1 m : MeasurableSpace α p : ℝ≥0∞ μ : Measure α E : Type u_2 𝕜 : Type u_3 inst✝² : IsROrC 𝕜 inst✝¹ : NormedAddCommGroup E inst✝ : InnerProductSpace 𝕜 E f : α → E c : E hf : Integrable f ⊢ Integrable fun x => inner c (f x) rw [← memℒp_one_iff_integrable] at hf ⊢ α : Type u_1 m : MeasurableSpace α p : ℝ≥0∞ μ : Measure α E : Type u_2 𝕜 : Type u_3 inst✝² : IsROrC 𝕜 inst✝¹ : NormedAddCommGroup E inst✝ : InnerProductSpace 𝕜 E f : α → E c : E hf : Memℒp f 1 ⊢ Memℒp (fun x => inner c (f x)) 1 exact hf.const_inner c ** Qed
MeasureTheory.Integrable.inner_const α : Type u_1 m : MeasurableSpace α p : ℝ≥0∞ μ : Measure α E : Type u_2 𝕜 : Type u_3 inst✝² : IsROrC 𝕜 inst✝¹ : NormedAddCommGroup E inst✝ : InnerProductSpace 𝕜 E f : α → E hf : Integrable f c : E ⊢ Integrable fun x => inner (f x) c rw [← memℒp_one_iff_integrable] at hf ⊢ α : Type u_1 m : MeasurableSpace α p : ℝ≥0∞ μ : Measure α E : Type u_2 𝕜 : Type u_3 inst✝² : IsROrC 𝕜 inst✝¹ : NormedAddCommGroup E inst✝ : InnerProductSpace 𝕜 E f : α → E hf : Memℒp f 1 c : E ⊢ Memℒp (fun x => inner (f x) c) 1 exact hf.inner_const c ** Qed
integral_eq_zero_of_forall_integral_inner_eq_zero α : Type u_1 m : MeasurableSpace α p : ℝ≥0∞ μ : Measure α E : Type u_2 𝕜 : Type u_3 inst✝⁴ : IsROrC 𝕜 inst✝³ : NormedAddCommGroup E inst✝² : InnerProductSpace 𝕜 E f✝ : α → E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f : α → E hf : Integrable f hf_int : ∀ (c : E), ∫ (x : α), inner c (f x) ∂μ = 0 ⊢ ∫ (x : α), f x ∂μ = 0 specialize hf_int (∫ x, f x ∂μ) α : Type u_1 m : MeasurableSpace α p : ℝ≥0∞ μ : Measure α E : Type u_2 𝕜 : Type u_3 inst✝⁴ : IsROrC 𝕜 inst✝³ : NormedAddCommGroup E inst✝² : InnerProductSpace 𝕜 E f✝ : α → E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f : α → E hf : Integrable f hf_int : ∫ (x : α), inner (∫ (x : α), f x ∂μ) (f x) ∂μ = 0 ⊢ ∫ (x : α), f x ∂μ = 0 rwa [integral_inner hf, inner_self_eq_zero] at hf_int ** Qed
MeasureTheory.L2.snorm_rpow_two_norm_lt_top α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f : { x // x ∈ Lp F 2 } ⊢ snorm (fun x => ‖↑↑f x‖ ^ 2) 1 μ < ⊤ have h_two : ENNReal.ofReal (2 : ℝ) = 2 := by simp [zero_le_one] α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f : { x // x ∈ Lp F 2 } h_two : ENNReal.ofReal 2 = 2 ⊢ snorm (fun x => ‖↑↑f x‖ ^ 2) 1 μ < ⊤ rw [snorm_norm_rpow f zero_lt_two, one_mul, h_two] α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f : { x // x ∈ Lp F 2 } h_two : ENNReal.ofReal 2 = 2 ⊢ snorm (↑↑f) 2 μ ^ 2 < ⊤ exact ENNReal.rpow_lt_top_of_nonneg zero_le_two (Lp.snorm_ne_top f) α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f : { x // x ∈ Lp F 2 } ⊢ ENNReal.ofReal 2 = 2 simp [zero_le_one] ** Qed
MeasureTheory.L2.integral_inner_eq_sq_snorm α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f : { x // x ∈ Lp E 2 } ⊢ ∫ (a : α), inner (↑↑f a) (↑↑f a) ∂μ = ↑(ENNReal.toReal (∫⁻ (a : α), ↑‖↑↑f a‖₊ ^ 2 ∂μ)) simp_rw [inner_self_eq_norm_sq_to_K] α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f : { x // x ∈ Lp E 2 } ⊢ ∫ (a : α), ↑‖↑↑f a‖ ^ 2 ∂μ = ↑(ENNReal.toReal (∫⁻ (a : α), ↑‖↑↑f a‖₊ ^ 2 ∂μ)) norm_cast α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f : { x // x ∈ Lp E 2 } ⊢ ∫ (a : α), ‖↑↑f a‖ ^ 2 ∂μ = ENNReal.toReal (∫⁻ (a : α), ↑(‖↑↑f a‖₊ ^ 2) ∂μ) rw [integral_eq_lintegral_of_nonneg_ae] α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f : { x // x ∈ Lp E 2 } ⊢ ENNReal.toReal (∫⁻ (a : α), ENNReal.ofReal (‖↑↑f a‖ ^ 2) ∂μ) = ENNReal.toReal (∫⁻ (a : α), ↑(‖↑↑f a‖₊ ^ 2) ∂μ) case hf α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f : { x // x ∈ Lp E 2 } ⊢ 0 ≤ᵐ[μ] fun a => ‖↑↑f a‖ ^ 2 case hfm α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f : { x // x ∈ Lp E 2 } ⊢ AEStronglyMeasurable (fun a => ‖↑↑f a‖ ^ 2) μ rotate_left α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f : { x // x ∈ Lp E 2 } ⊢ ENNReal.toReal (∫⁻ (a : α), ENNReal.ofReal (‖↑↑f a‖ ^ 2) ∂μ) = ENNReal.toReal (∫⁻ (a : α), ↑(‖↑↑f a‖₊ ^ 2) ∂μ) congr case e_a.e_f α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f : { x // x ∈ Lp E 2 } ⊢ (fun a => ENNReal.ofReal (‖↑↑f a‖ ^ 2)) = fun a => ↑(‖↑↑f a‖₊ ^ 2) ext1 x case e_a.e_f.h α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f : { x // x ∈ Lp E 2 } x : α ⊢ ENNReal.ofReal (‖↑↑f x‖ ^ 2) = ↑(‖↑↑f x‖₊ ^ 2) have h_two : (2 : ℝ) = ((2 : ℕ) : ℝ) := by simp case e_a.e_f.h α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f : { x // x ∈ Lp E 2 } x : α h_two : 2 = ↑2 ⊢ ENNReal.ofReal (‖↑↑f x‖ ^ 2) = ↑(‖↑↑f x‖₊ ^ 2) rw [← Real.rpow_nat_cast _ 2, ← h_two, ← ENNReal.ofReal_rpow_of_nonneg (norm_nonneg _) zero_le_two, ofReal_norm_eq_coe_nnnorm] case e_a.e_f.h α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f : { x // x ∈ Lp E 2 } x : α h_two : 2 = ↑2 ⊢ ↑‖↑↑f x‖₊ ^ 2 = ↑(‖↑↑f x‖₊ ^ 2) norm_cast case hf α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f : { x // x ∈ Lp E 2 } ⊢ 0 ≤ᵐ[μ] fun a => ‖↑↑f a‖ ^ 2 exact Filter.eventually_of_forall fun x => sq_nonneg _ case hfm α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f : { x // x ∈ Lp E 2 } ⊢ AEStronglyMeasurable (fun a => ‖↑↑f a‖ ^ 2) μ exact ((Lp.aestronglyMeasurable f).norm.aemeasurable.pow_const _).aestronglyMeasurable α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f : { x // x ∈ Lp E 2 } x : α ⊢ 2 = ↑2 simp ** Qed
MeasureTheory.L2.add_left' α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f f' g : { x // x ∈ Lp E 2 } ⊢ inner (f + f') g = inner f g + inner f' g simp_rw [inner_def, ← integral_add (integrable_inner f g) (integrable_inner f' g), ← inner_add_left] α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f f' g : { x // x ∈ Lp E 2 } ⊢ ∫ (a : α), inner (↑↑(f + f') a) (↑↑g a) ∂μ = ∫ (a : α), inner (↑↑f a + ↑↑f' a) (↑↑g a) ∂μ refine' integral_congr_ae ((coeFn_add f f').mono fun x hx => _) α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f f' g : { x // x ∈ Lp E 2 } x : α hx : ↑↑(f + f') x = (↑↑f + ↑↑f') x ⊢ (fun a => inner (↑↑(f + f') a) (↑↑g a)) x = (fun a => inner (↑↑f a + ↑↑f' a) (↑↑g a)) x simp only α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f f' g : { x // x ∈ Lp E 2 } x : α hx : ↑↑(f + f') x = (↑↑f + ↑↑f') x ⊢ inner (↑↑(f + f') x) (↑↑g x) = inner (↑↑f x + ↑↑f' x) (↑↑g x) rw [hx, Pi.add_apply] ** Qed
MeasureTheory.L2.smul_left' ** α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f g : { x // x ∈ Lp E 2 } r : 𝕜 ⊢ inner (r • f) g = ↑(starRingEnd 𝕜) r * inner f g rw [inner_def, inner_def, ← smul_eq_mul, ← integral_smul] α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f g : { x // x ∈ Lp E 2 } r : 𝕜 ⊢ ∫ (a : α), inner (↑↑(r • f) a) (↑↑g a) ∂μ = ∫ (a : α), ↑(starRingEnd 𝕜) r • inner (↑↑f a) (↑↑g a) ∂μ refine' integral_congr_ae ((coeFn_smul r f).mono fun x hx => _) α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f g : { x // x ∈ Lp E 2 } r : 𝕜 x : α hx : ↑↑(r • f) x = (r • ↑↑f) x ⊢ (fun a => inner (↑↑(r • f) a) (↑↑g a)) x = (fun a => ↑(starRingEnd 𝕜) r • inner (↑↑f a) (↑↑g a)) x simp only α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f g : { x // x ∈ Lp E 2 } r : 𝕜 x : α hx : ↑↑(r • f) x = (r • ↑↑f) x ⊢ inner (↑↑(r • f) x) (↑↑g x) = ↑(starRingEnd 𝕜) r • inner (↑↑f x) (↑↑g x) rw [smul_eq_mul, ← inner_smul_left, hx, Pi.smul_apply] Qed
MeasureTheory.L2.inner_indicatorConstLp_eq_inner_set_integral α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁶ : IsROrC 𝕜 inst✝⁵ : MeasurableSpace α μ : Measure α inst✝⁴ : NormedAddCommGroup E inst✝³ : InnerProductSpace 𝕜 E inst✝² : NormedAddCommGroup F s : Set α inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E hs : MeasurableSet s hμs : ↑↑μ s ≠ ⊤ c : E f : { x // x ∈ Lp E 2 } ⊢ inner (indicatorConstLp 2 hs hμs c) f = inner c (∫ (x : α) in s, ↑↑f x ∂μ) rw [← integral_inner (integrableOn_Lp_of_measure_ne_top f fact_one_le_two_ennreal.elim hμs), L2.inner_indicatorConstLp_eq_set_integral_inner] ** Qed
MeasureTheory.BoundedContinuousFunction.inner_toLp ** α : Type u_1 inst✝⁴ : TopologicalSpace α inst✝³ : MeasureSpace α inst✝² : BorelSpace α 𝕜 : Type u_2 inst✝¹ : IsROrC 𝕜 μ : Measure α inst✝ : IsFiniteMeasure μ f g : α →ᵇ 𝕜 ⊢ inner (↑(BoundedContinuousFunction.toLp 2 μ 𝕜) f) (↑(BoundedContinuousFunction.toLp 2 μ 𝕜) g) = ∫ (x : α), ↑(starRingEnd ((fun a => 𝕜) x)) (↑f x) * ↑g x ∂μ apply integral_congr_ae case h α : Type u_1 inst✝⁴ : TopologicalSpace α inst✝³ : MeasureSpace α inst✝² : BorelSpace α 𝕜 : Type u_2 inst✝¹ : IsROrC 𝕜 μ : Measure α inst✝ : IsFiniteMeasure μ f g : α →ᵇ 𝕜 ⊢ (fun a => inner (↑↑(↑(BoundedContinuousFunction.toLp 2 μ 𝕜) f) a) (↑↑(↑(BoundedContinuousFunction.toLp 2 μ 𝕜) g) a)) =ᵐ[μ] fun a => ↑(starRingEnd ((fun a => 𝕜) a)) (↑f a) * ↑g a have hf_ae := f.coeFn_toLp 2 μ 𝕜 case h α : Type u_1 inst✝⁴ : TopologicalSpace α inst✝³ : MeasureSpace α inst✝² : BorelSpace α 𝕜 : Type u_2 inst✝¹ : IsROrC 𝕜 μ : Measure α inst✝ : IsFiniteMeasure μ f g : α →ᵇ 𝕜 hf_ae : ↑↑(↑(BoundedContinuousFunction.toLp 2 μ 𝕜) f) =ᵐ[μ] ↑f ⊢ (fun a => inner (↑↑(↑(BoundedContinuousFunction.toLp 2 μ 𝕜) f) a) (↑↑(↑(BoundedContinuousFunction.toLp 2 μ 𝕜) g) a)) =ᵐ[μ] fun a => ↑(starRingEnd ((fun a => 𝕜) a)) (↑f a) * ↑g a have hg_ae := g.coeFn_toLp 2 μ 𝕜 case h α : Type u_1 inst✝⁴ : TopologicalSpace α inst✝³ : MeasureSpace α inst✝² : BorelSpace α 𝕜 : Type u_2 inst✝¹ : IsROrC 𝕜 μ : Measure α inst✝ : IsFiniteMeasure μ f g : α →ᵇ 𝕜 hf_ae : ↑↑(↑(BoundedContinuousFunction.toLp 2 μ 𝕜) f) =ᵐ[μ] ↑f hg_ae : ↑↑(↑(BoundedContinuousFunction.toLp 2 μ 𝕜) g) =ᵐ[μ] ↑g ⊢ (fun a => inner (↑↑(↑(BoundedContinuousFunction.toLp 2 μ 𝕜) f) a) (↑↑(↑(BoundedContinuousFunction.toLp 2 μ 𝕜) g) a)) =ᵐ[μ] fun a => ↑(starRingEnd ((fun a => 𝕜) a)) (↑f a) * ↑g a filter_upwards [hf_ae, hg_ae] with _ hf hg case h α : Type u_1 inst✝⁴ : TopologicalSpace α inst✝³ : MeasureSpace α inst✝² : BorelSpace α 𝕜 : Type u_2 inst✝¹ : IsROrC 𝕜 μ : Measure α inst✝ : IsFiniteMeasure μ f g : α →ᵇ 𝕜 hf_ae : ↑↑(↑(BoundedContinuousFunction.toLp 2 μ 𝕜) f) =ᵐ[μ] ↑f hg_ae : ↑↑(↑(BoundedContinuousFunction.toLp 2 μ 𝕜) g) =ᵐ[μ] ↑g a✝ : α hf : ↑↑(↑(BoundedContinuousFunction.toLp 2 μ 𝕜) f) a✝ = ↑f a✝ hg : ↑↑(↑(BoundedContinuousFunction.toLp 2 μ 𝕜) g) a✝ = ↑g a✝ ⊢ inner (↑↑(↑(BoundedContinuousFunction.toLp 2 μ 𝕜) f) a✝) (↑↑(↑(BoundedContinuousFunction.toLp 2 μ 𝕜) g) a✝) = ↑(starRingEnd 𝕜) (↑f a✝) * ↑g a✝ rw [hf, hg] case h α : Type u_1 inst✝⁴ : TopologicalSpace α inst✝³ : MeasureSpace α inst✝² : BorelSpace α 𝕜 : Type u_2 inst✝¹ : IsROrC 𝕜 μ : Measure α inst✝ : IsFiniteMeasure μ f g : α →ᵇ 𝕜 hf_ae : ↑↑(↑(BoundedContinuousFunction.toLp 2 μ 𝕜) f) =ᵐ[μ] ↑f hg_ae : ↑↑(↑(BoundedContinuousFunction.toLp 2 μ 𝕜) g) =ᵐ[μ] ↑g a✝ : α hf : ↑↑(↑(BoundedContinuousFunction.toLp 2 μ 𝕜) f) a✝ = ↑f a✝ hg : ↑↑(↑(BoundedContinuousFunction.toLp 2 μ 𝕜) g) a✝ = ↑g a✝ ⊢ inner (↑f a✝) (↑g a✝) = ↑(starRingEnd 𝕜) (↑f a✝) * ↑g a✝ simp Qed
MeasureTheory.ContinuousMap.inner_toLp ** α : Type u_1 inst✝⁵ : TopologicalSpace α inst✝⁴ : MeasureSpace α inst✝³ : BorelSpace α 𝕜 : Type u_2 inst✝² : IsROrC 𝕜 μ : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : CompactSpace α f g : C(α, 𝕜) ⊢ inner (↑(ContinuousMap.toLp 2 μ 𝕜) f) (↑(ContinuousMap.toLp 2 μ 𝕜) g) = ∫ (x : α), ↑(starRingEnd ((fun x => 𝕜) x)) (↑f x) * ↑g x ∂μ apply integral_congr_ae case h α : Type u_1 inst✝⁵ : TopologicalSpace α inst✝⁴ : MeasureSpace α inst✝³ : BorelSpace α 𝕜 : Type u_2 inst✝² : IsROrC 𝕜 μ : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : CompactSpace α f g : C(α, 𝕜) ⊢ (fun a => inner (↑↑(↑(ContinuousMap.toLp 2 μ 𝕜) f) a) (↑↑(↑(ContinuousMap.toLp 2 μ 𝕜) g) a)) =ᵐ[μ] fun a => ↑(starRingEnd ((fun x => 𝕜) a)) (↑f a) * ↑g a have hf_ae := f.coeFn_toLp (p := 2) (𝕜 := 𝕜) μ case h α : Type u_1 inst✝⁵ : TopologicalSpace α inst✝⁴ : MeasureSpace α inst✝³ : BorelSpace α 𝕜 : Type u_2 inst✝² : IsROrC 𝕜 μ : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : CompactSpace α f g : C(α, 𝕜) hf_ae : ↑↑(↑(ContinuousMap.toLp 2 μ 𝕜) f) =ᵐ[μ] ↑f ⊢ (fun a => inner (↑↑(↑(ContinuousMap.toLp 2 μ 𝕜) f) a) (↑↑(↑(ContinuousMap.toLp 2 μ 𝕜) g) a)) =ᵐ[μ] fun a => ↑(starRingEnd ((fun x => 𝕜) a)) (↑f a) * ↑g a have hg_ae := g.coeFn_toLp (p := 2) (𝕜 := 𝕜) μ case h α : Type u_1 inst✝⁵ : TopologicalSpace α inst✝⁴ : MeasureSpace α inst✝³ : BorelSpace α 𝕜 : Type u_2 inst✝² : IsROrC 𝕜 μ : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : CompactSpace α f g : C(α, 𝕜) hf_ae : ↑↑(↑(ContinuousMap.toLp 2 μ 𝕜) f) =ᵐ[μ] ↑f hg_ae : ↑↑(↑(ContinuousMap.toLp 2 μ 𝕜) g) =ᵐ[μ] ↑g ⊢ (fun a => inner (↑↑(↑(ContinuousMap.toLp 2 μ 𝕜) f) a) (↑↑(↑(ContinuousMap.toLp 2 μ 𝕜) g) a)) =ᵐ[μ] fun a => ↑(starRingEnd ((fun x => 𝕜) a)) (↑f a) * ↑g a filter_upwards [hf_ae, hg_ae] with _ hf hg case h α : Type u_1 inst✝⁵ : TopologicalSpace α inst✝⁴ : MeasureSpace α inst✝³ : BorelSpace α 𝕜 : Type u_2 inst✝² : IsROrC 𝕜 μ : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : CompactSpace α f g : C(α, 𝕜) hf_ae : ↑↑(↑(ContinuousMap.toLp 2 μ 𝕜) f) =ᵐ[μ] ↑f hg_ae : ↑↑(↑(ContinuousMap.toLp 2 μ 𝕜) g) =ᵐ[μ] ↑g a✝ : α hf : ↑↑(↑(ContinuousMap.toLp 2 μ 𝕜) f) a✝ = ↑f a✝ hg : ↑↑(↑(ContinuousMap.toLp 2 μ 𝕜) g) a✝ = ↑g a✝ ⊢ inner (↑↑(↑(ContinuousMap.toLp 2 μ 𝕜) f) a✝) (↑↑(↑(ContinuousMap.toLp 2 μ 𝕜) g) a✝) = ↑(starRingEnd 𝕜) (↑f a✝) * ↑g a✝ rw [hf, hg] case h α : Type u_1 inst✝⁵ : TopologicalSpace α inst✝⁴ : MeasureSpace α inst✝³ : BorelSpace α 𝕜 : Type u_2 inst✝² : IsROrC 𝕜 μ : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : CompactSpace α f g : C(α, 𝕜) hf_ae : ↑↑(↑(ContinuousMap.toLp 2 μ 𝕜) f) =ᵐ[μ] ↑f hg_ae : ↑↑(↑(ContinuousMap.toLp 2 μ 𝕜) g) =ᵐ[μ] ↑g a✝ : α hf : ↑↑(↑(ContinuousMap.toLp 2 μ 𝕜) f) a✝ = ↑f a✝ hg : ↑↑(↑(ContinuousMap.toLp 2 μ 𝕜) g) a✝ = ↑g a✝ ⊢ inner (↑f a✝) (↑g a✝) = ↑(starRingEnd 𝕜) (↑f a✝) * ↑g a✝ simp Qed
Turing.ToPartrec.Code.zero'_eval ⊢ eval zero' = fun v => pure (0 :: v) simp [eval] ** Qed
Turing.ToPartrec.Code.succ_eval ⊢ eval succ = fun v => pure [Nat.succ (List.headI v)] simp [eval] ** Qed
Turing.ToPartrec.Code.tail_eval ⊢ eval tail = fun v => pure (List.tail v) simp [eval] ** Qed
Turing.ToPartrec.Code.cons_eval f fs : Code ⊢ eval (cons f fs) = fun v => do let n ← eval f v let ns ← eval fs v pure (List.headI n :: ns) simp [eval] ** Qed
Turing.ToPartrec.Code.comp_eval f g : Code ⊢ eval (comp f g) = fun v => eval g v >>= eval f simp [eval] ** Qed
Turing.ToPartrec.Code.case_eval f g : Code ⊢ eval (case f g) = fun v => Nat.rec (eval f (List.tail v)) (fun y x => eval g (y :: List.tail v)) (List.headI v) simp [eval] ** Qed
Turing.ToPartrec.Code.fix_eval f : Code ⊢ eval (fix f) = PFun.fix fun v => Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (eval f v) simp [eval] ** Qed
Turing.ToPartrec.Code.nil_eval v : List ℕ ⊢ eval nil v = pure [] simp [nil] ** Qed
Turing.ToPartrec.Code.head_eval v : List ℕ ⊢ eval head v = pure [List.headI v] simp [head] ** Qed
Turing.ToPartrec.Code.exists_code.comp m n : ℕ f : Vector ℕ n →. ℕ g : Fin n → Vector ℕ m →. ℕ hf : ∃ c, ∀ (v : Vector ℕ n), eval c ↑v = pure <$> f v hg : ∀ (i : Fin n), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g i v ⊢ ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> ((Vector.mOfFn fun i => g i v) >>= f) rsuffices ⟨cg, hg⟩ : ∃ c : Code, ∀ v : Vector ℕ m, c.eval v.1 = Subtype.val <$> Vector.mOfFn fun i => g i v m n : ℕ f : Vector ℕ n →. ℕ g : Fin n → Vector ℕ m →. ℕ hf : ∃ c, ∀ (v : Vector ℕ n), eval c ↑v = pure <$> f v hg : ∀ (i : Fin n), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g i v ⊢ ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = Subtype.val <$> Vector.mOfFn fun i => g i v clear hf f m n : ℕ g : Fin n → Vector ℕ m →. ℕ hg : ∀ (i : Fin n), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g i v ⊢ ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = Subtype.val <$> Vector.mOfFn fun i => g i v induction' n with n IH case intro m n : ℕ f : Vector ℕ n →. ℕ g : Fin n → Vector ℕ m →. ℕ hf : ∃ c, ∀ (v : Vector ℕ n), eval c ↑v = pure <$> f v hg✝ : ∀ (i : Fin n), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g i v cg : Code hg : ∀ (v : Vector ℕ m), eval cg ↑v = Subtype.val <$> Vector.mOfFn fun i => g i v ⊢ ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> ((Vector.mOfFn fun i => g i v) >>= f) obtain ⟨cf, hf⟩ := hf case intro.intro m n : ℕ f : Vector ℕ n →. ℕ g : Fin n → Vector ℕ m →. ℕ hg✝ : ∀ (i : Fin n), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g i v cg : Code hg : ∀ (v : Vector ℕ m), eval cg ↑v = Subtype.val <$> Vector.mOfFn fun i => g i v cf : Code hf : ∀ (v : Vector ℕ n), eval cf ↑v = pure <$> f v ⊢ ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> ((Vector.mOfFn fun i => g i v) >>= f) exact ⟨cf.comp cg, fun v => by simp [hg, hf, map_bind, seq_bind_eq, Function.comp] rfl⟩ m n : ℕ f : Vector ℕ n →. ℕ g : Fin n → Vector ℕ m →. ℕ hg✝ : ∀ (i : Fin n), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g i v cg : Code hg : ∀ (v : Vector ℕ m), eval cg ↑v = Subtype.val <$> Vector.mOfFn fun i => g i v cf : Code hf : ∀ (v : Vector ℕ n), eval cf ↑v = pure <$> f v v : Vector ℕ m ⊢ eval (Code.comp cf cg) ↑v = pure <$> ((Vector.mOfFn fun i => g i v) >>= f) simp [hg, hf, map_bind, seq_bind_eq, Function.comp] m n : ℕ f : Vector ℕ n →. ℕ g : Fin n → Vector ℕ m →. ℕ hg✝ : ∀ (i : Fin n), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g i v cg : Code hg : ∀ (v : Vector ℕ m), eval cg ↑v = Subtype.val <$> Vector.mOfFn fun i => g i v cf : Code hf : ∀ (v : Vector ℕ n), eval cf ↑v = pure <$> f v v : Vector ℕ m ⊢ (do let x ← Vector.mOfFn fun i => g i v pure <$> f x) = do let a ← Vector.mOfFn fun i => g i v pure <$> f a rfl case zero m n : ℕ g✝ : Fin n → Vector ℕ m →. ℕ hg✝ : ∀ (i : Fin n), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g✝ i v g : Fin Nat.zero → Vector ℕ m →. ℕ hg : ∀ (i : Fin Nat.zero), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g i v ⊢ ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = Subtype.val <$> Vector.mOfFn fun i => g i v exact ⟨nil, fun v => by simp [Vector.mOfFn, Bind.bind]; rfl⟩ m n : ℕ g✝ : Fin n → Vector ℕ m →. ℕ hg✝ : ∀ (i : Fin n), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g✝ i v g : Fin Nat.zero → Vector ℕ m →. ℕ hg : ∀ (i : Fin Nat.zero), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g i v v : Vector ℕ m ⊢ eval nil ↑v = Subtype.val <$> Vector.mOfFn fun i => g i v simp [Vector.mOfFn, Bind.bind] m n : ℕ g✝ : Fin n → Vector ℕ m →. ℕ hg✝ : ∀ (i : Fin n), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g✝ i v g : Fin Nat.zero → Vector ℕ m →. ℕ hg : ∀ (i : Fin Nat.zero), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g i v v : Vector ℕ m ⊢ Part.some [] = Subtype.val <$> Part.some Vector.nil rfl case succ m n✝ : ℕ g✝ : Fin n✝ → Vector ℕ m →. ℕ hg✝ : ∀ (i : Fin n✝), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g✝ i v n : ℕ IH : ∀ {g : Fin n → Vector ℕ m →. ℕ}, (∀ (i : Fin n), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g i v) → ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = Subtype.val <$> Vector.mOfFn fun i => g i v g : Fin (Nat.succ n) → Vector ℕ m →. ℕ hg : ∀ (i : Fin (Nat.succ n)), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g i v ⊢ ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = Subtype.val <$> Vector.mOfFn fun i => g i v obtain ⟨cg, hg₁⟩ := hg 0 case succ.intro m n✝ : ℕ g✝ : Fin n✝ → Vector ℕ m →. ℕ hg✝ : ∀ (i : Fin n✝), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g✝ i v n : ℕ IH : ∀ {g : Fin n → Vector ℕ m →. ℕ}, (∀ (i : Fin n), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g i v) → ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = Subtype.val <$> Vector.mOfFn fun i => g i v g : Fin (Nat.succ n) → Vector ℕ m →. ℕ hg : ∀ (i : Fin (Nat.succ n)), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g i v cg : Code hg₁ : ∀ (v : Vector ℕ m), eval cg ↑v = pure <$> g 0 v ⊢ ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = Subtype.val <$> Vector.mOfFn fun i => g i v obtain ⟨cl, hl⟩ := IH fun i => hg i.succ m n✝ : ℕ g✝ : Fin n✝ → Vector ℕ m →. ℕ hg✝ : ∀ (i : Fin n✝), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g✝ i v n : ℕ IH : ∀ {g : Fin n → Vector ℕ m →. ℕ}, (∀ (i : Fin n), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g i v) → ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = Subtype.val <$> Vector.mOfFn fun i => g i v g : Fin (Nat.succ n) → Vector ℕ m →. ℕ hg : ∀ (i : Fin (Nat.succ n)), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g i v cg : Code hg₁ : ∀ (v : Vector ℕ m), eval cg ↑v = pure <$> g 0 v cl : Code hl : ∀ (v : Vector ℕ m), eval cl ↑v = Subtype.val <$> Vector.mOfFn fun i => g (Fin.succ i) v v : Vector ℕ m ⊢ (do let x ← g 0 v let x_1 ← Vector.mOfFn fun i => g (Fin.succ i) v Part.some (List.headI (pure x) :: ↑x_1)) = do let a ← g 0 v let a_1 ← Vector.mOfFn fun i => g (Fin.succ i) v Subtype.val <$> Part.some (a ::ᵥ a_1) rfl ** Qed
Turing.ToPartrec.Cont.then_eval k k' : Cont v : List ℕ ⊢ eval (then k k') v = eval k v >>= eval k' induction' k with _ _ _ _ _ _ _ _ _ k_ih _ _ k_ih generalizing v <;> simp only [Cont.eval, Cont.then, bind_assoc, pure_bind, ] * case comp k' : Cont v✝ : List ℕ a✝¹ : Code a✝ : Cont k_ih : ∀ {v : List ℕ}, eval (then a✝ k') v = eval a✝ v >>= eval k' v : List ℕ ⊢ Code.eval a✝¹ v >>= eval (then a✝ k') = Code.eval a✝¹ v >>= fun x => eval a✝ x >>= eval k' simp only [← k_ih] case fix k' : Cont v✝ : List ℕ a✝¹ : Code a✝ : Cont k_ih : ∀ {v : List ℕ}, eval (then a✝ k') v = eval a✝ v >>= eval k' v : List ℕ ⊢ (if List.headI v = 0 then eval a✝ (List.tail v) >>= eval k' else Code.eval (Code.fix a✝¹) (List.tail v) >>= eval (then a✝ k')) = (if List.headI v = 0 then eval a✝ (List.tail v) else Code.eval (Code.fix a✝¹) (List.tail v) >>= eval a✝) >>= eval k' split_ifs <;> [rfl; simp only [← k_ih, bind_assoc]] Qed
Turing.ToPartrec.stepNormal_then c : Code k k' : Cont v : List ℕ ⊢ stepNormal c (Cont.then k k') v = Cfg.then (stepNormal c k v) k' induction c generalizing k v <;> simp only [Cont.then, stepNormal, ] <;> try { simp only [Cfg.then]; done } * case cons k' : Cont a✝¹ a✝ : Code a_ih✝¹ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝¹ (Cont.then k k') v = Cfg.then (stepNormal a✝¹ k v) k' a_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝ (Cont.then k k') v = Cfg.then (stepNormal a✝ k v) k' k : Cont v : List ℕ ⊢ stepNormal a✝¹ (Cont.cons₁ a✝ v (Cont.then k k')) v = Cfg.then (stepNormal a✝¹ (Cont.cons₁ a✝ v k) v) k' case comp k' : Cont a✝¹ a✝ : Code a_ih✝¹ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝¹ (Cont.then k k') v = Cfg.then (stepNormal a✝¹ k v) k' a_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝ (Cont.then k k') v = Cfg.then (stepNormal a✝ k v) k' k : Cont v : List ℕ ⊢ stepNormal a✝ (Cont.comp a✝¹ (Cont.then k k')) v = Cfg.then (stepNormal a✝ (Cont.comp a✝¹ k) v) k' case case k' : Cont a✝¹ a✝ : Code a_ih✝¹ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝¹ (Cont.then k k') v = Cfg.then (stepNormal a✝¹ k v) k' a_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝ (Cont.then k k') v = Cfg.then (stepNormal a✝ k v) k' k : Cont v : List ℕ ⊢ Nat.rec (Cfg.then (stepNormal a✝¹ k (List.tail v)) k') (fun y x => Cfg.then (stepNormal a✝ k (y :: List.tail v)) k') (List.headI v) = Cfg.then (Nat.rec (stepNormal a✝¹ k (List.tail v)) (fun y x => stepNormal a✝ k (y :: List.tail v)) (List.headI v)) k' case fix k' : Cont a✝ : Code a_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝ (Cont.then k k') v = Cfg.then (stepNormal a✝ k v) k' k : Cont v : List ℕ ⊢ stepNormal a✝ (Cont.fix a✝ (Cont.then k k')) v = Cfg.then (stepNormal a✝ (Cont.fix a✝ k) v) k' case cons c c' ih _ => rw [← ih, Cont.then] case comp k' : Cont a✝¹ a✝ : Code a_ih✝¹ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝¹ (Cont.then k k') v = Cfg.then (stepNormal a✝¹ k v) k' a_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝ (Cont.then k k') v = Cfg.then (stepNormal a✝ k v) k' k : Cont v : List ℕ ⊢ stepNormal a✝ (Cont.comp a✝¹ (Cont.then k k')) v = Cfg.then (stepNormal a✝ (Cont.comp a✝¹ k) v) k' case case k' : Cont a✝¹ a✝ : Code a_ih✝¹ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝¹ (Cont.then k k') v = Cfg.then (stepNormal a✝¹ k v) k' a_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝ (Cont.then k k') v = Cfg.then (stepNormal a✝ k v) k' k : Cont v : List ℕ ⊢ Nat.rec (Cfg.then (stepNormal a✝¹ k (List.tail v)) k') (fun y x => Cfg.then (stepNormal a✝ k (y :: List.tail v)) k') (List.headI v) = Cfg.then (Nat.rec (stepNormal a✝¹ k (List.tail v)) (fun y x => stepNormal a✝ k (y :: List.tail v)) (List.headI v)) k' case fix k' : Cont a✝ : Code a_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝ (Cont.then k k') v = Cfg.then (stepNormal a✝ k v) k' k : Cont v : List ℕ ⊢ stepNormal a✝ (Cont.fix a✝ (Cont.then k k')) v = Cfg.then (stepNormal a✝ (Cont.fix a✝ k) v) k' case comp c c' _ ih' => rw [← ih', Cont.then] case fix k' : Cont a✝ : Code a_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝ (Cont.then k k') v = Cfg.then (stepNormal a✝ k v) k' k : Cont v : List ℕ ⊢ stepNormal a✝ (Cont.fix a✝ (Cont.then k k')) v = Cfg.then (stepNormal a✝ (Cont.fix a✝ k) v) k' case fix c ih => rw [← ih, Cont.then] k' : Cont c c' : Code ih : ∀ (k : Cont) (v : List ℕ), stepNormal c (Cont.then k k') v = Cfg.then (stepNormal c k v) k' a_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal c' (Cont.then k k') v = Cfg.then (stepNormal c' k v) k' k : Cont v : List ℕ ⊢ stepNormal c (Cont.cons₁ c' v (Cont.then k k')) v = Cfg.then (stepNormal c (Cont.cons₁ c' v k) v) k' rw [← ih, Cont.then] k' : Cont c c' : Code a_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal c (Cont.then k k') v = Cfg.then (stepNormal c k v) k' ih' : ∀ (k : Cont) (v : List ℕ), stepNormal c' (Cont.then k k') v = Cfg.then (stepNormal c' k v) k' k : Cont v : List ℕ ⊢ stepNormal c' (Cont.comp c (Cont.then k k')) v = Cfg.then (stepNormal c' (Cont.comp c k) v) k' rw [← ih', Cont.then] case case k' : Cont a✝¹ a✝ : Code a_ih✝¹ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝¹ (Cont.then k k') v = Cfg.then (stepNormal a✝¹ k v) k' a_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝ (Cont.then k k') v = Cfg.then (stepNormal a✝ k v) k' k : Cont v : List ℕ ⊢ Nat.rec (Cfg.then (stepNormal a✝¹ k (List.tail v)) k') (fun y x => Cfg.then (stepNormal a✝ k (y :: List.tail v)) k') (List.headI v) = Cfg.then (Nat.rec (stepNormal a✝¹ k (List.tail v)) (fun y x => stepNormal a✝ k (y :: List.tail v)) (List.headI v)) k' cases v.headI <;> simp only [Nat.rec] k' : Cont c : Code ih : ∀ (k : Cont) (v : List ℕ), stepNormal c (Cont.then k k') v = Cfg.then (stepNormal c k v) k' k : Cont v : List ℕ ⊢ stepNormal c (Cont.fix c (Cont.then k k')) v = Cfg.then (stepNormal c (Cont.fix c k) v) k' rw [← ih, Cont.then] Qed
Turing.ToPartrec.stepRet_then k k' : Cont v : List ℕ ⊢ stepRet (Cont.then k k') v = Cfg.then (stepRet k v) k' induction k generalizing v <;> simp only [Cont.then, stepRet, ] <;> try { simp only [Cfg.then]; done } * case cons₁ k' : Cont a✝² : Code a✝¹ : List ℕ a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, stepRet (Cont.then a✝ k') v = Cfg.then (stepRet a✝ v) k' v : List ℕ ⊢ stepNormal a✝² (Cont.cons₂ v (Cont.then a✝ k')) a✝¹ = Cfg.then (stepNormal a✝² (Cont.cons₂ v a✝) a✝¹) k' case comp k' : Cont a✝¹ : Code a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, stepRet (Cont.then a✝ k') v = Cfg.then (stepRet a✝ v) k' v : List ℕ ⊢ stepNormal a✝¹ (Cont.then a✝ k') v = Cfg.then (stepNormal a✝¹ a✝ v) k' case fix k' : Cont a✝¹ : Code a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, stepRet (Cont.then a✝ k') v = Cfg.then (stepRet a✝ v) k' v : List ℕ ⊢ (if List.headI v = 0 then Cfg.then (stepRet a✝ (List.tail v)) k' else stepNormal a✝¹ (Cont.fix a✝¹ (Cont.then a✝ k')) (List.tail v)) = Cfg.then (if List.headI v = 0 then stepRet a✝ (List.tail v) else stepNormal a✝¹ (Cont.fix a✝¹ a✝) (List.tail v)) k' case cons₁ => rw [← stepNormal_then] rfl case comp k' : Cont a✝¹ : Code a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, stepRet (Cont.then a✝ k') v = Cfg.then (stepRet a✝ v) k' v : List ℕ ⊢ stepNormal a✝¹ (Cont.then a✝ k') v = Cfg.then (stepNormal a✝¹ a✝ v) k' case fix k' : Cont a✝¹ : Code a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, stepRet (Cont.then a✝ k') v = Cfg.then (stepRet a✝ v) k' v : List ℕ ⊢ (if List.headI v = 0 then Cfg.then (stepRet a✝ (List.tail v)) k' else stepNormal a✝¹ (Cont.fix a✝¹ (Cont.then a✝ k')) (List.tail v)) = Cfg.then (if List.headI v = 0 then stepRet a✝ (List.tail v) else stepNormal a✝¹ (Cont.fix a✝¹ a✝) (List.tail v)) k' case comp => rw [← stepNormal_then] k' : Cont a✝² : Code a✝¹ : List ℕ a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, stepRet (Cont.then a✝ k') v = Cfg.then (stepRet a✝ v) k' v : List ℕ ⊢ stepNormal a✝² (Cont.cons₂ v (Cont.then a✝ k')) a✝¹ = Cfg.then (stepNormal a✝² (Cont.cons₂ v a✝) a✝¹) k' rw [← stepNormal_then] k' : Cont a✝² : Code a✝¹ : List ℕ a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, stepRet (Cont.then a✝ k') v = Cfg.then (stepRet a✝ v) k' v : List ℕ ⊢ stepNormal a✝² (Cont.cons₂ v (Cont.then a✝ k')) a✝¹ = stepNormal a✝² (Cont.then (Cont.cons₂ v a✝) k') a✝¹ rfl k' : Cont a✝¹ : Code a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, stepRet (Cont.then a✝ k') v = Cfg.then (stepRet a✝ v) k' v : List ℕ ⊢ stepNormal a✝¹ (Cont.then a✝ k') v = Cfg.then (stepNormal a✝¹ a✝ v) k' rw [← stepNormal_then] k' : Cont a✝¹ : Code a✝ : Cont k_ih : ∀ {v : List ℕ}, stepRet (Cont.then a✝ k') v = Cfg.then (stepRet a✝ v) k' v : List ℕ ⊢ (if List.headI v = 0 then Cfg.then (stepRet a✝ (List.tail v)) k' else stepNormal a✝¹ (Cont.fix a✝¹ (Cont.then a✝ k')) (List.tail v)) = Cfg.then (if List.headI v = 0 then stepRet a✝ (List.tail v) else stepNormal a✝¹ (Cont.fix a✝¹ a✝) (List.tail v)) k' split_ifs case pos k' : Cont a✝¹ : Code a✝ : Cont k_ih : ∀ {v : List ℕ}, stepRet (Cont.then a✝ k') v = Cfg.then (stepRet a✝ v) k' v : List ℕ h✝ : List.headI v = 0 ⊢ Cfg.then (stepRet a✝ (List.tail v)) k' = Cfg.then (stepRet a✝ (List.tail v)) k' rw [← k_ih] case neg k' : Cont a✝¹ : Code a✝ : Cont k_ih : ∀ {v : List ℕ}, stepRet (Cont.then a✝ k') v = Cfg.then (stepRet a✝ v) k' v : List ℕ h✝ : ¬List.headI v = 0 ⊢ stepNormal a✝¹ (Cont.fix a✝¹ (Cont.then a✝ k')) (List.tail v) = Cfg.then (stepNormal a✝¹ (Cont.fix a✝¹ a✝) (List.tail v)) k' rw [← stepNormal_then] case neg k' : Cont a✝¹ : Code a✝ : Cont k_ih : ∀ {v : List ℕ}, stepRet (Cont.then a✝ k') v = Cfg.then (stepRet a✝ v) k' v : List ℕ h✝ : ¬List.headI v = 0 ⊢ stepNormal a✝¹ (Cont.fix a✝¹ (Cont.then a✝ k')) (List.tail v) = stepNormal a✝¹ (Cont.then (Cont.fix a✝¹ a✝) k') (List.tail v) rfl Qed
Turing.ToPartrec.Code.Ok.zero c : Code h : Ok c v : List ℕ ⊢ Turing.eval step (stepNormal c Cont.halt v) = Cfg.halt <$> eval c v rw [h, ← bind_pure_comp] c : Code h : Ok c v : List ℕ ⊢ (do let v ← eval c v Turing.eval step (Cfg.ret Cont.halt v)) = do let a ← eval c v pure (Cfg.halt a) congr case e_a c : Code h : Ok c v : List ℕ ⊢ (fun v => Turing.eval step (Cfg.ret Cont.halt v)) = fun a => pure (Cfg.halt a) funext v case e_a.h c : Code h : Ok c v✝ v : List ℕ ⊢ Turing.eval step (Cfg.ret Cont.halt v) = pure (Cfg.halt v) exact Part.eq_some_iff.2 (mem_eval.2 ⟨ReflTransGen.single rfl, rfl⟩) ** Qed
Turing.ToPartrec.stepNormal.is_ret c : Code k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal c k v = Cfg.ret k' v' induction c generalizing k v case zero' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal Code.zero' k v = Cfg.ret k' v' case succ k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal Code.succ k v = Cfg.ret k' v' case tail k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal Code.tail k v = Cfg.ret k' v' case cons a✝¹ a✝ : Code a_ih✝¹ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝¹ k v = Cfg.ret k' v' a_ih✝ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝ k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.cons a✝¹ a✝) k v = Cfg.ret k' v' case comp a✝¹ a✝ : Code a_ih✝¹ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝¹ k v = Cfg.ret k' v' a_ih✝ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝ k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.comp a✝¹ a✝) k v = Cfg.ret k' v' case case a✝¹ a✝ : Code a_ih✝¹ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝¹ k v = Cfg.ret k' v' a_ih✝ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝ k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.case a✝¹ a✝) k v = Cfg.ret k' v' case fix a✝ : Code a_ih✝ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝ k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.fix a✝) k v = Cfg.ret k' v' iterate 3 exact ⟨_, _, rfl⟩ case cons a✝¹ a✝ : Code a_ih✝¹ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝¹ k v = Cfg.ret k' v' a_ih✝ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝ k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.cons a✝¹ a✝) k v = Cfg.ret k' v' case comp a✝¹ a✝ : Code a_ih✝¹ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝¹ k v = Cfg.ret k' v' a_ih✝ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝ k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.comp a✝¹ a✝) k v = Cfg.ret k' v' case case a✝¹ a✝ : Code a_ih✝¹ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝¹ k v = Cfg.ret k' v' a_ih✝ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝ k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.case a✝¹ a✝) k v = Cfg.ret k' v' case fix a✝ : Code a_ih✝ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝ k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.fix a✝) k v = Cfg.ret k' v' case cons _f fs IHf _IHfs => apply IHf case comp a✝¹ a✝ : Code a_ih✝¹ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝¹ k v = Cfg.ret k' v' a_ih✝ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝ k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.comp a✝¹ a✝) k v = Cfg.ret k' v' case case a✝¹ a✝ : Code a_ih✝¹ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝¹ k v = Cfg.ret k' v' a_ih✝ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝ k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.case a✝¹ a✝) k v = Cfg.ret k' v' case fix a✝ : Code a_ih✝ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝ k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.fix a✝) k v = Cfg.ret k' v' case comp f _g _IHf IHg => apply IHg case case a✝¹ a✝ : Code a_ih✝¹ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝¹ k v = Cfg.ret k' v' a_ih✝ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝ k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.case a✝¹ a✝) k v = Cfg.ret k' v' case fix a✝ : Code a_ih✝ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝ k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.fix a✝) k v = Cfg.ret k' v' case case f g IHf IHg => rw [stepNormal] simp only [] cases v.headI <;> simp only [] <;> [apply IHf; apply IHg] case fix a✝ : Code a_ih✝ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝ k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.fix a✝) k v = Cfg.ret k' v' case fix f IHf => apply IHf case tail k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal Code.tail k v = Cfg.ret k' v' case cons a✝¹ a✝ : Code a_ih✝¹ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝¹ k v = Cfg.ret k' v' a_ih✝ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝ k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.cons a✝¹ a✝) k v = Cfg.ret k' v' case comp a✝¹ a✝ : Code a_ih✝¹ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝¹ k v = Cfg.ret k' v' a_ih✝ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝ k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.comp a✝¹ a✝) k v = Cfg.ret k' v' case case a✝¹ a✝ : Code a_ih✝¹ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝¹ k v = Cfg.ret k' v' a_ih✝ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝ k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.case a✝¹ a✝) k v = Cfg.ret k' v' case fix a✝ : Code a_ih✝ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝ k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.fix a✝) k v = Cfg.ret k' v' exact ⟨_, _, rfl⟩ _f fs : Code IHf : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal _f k v = Cfg.ret k' v' _IHfs : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal fs k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.cons _f fs) k v = Cfg.ret k' v' apply IHf f _g : Code _IHf : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal f k v = Cfg.ret k' v' IHg : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal _g k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.comp f _g) k v = Cfg.ret k' v' apply IHg f g : Code IHf : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal f k v = Cfg.ret k' v' IHg : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal g k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.case f g) k v = Cfg.ret k' v' rw [stepNormal] f g : Code IHf : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal f k v = Cfg.ret k' v' IHg : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal g k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', (fun k v => Nat.rec (stepNormal f k (List.tail v)) (fun y x => stepNormal g k (y :: List.tail v)) (List.headI v)) k v = Cfg.ret k' v' simp only [] f g : Code IHf : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal f k v = Cfg.ret k' v' IHg : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal g k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', Nat.rec (stepNormal f k (List.tail v)) (fun y x => stepNormal g k (y :: List.tail v)) (List.headI v) = Cfg.ret k' v' cases v.headI <;> simp only [] <;> [apply IHf; apply IHg] f : Code IHf : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal f k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.fix f) k v = Cfg.ret k' v' apply IHf ** Qed
Turing.ToPartrec.code_is_ok c : Code ⊢ Code.Ok c induction c <;> intro k v <;> rw [stepNormal] case zero' k : Cont v : List ℕ ⊢ eval step ((fun k v => Cfg.ret k (0 :: v)) k v) = do let v ← Code.eval Code.zero' v eval step (Cfg.ret k v) case succ k : Cont v : List ℕ ⊢ eval step ((fun k v => Cfg.ret k [Nat.succ (List.headI v)]) k v) = do let v ← Code.eval Code.succ v eval step (Cfg.ret k v) case tail k : Cont v : List ℕ ⊢ eval step ((fun k v => Cfg.ret k (List.tail v)) k v) = do let v ← Code.eval Code.tail v eval step (Cfg.ret k v) case cons a✝¹ a✝ : Code a_ih✝¹ : Code.Ok a✝¹ a_ih✝ : Code.Ok a✝ k : Cont v : List ℕ ⊢ eval step ((fun k v => stepNormal a✝¹ (Cont.cons₁ a✝ v k) v) k v) = do let v ← Code.eval (Code.cons a✝¹ a✝) v eval step (Cfg.ret k v) case comp a✝¹ a✝ : Code a_ih✝¹ : Code.Ok a✝¹ a_ih✝ : Code.Ok a✝ k : Cont v : List ℕ ⊢ eval step ((fun k v => stepNormal a✝ (Cont.comp a✝¹ k) v) k v) = do let v ← Code.eval (Code.comp a✝¹ a✝) v eval step (Cfg.ret k v) case case a✝¹ a✝ : Code a_ih✝¹ : Code.Ok a✝¹ a_ih✝ : Code.Ok a✝ k : Cont v : List ℕ ⊢ eval step ((fun k v => Nat.rec (stepNormal a✝¹ k (List.tail v)) (fun y x => stepNormal a✝ k (y :: List.tail v)) (List.headI v)) k v) = do let v ← Code.eval (Code.case a✝¹ a✝) v eval step (Cfg.ret k v) case fix a✝ : Code a_ih✝ : Code.Ok a✝ k : Cont v : List ℕ ⊢ eval step ((fun k v => stepNormal a✝ (Cont.fix a✝ k) v) k v) = do let v ← Code.eval (Code.fix a✝) v eval step (Cfg.ret k v) iterate 3 simp only [Code.eval, pure_bind] case cons a✝¹ a✝ : Code a_ih✝¹ : Code.Ok a✝¹ a_ih✝ : Code.Ok a✝ k : Cont v : List ℕ ⊢ eval step ((fun k v => stepNormal a✝¹ (Cont.cons₁ a✝ v k) v) k v) = do let v ← Code.eval (Code.cons a✝¹ a✝) v eval step (Cfg.ret k v) case comp a✝¹ a✝ : Code a_ih✝¹ : Code.Ok a✝¹ a_ih✝ : Code.Ok a✝ k : Cont v : List ℕ ⊢ eval step ((fun k v => stepNormal a✝ (Cont.comp a✝¹ k) v) k v) = do let v ← Code.eval (Code.comp a✝¹ a✝) v eval step (Cfg.ret k v) case case a✝¹ a✝ : Code a_ih✝¹ : Code.Ok a✝¹ a_ih✝ : Code.Ok a✝ k : Cont v : List ℕ ⊢ eval step ((fun k v => Nat.rec (stepNormal a✝¹ k (List.tail v)) (fun y x => stepNormal a✝ k (y :: List.tail v)) (List.headI v)) k v) = do let v ← Code.eval (Code.case a✝¹ a✝) v eval step (Cfg.ret k v) case fix a✝ : Code a_ih✝ : Code.Ok a✝ k : Cont v : List ℕ ⊢ eval step ((fun k v => stepNormal a✝ (Cont.fix a✝ k) v) k v) = do let v ← Code.eval (Code.fix a✝) v eval step (Cfg.ret k v) case cons f fs IHf IHfs => rw [Code.eval, IHf] simp only [bind_assoc, Cont.eval, pure_bind]; congr; funext v rw [reaches_eval]; swap; exact ReflTransGen.single rfl rw [stepRet, IHfs]; congr; funext v' refine' Eq.trans _ (Eq.symm _) <;> try exact reaches_eval (ReflTransGen.single rfl) case comp a✝¹ a✝ : Code a_ih✝¹ : Code.Ok a✝¹ a_ih✝ : Code.Ok a✝ k : Cont v : List ℕ ⊢ eval step ((fun k v => stepNormal a✝ (Cont.comp a✝¹ k) v) k v) = do let v ← Code.eval (Code.comp a✝¹ a✝) v eval step (Cfg.ret k v) case case a✝¹ a✝ : Code a_ih✝¹ : Code.Ok a✝¹ a_ih✝ : Code.Ok a✝ k : Cont v : List ℕ ⊢ eval step ((fun k v => Nat.rec (stepNormal a✝¹ k (List.tail v)) (fun y x => stepNormal a✝ k (y :: List.tail v)) (List.headI v)) k v) = do let v ← Code.eval (Code.case a✝¹ a✝) v eval step (Cfg.ret k v) case fix a✝ : Code a_ih✝ : Code.Ok a✝ k : Cont v : List ℕ ⊢ eval step ((fun k v => stepNormal a✝ (Cont.fix a✝ k) v) k v) = do let v ← Code.eval (Code.fix a✝) v eval step (Cfg.ret k v) case comp f g IHf IHg => rw [Code.eval, IHg] simp only [bind_assoc, Cont.eval, pure_bind]; congr; funext v rw [reaches_eval]; swap; exact ReflTransGen.single rfl rw [stepRet, IHf] case case a✝¹ a✝ : Code a_ih✝¹ : Code.Ok a✝¹ a_ih✝ : Code.Ok a✝ k : Cont v : List ℕ ⊢ eval step ((fun k v => Nat.rec (stepNormal a✝¹ k (List.tail v)) (fun y x => stepNormal a✝ k (y :: List.tail v)) (List.headI v)) k v) = do let v ← Code.eval (Code.case a✝¹ a✝) v eval step (Cfg.ret k v) case fix a✝ : Code a_ih✝ : Code.Ok a✝ k : Cont v : List ℕ ⊢ eval step ((fun k v => stepNormal a✝ (Cont.fix a✝ k) v) k v) = do let v ← Code.eval (Code.fix a✝) v eval step (Cfg.ret k v) case case f g IHf IHg => simp only [Code.eval] cases v.headI <;> simp only [Code.eval] <;> [apply IHf; apply IHg] case fix a✝ : Code a_ih✝ : Code.Ok a✝ k : Cont v : List ℕ ⊢ eval step ((fun k v => stepNormal a✝ (Cont.fix a✝ k) v) k v) = do let v ← Code.eval (Code.fix a✝) v eval step (Cfg.ret k v) case fix f IHf => rw [cont_eval_fix IHf] case tail k : Cont v : List ℕ ⊢ eval step ((fun k v => Cfg.ret k (List.tail v)) k v) = do let v ← Code.eval Code.tail v eval step (Cfg.ret k v) case cons a✝¹ a✝ : Code a_ih✝¹ : Code.Ok a✝¹ a_ih✝ : Code.Ok a✝ k : Cont v : List ℕ ⊢ eval step ((fun k v => stepNormal a✝¹ (Cont.cons₁ a✝ v k) v) k v) = do let v ← Code.eval (Code.cons a✝¹ a✝) v eval step (Cfg.ret k v) case comp a✝¹ a✝ : Code a_ih✝¹ : Code.Ok a✝¹ a_ih✝ : Code.Ok a✝ k : Cont v : List ℕ ⊢ eval step ((fun k v => stepNormal a✝ (Cont.comp a✝¹ k) v) k v) = do let v ← Code.eval (Code.comp a✝¹ a✝) v eval step (Cfg.ret k v) case case a✝¹ a✝ : Code a_ih✝¹ : Code.Ok a✝¹ a_ih✝ : Code.Ok a✝ k : Cont v : List ℕ ⊢ eval step ((fun k v => Nat.rec (stepNormal a✝¹ k (List.tail v)) (fun y x => stepNormal a✝ k (y :: List.tail v)) (List.headI v)) k v) = do let v ← Code.eval (Code.case a✝¹ a✝) v eval step (Cfg.ret k v) case fix a✝ : Code a_ih✝ : Code.Ok a✝ k : Cont v : List ℕ ⊢ eval step ((fun k v => stepNormal a✝ (Cont.fix a✝ k) v) k v) = do let v ← Code.eval (Code.fix a✝) v eval step (Cfg.ret k v) simp only [Code.eval, pure_bind] f fs : Code IHf : Code.Ok f IHfs : Code.Ok fs k : Cont v : List ℕ ⊢ eval step ((fun k v => stepNormal f (Cont.cons₁ fs v k) v) k v) = do let v ← Code.eval (Code.cons f fs) v eval step (Cfg.ret k v) rw [Code.eval, IHf] f fs : Code IHf : Code.Ok f IHfs : Code.Ok fs k : Cont v : List ℕ ⊢ (do let v_1 ← Code.eval f v eval step (Cfg.ret (Cont.cons₁ fs v k) v_1)) = do let v ← (fun v => do let n ← Code.eval f v let ns ← Code.eval fs v pure (List.headI n :: ns)) v eval step (Cfg.ret k v) simp only [bind_assoc, Cont.eval, pure_bind] f fs : Code IHf : Code.Ok f IHfs : Code.Ok fs k : Cont v : List ℕ ⊢ (do let v_1 ← Code.eval f v eval step (Cfg.ret (Cont.cons₁ fs v k) v_1)) = do let x ← Code.eval f v let x_1 ← Code.eval fs v eval step (Cfg.ret k (List.headI x :: x_1)) congr case e_a f fs : Code IHf : Code.Ok f IHfs : Code.Ok fs k : Cont v : List ℕ ⊢ (fun v_1 => eval step (Cfg.ret (Cont.cons₁ fs v k) v_1)) = fun x => do let x_1 ← Code.eval fs v eval step (Cfg.ret k (List.headI x :: x_1)) funext v case e_a.h f fs : Code IHf : Code.Ok f IHfs : Code.Ok fs k : Cont v✝ v : List ℕ ⊢ eval step (Cfg.ret (Cont.cons₁ fs v✝ k) v) = do let x ← Code.eval fs v✝ eval step (Cfg.ret k (List.headI v :: x)) rw [reaches_eval] case e_a.h f fs : Code IHf : Code.Ok f IHfs : Code.Ok fs k : Cont v✝ v : List ℕ ⊢ eval step ?m.200354 = do let x ← Code.eval fs v✝ eval step (Cfg.ret k (List.headI v :: x)) case e_a.h f fs : Code IHf : Code.Ok f IHfs : Code.Ok fs k : Cont v✝ v : List ℕ ⊢ Reaches step (Cfg.ret (Cont.cons₁ fs v✝ k) v) ?m.200354 f fs : Code IHf : Code.Ok f IHfs : Code.Ok fs k : Cont v✝ v : List ℕ ⊢ Cfg swap case e_a.h f fs : Code IHf : Code.Ok f IHfs : Code.Ok fs k : Cont v✝ v : List ℕ ⊢ Reaches step (Cfg.ret (Cont.cons₁ fs v✝ k) v) ?m.200354 case e_a.h f fs : Code IHf : Code.Ok f IHfs : Code.Ok fs k : Cont v✝ v : List ℕ ⊢ eval step ?m.200354 = do let x ← Code.eval fs v✝ eval step (Cfg.ret k (List.headI v :: x)) f fs : Code IHf : Code.Ok f IHfs : Code.Ok fs k : Cont v✝ v : List ℕ ⊢ Cfg exact ReflTransGen.single rfl case e_a.h f fs : Code IHf : Code.Ok f IHfs : Code.Ok fs k : Cont v✝ v : List ℕ ⊢ eval step (stepRet (Cont.cons₁ fs v✝ k) v) = do let x ← Code.eval fs v✝ eval step (Cfg.ret k (List.headI v :: x)) rw [stepRet, IHfs] case e_a.h f fs : Code IHf : Code.Ok f IHfs : Code.Ok fs k : Cont v✝ v : List ℕ ⊢ (do let v_1 ← Code.eval fs v✝ eval step (Cfg.ret (Cont.cons₂ v k) v_1)) = do let x ← Code.eval fs v✝ eval step (Cfg.ret k (List.headI v :: x)) congr case e_a.h.e_a f fs : Code IHf : Code.Ok f IHfs : Code.Ok fs k : Cont v✝ v : List ℕ ⊢ (fun v_1 => eval step (Cfg.ret (Cont.cons₂ v k) v_1)) = fun x => eval step (Cfg.ret k (List.headI v :: x)) funext v' case e_a.h.e_a.h f fs : Code IHf : Code.Ok f IHfs : Code.Ok fs k : Cont v✝ v v' : List ℕ ⊢ eval step (Cfg.ret (Cont.cons₂ v k) v') = eval step (Cfg.ret k (List.headI v :: v')) refine' Eq.trans _ (Eq.symm _) <;> try exact reaches_eval (ReflTransGen.single rfl) case e_a.h.e_a.h.refine'_3 f fs : Code IHf : Code.Ok f IHfs : Code.Ok fs k : Cont v✝ v v' : List ℕ ⊢ eval step (Cfg.ret k (List.headI v :: v')) = eval step (stepRet (Cont.cons₂ v k) v') exact reaches_eval (ReflTransGen.single rfl) f g : Code IHf : Code.Ok f IHg : Code.Ok g k : Cont v : List ℕ ⊢ eval step ((fun k v => stepNormal g (Cont.comp f k) v) k v) = do let v ← Code.eval (Code.comp f g) v eval step (Cfg.ret k v) rw [Code.eval, IHg] f g : Code IHf : Code.Ok f IHg : Code.Ok g k : Cont v : List ℕ ⊢ (do let v ← Code.eval g v eval step (Cfg.ret (Cont.comp f k) v)) = do let v ← (fun v => Code.eval g v >>= Code.eval f) v eval step (Cfg.ret k v) simp only [bind_assoc, Cont.eval, pure_bind] f g : Code IHf : Code.Ok f IHg : Code.Ok g k : Cont v : List ℕ ⊢ (do let v ← Code.eval g v eval step (Cfg.ret (Cont.comp f k) v)) = do let x ← Code.eval g v let v ← Code.eval f x eval step (Cfg.ret k v) congr case e_a f g : Code IHf : Code.Ok f IHg : Code.Ok g k : Cont v : List ℕ ⊢ (fun v => eval step (Cfg.ret (Cont.comp f k) v)) = fun x => do let v ← Code.eval f x eval step (Cfg.ret k v) funext v case e_a.h f g : Code IHf : Code.Ok f IHg : Code.Ok g k : Cont v✝ v : List ℕ ⊢ eval step (Cfg.ret (Cont.comp f k) v) = do let v ← Code.eval f v eval step (Cfg.ret k v) rw [reaches_eval] case e_a.h f g : Code IHf : Code.Ok f IHg : Code.Ok g k : Cont v✝ v : List ℕ ⊢ eval step ?m.201128 = do let v ← Code.eval f v eval step (Cfg.ret k v) case e_a.h f g : Code IHf : Code.Ok f IHg : Code.Ok g k : Cont v✝ v : List ℕ ⊢ Reaches step (Cfg.ret (Cont.comp f k) v) ?m.201128 f g : Code IHf : Code.Ok f IHg : Code.Ok g k : Cont v✝ v : List ℕ ⊢ Cfg swap case e_a.h f g : Code IHf : Code.Ok f IHg : Code.Ok g k : Cont v✝ v : List ℕ ⊢ Reaches step (Cfg.ret (Cont.comp f k) v) ?m.201128 case e_a.h f g : Code IHf : Code.Ok f IHg : Code.Ok g k : Cont v✝ v : List ℕ ⊢ eval step ?m.201128 = do let v ← Code.eval f v eval step (Cfg.ret k v) f g : Code IHf : Code.Ok f IHg : Code.Ok g k : Cont v✝ v : List ℕ ⊢ Cfg exact ReflTransGen.single rfl case e_a.h f g : Code IHf : Code.Ok f IHg : Code.Ok g k : Cont v✝ v : List ℕ ⊢ eval step (stepRet (Cont.comp f k) v) = do let v ← Code.eval f v eval step (Cfg.ret k v) rw [stepRet, IHf] f g : Code IHf : Code.Ok f IHg : Code.Ok g k : Cont v : List ℕ ⊢ eval step ((fun k v => Nat.rec (stepNormal f k (List.tail v)) (fun y x => stepNormal g k (y :: List.tail v)) (List.headI v)) k v) = do let v ← Code.eval (Code.case f g) v eval step (Cfg.ret k v) simp only [Code.eval] f g : Code IHf : Code.Ok f IHg : Code.Ok g k : Cont v : List ℕ ⊢ eval step (Nat.rec (stepNormal f k (List.tail v)) (fun y x => stepNormal g k (y :: List.tail v)) (List.headI v)) = do let v ← Nat.rec (Code.eval f (List.tail v)) (fun y x => Code.eval g (y :: List.tail v)) (List.headI v) eval step (Cfg.ret k v) cases v.headI <;> simp only [Code.eval] <;> [apply IHf; apply IHg] f : Code IHf : Code.Ok f k : Cont v : List ℕ ⊢ eval step ((fun k v => stepNormal f (Cont.fix f k) v) k v) = do let v ← Code.eval (Code.fix f) v eval step (Cfg.ret k v) rw [cont_eval_fix IHf] ** Qed
Turing.ToPartrec.stepRet_eval k : Cont v : List ℕ ⊢ eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v induction k generalizing v case halt v : List ℕ ⊢ eval step (stepRet Cont.halt v) = Cfg.halt <$> Cont.eval Cont.halt v case cons₁ a✝² : Code a✝¹ : List ℕ a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, eval step (stepRet a✝ v) = Cfg.halt <$> Cont.eval a✝ v v : List ℕ ⊢ eval step (stepRet (Cont.cons₁ a✝² a✝¹ a✝) v) = Cfg.halt <$> Cont.eval (Cont.cons₁ a✝² a✝¹ a✝) v case cons₂ a✝¹ : List ℕ a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, eval step (stepRet a✝ v) = Cfg.halt <$> Cont.eval a✝ v v : List ℕ ⊢ eval step (stepRet (Cont.cons₂ a✝¹ a✝) v) = Cfg.halt <$> Cont.eval (Cont.cons₂ a✝¹ a✝) v case comp a✝¹ : Code a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, eval step (stepRet a✝ v) = Cfg.halt <$> Cont.eval a✝ v v : List ℕ ⊢ eval step (stepRet (Cont.comp a✝¹ a✝) v) = Cfg.halt <$> Cont.eval (Cont.comp a✝¹ a✝) v case fix a✝¹ : Code a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, eval step (stepRet a✝ v) = Cfg.halt <$> Cont.eval a✝ v v : List ℕ ⊢ eval step (stepRet (Cont.fix a✝¹ a✝) v) = Cfg.halt <$> Cont.eval (Cont.fix a✝¹ a✝) v case halt => simp only [mem_eval, Cont.eval, map_pure] exact Part.eq_some_iff.2 (mem_eval.2 ⟨ReflTransGen.refl, rfl⟩) case cons₁ a✝² : Code a✝¹ : List ℕ a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, eval step (stepRet a✝ v) = Cfg.halt <$> Cont.eval a✝ v v : List ℕ ⊢ eval step (stepRet (Cont.cons₁ a✝² a✝¹ a✝) v) = Cfg.halt <$> Cont.eval (Cont.cons₁ a✝² a✝¹ a✝) v case cons₂ a✝¹ : List ℕ a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, eval step (stepRet a✝ v) = Cfg.halt <$> Cont.eval a✝ v v : List ℕ ⊢ eval step (stepRet (Cont.cons₂ a✝¹ a✝) v) = Cfg.halt <$> Cont.eval (Cont.cons₂ a✝¹ a✝) v case comp a✝¹ : Code a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, eval step (stepRet a✝ v) = Cfg.halt <$> Cont.eval a✝ v v : List ℕ ⊢ eval step (stepRet (Cont.comp a✝¹ a✝) v) = Cfg.halt <$> Cont.eval (Cont.comp a✝¹ a✝) v case fix a✝¹ : Code a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, eval step (stepRet a✝ v) = Cfg.halt <$> Cont.eval a✝ v v : List ℕ ⊢ eval step (stepRet (Cont.fix a✝¹ a✝) v) = Cfg.halt <$> Cont.eval (Cont.fix a✝¹ a✝) v case cons₁ fs as k IH => rw [Cont.eval, stepRet, code_is_ok] simp only [← bind_pure_comp, bind_assoc]; congr; funext v' rw [reaches_eval]; swap; exact ReflTransGen.single rfl rw [stepRet, IH, bind_pure_comp] case cons₂ a✝¹ : List ℕ a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, eval step (stepRet a✝ v) = Cfg.halt <$> Cont.eval a✝ v v : List ℕ ⊢ eval step (stepRet (Cont.cons₂ a✝¹ a✝) v) = Cfg.halt <$> Cont.eval (Cont.cons₂ a✝¹ a✝) v case comp a✝¹ : Code a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, eval step (stepRet a✝ v) = Cfg.halt <$> Cont.eval a✝ v v : List ℕ ⊢ eval step (stepRet (Cont.comp a✝¹ a✝) v) = Cfg.halt <$> Cont.eval (Cont.comp a✝¹ a✝) v case fix a✝¹ : Code a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, eval step (stepRet a✝ v) = Cfg.halt <$> Cont.eval a✝ v v : List ℕ ⊢ eval step (stepRet (Cont.fix a✝¹ a✝) v) = Cfg.halt <$> Cont.eval (Cont.fix a✝¹ a✝) v case cons₂ ns k IH => rw [Cont.eval, stepRet]; exact IH case comp a✝¹ : Code a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, eval step (stepRet a✝ v) = Cfg.halt <$> Cont.eval a✝ v v : List ℕ ⊢ eval step (stepRet (Cont.comp a✝¹ a✝) v) = Cfg.halt <$> Cont.eval (Cont.comp a✝¹ a✝) v case fix a✝¹ : Code a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, eval step (stepRet a✝ v) = Cfg.halt <$> Cont.eval a✝ v v : List ℕ ⊢ eval step (stepRet (Cont.fix a✝¹ a✝) v) = Cfg.halt <$> Cont.eval (Cont.fix a✝¹ a✝) v case comp f k IH => rw [Cont.eval, stepRet, code_is_ok] simp only [← bind_pure_comp, bind_assoc]; congr; funext v' rw [reaches_eval]; swap; exact ReflTransGen.single rfl rw [IH, bind_pure_comp] v : List ℕ ⊢ eval step (stepRet Cont.halt v) = Cfg.halt <$> Cont.eval Cont.halt v simp only [mem_eval, Cont.eval, map_pure] v : List ℕ ⊢ eval step (stepRet Cont.halt v) = pure (Cfg.halt v) exact Part.eq_some_iff.2 (mem_eval.2 ⟨ReflTransGen.refl, rfl⟩) fs : Code as : List ℕ k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v : List ℕ ⊢ eval step (stepRet (Cont.cons₁ fs as k) v) = Cfg.halt <$> Cont.eval (Cont.cons₁ fs as k) v rw [Cont.eval, stepRet, code_is_ok] fs : Code as : List ℕ k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v : List ℕ ⊢ (do let v_1 ← Code.eval fs as eval step (Cfg.ret (Cont.cons₂ v k) v_1)) = Cfg.halt <$> (fun v => do let ns ← Code.eval fs as Cont.eval k (List.headI v :: ns)) v simp only [← bind_pure_comp, bind_assoc] fs : Code as : List ℕ k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v : List ℕ ⊢ (do let v_1 ← Code.eval fs as eval step (Cfg.ret (Cont.cons₂ v k) v_1)) = do let x ← Code.eval fs as let a ← Cont.eval k (List.headI v :: x) pure (Cfg.halt a) congr case e_a fs : Code as : List ℕ k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v : List ℕ ⊢ (fun v_1 => eval step (Cfg.ret (Cont.cons₂ v k) v_1)) = fun x => do let a ← Cont.eval k (List.headI v :: x) pure (Cfg.halt a) funext v' case e_a.h fs : Code as : List ℕ k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v v' : List ℕ ⊢ eval step (Cfg.ret (Cont.cons₂ v k) v') = do let a ← Cont.eval k (List.headI v :: v') pure (Cfg.halt a) rw [reaches_eval] case e_a.h fs : Code as : List ℕ k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v v' : List ℕ ⊢ eval step ?m.202867 = do let a ← Cont.eval k (List.headI v :: v') pure (Cfg.halt a) case e_a.h fs : Code as : List ℕ k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v v' : List ℕ ⊢ Reaches step (Cfg.ret (Cont.cons₂ v k) v') ?m.202867 fs : Code as : List ℕ k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v v' : List ℕ ⊢ Cfg swap case e_a.h fs : Code as : List ℕ k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v v' : List ℕ ⊢ Reaches step (Cfg.ret (Cont.cons₂ v k) v') ?m.202867 case e_a.h fs : Code as : List ℕ k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v v' : List ℕ ⊢ eval step ?m.202867 = do let a ← Cont.eval k (List.headI v :: v') pure (Cfg.halt a) fs : Code as : List ℕ k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v v' : List ℕ ⊢ Cfg exact ReflTransGen.single rfl case e_a.h fs : Code as : List ℕ k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v v' : List ℕ ⊢ eval step (stepRet (Cont.cons₂ v k) v') = do let a ← Cont.eval k (List.headI v :: v') pure (Cfg.halt a) rw [stepRet, IH, bind_pure_comp] ns : List ℕ k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v : List ℕ ⊢ eval step (stepRet (Cont.cons₂ ns k) v) = Cfg.halt <$> Cont.eval (Cont.cons₂ ns k) v rw [Cont.eval, stepRet] ns : List ℕ k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v : List ℕ ⊢ eval step (stepRet k (List.headI ns :: v)) = Cfg.halt <$> (fun v => Cont.eval k (List.headI ns :: v)) v exact IH f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v : List ℕ ⊢ eval step (stepRet (Cont.comp f k) v) = Cfg.halt <$> Cont.eval (Cont.comp f k) v rw [Cont.eval, stepRet, code_is_ok] f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v : List ℕ ⊢ (do let v ← Code.eval f v eval step (Cfg.ret k v)) = Cfg.halt <$> (fun v => Code.eval f v >>= Cont.eval k) v simp only [← bind_pure_comp, bind_assoc] f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v : List ℕ ⊢ (do let v ← Code.eval f v eval step (Cfg.ret k v)) = do let x ← Code.eval f v let a ← Cont.eval k x pure (Cfg.halt a) congr case e_a f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v : List ℕ ⊢ (fun v => eval step (Cfg.ret k v)) = fun x => do let a ← Cont.eval k x pure (Cfg.halt a) funext v' case e_a.h f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v v' : List ℕ ⊢ eval step (Cfg.ret k v') = do let a ← Cont.eval k v' pure (Cfg.halt a) rw [reaches_eval] case e_a.h f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v v' : List ℕ ⊢ eval step ?m.203559 = do let a ← Cont.eval k v' pure (Cfg.halt a) case e_a.h f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v v' : List ℕ ⊢ Reaches step (Cfg.ret k v') ?m.203559 f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v v' : List ℕ ⊢ Cfg swap case e_a.h f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v v' : List ℕ ⊢ Reaches step (Cfg.ret k v') ?m.203559 case e_a.h f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v v' : List ℕ ⊢ eval step ?m.203559 = do let a ← Cont.eval k v' pure (Cfg.halt a) f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v v' : List ℕ ⊢ Cfg exact ReflTransGen.single rfl case e_a.h f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v v' : List ℕ ⊢ eval step (stepRet k v') = do let a ← Cont.eval k v' pure (Cfg.halt a) rw [IH, bind_pure_comp] f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v : List ℕ ⊢ eval step (stepRet (Cont.fix f k) v) = Cfg.halt <$> Cont.eval (Cont.fix f k) v rw [Cont.eval, stepRet] f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v : List ℕ ⊢ eval step (if List.headI v = 0 then stepRet k (List.tail v) else stepNormal f (Cont.fix f k) (List.tail v)) = Cfg.halt <$> (fun v => if List.headI v = 0 then Cont.eval k (List.tail v) else Code.eval (Code.fix f) (List.tail v) >>= Cont.eval k) v simp only [bind_pure_comp] f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v : List ℕ ⊢ eval step (if List.headI v = 0 then stepRet k (List.tail v) else stepNormal f (Cont.fix f k) (List.tail v)) = Cfg.halt <$> if List.headI v = 0 then Cont.eval k (List.tail v) else Code.eval (Code.fix f) (List.tail v) >>= Cont.eval k split_ifs case neg f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v : List ℕ h✝ : ¬List.headI v = 0 ⊢ eval step (stepNormal f (Cont.fix f k) (List.tail v)) = Cfg.halt <$> (Code.eval (Code.fix f) (List.tail v) >>= Cont.eval k) simp only [← bind_pure_comp, bind_assoc, cont_eval_fix (code_is_ok _)] case neg f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v : List ℕ h✝ : ¬List.headI v = 0 ⊢ (do let v ← Code.eval (Code.fix f) (List.tail v) eval step (Cfg.ret k v)) = do let x ← Code.eval (Code.fix f) (List.tail v) let a ← Cont.eval k x pure (Cfg.halt a) congr case neg.e_a f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v : List ℕ h✝ : ¬List.headI v = 0 ⊢ (fun v => eval step (Cfg.ret k v)) = fun x => do let a ← Cont.eval k x pure (Cfg.halt a) funext case neg.e_a.h f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v : List ℕ h✝ : ¬List.headI v = 0 x✝ : List ℕ ⊢ eval step (Cfg.ret k x✝) = do let a ← Cont.eval k x✝ pure (Cfg.halt a) rw [bind_pure_comp, ← IH] case neg.e_a.h f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v : List ℕ h✝ : ¬List.headI v = 0 x✝ : List ℕ ⊢ eval step (Cfg.ret k x✝) = eval step (stepRet k x✝) exact reaches_eval (ReflTransGen.single rfl) case pos f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v : List ℕ h✝ : List.headI v = 0 ⊢ eval step (stepRet k (List.tail v)) = Cfg.halt <$> Cont.eval k (List.tail v) exact IH ** Qed
Turing.PartrecToTM2.trNat_zero ⊢ trNat 0 = [] rw [trNat, Nat.cast_zero] ⊢ trNum 0 = [] rfl ** Qed
Turing.PartrecToTM2.K'.elim_update_main a b c d a' : List Γ' ⊢ update (elim a b c d) main a' = elim a' b c d funext x case h a b c d a' : List Γ' x : K' ⊢ update (elim a b c d) main a' x = elim a' b c d x cases x <;> rfl ** Qed
Turing.PartrecToTM2.K'.elim_update_rev a b c d b' : List Γ' ⊢ update (elim a b c d) rev b' = elim a b' c d funext x case h a b c d b' : List Γ' x : K' ⊢ update (elim a b c d) rev b' x = elim a b' c d x cases x <;> rfl ** Qed
Turing.PartrecToTM2.K'.elim_update_aux a b c d c' : List Γ' ⊢ update (elim a b c d) aux c' = elim a b c' d funext x case h a b c d c' : List Γ' x : K' ⊢ update (elim a b c d) aux c' x = elim a b c' d x cases x <;> rfl ** Qed
Turing.PartrecToTM2.K'.elim_update_stack a b c d d' : List Γ' ⊢ update (elim a b c d) stack d' = elim a b c d' funext x case h a b c d d' : List Γ' x : K' ⊢ update (elim a b c d) stack d' x = elim a b c d' x cases x <;> rfl ** Qed
Turing.PartrecToTM2.splitAtPred_eq α : Type u_1 p : α → Bool l₁ : List α o : α l₂ : List α x✝ : ∀ (x : α), x ∈ l₁ → p x = false left✝ : p o = true h₃ : [] = l₁ ++ o :: l₂ ⊢ splitAtPred p [] = (l₁, some o, l₂) simp at h₃ α : Type u_1 p : α → Bool a : α L l₁ : List α o : Option α l₂ : List α h₁ : ∀ (x : α), x ∈ l₁ → p x = false h₂ : Option.elim' (a :: L = l₁ ∧ l₂ = []) (fun a_1 => p a_1 = true ∧ a :: L = l₁ ++ a_1 :: l₂) o ⊢ splitAtPred p (a :: L) = (l₁, o, l₂) rw [splitAtPred] α : Type u_1 p : α → Bool a : α L l₁ : List α o : Option α l₂ : List α h₁ : ∀ (x : α), x ∈ l₁ → p x = false h₂ : Option.elim' (a :: L = l₁ ∧ l₂ = []) (fun a_1 => p a_1 = true ∧ a :: L = l₁ ++ a_1 :: l₂) o ⊢ (bif p a then ([], some a, L) else match splitAtPred p L with \| (l₁, o, l₂) => (a :: l₁, o, l₂)) = (l₁, o, l₂) have IH := splitAtPred_eq p L α : Type u_1 p : α → Bool a : α L l₁ : List α o : Option α l₂ : List α h₁ : ∀ (x : α), x ∈ l₁ → p x = false h₂ : Option.elim' (a :: L = l₁ ∧ l₂ = []) (fun a_1 => p a_1 = true ∧ a :: L = l₁ ++ a_1 :: l₂) o IH : ∀ (l₁ : List α) (o : Option α) (l₂ : List α), (∀ (x : α), x ∈ l₁ → p x = false) → Option.elim' (L = l₁ ∧ l₂ = []) (fun a => p a = true ∧ L = l₁ ++ a :: l₂) o → splitAtPred p L = (l₁, o, l₂) ⊢ (bif p a then ([], some a, L) else match splitAtPred p L with \| (l₁, o, l₂) => (a :: l₁, o, l₂)) = (l₁, o, l₂) cases' o with o case none α : Type u_1 p : α → Bool a : α L l₁ l₂ : List α h₁ : ∀ (x : α), x ∈ l₁ → p x = false IH : ∀ (l₁ : List α) (o : Option α) (l₂ : List α), (∀ (x : α), x ∈ l₁ → p x = false) → Option.elim' (L = l₁ ∧ l₂ = []) (fun a => p a = true ∧ L = l₁ ++ a :: l₂) o → splitAtPred p L = (l₁, o, l₂) h₂ : Option.elim' (a :: L = l₁ ∧ l₂ = []) (fun a_1 => p a_1 = true ∧ a :: L = l₁ ++ a_1 :: l₂) none ⊢ (bif p a then ([], some a, L) else match splitAtPred p L with \| (l₁, o, l₂) => (a :: l₁, o, l₂)) = (l₁, none, l₂) cases' l₁ with a' l₁ <;> rcases h₂ with ⟨⟨⟩, rfl⟩ case none.cons.intro.refl α : Type u_1 p : α → Bool a : α L : List α IH : ∀ (l₁ : List α) (o : Option α) (l₂ : List α), (∀ (x : α), x ∈ l₁ → p x = false) → Option.elim' (L = l₁ ∧ l₂ = []) (fun a => p a = true ∧ L = l₁ ++ a :: l₂) o → splitAtPred p L = (l₁, o, l₂) h₁ : ∀ (x : α), x ∈ a :: L → p x = false ⊢ (bif p a then ([], some a, L) else match splitAtPred p L with \| (l₁, o, l₂) => (a :: l₁, o, l₂)) = (a :: L, none, []) rw [h₁ a (List.Mem.head _), cond, IH L none [] _ ⟨rfl, rfl⟩] α : Type u_1 p : α → Bool a : α L : List α IH : ∀ (l₁ : List α) (o : Option α) (l₂ : List α), (∀ (x : α), x ∈ l₁ → p x = false) → Option.elim' (L = l₁ ∧ l₂ = []) (fun a => p a = true ∧ L = l₁ ++ a :: l₂) o → splitAtPred p L = (l₁, o, l₂) h₁ : ∀ (x : α), x ∈ a :: L → p x = false ⊢ ∀ (x : α), x ∈ L → p x = false exact fun x h => h₁ x (List.Mem.tail _ h) case some α : Type u_1 p : α → Bool a : α L l₁ l₂ : List α h₁ : ∀ (x : α), x ∈ l₁ → p x = false IH : ∀ (l₁ : List α) (o : Option α) (l₂ : List α), (∀ (x : α), x ∈ l₁ → p x = false) → Option.elim' (L = l₁ ∧ l₂ = []) (fun a => p a = true ∧ L = l₁ ++ a :: l₂) o → splitAtPred p L = (l₁, o, l₂) o : α h₂ : Option.elim' (a :: L = l₁ ∧ l₂ = []) (fun a_1 => p a_1 = true ∧ a :: L = l₁ ++ a_1 :: l₂) (some o) ⊢ (bif p a then ([], some a, L) else match splitAtPred p L with \| (l₁, o, l₂) => (a :: l₁, o, l₂)) = (l₁, some o, l₂) cases' l₁ with a' l₁ <;> rcases h₂ with ⟨h₂, ⟨⟩⟩ case some.cons.intro.refl α : Type u_1 p : α → Bool a : α l₂ : List α o : α l₁ : List α h₂ : p o = true h₁ : ∀ (x : α), x ∈ a :: l₁ → p x = false IH : ∀ (l₁_1 : List α) (o_1 : Option α) (l₂_1 : List α), (∀ (x : α), x ∈ l₁_1 → p x = false) → Option.elim' (List.append l₁ (o :: l₂) = l₁_1 ∧ l₂_1 = []) (fun a => p a = true ∧ List.append l₁ (o :: l₂) = l₁_1 ++ a :: l₂_1) o_1 → splitAtPred p (List.append l₁ (o :: l₂)) = (l₁_1, o_1, l₂_1) ⊢ (bif p a then ([], some a, List.append l₁ (o :: l₂)) else match splitAtPred p (List.append l₁ (o :: l₂)) with \| (l₁, o, l₂) => (a :: l₁, o, l₂)) = (a :: l₁, some o, l₂) rw [h₁ a (List.Mem.head _), cond, IH l₁ (some o) l₂ _ ⟨h₂, _⟩] <;> try rfl α : Type u_1 p : α → Bool a : α l₂ : List α o : α l₁ : List α h₂ : p o = true h₁ : ∀ (x : α), x ∈ a :: l₁ → p x = false IH : ∀ (l₁_1 : List α) (o_1 : Option α) (l₂_1 : List α), (∀ (x : α), x ∈ l₁_1 → p x = false) → Option.elim' (List.append l₁ (o :: l₂) = l₁_1 ∧ l₂_1 = []) (fun a => p a = true ∧ List.append l₁ (o :: l₂) = l₁_1 ++ a :: l₂_1) o_1 → splitAtPred p (List.append l₁ (o :: l₂)) = (l₁_1, o_1, l₂_1) ⊢ ∀ (x : α), x ∈ l₁ → p x = false exact fun x h => h₁ x (List.Mem.tail _ h) case some.nil.intro.refl α : Type u_1 p : α → Bool a : α L : List α IH : ∀ (l₁ : List α) (o : Option α) (l₂ : List α), (∀ (x : α), x ∈ l₁ → p x = false) → Option.elim' (L = l₁ ∧ l₂ = []) (fun a => p a = true ∧ L = l₁ ++ a :: l₂) o → splitAtPred p L = (l₁, o, l₂) h₁ : ∀ (x : α), x ∈ [] → p x = false h₂ : p a = true ⊢ (bif p a then ([], some a, L) else match splitAtPred p L with \| (l₁, o, l₂) => (a :: l₁, o, l₂)) = ([], some a, L) rw [h₂, cond] α : Type u_1 p : α → Bool a : α l₂ : List α o : α l₁ : List α h₂ : p o = true h₁ : ∀ (x : α), x ∈ a :: l₁ → p x = false IH : ∀ (l₁_1 : List α) (o_1 : Option α) (l₂_1 : List α), (∀ (x : α), x ∈ l₁_1 → p x = false) → Option.elim' (List.append l₁ (o :: l₂) = l₁_1 ∧ l₂_1 = []) (fun a => p a = true ∧ List.append l₁ (o :: l₂) = l₁_1 ++ a :: l₂_1) o_1 → splitAtPred p (List.append l₁ (o :: l₂)) = (l₁_1, o_1, l₂_1) ⊢ List.append l₁ (o :: l₂) = l₁ ++ o :: l₂ rfl ** Qed
Turing.PartrecToTM2.unrev_ok q : Λ' s : Option Γ' S : K' → List Γ' ⊢ rev ≠ main decide ** Qed
Turing.PartrecToTM2.move₂_ok p : Γ' → Bool k₁ k₂ : K' q : Λ' s : Option Γ' L₁ : List Γ' o : Option Γ' L₂ : List Γ' S : K' → List Γ' h₁ : k₁ ≠ rev ∧ k₂ ≠ rev ∧ k₁ ≠ k₂ h₂ : S rev = [] e : splitAtPred p (S k₁) = (L₁, o, L₂) ⊢ Reaches₁ (TM2.step tr) { l := some (move₂ p k₁ k₂ q), var := s, stk := S } { l := some q, var := none, stk := update (update S k₁ (Option.elim o id List.cons L₂)) k₂ (L₁ ++ S k₂) } refine' (move_ok h₁.1 e).trans (TransGen.head rfl _) p : Γ' → Bool k₁ k₂ : K' q : Λ' s : Option Γ' L₁ : List Γ' o : Option Γ' L₂ : List Γ' S : K' → List Γ' h₁ : k₁ ≠ rev ∧ k₂ ≠ rev ∧ k₁ ≠ k₂ h₂ : S rev = [] e : splitAtPred p (S k₁) = (L₁, o, L₂) ⊢ TransGen (fun a b => b ∈ TM2.step tr a) (TM2.stepAux (tr (Λ'.push k₁ id (Λ'.move (fun x => false) rev k₂ q))) o (update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev)))) { l := some q, var := none, stk := update (update S k₁ (Option.elim o id List.cons L₂)) k₂ (L₁ ++ S k₂) } simp only [TM2.step, Option.mem_def, TM2.stepAux, id_eq, ne_eq, Option.elim] p : Γ' → Bool k₁ k₂ : K' q : Λ' s : Option Γ' L₁ : List Γ' o : Option Γ' L₂ : List Γ' S : K' → List Γ' h₁ : k₁ ≠ rev ∧ k₂ ≠ rev ∧ k₁ ≠ k₂ h₂ : S rev = [] e : splitAtPred p (S k₁) = (L₁, o, L₂) ⊢ TransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif Option.isSome o then { l := some (Λ'.move (fun x => false) rev k₂ q), var := o, stk := update (update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev))) k₁ (Option.iget o :: update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev)) k₁) } else { l := some (Λ'.move (fun x => false) rev k₂ q), var := o, stk := update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev)) }) { l := some q, var := none, stk := update (update S k₁ ((match (motive := Option Γ' → (List Γ' → List Γ') → (Γ' → List Γ' → List Γ') → List Γ' → List Γ') o, id, List.cons with \| some x, x_1, f => f x \| none, y, x => y) L₂)) k₂ (L₁ ++ S k₂) } cases o <;> simp only [Option.elim, id.def] case none p : Γ' → Bool k₁ k₂ : K' q : Λ' s : Option Γ' L₁ L₂ : List Γ' S : K' → List Γ' h₁ : k₁ ≠ rev ∧ k₂ ≠ rev ∧ k₁ ≠ k₂ h₂ : S rev = [] e : splitAtPred p (S k₁) = (L₁, none, L₂) ⊢ TransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif Option.isSome none then { l := some (Λ'.move (fun x => false) rev k₂ q), var := none, stk := update (update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev))) k₁ (Option.iget none :: update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev)) k₁) } else { l := some (Λ'.move (fun x => false) rev k₂ q), var := none, stk := update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev)) }) { l := some q, var := none, stk := update (update S k₁ L₂) k₂ (L₁ ++ S k₂) } simp only [TM2.stepAux, Option.isSome, cond_false] case none p : Γ' → Bool k₁ k₂ : K' q : Λ' s : Option Γ' L₁ L₂ : List Γ' S : K' → List Γ' h₁ : k₁ ≠ rev ∧ k₂ ≠ rev ∧ k₁ ≠ k₂ h₂ : S rev = [] e : splitAtPred p (S k₁) = (L₁, none, L₂) ⊢ TransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) { l := some (Λ'.move (fun x => false) rev k₂ q), var := none, stk := update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev)) } { l := some q, var := none, stk := update (update S k₁ L₂) k₂ (L₁ ++ S k₂) } convert move_ok h₁.2.1.symm (splitAtPred_false _) using 2 case h.e'_2.h.e'_7 p : Γ' → Bool k₁ k₂ : K' q : Λ' s : Option Γ' L₁ L₂ : List Γ' S : K' → List Γ' h₁ : k₁ ≠ rev ∧ k₂ ≠ rev ∧ k₁ ≠ k₂ h₂ : S rev = [] e : splitAtPred p (S k₁) = (L₁, none, L₂) ⊢ update (update S k₁ L₂) k₂ (L₁ ++ S k₂) = update (update (update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev))) rev []) k₂ (List.reverseAux (update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev)) rev) (update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev)) k₂)) simp only [Function.update_comm h₁.1, Function.update_idem] case h.e'_2.h.e'_7 p : Γ' → Bool k₁ k₂ : K' q : Λ' s : Option Γ' L₁ L₂ : List Γ' S : K' → List Γ' h₁ : k₁ ≠ rev ∧ k₂ ≠ rev ∧ k₁ ≠ k₂ h₂ : S rev = [] e : splitAtPred p (S k₁) = (L₁, none, L₂) ⊢ update (update S k₁ L₂) k₂ (L₁ ++ S k₂) = update (update (update S rev []) k₁ L₂) k₂ (List.reverseAux (update (update S rev (List.reverseAux L₁ (S rev))) k₁ L₂ rev) (update (update S rev (List.reverseAux L₁ (S rev))) k₁ L₂ k₂)) rw [show update S rev [] = S by rw [← h₂, Function.update_eq_self]] case h.e'_2.h.e'_7 p : Γ' → Bool k₁ k₂ : K' q : Λ' s : Option Γ' L₁ L₂ : List Γ' S : K' → List Γ' h₁ : k₁ ≠ rev ∧ k₂ ≠ rev ∧ k₁ ≠ k₂ h₂ : S rev = [] e : splitAtPred p (S k₁) = (L₁, none, L₂) ⊢ update (update S k₁ L₂) k₂ (L₁ ++ S k₂) = update (update S k₁ L₂) k₂ (List.reverseAux (update (update S rev (List.reverseAux L₁ (S rev))) k₁ L₂ rev) (update (update S rev (List.reverseAux L₁ (S rev))) k₁ L₂ k₂)) simp only [Function.update_noteq h₁.2.2.symm, Function.update_noteq h₁.2.1, Function.update_noteq h₁.1.symm, List.reverseAux_eq, h₂, Function.update_same, List.append_nil, List.reverse_reverse] p : Γ' → Bool k₁ k₂ : K' q : Λ' s : Option Γ' L₁ L₂ : List Γ' S : K' → List Γ' h₁ : k₁ ≠ rev ∧ k₂ ≠ rev ∧ k₁ ≠ k₂ h₂ : S rev = [] e : splitAtPred p (S k₁) = (L₁, none, L₂) ⊢ update S rev [] = S rw [← h₂, Function.update_eq_self] case some p : Γ' → Bool k₁ k₂ : K' q : Λ' s : Option Γ' L₁ L₂ : List Γ' S : K' → List Γ' h₁ : k₁ ≠ rev ∧ k₂ ≠ rev ∧ k₁ ≠ k₂ h₂ : S rev = [] val✝ : Γ' e : splitAtPred p (S k₁) = (L₁, some val✝, L₂) ⊢ TransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif Option.isSome (some val✝) then { l := some (Λ'.move (fun x => false) rev k₂ q), var := some val✝, stk := update (update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev))) k₁ (Option.iget (some val✝) :: update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev)) k₁) } else { l := some (Λ'.move (fun x => false) rev k₂ q), var := some val✝, stk := update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev)) }) { l := some q, var := none, stk := update (update S k₁ (val✝ :: L₂)) k₂ (L₁ ++ S k₂) } simp only [TM2.stepAux, Option.isSome, cond_true] case some p : Γ' → Bool k₁ k₂ : K' q : Λ' s : Option Γ' L₁ L₂ : List Γ' S : K' → List Γ' h₁ : k₁ ≠ rev ∧ k₂ ≠ rev ∧ k₁ ≠ k₂ h₂ : S rev = [] val✝ : Γ' e : splitAtPred p (S k₁) = (L₁, some val✝, L₂) ⊢ TransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) { l := some (Λ'.move (fun x => false) rev k₂ q), var := some val✝, stk := update (update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev))) k₁ (Option.iget (some val✝) :: update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev)) k₁) } { l := some q, var := none, stk := update (update S k₁ (val✝ :: L₂)) k₂ (L₁ ++ S k₂) } convert move_ok h₁.2.1.symm (splitAtPred_false _) using 2 case h.e'_2.h.e'_7 p : Γ' → Bool k₁ k₂ : K' q : Λ' s : Option Γ' L₁ L₂ : List Γ' S : K' → List Γ' h₁ : k₁ ≠ rev ∧ k₂ ≠ rev ∧ k₁ ≠ k₂ h₂ : S rev = [] val✝ : Γ' e : splitAtPred p (S k₁) = (L₁, some val✝, L₂) ⊢ update (update S k₁ (val✝ :: L₂)) k₂ (L₁ ++ S k₂) = update (update (update (update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev))) k₁ (Option.iget (some val✝) :: update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev)) k₁)) rev []) k₂ (List.reverseAux (update (update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev))) k₁ (Option.iget (some val✝) :: update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev)) k₁) rev) (update (update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev))) k₁ (Option.iget (some val✝) :: update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev)) k₁) k₂)) simp only [h₂, Function.update_comm h₁.1, List.reverseAux_eq, Function.update_same, List.append_nil, Function.update_idem] case h.e'_2.h.e'_7 p : Γ' → Bool k₁ k₂ : K' q : Λ' s : Option Γ' L₁ L₂ : List Γ' S : K' → List Γ' h₁ : k₁ ≠ rev ∧ k₂ ≠ rev ∧ k₁ ≠ k₂ h₂ : S rev = [] val✝ : Γ' e : splitAtPred p (S k₁) = (L₁, some val✝, L₂) ⊢ update (update S k₁ (val✝ :: L₂)) k₂ (L₁ ++ S k₂) = update (update (update S rev []) k₁ (Option.iget (some val✝) :: L₂)) k₂ (List.reverse (update (update S rev (List.reverse L₁)) k₁ (Option.iget (some val✝) :: L₂) rev) ++ update (update S rev (List.reverse L₁)) k₁ (Option.iget (some val✝) :: L₂) k₂) rw [show update S rev [] = S by rw [← h₂, Function.update_eq_self]] case h.e'_2.h.e'_7 p : Γ' → Bool k₁ k₂ : K' q : Λ' s : Option Γ' L₁ L₂ : List Γ' S : K' → List Γ' h₁ : k₁ ≠ rev ∧ k₂ ≠ rev ∧ k₁ ≠ k₂ h₂ : S rev = [] val✝ : Γ' e : splitAtPred p (S k₁) = (L₁, some val✝, L₂) ⊢ update (update S k₁ (val✝ :: L₂)) k₂ (L₁ ++ S k₂) = update (update S k₁ (Option.iget (some val✝) :: L₂)) k₂ (List.reverse (update (update S rev (List.reverse L₁)) k₁ (Option.iget (some val✝) :: L₂) rev) ++ update (update S rev (List.reverse L₁)) k₁ (Option.iget (some val✝) :: L₂) k₂) simp only [Function.update_noteq h₁.1.symm, Function.update_noteq h₁.2.2.symm, Function.update_noteq h₁.2.1, Function.update_same, List.reverse_reverse] p : Γ' → Bool k₁ k₂ : K' q : Λ' s : Option Γ' L₁ L₂ : List Γ' S : K' → List Γ' h₁ : k₁ ≠ rev ∧ k₂ ≠ rev ∧ k₁ ≠ k₂ h₂ : S rev = [] val✝ : Γ' e : splitAtPred p (S k₁) = (L₁, some val✝, L₂) ⊢ update S rev [] = S rw [← h₂, Function.update_eq_self] ** Qed
Turing.PartrecToTM2.clear_ok p : Γ' → Bool k : K' q : Λ' s : Option Γ' L₁ : List Γ' o : Option Γ' L₂ : List Γ' S : K' → List Γ' e : splitAtPred p (S k) = (L₁, o, L₂) ⊢ Reaches₁ (TM2.step tr) { l := some (Λ'.clear p k q), var := s, stk := S } { l := some q, var := o, stk := update S k L₂ } induction' L₁ with a L₁ IH generalizing S s case nil p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e✝ : splitAtPred p (S✝ k) = (L₁, o, L₂) s : Option Γ' S : K' → List Γ' e : splitAtPred p (S k) = ([], o, L₂) ⊢ Reaches₁ (TM2.step tr) { l := some (Λ'.clear p k q), var := s, stk := S } { l := some q, var := o, stk := update S k L₂ } refine' TransGen.head' rfl _ case nil p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e✝ : splitAtPred p (S✝ k) = (L₁, o, L₂) s : Option Γ' S : K' → List Γ' e : splitAtPred p (S k) = ([], o, L₂) ⊢ ReflTransGen (fun a b => b ∈ TM2.step tr a) (TM2.stepAux (tr (Λ'.clear p k q)) s S) { l := some q, var := o, stk := update S k L₂ } simp only [TM2.step, Option.mem_def, TM2.stepAux, Option.elim] case nil p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e✝ : splitAtPred p (S✝ k) = (L₁, o, L₂) s : Option Γ' S : K' → List Γ' e : splitAtPred p (S k) = ([], o, L₂) ⊢ ReflTransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif match List.head? (S k), true, p with \| some x, x_1, f => f x \| none, y, x => y then { l := some q, var := List.head? (S k), stk := update S k (List.tail (S k)) } else { l := some (Λ'.clear p k q), var := List.head? (S k), stk := update S k (List.tail (S k)) }) { l := some q, var := o, stk := update S k L₂ } revert e case nil p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e : splitAtPred p (S✝ k) = (L₁, o, L₂) s : Option Γ' S : K' → List Γ' ⊢ splitAtPred p (S k) = ([], o, L₂) → ReflTransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif match List.head? (S k), true, p with \| some x, x_1, f => f x \| none, y, x => y then { l := some q, var := List.head? (S k), stk := update S k (List.tail (S k)) } else { l := some (Λ'.clear p k q), var := List.head? (S k), stk := update S k (List.tail (S k)) }) { l := some q, var := o, stk := update S k L₂ } cases' S k with a Sk <;> intro e case nil.cons p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e✝ : splitAtPred p (S✝ k) = (L₁, o, L₂) s : Option Γ' S : K' → List Γ' a : Γ' Sk : List Γ' e : splitAtPred p (a :: Sk) = ([], o, L₂) ⊢ ReflTransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif match List.head? (a :: Sk), true, p with \| some x, x_1, f => f x \| none, y, x => y then { l := some q, var := List.head? (a :: Sk), stk := update S k (List.tail (a :: Sk)) } else { l := some (Λ'.clear p k q), var := List.head? (a :: Sk), stk := update S k (List.tail (a :: Sk)) }) { l := some q, var := o, stk := update S k L₂ } simp only [splitAtPred, Option.elim, List.head?, List.tail_cons] at e ⊢ case nil.cons p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e✝ : splitAtPred p (S✝ k) = (L₁, o, L₂) s : Option Γ' S : K' → List Γ' a : Γ' Sk : List Γ' e : (bif p a then ([], some a, Sk) else (a :: (splitAtPred p Sk).1, (splitAtPred p Sk).2.1, (splitAtPred p Sk).2.2)) = ([], o, L₂) ⊢ ReflTransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif p a then { l := some q, var := some a, stk := update S k Sk } else { l := some (Λ'.clear p k q), var := some a, stk := update S k Sk }) { l := some q, var := o, stk := update S k L₂ } revert e case nil.cons p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e : splitAtPred p (S✝ k) = (L₁, o, L₂) s : Option Γ' S : K' → List Γ' a : Γ' Sk : List Γ' ⊢ (bif p a then ([], some a, Sk) else (a :: (splitAtPred p Sk).1, (splitAtPred p Sk).2.1, (splitAtPred p Sk).2.2)) = ([], o, L₂) → ReflTransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif p a then { l := some q, var := some a, stk := update S k Sk } else { l := some (Λ'.clear p k q), var := some a, stk := update S k Sk }) { l := some q, var := o, stk := update S k L₂ } cases p a <;> intro e <;> simp only [cond_false, cond_true, Prod.mk.injEq, true_and, false_and] at e ⊢ case nil.nil p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e✝ : splitAtPred p (S✝ k) = (L₁, o, L₂) s : Option Γ' S : K' → List Γ' e : splitAtPred p [] = ([], o, L₂) ⊢ ReflTransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif match List.head? [], true, p with \| some x, x_1, f => f x \| none, y, x => y then { l := some q, var := List.head? [], stk := update S k (List.tail []) } else { l := some (Λ'.clear p k q), var := List.head? [], stk := update S k (List.tail []) }) { l := some q, var := o, stk := update S k L₂ } cases e case nil.nil.refl p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁ : List Γ' S✝ : K' → List Γ' s : Option Γ' S : K' → List Γ' e : splitAtPred p (S✝ k) = (L₁, none, []) ⊢ ReflTransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif match List.head? [], true, p with \| some x, x_1, f => f x \| none, y, x => y then { l := some q, var := List.head? [], stk := update S k (List.tail []) } else { l := some (Λ'.clear p k q), var := List.head? [], stk := update S k (List.tail []) }) { l := some q, var := none, stk := update S k [] } rfl case nil.cons.true p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e✝ : splitAtPred p (S✝ k) = (L₁, o, L₂) s : Option Γ' S : K' → List Γ' a : Γ' Sk : List Γ' e : some a = o ∧ Sk = L₂ ⊢ ReflTransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) { l := some q, var := some a, stk := update S k Sk } { l := some q, var := o, stk := update S k L₂ } rcases e with ⟨e₁, e₂⟩ case nil.cons.true.intro p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e : splitAtPred p (S✝ k) = (L₁, o, L₂) s : Option Γ' S : K' → List Γ' a : Γ' Sk : List Γ' e₁ : some a = o e₂ : Sk = L₂ ⊢ ReflTransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) { l := some q, var := some a, stk := update S k Sk } { l := some q, var := o, stk := update S k L₂ } rw [e₁, e₂] case cons p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁✝ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e✝ : splitAtPred p (S✝ k) = (L₁✝, o, L₂) a : Γ' L₁ : List Γ' IH : ∀ {s : Option Γ'} {S : K' → List Γ'}, splitAtPred p (S k) = (L₁, o, L₂) → Reaches₁ (TM2.step tr) { l := some (Λ'.clear p k q), var := s, stk := S } { l := some q, var := o, stk := update S k L₂ } s : Option Γ' S : K' → List Γ' e : splitAtPred p (S k) = (a :: L₁, o, L₂) ⊢ Reaches₁ (TM2.step tr) { l := some (Λ'.clear p k q), var := s, stk := S } { l := some q, var := o, stk := update S k L₂ } refine' TransGen.head rfl _ case cons p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁✝ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e✝ : splitAtPred p (S✝ k) = (L₁✝, o, L₂) a : Γ' L₁ : List Γ' IH : ∀ {s : Option Γ'} {S : K' → List Γ'}, splitAtPred p (S k) = (L₁, o, L₂) → Reaches₁ (TM2.step tr) { l := some (Λ'.clear p k q), var := s, stk := S } { l := some q, var := o, stk := update S k L₂ } s : Option Γ' S : K' → List Γ' e : splitAtPred p (S k) = (a :: L₁, o, L₂) ⊢ TransGen (fun a b => b ∈ TM2.step tr a) (TM2.stepAux (tr (Λ'.clear p k q)) s S) { l := some q, var := o, stk := update S k L₂ } simp only [TM2.step, Option.mem_def, TM2.stepAux, Option.elim] case cons p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁✝ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e✝ : splitAtPred p (S✝ k) = (L₁✝, o, L₂) a : Γ' L₁ : List Γ' IH : ∀ {s : Option Γ'} {S : K' → List Γ'}, splitAtPred p (S k) = (L₁, o, L₂) → Reaches₁ (TM2.step tr) { l := some (Λ'.clear p k q), var := s, stk := S } { l := some q, var := o, stk := update S k L₂ } s : Option Γ' S : K' → List Γ' e : splitAtPred p (S k) = (a :: L₁, o, L₂) ⊢ TransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif match List.head? (S k), true, p with \| some x, x_1, f => f x \| none, y, x => y then { l := some q, var := List.head? (S k), stk := update S k (List.tail (S k)) } else { l := some (Λ'.clear p k q), var := List.head? (S k), stk := update S k (List.tail (S k)) }) { l := some q, var := o, stk := update S k L₂ } cases' e₁ : S k with a' Sk <;> rw [e₁, splitAtPred] at e case cons.cons p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁✝ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e✝ : splitAtPred p (S✝ k) = (L₁✝, o, L₂) a : Γ' L₁ : List Γ' IH : ∀ {s : Option Γ'} {S : K' → List Γ'}, splitAtPred p (S k) = (L₁, o, L₂) → Reaches₁ (TM2.step tr) { l := some (Λ'.clear p k q), var := s, stk := S } { l := some q, var := o, stk := update S k L₂ } s : Option Γ' S : K' → List Γ' a' : Γ' Sk : List Γ' e : (bif p a' then ([], some a', Sk) else match splitAtPred p Sk with \| (l₁, o, l₂) => (a' :: l₁, o, l₂)) = (a :: L₁, o, L₂) e₁ : S k = a' :: Sk ⊢ TransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif match List.head? (a' :: Sk), true, p with \| some x, x_1, f => f x \| none, y, x => y then { l := some q, var := List.head? (a' :: Sk), stk := update S k (List.tail (a' :: Sk)) } else { l := some (Λ'.clear p k q), var := List.head? (a' :: Sk), stk := update S k (List.tail (a' :: Sk)) }) { l := some q, var := o, stk := update S k L₂ } cases e₂ : p a' <;> simp only [e₂, cond] at e case cons.cons.false p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁✝ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e✝ : splitAtPred p (S✝ k) = (L₁✝, o, L₂) a : Γ' L₁ : List Γ' IH : ∀ {s : Option Γ'} {S : K' → List Γ'}, splitAtPred p (S k) = (L₁, o, L₂) → Reaches₁ (TM2.step tr) { l := some (Λ'.clear p k q), var := s, stk := S } { l := some q, var := o, stk := update S k L₂ } s : Option Γ' S : K' → List Γ' a' : Γ' Sk : List Γ' e₁ : S k = a' :: Sk e₂ : p a' = false e : (a' :: (splitAtPred p Sk).1, (splitAtPred p Sk).2.1, (splitAtPred p Sk).2.2) = (a :: L₁, o, L₂) ⊢ TransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif match List.head? (a' :: Sk), true, p with \| some x, x_1, f => f x \| none, y, x => y then { l := some q, var := List.head? (a' :: Sk), stk := update S k (List.tail (a' :: Sk)) } else { l := some (Λ'.clear p k q), var := List.head? (a' :: Sk), stk := update S k (List.tail (a' :: Sk)) }) { l := some q, var := o, stk := update S k L₂ } case cons.cons.true p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁✝ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e✝ : splitAtPred p (S✝ k) = (L₁✝, o, L₂) a : Γ' L₁ : List Γ' IH : ∀ {s : Option Γ'} {S : K' → List Γ'}, splitAtPred p (S k) = (L₁, o, L₂) → Reaches₁ (TM2.step tr) { l := some (Λ'.clear p k q), var := s, stk := S } { l := some q, var := o, stk := update S k L₂ } s : Option Γ' S : K' → List Γ' a' : Γ' Sk : List Γ' e₁ : S k = a' :: Sk e₂ : p a' = true e : ([], some a', Sk) = (a :: L₁, o, L₂) ⊢ TransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif match List.head? (a' :: Sk), true, p with \| some x, x_1, f => f x \| none, y, x => y then { l := some q, var := List.head? (a' :: Sk), stk := update S k (List.tail (a' :: Sk)) } else { l := some (Λ'.clear p k q), var := List.head? (a' :: Sk), stk := update S k (List.tail (a' :: Sk)) }) { l := some q, var := o, stk := update S k L₂ } swap case cons.cons.false p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁✝ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e✝ : splitAtPred p (S✝ k) = (L₁✝, o, L₂) a : Γ' L₁ : List Γ' IH : ∀ {s : Option Γ'} {S : K' → List Γ'}, splitAtPred p (S k) = (L₁, o, L₂) → Reaches₁ (TM2.step tr) { l := some (Λ'.clear p k q), var := s, stk := S } { l := some q, var := o, stk := update S k L₂ } s : Option Γ' S : K' → List Γ' a' : Γ' Sk : List Γ' e₁ : S k = a' :: Sk e₂ : p a' = false e : (a' :: (splitAtPred p Sk).1, (splitAtPred p Sk).2.1, (splitAtPred p Sk).2.2) = (a :: L₁, o, L₂) ⊢ TransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif match List.head? (a' :: Sk), true, p with \| some x, x_1, f => f x \| none, y, x => y then { l := some q, var := List.head? (a' :: Sk), stk := update S k (List.tail (a' :: Sk)) } else { l := some (Λ'.clear p k q), var := List.head? (a' :: Sk), stk := update S k (List.tail (a' :: Sk)) }) { l := some q, var := o, stk := update S k L₂ } rcases e₃ : splitAtPred p Sk with ⟨_, _, _⟩ case cons.cons.false.mk.mk p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁✝ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e✝ : splitAtPred p (S✝ k) = (L₁✝, o, L₂) a : Γ' L₁ : List Γ' IH : ∀ {s : Option Γ'} {S : K' → List Γ'}, splitAtPred p (S k) = (L₁, o, L₂) → Reaches₁ (TM2.step tr) { l := some (Λ'.clear p k q), var := s, stk := S } { l := some q, var := o, stk := update S k L₂ } s : Option Γ' S : K' → List Γ' a' : Γ' Sk : List Γ' e₁ : S k = a' :: Sk e₂ : p a' = false e : (a' :: (splitAtPred p Sk).1, (splitAtPred p Sk).2.1, (splitAtPred p Sk).2.2) = (a :: L₁, o, L₂) fst✝¹ : List Γ' fst✝ : Option Γ' snd✝ : List Γ' e₃ : splitAtPred p Sk = (fst✝¹, fst✝, snd✝) ⊢ TransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif match List.head? (a' :: Sk), true, p with \| some x, x_1, f => f x \| none, y, x => y then { l := some q, var := List.head? (a' :: Sk), stk := update S k (List.tail (a' :: Sk)) } else { l := some (Λ'.clear p k q), var := List.head? (a' :: Sk), stk := update S k (List.tail (a' :: Sk)) }) { l := some q, var := o, stk := update S k L₂ } rw [e₃] at e case cons.cons.false.mk.mk p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁✝ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e✝ : splitAtPred p (S✝ k) = (L₁✝, o, L₂) a : Γ' L₁ : List Γ' IH : ∀ {s : Option Γ'} {S : K' → List Γ'}, splitAtPred p (S k) = (L₁, o, L₂) → Reaches₁ (TM2.step tr) { l := some (Λ'.clear p k q), var := s, stk := S } { l := some q, var := o, stk := update S k L₂ } s : Option Γ' S : K' → List Γ' a' : Γ' Sk : List Γ' e₁ : S k = a' :: Sk e₂ : p a' = false fst✝¹ : List Γ' fst✝ : Option Γ' snd✝ : List Γ' e : (a' :: (fst✝¹, fst✝, snd✝).1, (fst✝¹, fst✝, snd✝).2.1, (fst✝¹, fst✝, snd✝).2.2) = (a :: L₁, o, L₂) e₃ : splitAtPred p Sk = (fst✝¹, fst✝, snd✝) ⊢ TransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif match List.head? (a' :: Sk), true, p with \| some x, x_1, f => f x \| none, y, x => y then { l := some q, var := List.head? (a' :: Sk), stk := update S k (List.tail (a' :: Sk)) } else { l := some (Λ'.clear p k q), var := List.head? (a' :: Sk), stk := update S k (List.tail (a' :: Sk)) }) { l := some q, var := o, stk := update S k L₂ } cases e case cons.cons.false.mk.mk.refl p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁ : List Γ' S✝ : K' → List Γ' a : Γ' s : Option Γ' S : K' → List Γ' Sk fst✝¹ : List Γ' fst✝ : Option Γ' snd✝ : List Γ' e₃ : splitAtPred p Sk = (fst✝¹, fst✝, snd✝) e₁ : S k = a :: Sk e₂ : p a = false e : splitAtPred p (S✝ k) = (L₁, (fst✝¹, fst✝, snd✝).2.1, (fst✝¹, fst✝, snd✝).2.2) IH : ∀ {s : Option Γ'} {S : K' → List Γ'}, splitAtPred p (S k) = ((fst✝¹, fst✝, snd✝).1, (fst✝¹, fst✝, snd✝).2.1, (fst✝¹, fst✝, snd✝).2.2) → Reaches₁ (TM2.step tr) { l := some (Λ'.clear p k q), var := s, stk := S } { l := some q, var := (fst✝¹, fst✝, snd✝).2.1, stk := update S k (fst✝¹, fst✝, snd✝).2.2 } ⊢ TransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif match List.head? (a :: Sk), true, p with \| some x, x_1, f => f x \| none, y, x => y then { l := some q, var := List.head? (a :: Sk), stk := update S k (List.tail (a :: Sk)) } else { l := some (Λ'.clear p k q), var := List.head? (a :: Sk), stk := update S k (List.tail (a :: Sk)) }) { l := some q, var := (fst✝¹, fst✝, snd✝).2.1, stk := update S k (fst✝¹, fst✝, snd✝).2.2 } simp only [List.head?_cons, e₂, List.tail_cons, cond_false] case cons.cons.false.mk.mk.refl p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁ : List Γ' S✝ : K' → List Γ' a : Γ' s : Option Γ' S : K' → List Γ' Sk fst✝¹ : List Γ' fst✝ : Option Γ' snd✝ : List Γ' e₃ : splitAtPred p Sk = (fst✝¹, fst✝, snd✝) e₁ : S k = a :: Sk e₂ : p a = false e : splitAtPred p (S✝ k) = (L₁, (fst✝¹, fst✝, snd✝).2.1, (fst✝¹, fst✝, snd✝).2.2) IH : ∀ {s : Option Γ'} {S : K' → List Γ'}, splitAtPred p (S k) = ((fst✝¹, fst✝, snd✝).1, (fst✝¹, fst✝, snd✝).2.1, (fst✝¹, fst✝, snd✝).2.2) → Reaches₁ (TM2.step tr) { l := some (Λ'.clear p k q), var := s, stk := S } { l := some q, var := (fst✝¹, fst✝, snd✝).2.1, stk := update S k (fst✝¹, fst✝, snd✝).2.2 } ⊢ TransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) { l := some (Λ'.clear p k q), var := some a, stk := update S k Sk } { l := some q, var := fst✝, stk := update S k snd✝ } convert @IH _ (update S k Sk) _ using 2 <;> simp [e₃] case cons.nil p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁✝ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e✝ : splitAtPred p (S✝ k) = (L₁✝, o, L₂) a : Γ' L₁ : List Γ' IH : ∀ {s : Option Γ'} {S : K' → List Γ'}, splitAtPred p (S k) = (L₁, o, L₂) → Reaches₁ (TM2.step tr) { l := some (Λ'.clear p k q), var := s, stk := S } { l := some q, var := o, stk := update S k L₂ } s : Option Γ' S : K' → List Γ' e : ([], none, []) = (a :: L₁, o, L₂) e₁ : S k = [] ⊢ TransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif match List.head? [], true, p with \| some x, x_1, f => f x \| none, y, x => y then { l := some q, var := List.head? [], stk := update S k (List.tail []) } else { l := some (Λ'.clear p k q), var := List.head? [], stk := update S k (List.tail []) }) { l := some q, var := o, stk := update S k L₂ } cases e case cons.cons.true p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁✝ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e✝ : splitAtPred p (S✝ k) = (L₁✝, o, L₂) a : Γ' L₁ : List Γ' IH : ∀ {s : Option Γ'} {S : K' → List Γ'}, splitAtPred p (S k) = (L₁, o, L₂) → Reaches₁ (TM2.step tr) { l := some (Λ'.clear p k q), var := s, stk := S } { l := some q, var := o, stk := update S k L₂ } s : Option Γ' S : K' → List Γ' a' : Γ' Sk : List Γ' e₁ : S k = a' :: Sk e₂ : p a' = true e : ([], some a', Sk) = (a :: L₁, o, L₂) ⊢ TransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif match List.head? (a' :: Sk), true, p with \| some x, x_1, f => f x \| none, y, x => y then { l := some q, var := List.head? (a' :: Sk), stk := update S k (List.tail (a' :: Sk)) } else { l := some (Λ'.clear p k q), var := List.head? (a' :: Sk), stk := update S k (List.tail (a' :: Sk)) }) { l := some q, var := o, stk := update S k L₂ } cases e ** Qed

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StieltjesFunction.measure_singleton f : StieltjesFunction a : ℝ ⊢ ↑↑(StieltjesFunction.measure f) {a} = ofReal (↑f a - leftLim (↑f) a) obtain ⟨u, u_mono, u_lt_a, u_lim⟩ : ∃ u : ℕ → ℝ, StrictMono u ∧ (∀ n : ℕ, u n < a) ∧ Tendsto u atTop (𝓝 a) := exists_seq_strictMono_tendsto a case intro.intro.intro f : StieltjesFunction a : ℝ u : ℕ → ℝ u_mono : StrictMono u u_lt_a : ∀ (n : ℕ), u n < a u_lim : Tendsto u atTop (𝓝 a) ⊢ ↑↑(StieltjesFunction.measure f) {a} = ofReal (↑f a - leftLim (↑f) a) have A : {a} = ⋂ n, Ioc (u n) a := by refine' Subset.antisymm (fun x hx => by simp [mem_singleton_iff.1 hx, u_lt_a]) fun x hx => _ simp at hx have : a ≤ x := le_of_tendsto' u_lim fun n => (hx n).1.le simp [le_antisymm this (hx 0).2] case intro.intro.intro f : StieltjesFunction a : ℝ u : ℕ → ℝ u_mono : StrictMono u u_lt_a : ∀ (n : ℕ), u n < a u_lim : Tendsto u atTop (𝓝 a) A : {a} = ⋂ n, Ioc (u n) a L1 : Tendsto (fun n => ↑↑(StieltjesFunction.measure f) (Ioc (u n) a)) atTop (𝓝 (↑↑(StieltjesFunction.measure f) {a})) ⊢ ↑↑(StieltjesFunction.measure f) {a} = ofReal (↑f a - leftLim (↑f) a) have L2 : Tendsto (fun n => f.measure (Ioc (u n) a)) atTop (𝓝 (ofReal (f a - leftLim f a))) := by simp only [measure_Ioc] have : Tendsto (fun n => f (u n)) atTop (𝓝 (leftLim f a)) := by apply (f.mono.tendsto_leftLim a).comp exact tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _ u_lim (eventually_of_forall fun n => u_lt_a n) exact ENNReal.continuous_ofReal.continuousAt.tendsto.comp (tendsto_const_nhds.sub this) case intro.intro.intro f : StieltjesFunction a : ℝ u : ℕ → ℝ u_mono : StrictMono u u_lt_a : ∀ (n : ℕ), u n < a u_lim : Tendsto u atTop (𝓝 a) A : {a} = ⋂ n, Ioc (u n) a L1 : Tendsto (fun n => ↑↑(StieltjesFunction.measure f) (Ioc (u n) a)) atTop (𝓝 (↑↑(StieltjesFunction.measure f) {a})) L2 : Tendsto (fun n => ↑↑(StieltjesFunction.measure f) (Ioc (u n) a)) atTop (𝓝 (ofReal (↑f a - leftLim (↑f) a))) ⊢ ↑↑(StieltjesFunction.measure f) {a} = ofReal (↑f a - leftLim (↑f) a) exact tendsto_nhds_unique L1 L2 f : StieltjesFunction a : ℝ u : ℕ → ℝ u_mono : StrictMono u u_lt_a : ∀ (n : ℕ), u n < a u_lim : Tendsto u atTop (𝓝 a) ⊢ {a} = ⋂ n, Ioc (u n) a refine' Subset.antisymm (fun x hx => by simp [mem_singleton_iff.1 hx, u_lt_a]) fun x hx => _ f : StieltjesFunction a : ℝ u : ℕ → ℝ u_mono : StrictMono u u_lt_a : ∀ (n : ℕ), u n < a u_lim : Tendsto u atTop (𝓝 a) x : ℝ hx : x ∈ ⋂ n, Ioc (u n) a ⊢ x ∈ {a} simp at hx f : StieltjesFunction a : ℝ u : ℕ → ℝ u_mono : StrictMono u u_lt_a : ∀ (n : ℕ), u n < a u_lim : Tendsto u atTop (𝓝 a) x : ℝ hx : ∀ (i : ℕ), u i < x ∧ x ≤ a ⊢ x ∈ {a} have : a ≤ x := le_of_tendsto' u_lim fun n => (hx n).1.le f : StieltjesFunction a : ℝ u : ℕ → ℝ u_mono : StrictMono u u_lt_a : ∀ (n : ℕ), u n < a u_lim : Tendsto u atTop (𝓝 a) x : ℝ hx : ∀ (i : ℕ), u i < x ∧ x ≤ a this : a ≤ x ⊢ x ∈ {a} simp [le_antisymm this (hx 0).2] f : StieltjesFunction a : ℝ u : ℕ → ℝ u_mono : StrictMono u u_lt_a : ∀ (n : ℕ), u n < a u_lim : Tendsto u atTop (𝓝 a) x : ℝ hx : x ∈ {a} ⊢ x ∈ ⋂ n, Ioc (u n) a simp [mem_singleton_iff.1 hx, u_lt_a] f : StieltjesFunction a : ℝ u : ℕ → ℝ u_mono : StrictMono u u_lt_a : ∀ (n : ℕ), u n < a u_lim : Tendsto u atTop (𝓝 a) A : {a} = ⋂ n, Ioc (u n) a ⊢ Tendsto (fun n => ↑↑(StieltjesFunction.measure f) (Ioc (u n) a)) atTop (𝓝 (↑↑(StieltjesFunction.measure f) {a})) rw [A] f : StieltjesFunction a : ℝ u : ℕ → ℝ u_mono : StrictMono u u_lt_a : ∀ (n : ℕ), u n < a u_lim : Tendsto u atTop (𝓝 a) A : {a} = ⋂ n, Ioc (u n) a ⊢ Tendsto (fun n => ↑↑(StieltjesFunction.measure f) (Ioc (u n) a)) atTop (𝓝 (↑↑(StieltjesFunction.measure f) (⋂ n, Ioc (u n) a))) refine' tendsto_measure_iInter (fun n => measurableSet_Ioc) (fun m n hmn => _) _ case refine'_1 f : StieltjesFunction a : ℝ u : ℕ → ℝ u_mono : StrictMono u u_lt_a : ∀ (n : ℕ), u n < a u_lim : Tendsto u atTop (𝓝 a) A : {a} = ⋂ n, Ioc (u n) a m n : ℕ hmn : m ≤ n ⊢ Ioc (u n) a ≤ Ioc (u m) a exact Ioc_subset_Ioc (u_mono.monotone hmn) le_rfl case refine'_2 f : StieltjesFunction a : ℝ u : ℕ → ℝ u_mono : StrictMono u u_lt_a : ∀ (n : ℕ), u n < a u_lim : Tendsto u atTop (𝓝 a) A : {a} = ⋂ n, Ioc (u n) a ⊢ ∃ i, ↑↑(StieltjesFunction.measure f) (Ioc (u i) a) ≠ ⊤ exact ⟨0, by simpa only [measure_Ioc] using ENNReal.ofReal_ne_top⟩ f : StieltjesFunction a : ℝ u : ℕ → ℝ u_mono : StrictMono u u_lt_a : ∀ (n : ℕ), u n < a u_lim : Tendsto u atTop (𝓝 a) A : {a} = ⋂ n, Ioc (u n) a ⊢ ↑↑(StieltjesFunction.measure f) (Ioc (u 0) a) ≠ ⊤ simpa only [measure_Ioc] using ENNReal.ofReal_ne_top f : StieltjesFunction a : ℝ u : ℕ → ℝ u_mono : StrictMono u u_lt_a : ∀ (n : ℕ), u n < a u_lim : Tendsto u atTop (𝓝 a) A : {a} = ⋂ n, Ioc (u n) a L1 : Tendsto (fun n => ↑↑(StieltjesFunction.measure f) (Ioc (u n) a)) atTop (𝓝 (↑↑(StieltjesFunction.measure f) {a})) ⊢ Tendsto (fun n => ↑↑(StieltjesFunction.measure f) (Ioc (u n) a)) atTop (𝓝 (ofReal (↑f a - leftLim (↑f) a))) simp only [measure_Ioc] f : StieltjesFunction a : ℝ u : ℕ → ℝ u_mono : StrictMono u u_lt_a : ∀ (n : ℕ), u n < a u_lim : Tendsto u atTop (𝓝 a) A : {a} = ⋂ n, Ioc (u n) a L1 : Tendsto (fun n => ↑↑(StieltjesFunction.measure f) (Ioc (u n) a)) atTop (𝓝 (↑↑(StieltjesFunction.measure f) {a})) ⊢ Tendsto (fun n => ofReal (↑f a - ↑f (u n))) atTop (𝓝 (ofReal (↑f a - leftLim (↑f) a))) have : Tendsto (fun n => f (u n)) atTop (𝓝 (leftLim f a)) := by apply (f.mono.tendsto_leftLim a).comp exact tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _ u_lim (eventually_of_forall fun n => u_lt_a n) f : StieltjesFunction a : ℝ u : ℕ → ℝ u_mono : StrictMono u u_lt_a : ∀ (n : ℕ), u n < a u_lim : Tendsto u atTop (𝓝 a) A : {a} = ⋂ n, Ioc (u n) a L1 : Tendsto (fun n => ↑↑(StieltjesFunction.measure f) (Ioc (u n) a)) atTop (𝓝 (↑↑(StieltjesFunction.measure f) {a})) this : Tendsto (fun n => ↑f (u n)) atTop (𝓝 (leftLim (↑f) a)) ⊢ Tendsto (fun n => ofReal (↑f a - ↑f (u n))) atTop (𝓝 (ofReal (↑f a - leftLim (↑f) a))) exact ENNReal.continuous_ofReal.continuousAt.tendsto.comp (tendsto_const_nhds.sub this) f : StieltjesFunction a : ℝ u : ℕ → ℝ u_mono : StrictMono u u_lt_a : ∀ (n : ℕ), u n < a u_lim : Tendsto u atTop (𝓝 a) A : {a} = ⋂ n, Ioc (u n) a L1 : Tendsto (fun n => ↑↑(StieltjesFunction.measure f) (Ioc (u n) a)) atTop (𝓝 (↑↑(StieltjesFunction.measure f) {a})) ⊢ Tendsto (fun n => ↑f (u n)) atTop (𝓝 (leftLim (↑f) a)) apply (f.mono.tendsto_leftLim a).comp f : StieltjesFunction a : ℝ u : ℕ → ℝ u_mono : StrictMono u u_lt_a : ∀ (n : ℕ), u n < a u_lim : Tendsto u atTop (𝓝 a) A : {a} = ⋂ n, Ioc (u n) a L1 : Tendsto (fun n => ↑↑(StieltjesFunction.measure f) (Ioc (u n) a)) atTop (𝓝 (↑↑(StieltjesFunction.measure f) {a})) ⊢ Tendsto (fun n => u n) atTop (𝓝[Iio a] a) exact tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _ u_lim (eventually_of_forall fun n => u_lt_a n) ** Qed
StieltjesFunction.measure_Ico f : StieltjesFunction a b : ℝ ⊢ ↑↑(StieltjesFunction.measure f) (Ico a b) = ofReal (leftLim (↑f) b - leftLim (↑f) a) rcases le_or_lt b a with (hab \| hab) case inl f : StieltjesFunction a b : ℝ hab : b ≤ a ⊢ ↑↑(StieltjesFunction.measure f) (Ico a b) = ofReal (leftLim (↑f) b - leftLim (↑f) a) simp only [hab, measure_empty, Ico_eq_empty, not_lt] case inl f : StieltjesFunction a b : ℝ hab : b ≤ a ⊢ 0 = ofReal (leftLim (↑f) b - leftLim (↑f) a) symm case inl f : StieltjesFunction a b : ℝ hab : b ≤ a ⊢ ofReal (leftLim (↑f) b - leftLim (↑f) a) = 0 simp [ENNReal.ofReal_eq_zero, f.mono.leftLim hab] case inr f : StieltjesFunction a b : ℝ hab : a < b ⊢ ↑↑(StieltjesFunction.measure f) (Ico a b) = ofReal (leftLim (↑f) b - leftLim (↑f) a) have A : Disjoint {a} (Ioo a b) := by simp case inr f : StieltjesFunction a b : ℝ hab : a < b A : Disjoint {a} (Ioo a b) ⊢ ↑↑(StieltjesFunction.measure f) (Ico a b) = ofReal (leftLim (↑f) b - leftLim (↑f) a) simp [← Icc_union_Ioo_eq_Ico le_rfl hab, -singleton_union, hab.ne, f.mono.leftLim_le, measure_union A measurableSet_Ioo, f.mono.le_leftLim hab, ← ENNReal.ofReal_add] f : StieltjesFunction a b : ℝ hab : a < b ⊢ Disjoint {a} (Ioo a b) simp ** Qed
StieltjesFunction.measure_univ f : StieltjesFunction l u : ℝ hfl : Tendsto (↑f) atBot (𝓝 l) hfu : Tendsto (↑f) atTop (𝓝 u) ⊢ ↑↑(StieltjesFunction.measure f) univ = ofReal (u - l) refine' tendsto_nhds_unique (tendsto_measure_Iic_atTop _) _ f : StieltjesFunction l u : ℝ hfl : Tendsto (↑f) atBot (𝓝 l) hfu : Tendsto (↑f) atTop (𝓝 u) ⊢ Tendsto (fun x => ↑↑(StieltjesFunction.measure f) (Iic x)) atTop (𝓝 (ofReal (u - l))) simp_rw [measure_Iic f hfl] f : StieltjesFunction l u : ℝ hfl : Tendsto (↑f) atBot (𝓝 l) hfu : Tendsto (↑f) atTop (𝓝 u) ⊢ Tendsto (fun x => ofReal (↑f x - l)) atTop (𝓝 (ofReal (u - l))) exact ENNReal.tendsto_ofReal (Tendsto.sub_const hfu _) ** Qed
MeasureTheory.NullMeasurableSet.sUnion ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 m0 : MeasurableSpace α μ : Measure α s✝ t : Set α s : Set (Set α) hs : Set.Countable s h : ∀ (t : Set α), t ∈ s → NullMeasurableSet t ⊢ NullMeasurableSet (⋃₀ s) rw [sUnion_eq_biUnion] ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 m0 : MeasurableSpace α μ : Measure α s✝ t : Set α s : Set (Set α) hs : Set.Countable s h : ∀ (t : Set α), t ∈ s → NullMeasurableSet t ⊢ NullMeasurableSet (⋃ i ∈ s, i) exact MeasurableSet.biUnion hs h ** Qed
MeasureTheory.NullMeasurableSet.exists_measurable_superset_ae_eq ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 m0 : MeasurableSpace α μ : Measure α s t : Set α h : NullMeasurableSet s ⊢ ∃ t, t ⊇ s ∧ MeasurableSet t ∧ t =ᵐ[μ] s rcases h with ⟨t, htm, hst⟩ case intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 m0 : MeasurableSpace α μ : Measure α s t✝ t : Set α htm : MeasurableSet t hst : s =ᵐ[μ] t ⊢ ∃ t, t ⊇ s ∧ MeasurableSet t ∧ t =ᵐ[μ] s refine' ⟨t ∪ toMeasurable μ (s \ t), _, htm.union (measurableSet_toMeasurable _ _), _⟩ case intro.intro.refine'_1 ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 m0 : MeasurableSpace α μ : Measure α s t✝ t : Set α htm : MeasurableSet t hst : s =ᵐ[μ] t ⊢ t ∪ toMeasurable μ (s \ t) ⊇ s exact diff_subset_iff.1 (subset_toMeasurable _ _) case intro.intro.refine'_2 ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 m0 : MeasurableSpace α μ : Measure α s t✝ t : Set α htm : MeasurableSet t hst : s =ᵐ[μ] t ⊢ t ∪ toMeasurable μ (s \ t) =ᵐ[μ] s have : toMeasurable μ (s \ t) =ᵐ[μ] (∅ : Set α) := by simp [ae_le_set.1 hst.le] case intro.intro.refine'_2 ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 m0 : MeasurableSpace α μ : Measure α s t✝ t : Set α htm : MeasurableSet t hst : s =ᵐ[μ] t this : toMeasurable μ (s \ t) =ᵐ[μ] ∅ ⊢ t ∪ toMeasurable μ (s \ t) =ᵐ[μ] s simpa only [union_empty] using hst.symm.union this ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 m0 : MeasurableSpace α μ : Measure α s t✝ t : Set α htm : MeasurableSet t hst : s =ᵐ[μ] t ⊢ toMeasurable μ (s \ t) =ᵐ[μ] ∅ simp [ae_le_set.1 hst.le] ** Qed
MeasureTheory.measure_iUnion ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 m0✝ : MeasurableSpace α μ✝ : Measure α s t : Set α m0 : MeasurableSpace α μ : Measure α inst✝ : Countable ι f : ι → Set α hn : Pairwise (Disjoint on f) h : ∀ (i : ι), MeasurableSet (f i) ⊢ ↑↑μ (⋃ i, f i) = ∑' (i : ι), ↑↑μ (f i) rw [measure_eq_extend (MeasurableSet.iUnion h), extend_iUnion MeasurableSet.empty _ MeasurableSet.iUnion _ hn h] ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 m0✝ : MeasurableSpace α μ✝ : Measure α s t : Set α m0 : MeasurableSpace α μ : Measure α inst✝ : Countable ι f : ι → Set α hn : Pairwise (Disjoint on f) h : ∀ (i : ι), MeasurableSet (f i) ⊢ ∑' (i : ι), extend (fun t _ht => ↑↑μ t) (f i) = ∑' (i : ι), ↑↑μ (f i) simp [measure_eq_extend, h] ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 m0✝ : MeasurableSpace α μ✝ : Measure α s t : Set α m0 : MeasurableSpace α μ : Measure α inst✝ : Countable ι f : ι → Set α hn : Pairwise (Disjoint on f) h : ∀ (i : ι), MeasurableSet (f i) ⊢ ↑↑μ ∅ = 0 exact μ.empty ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 m0✝ : MeasurableSpace α μ✝ : Measure α s t : Set α m0 : MeasurableSpace α μ : Measure α inst✝ : Countable ι f : ι → Set α hn : Pairwise (Disjoint on f) h : ∀ (i : ι), MeasurableSet (f i) ⊢ ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), MeasurableSet (f i)) → Pairwise (Disjoint on f) → ↑↑μ (⋃ i, f i) = ∑' (i : ℕ), ↑↑μ (f i) exact μ.m_iUnion ** Qed
MeasureTheory.measure_union_add_inter₀ ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 m0 : MeasurableSpace α μ : Measure α s✝ t s : Set α ht : NullMeasurableSet t ⊢ ↑↑μ (s ∪ t) + ↑↑μ (s ∩ t) = ↑↑μ s + ↑↑μ t rw [← measure_inter_add_diff₀ (s ∪ t) ht, union_inter_cancel_right, union_diff_right, ← measure_inter_add_diff₀ s ht, add_comm, ← add_assoc, add_right_comm] ** Qed
MeasureTheory.measure_union_add_inter₀' ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 m0 : MeasurableSpace α μ : Measure α s t✝ : Set α hs : NullMeasurableSet s t : Set α ⊢ ↑↑μ (s ∪ t) + ↑↑μ (s ∩ t) = ↑↑μ s + ↑↑μ t rw [union_comm, inter_comm, measure_union_add_inter₀ t hs, add_comm] ** Qed
MeasureTheory.measure_union₀ ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 m0 : MeasurableSpace α μ : Measure α s t : Set α ht : NullMeasurableSet t hd : AEDisjoint μ s t ⊢ ↑↑μ (s ∪ t) = ↑↑μ s + ↑↑μ t rw [← measure_union_add_inter₀ s ht, hd, add_zero] ** Qed
MeasureTheory.measure_union₀' ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 m0 : MeasurableSpace α μ : Measure α s t : Set α hs : NullMeasurableSet s hd : AEDisjoint μ s t ⊢ ↑↑μ (s ∪ t) = ↑↑μ s + ↑↑μ t rw [union_comm, measure_union₀ hs (AEDisjoint.symm hd), add_comm] ** Qed
MeasureTheory.measure_add_measure_compl₀ ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 m0 : MeasurableSpace α μ : Measure α s✝ t s : Set α hs : NullMeasurableSet s ⊢ ↑↑μ s + ↑↑μ sᶜ = ↑↑μ univ rw [← measure_union₀' hs aedisjoint_compl_right, union_compl_self] ** Qed
MeasureTheory.trim_eq_self α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α ⊢ Measure.trim μ (_ : inst✝ ≤ inst✝) = μ simp [Measure.trim] ** Qed
MeasureTheory.toOuterMeasure_trim_eq_trim_toOuterMeasure α : Type u_1 m m0 : MeasurableSpace α μ✝ : Measure α s : Set α μ : Measure α hm : m ≤ m0 ⊢ ↑(Measure.trim μ hm) = OuterMeasure.trim ↑μ rw [Measure.trim, toMeasure_toOuterMeasure (ms := m)] ** Qed
MeasureTheory.trim_measurableSet_eq α : Type u_1 m m0 : MeasurableSpace α μ : Measure α s : Set α hm : m ≤ m0 hs : MeasurableSet s ⊢ ↑↑(Measure.trim μ hm) s = ↑↑μ s rw [Measure.trim, toMeasure_apply (ms := m) _ _ hs] ** Qed
MeasureTheory.le_trim α : Type u_1 m m0 : MeasurableSpace α μ : Measure α s : Set α hm : m ≤ m0 ⊢ ↑↑μ s ≤ ↑↑(Measure.trim μ hm) s simp_rw [Measure.trim] α : Type u_1 m m0 : MeasurableSpace α μ : Measure α s : Set α hm : m ≤ m0 ⊢ ↑↑μ s ≤ ↑↑(OuterMeasure.toMeasure ↑μ (_ : m ≤ OuterMeasure.caratheodory ↑μ)) s exact @le_toMeasure_apply _ m _ _ _ ** Qed
MeasureTheory.measure_trim_toMeasurable_eq_zero α : Type u_1 m m0 : MeasurableSpace α μ : Measure α s : Set α hm : m ≤ m0 hs : ↑↑(Measure.trim μ hm) s = 0 ⊢ ↑↑(Measure.trim μ hm) (toMeasurable (Measure.trim μ hm) s) = 0 rwa [@measure_toMeasurable _ m] ** Qed
MeasureTheory.trim_trim α : Type u_1 m m0 : MeasurableSpace α μ : Measure α s : Set α m₁ m₂ : MeasurableSpace α hm₁₂ : m₁ ≤ m₂ hm₂ : m₂ ≤ m0 ⊢ Measure.trim (Measure.trim μ hm₂) hm₁₂ = Measure.trim μ (_ : m₁ ≤ m0) refine @Measure.ext _ m₁ _ _ (fun t ht => ?_) α : Type u_1 m m0 : MeasurableSpace α μ : Measure α s : Set α m₁ m₂ : MeasurableSpace α hm₁₂ : m₁ ≤ m₂ hm₂ : m₂ ≤ m0 t : Set α ht : MeasurableSet t ⊢ ↑↑(Measure.trim (Measure.trim μ hm₂) hm₁₂) t = ↑↑(Measure.trim μ (_ : m₁ ≤ m0)) t rw [trim_measurableSet_eq hm₁₂ ht, trim_measurableSet_eq (hm₁₂.trans hm₂) ht, trim_measurableSet_eq hm₂ (hm₁₂ t ht)] ** Qed
MeasureTheory.restrict_trim α : Type u_1 m m0 : MeasurableSpace α μ✝ : Measure α s : Set α hm : m ≤ m0 μ : Measure α hs : MeasurableSet s ⊢ Measure.restrict (Measure.trim μ hm) s = Measure.trim (Measure.restrict μ s) hm refine @Measure.ext _ m _ _ (fun t ht => ?_) α : Type u_1 m m0 : MeasurableSpace α μ✝ : Measure α s : Set α hm : m ≤ m0 μ : Measure α hs : MeasurableSet s t : Set α ht : MeasurableSet t ⊢ ↑↑(Measure.restrict (Measure.trim μ hm) s) t = ↑↑(Measure.trim (Measure.restrict μ s) hm) t rw [@Measure.restrict_apply α m _ _ _ ht, trim_measurableSet_eq hm ht, Measure.restrict_apply (hm t ht), trim_measurableSet_eq hm (@MeasurableSet.inter α m t s ht hs)] ** Qed
MeasureTheory.exists_null_pairwise_disjoint_diff ι : Type u_1 α : Type u_2 m : MeasurableSpace α μ : Measure α s✝ t u v : Set α inst✝ : Countable ι s : ι → Set α hd : Pairwise (AEDisjoint μ on s) ⊢ ∃ t, (∀ (i : ι), MeasurableSet (t i)) ∧ (∀ (i : ι), ↑↑μ (t i) = 0) ∧ Pairwise (Disjoint on fun i => s i \ t i) refine' ⟨fun i => toMeasurable μ (s i ∩ ⋃ j ∈ ({i}ᶜ : Set ι), s j), fun i => measurableSet_toMeasurable _ _, fun i => _, _⟩ case refine'_1 ι : Type u_1 α : Type u_2 m : MeasurableSpace α μ : Measure α s✝ t u v : Set α inst✝ : Countable ι s : ι → Set α hd : Pairwise (AEDisjoint μ on s) i : ι ⊢ ↑↑μ ((fun i => toMeasurable μ (s i ∩ ⋃ j ∈ {i}ᶜ, s j)) i) = 0 simp only [measure_toMeasurable, inter_iUnion] case refine'_1 ι : Type u_1 α : Type u_2 m : MeasurableSpace α μ : Measure α s✝ t u v : Set α inst✝ : Countable ι s : ι → Set α hd : Pairwise (AEDisjoint μ on s) i : ι ⊢ ↑↑μ (⋃ i_1 ∈ {i}ᶜ, s i ∩ s i_1) = 0 exact (measure_biUnion_null_iff <\| to_countable _).2 fun j hj => hd (Ne.symm hj) case refine'_2 ι : Type u_1 α : Type u_2 m : MeasurableSpace α μ : Measure α s✝ t u v : Set α inst✝ : Countable ι s : ι → Set α hd : Pairwise (AEDisjoint μ on s) ⊢ Pairwise (Disjoint on fun i => s i \ (fun i => toMeasurable μ (s i ∩ ⋃ j ∈ {i}ᶜ, s j)) i) simp only [Pairwise, disjoint_left, onFun, mem_diff, not_and, and_imp, Classical.not_not] case refine'_2 ι : Type u_1 α : Type u_2 m : MeasurableSpace α μ : Measure α s✝ t u v : Set α inst✝ : Countable ι s : ι → Set α hd : Pairwise (AEDisjoint μ on s) ⊢ ∀ ⦃i j : ι⦄, i ≠ j → ∀ ⦃a : α⦄, a ∈ s i → ¬a ∈ toMeasurable μ (s i ∩ ⋃ j ∈ {i}ᶜ, s j) → a ∈ s j → a ∈ toMeasurable μ (s j ∩ ⋃ j_1 ∈ {j}ᶜ, s j_1) intro i j hne x hi hU hj case refine'_2 ι : Type u_1 α : Type u_2 m : MeasurableSpace α μ : Measure α s✝ t u v : Set α inst✝ : Countable ι s : ι → Set α hd : Pairwise (AEDisjoint μ on s) i j : ι hne : i ≠ j x : α hi : x ∈ s i hU : ¬x ∈ toMeasurable μ (s i ∩ ⋃ j ∈ {i}ᶜ, s j) hj : x ∈ s j ⊢ x ∈ toMeasurable μ (s j ∩ ⋃ j_1 ∈ {j}ᶜ, s j_1) replace hU : x ∉ s i ∩ iUnion λ j => iUnion λ _ => s j := λ h => hU (subset_toMeasurable _ _ h) case refine'_2 ι : Type u_1 α : Type u_2 m : MeasurableSpace α μ : Measure α s✝ t u v : Set α inst✝ : Countable ι s : ι → Set α hd : Pairwise (AEDisjoint μ on s) i j : ι hne : i ≠ j x : α hi : x ∈ s i hj : x ∈ s j hU : ¬x ∈ s i ∩ ⋃ j ∈ {i}ᶜ, s j ⊢ x ∈ toMeasurable μ (s j ∩ ⋃ j_1 ∈ {j}ᶜ, s j_1) simp only [mem_inter_iff, mem_iUnion, not_and, not_exists] at hU case refine'_2 ι : Type u_1 α : Type u_2 m : MeasurableSpace α μ : Measure α s✝ t u v : Set α inst✝ : Countable ι s : ι → Set α hd : Pairwise (AEDisjoint μ on s) i j : ι hne : i ≠ j x : α hi : x ∈ s i hj : x ∈ s j hU : x ∈ s i → ∀ (x_1 : ι), x_1 ∈ {i}ᶜ → ¬x ∈ s x_1 ⊢ x ∈ toMeasurable μ (s j ∩ ⋃ j_1 ∈ {j}ᶜ, s j_1) exact (hU hi j hne.symm hj).elim ** Qed
MeasureTheory.AEDisjoint.symm ι : Type u_1 α : Type u_2 m : MeasurableSpace α μ : Measure α s t u v : Set α h : AEDisjoint μ s t ⊢ AEDisjoint μ t s rwa [AEDisjoint, inter_comm] ** Qed
Disjoint.aedisjoint ι : Type u_1 α : Type u_2 m : MeasurableSpace α μ : Measure α s t u v : Set α h : Disjoint s t ⊢ AEDisjoint μ s t rw [AEDisjoint, disjoint_iff_inter_eq_empty.1 h, measure_empty] ** Qed
MeasureTheory.AEDisjoint.iUnion_left_iff ι : Type u_1 α : Type u_2 m : MeasurableSpace α μ : Measure α s✝ t u v : Set α inst✝ : Countable ι s : ι → Set α ⊢ AEDisjoint μ (⋃ i, s i) t ↔ ∀ (i : ι), AEDisjoint μ (s i) t simp only [AEDisjoint, iUnion_inter, measure_iUnion_null_iff] ** Qed
MeasureTheory.AEDisjoint.iUnion_right_iff ι : Type u_1 α : Type u_2 m : MeasurableSpace α μ : Measure α s t✝ u v : Set α inst✝ : Countable ι t : ι → Set α ⊢ AEDisjoint μ s (⋃ i, t i) ↔ ∀ (i : ι), AEDisjoint μ s (t i) simp only [AEDisjoint, inter_iUnion, measure_iUnion_null_iff] ** Qed
MeasureTheory.AEDisjoint.union_left_iff ι : Type u_1 α : Type u_2 m : MeasurableSpace α μ : Measure α s t u v : Set α ⊢ AEDisjoint μ (s ∪ t) u ↔ AEDisjoint μ s u ∧ AEDisjoint μ t u simp [union_eq_iUnion, and_comm] ** Qed
MeasureTheory.AEDisjoint.union_right_iff ι : Type u_1 α : Type u_2 m : MeasurableSpace α μ : Measure α s t u v : Set α ⊢ AEDisjoint μ s (t ∪ u) ↔ AEDisjoint μ s t ∧ AEDisjoint μ s u simp [union_eq_iUnion, and_comm] ** Qed
MeasureTheory.AEDisjoint.exists_disjoint_diff ι : Type u_1 α : Type u_2 m : MeasurableSpace α μ : Measure α s t u v : Set α h : AEDisjoint μ s t x : α hx : x ∈ s \ toMeasurable μ (s ∩ t) ⊢ x ∈ ?m.9910 h \ t simp ι : Type u_1 α : Type u_2 m : MeasurableSpace α μ : Measure α s t u v : Set α h : AEDisjoint μ s t x : α hx : x ∈ s \ toMeasurable μ (s ∩ t) ⊢ x ∈ ?m.9910 h ∧ ¬x ∈ t exact ⟨hx.1, fun hxt => hx.2 <\| subset_toMeasurable _ _ ⟨hx.1, hxt⟩⟩ ** Qed
LinearMap.exists_map_addHaar_eq_smul_addHaar 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : CompleteSpace 𝕜 inst✝¹⁰ : NormedAddCommGroup E inst✝⁹ : MeasurableSpace E inst✝⁸ : BorelSpace E inst✝⁷ : NormedSpace 𝕜 E inst✝⁶ : NormedAddCommGroup F inst✝⁵ : MeasurableSpace F inst✝⁴ : BorelSpace F inst✝³ : NormedSpace 𝕜 F L : E →ₗ[𝕜] F μ : Measure E ν : Measure F inst✝² : IsAddHaarMeasure μ inst✝¹ : IsAddHaarMeasure ν inst✝ : LocallyCompactSpace E h : Function.Surjective ↑L ⊢ ∃ c, 0 < c ∧ map (↑L) μ = c • ν rcases L.exists_map_addHaar_eq_smul_addHaar' μ ν h with ⟨c, c_pos, -, hc⟩ ** case intro.intro.intro 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : CompleteSpace 𝕜 inst✝¹⁰ : NormedAddCommGroup E inst✝⁹ : MeasurableSpace E inst✝⁸ : BorelSpace E inst✝⁷ : NormedSpace 𝕜 E inst✝⁶ : NormedAddCommGroup F inst✝⁵ : MeasurableSpace F inst✝⁴ : BorelSpace F inst✝³ : NormedSpace 𝕜 F L : E →ₗ[𝕜] F μ : Measure E ν : Measure F inst✝² : IsAddHaarMeasure μ inst✝¹ : IsAddHaarMeasure ν inst✝ : LocallyCompactSpace E h : Function.Surjective ↑L c : ℝ≥0∞ c_pos : 0 < c hc : map (↑L) μ = (c * ↑↑addHaar univ) • ν ⊢ ∃ c, 0 < c ∧ map (↑L) μ = c • ν exact ⟨_, by simp [c_pos, NeZero.ne addHaar], hc⟩ 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹² : NontriviallyNormedField 𝕜 inst✝¹¹ : CompleteSpace 𝕜 inst✝¹⁰ : NormedAddCommGroup E inst✝⁹ : MeasurableSpace E inst✝⁸ : BorelSpace E inst✝⁷ : NormedSpace 𝕜 E inst✝⁶ : NormedAddCommGroup F inst✝⁵ : MeasurableSpace F inst✝⁴ : BorelSpace F inst✝³ : NormedSpace 𝕜 F L : E →ₗ[𝕜] F μ : Measure E ν : Measure F inst✝² : IsAddHaarMeasure μ inst✝¹ : IsAddHaarMeasure ν inst✝ : LocallyCompactSpace E h : Function.Surjective ↑L c : ℝ≥0∞ c_pos : 0 < c hc : map (↑L) μ = (c * ↑↑addHaar univ) • ν ⊢ 0 < c * ↑↑addHaar univ simp [c_pos, NeZero.ne addHaar] Qed
mem_parallelepiped_iff ι : Type u_1 ι' : Type u_2 E : Type u_3 F : Type u_4 inst✝⁵ : Fintype ι inst✝⁴ : Fintype ι' inst✝³ : AddCommGroup E inst✝² : Module ℝ E inst✝¹ : AddCommGroup F inst✝ : Module ℝ F v : ι → E x : E ⊢ x ∈ parallelepiped v ↔ ∃ t _ht, x = ∑ i : ι, t i • v i simp [parallelepiped, eq_comm] ** Qed
image_parallelepiped ι : Type u_1 ι' : Type u_2 E : Type u_3 F : Type u_4 inst✝⁵ : Fintype ι inst✝⁴ : Fintype ι' inst✝³ : AddCommGroup E inst✝² : Module ℝ E inst✝¹ : AddCommGroup F inst✝ : Module ℝ F f : E →ₗ[ℝ] F v : ι → E ⊢ ↑f '' parallelepiped v = parallelepiped (↑f ∘ v) simp only [parallelepiped, ← image_comp] ι : Type u_1 ι' : Type u_2 E : Type u_3 F : Type u_4 inst✝⁵ : Fintype ι inst✝⁴ : Fintype ι' inst✝³ : AddCommGroup E inst✝² : Module ℝ E inst✝¹ : AddCommGroup F inst✝ : Module ℝ F f : E →ₗ[ℝ] F v : ι → E ⊢ ((fun a => ↑f a) ∘ fun t => ∑ i : ι, t i • v i) '' Icc 0 1 = (fun a => ∑ x : ι, a x • (↑f ∘ v) x) '' Icc 0 1 congr 1 with t case h ι : Type u_1 ι' : Type u_2 E : Type u_3 F : Type u_4 inst✝⁵ : Fintype ι inst✝⁴ : Fintype ι' inst✝³ : AddCommGroup E inst✝² : Module ℝ E inst✝¹ : AddCommGroup F inst✝ : Module ℝ F f : E →ₗ[ℝ] F v : ι → E t : F ⊢ t ∈ ((fun a => ↑f a) ∘ fun t => ∑ i : ι, t i • v i) '' Icc 0 1 ↔ t ∈ (fun a => ∑ x : ι, a x • (↑f ∘ v) x) '' Icc 0 1 simp only [Function.comp_apply, map_sum, LinearMap.map_smulₛₗ, RingHom.id_apply] ** Qed
Basis.addHaar_reindex ι : Type u_1 ι' : Type u_2 E : Type u_3 F : Type u_4 inst✝⁷ : Fintype ι inst✝⁶ : Fintype ι' inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ E inst✝² : NormedSpace ℝ F inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E b : Basis ι ℝ E e : ι ≃ ι' ⊢ addHaar (reindex b e) = addHaar b rw [Basis.addHaar, b.parallelepiped_reindex e, ← Basis.addHaar] ** Qed
Basis.addHaar_self ι : Type u_1 ι' : Type u_2 E : Type u_3 F : Type u_4 inst✝⁷ : Fintype ι inst✝⁶ : Fintype ι' inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ E inst✝² : NormedSpace ℝ F inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E b : Basis ι ℝ E ⊢ ↑↑(addHaar b) (_root_.parallelepiped ↑b) = 1 rw [Basis.addHaar] ι : Type u_1 ι' : Type u_2 E : Type u_3 F : Type u_4 inst✝⁷ : Fintype ι inst✝⁶ : Fintype ι' inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedAddCommGroup F inst✝³ : NormedSpace ℝ E inst✝² : NormedSpace ℝ F inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E b : Basis ι ℝ E ⊢ ↑↑(addHaarMeasure (parallelepiped b)) (_root_.parallelepiped ↑b) = 1 exact addHaarMeasure_self ** Qed
MeasureTheory.Measure.dirac_apply_of_mem α : Type u_1 β : Type ?u.2369 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β s : Set α a : α h : a ∈ s ⊢ ↑↑(dirac a) s = 1 have : ∀ t : Set α, a ∈ t → t.indicator (1 : α → ℝ≥0∞) a = 1 := fun t ht => indicator_of_mem ht 1 α : Type u_1 β : Type ?u.2369 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β s : Set α a : α h : a ∈ s this : ∀ (t : Set α), a ∈ t → indicator t 1 a = 1 ⊢ ↑↑(dirac a) s = 1 refine' le_antisymm (this univ trivial ▸ _) (this s h ▸ le_dirac_apply) α : Type u_1 β : Type ?u.2369 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β s : Set α a : α h : a ∈ s this : ∀ (t : Set α), a ∈ t → indicator t 1 a = 1 ⊢ ↑↑(dirac a) s ≤ indicator univ 1 a rw [← dirac_apply' a MeasurableSet.univ] α : Type u_1 β : Type ?u.2369 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β s : Set α a : α h : a ∈ s this : ∀ (t : Set α), a ∈ t → indicator t 1 a = 1 ⊢ ↑↑(dirac a) s ≤ ↑↑(dirac a) univ exact measure_mono (subset_univ s) ** Qed
MeasureTheory.Measure.dirac_apply α : Type u_1 β : Type ?u.3265 inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β s✝ : Set α inst✝ : MeasurableSingletonClass α a : α s : Set α ⊢ ↑↑(dirac a) s = indicator s 1 a by_cases h : a ∈ s case neg α : Type u_1 β : Type ?u.3265 inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β s✝ : Set α inst✝ : MeasurableSingletonClass α a : α s : Set α h : ¬a ∈ s ⊢ ↑↑(dirac a) s = indicator s 1 a rw [indicator_of_not_mem h, ← nonpos_iff_eq_zero] case neg α : Type u_1 β : Type ?u.3265 inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β s✝ : Set α inst✝ : MeasurableSingletonClass α a : α s : Set α h : ¬a ∈ s ⊢ ↑↑(dirac a) s ≤ 0 calc dirac a s ≤ dirac a {a}ᶜ := measure_mono (subset_compl_comm.1 <\| singleton_subset_iff.2 h) _ = 0 := by simp [dirac_apply' _ (measurableSet_singleton _).compl] case pos α : Type u_1 β : Type ?u.3265 inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β s✝ : Set α inst✝ : MeasurableSingletonClass α a : α s : Set α h : a ∈ s ⊢ ↑↑(dirac a) s = indicator s 1 a rw [dirac_apply_of_mem h, indicator_of_mem h, Pi.one_apply] α : Type u_1 β : Type ?u.3265 inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β s✝ : Set α inst✝ : MeasurableSingletonClass α a : α s : Set α h : ¬a ∈ s ⊢ ↑↑(dirac a) {a}ᶜ = 0 simp [dirac_apply' _ (measurableSet_singleton _).compl] ** Qed
MeasureTheory.Measure.map_dirac α : Type u_2 β : Type u_1 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β s✝ : Set α f : α → β hf : Measurable f a : α s : Set β hs : MeasurableSet s ⊢ ↑↑(map f (dirac a)) s = ↑↑(dirac (f a)) s simp [hs, map_apply hf hs, hf hs, indicator_apply] ** Qed
MeasureTheory.Measure.tsum_indicator_apply_singleton α : Type u_1 β : Type ?u.19731 inst✝³ : MeasurableSpace α inst✝² : MeasurableSpace β s✝ : Set α inst✝¹ : Countable α inst✝ : MeasurableSingletonClass α μ : Measure α s : Set α hs : MeasurableSet s ⊢ ∑' (x : α), indicator s (fun x => ↑↑μ {x}) x = ↑↑(sum fun a => ↑↑μ {a} • dirac a) s simp only [Measure.sum_apply _ hs, Measure.smul_apply, smul_eq_mul, Measure.dirac_apply, Set.indicator_apply, mul_ite, Pi.one_apply, mul_one, mul_zero] α : Type u_1 β : Type ?u.19731 inst✝³ : MeasurableSpace α inst✝² : MeasurableSpace β s✝ : Set α inst✝¹ : Countable α inst✝ : MeasurableSingletonClass α μ : Measure α s : Set α hs : MeasurableSet s ⊢ ↑↑(sum fun a => ↑↑μ {a} • dirac a) s = ↑↑μ s rw [μ.sum_smul_dirac] ** Qed
MeasureTheory.mem_ae_dirac_iff α : Type u_1 β : Type ?u.22669 inst✝¹ : MeasurableSpace α inst✝ : MeasurableSpace β s : Set α a : α hs : MeasurableSet s ⊢ s ∈ ae (dirac a) ↔ a ∈ s by_cases a ∈ s <;> simp [mem_ae_iff, dirac_apply', hs.compl, indicator_apply, ] * Qed
MeasureTheory.ae_dirac_eq α : Type u_1 β : Type ?u.24458 inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β s : Set α inst✝ : MeasurableSingletonClass α a : α ⊢ ae (dirac a) = pure a ext s case a α : Type u_1 β : Type ?u.24458 inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β s✝ : Set α inst✝ : MeasurableSingletonClass α a : α s : Set α ⊢ s ∈ ae (dirac a) ↔ s ∈ pure a simp [mem_ae_iff, imp_false] ** Qed
MeasureTheory.ae_eq_dirac α : Type u_2 β : Type ?u.26627 inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β s : Set α δ : Type u_1 inst✝ : MeasurableSingletonClass α a : α f : α → δ ⊢ f =ᶠ[ae (dirac a)] const α (f a) simp [Filter.EventuallyEq] ** Qed
MeasureTheory.restrict_dirac' α : Type u_1 β : Type ?u.27642 inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β s : Set α a : α hs : MeasurableSet s inst✝ : Decidable (a ∈ s) ⊢ Measure.restrict (dirac a) s = if a ∈ s then dirac a else 0 split_ifs with has case pos α : Type u_1 β : Type ?u.27642 inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β s : Set α a : α hs : MeasurableSet s inst✝ : Decidable (a ∈ s) has : a ∈ s ⊢ Measure.restrict (dirac a) s = dirac a apply restrict_eq_self_of_ae_mem case pos.hs α : Type u_1 β : Type ?u.27642 inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β s : Set α a : α hs : MeasurableSet s inst✝ : Decidable (a ∈ s) has : a ∈ s ⊢ ∀ᵐ (x : α) ∂dirac a, x ∈ s rw [ae_dirac_iff] <;> assumption case neg α : Type u_1 β : Type ?u.27642 inst✝² : MeasurableSpace α inst✝¹ : MeasurableSpace β s : Set α a : α hs : MeasurableSet s inst✝ : Decidable (a ∈ s) has : ¬a ∈ s ⊢ Measure.restrict (dirac a) s = 0 rw [restrict_eq_zero, dirac_apply' _ hs, indicator_of_not_mem has] ** Qed
MeasureTheory.restrict_dirac α : Type u_1 β : Type ?u.28539 inst✝³ : MeasurableSpace α inst✝² : MeasurableSpace β s : Set α a : α inst✝¹ : MeasurableSingletonClass α inst✝ : Decidable (a ∈ s) ⊢ Measure.restrict (dirac a) s = if a ∈ s then dirac a else 0 split_ifs with has case pos α : Type u_1 β : Type ?u.28539 inst✝³ : MeasurableSpace α inst✝² : MeasurableSpace β s : Set α a : α inst✝¹ : MeasurableSingletonClass α inst✝ : Decidable (a ∈ s) has : a ∈ s ⊢ Measure.restrict (dirac a) s = dirac a apply restrict_eq_self_of_ae_mem case pos.hs α : Type u_1 β : Type ?u.28539 inst✝³ : MeasurableSpace α inst✝² : MeasurableSpace β s : Set α a : α inst✝¹ : MeasurableSingletonClass α inst✝ : Decidable (a ∈ s) has : a ∈ s ⊢ ∀ᵐ (x : α) ∂dirac a, x ∈ s rwa [ae_dirac_eq] case neg α : Type u_1 β : Type ?u.28539 inst✝³ : MeasurableSpace α inst✝² : MeasurableSpace β s : Set α a : α inst✝¹ : MeasurableSingletonClass α inst✝ : Decidable (a ∈ s) has : ¬a ∈ s ⊢ Measure.restrict (dirac a) s = 0 rw [restrict_eq_zero, dirac_apply, indicator_of_not_mem has] ** Qed
MeasureTheory.FiniteMeasure.apply_mono Ω : Type u_1 inst✝ : MeasurableSpace Ω μ : FiniteMeasure Ω s₁ s₂ : Set Ω h : s₁ ⊆ s₂ ⊢ (fun s => ENNReal.toNNReal (↑↑↑μ s)) s₁ ≤ (fun s => ENNReal.toNNReal (↑↑↑μ s)) s₂ change ((μ : Measure Ω) s₁).toNNReal ≤ ((μ : Measure Ω) s₂).toNNReal Ω : Type u_1 inst✝ : MeasurableSpace Ω μ : FiniteMeasure Ω s₁ s₂ : Set Ω h : s₁ ⊆ s₂ ⊢ ENNReal.toNNReal (↑↑↑μ s₁) ≤ ENNReal.toNNReal (↑↑↑μ s₂) have key : (μ : Measure Ω) s₁ ≤ (μ : Measure Ω) s₂ := (μ : Measure Ω).mono h Ω : Type u_1 inst✝ : MeasurableSpace Ω μ : FiniteMeasure Ω s₁ s₂ : Set Ω h : s₁ ⊆ s₂ key : ↑↑↑μ s₁ ≤ ↑↑↑μ s₂ ⊢ ENNReal.toNNReal (↑↑↑μ s₁) ≤ ENNReal.toNNReal (↑↑↑μ s₂) apply (ENNReal.toNNReal_le_toNNReal (measure_ne_top _ s₁) (measure_ne_top _ s₂)).mpr key ** Qed
MeasureTheory.FiniteMeasure.apply_le_mass Ω : Type u_1 inst✝ : MeasurableSpace Ω μ : FiniteMeasure Ω s : Set Ω ⊢ (fun s => ENNReal.toNNReal (↑↑↑μ s)) s ≤ mass μ simpa using apply_mono μ (subset_univ s) ** Qed
MeasureTheory.FiniteMeasure.mass_nonzero_iff Ω : Type u_1 inst✝ : MeasurableSpace Ω μ : FiniteMeasure Ω ⊢ mass μ ≠ 0 ↔ μ ≠ 0 rw [not_iff_not] Ω : Type u_1 inst✝ : MeasurableSpace Ω μ : FiniteMeasure Ω ⊢ mass μ = 0 ↔ μ = 0 exact FiniteMeasure.mass_zero_iff μ ** Qed
MeasureTheory.FiniteMeasure.eq_of_forall_toMeasure_apply_eq Ω : Type u_1 inst✝ : MeasurableSpace Ω μ ν : FiniteMeasure Ω h : ∀ (s : Set Ω), MeasurableSet s → ↑↑↑μ s = ↑↑↑ν s ⊢ μ = ν apply Subtype.ext case a Ω : Type u_1 inst✝ : MeasurableSpace Ω μ ν : FiniteMeasure Ω h : ∀ (s : Set Ω), MeasurableSet s → ↑↑↑μ s = ↑↑↑ν s ⊢ ↑μ = ↑ν ext1 s s_mble case a.h Ω : Type u_1 inst✝ : MeasurableSpace Ω μ ν : FiniteMeasure Ω h : ∀ (s : Set Ω), MeasurableSet s → ↑↑↑μ s = ↑↑↑ν s s : Set Ω s_mble : MeasurableSet s ⊢ ↑↑↑μ s = ↑↑↑ν s exact h s s_mble ** Qed
MeasureTheory.FiniteMeasure.eq_of_forall_apply_eq Ω : Type u_1 inst✝ : MeasurableSpace Ω μ ν : FiniteMeasure Ω h : ∀ (s : Set Ω), MeasurableSet s → (fun s => ENNReal.toNNReal (↑↑↑μ s)) s = (fun s => ENNReal.toNNReal (↑↑↑ν s)) s ⊢ μ = ν ext1 s s_mble case h Ω : Type u_1 inst✝ : MeasurableSpace Ω μ ν : FiniteMeasure Ω h : ∀ (s : Set Ω), MeasurableSet s → (fun s => ENNReal.toNNReal (↑↑↑μ s)) s = (fun s => ENNReal.toNNReal (↑↑↑ν s)) s s : Set Ω s_mble : MeasurableSet s ⊢ ↑↑↑μ s = ↑↑↑ν s simpa [ennreal_coeFn_eq_coeFn_toMeasure] using congr_arg ((↑) : ℝ≥0 → ℝ≥0∞) (h s s_mble) ** Qed
MeasureTheory.FiniteMeasure.coeFn_zero Ω : Type u_1 inst✝⁴ : MeasurableSpace Ω R : Type u_2 inst✝³ : SMul R ℝ≥0 inst✝² : SMul R ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ ⊢ (fun s => ENNReal.toNNReal (↑↑↑0 s)) = 0 funext case h Ω : Type u_1 inst✝⁴ : MeasurableSpace Ω R : Type u_2 inst✝³ : SMul R ℝ≥0 inst✝² : SMul R ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ x✝ : Set Ω ⊢ ENNReal.toNNReal (↑↑↑0 x✝) = OfNat.ofNat 0 x✝ rfl ** Qed
MeasureTheory.FiniteMeasure.coeFn_add Ω : Type u_1 inst✝⁴ : MeasurableSpace Ω R : Type u_2 inst✝³ : SMul R ℝ≥0 inst✝² : SMul R ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ μ ν : FiniteMeasure Ω ⊢ (fun s => ENNReal.toNNReal (↑↑↑(μ + ν) s)) = (fun s => ENNReal.toNNReal (↑↑↑μ s)) + fun s => ENNReal.toNNReal (↑↑↑ν s) funext case h Ω : Type u_1 inst✝⁴ : MeasurableSpace Ω R : Type u_2 inst✝³ : SMul R ℝ≥0 inst✝² : SMul R ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ μ ν : FiniteMeasure Ω x✝ : Set Ω ⊢ ENNReal.toNNReal (↑↑↑(μ + ν) x✝) = ((fun s => ENNReal.toNNReal (↑↑↑μ s)) + fun s => ENNReal.toNNReal (↑↑↑ν s)) x✝ simp only [Pi.add_apply, ← ENNReal.coe_eq_coe, ne_eq, ennreal_coeFn_eq_coeFn_toMeasure, ENNReal.coe_add] case h Ω : Type u_1 inst✝⁴ : MeasurableSpace Ω R : Type u_2 inst✝³ : SMul R ℝ≥0 inst✝² : SMul R ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ μ ν : FiniteMeasure Ω x✝ : Set Ω ⊢ ↑↑↑(μ + ν) x✝ = ↑↑↑μ x✝ + ↑↑↑ν x✝ norm_cast ** Qed
MeasureTheory.FiniteMeasure.coeFn_smul Ω : Type u_1 inst✝⁵ : MeasurableSpace Ω R : Type u_2 inst✝⁴ : SMul R ℝ≥0 inst✝³ : SMul R ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝ : IsScalarTower R ℝ≥0 ℝ≥0 c : R μ : FiniteMeasure Ω ⊢ (fun s => ENNReal.toNNReal (↑↑↑(c • μ) s)) = c • fun s => ENNReal.toNNReal (↑↑↑μ s) funext case h Ω : Type u_1 inst✝⁵ : MeasurableSpace Ω R : Type u_2 inst✝⁴ : SMul R ℝ≥0 inst✝³ : SMul R ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝ : IsScalarTower R ℝ≥0 ℝ≥0 c : R μ : FiniteMeasure Ω x✝ : Set Ω ⊢ ENNReal.toNNReal (↑↑↑(c • μ) x✝) = (c • fun s => ENNReal.toNNReal (↑↑↑μ s)) x✝ simp only [Pi.smul_apply, ← ENNReal.coe_eq_coe, ne_eq, ennreal_coeFn_eq_coeFn_toMeasure, ENNReal.coe_smul] case h Ω : Type u_1 inst✝⁵ : MeasurableSpace Ω R : Type u_2 inst✝⁴ : SMul R ℝ≥0 inst✝³ : SMul R ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝ : IsScalarTower R ℝ≥0 ℝ≥0 c : R μ : FiniteMeasure Ω x✝ : Set Ω ⊢ ↑↑↑(c • μ) x✝ = c • ↑↑↑μ x✝ norm_cast ** Qed
MeasureTheory.FiniteMeasure.restrict_apply Ω : Type u_1 inst✝⁴ : MeasurableSpace Ω R : Type u_2 inst✝³ : SMul R ℝ≥0 inst✝² : SMul R ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ μ : FiniteMeasure Ω A s : Set Ω s_mble : MeasurableSet s ⊢ (fun s => ENNReal.toNNReal (↑↑↑(restrict μ A) s)) s = (fun s => ENNReal.toNNReal (↑↑↑μ s)) (s ∩ A) apply congr_arg ENNReal.toNNReal Ω : Type u_1 inst✝⁴ : MeasurableSpace Ω R : Type u_2 inst✝³ : SMul R ℝ≥0 inst✝² : SMul R ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ μ : FiniteMeasure Ω A s : Set Ω s_mble : MeasurableSet s ⊢ ↑↑↑(restrict μ A) s = ↑↑↑μ (s ∩ A) exact Measure.restrict_apply s_mble ** Qed
MeasureTheory.FiniteMeasure.restrict_eq_zero_iff Ω : Type u_1 inst✝⁴ : MeasurableSpace Ω R : Type u_2 inst✝³ : SMul R ℝ≥0 inst✝² : SMul R ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ μ : FiniteMeasure Ω A : Set Ω ⊢ restrict μ A = 0 ↔ (fun s => ENNReal.toNNReal (↑↑↑μ s)) A = 0 rw [← mass_zero_iff, restrict_mass] ** Qed
MeasureTheory.FiniteMeasure.restrict_nonzero_iff Ω : Type u_1 inst✝⁴ : MeasurableSpace Ω R : Type u_2 inst✝³ : SMul R ℝ≥0 inst✝² : SMul R ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ μ : FiniteMeasure Ω A : Set Ω ⊢ restrict μ A ≠ 0 ↔ (fun s => ENNReal.toNNReal (↑↑↑μ s)) A ≠ 0 rw [← mass_nonzero_iff, restrict_mass] ** Qed
MeasureTheory.FiniteMeasure.testAgainstNN_const ** Ω : Type u_1 inst✝⁵ : MeasurableSpace Ω R : Type u_2 inst✝⁴ : SMul R ℝ≥0 inst✝³ : SMul R ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝ : TopologicalSpace Ω μ : FiniteMeasure Ω c : ℝ≥0 ⊢ testAgainstNN μ (const Ω c) = c * mass μ simp [← ENNReal.coe_eq_coe] Qed
MeasureTheory.FiniteMeasure.testAgainstNN_mono Ω : Type u_1 inst✝⁵ : MeasurableSpace Ω R : Type u_2 inst✝⁴ : SMul R ℝ≥0 inst✝³ : SMul R ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝ : TopologicalSpace Ω μ : FiniteMeasure Ω f g : Ω →ᵇ ℝ≥0 f_le_g : ↑f ≤ ↑g ⊢ testAgainstNN μ f ≤ testAgainstNN μ g simp only [← ENNReal.coe_le_coe, testAgainstNN_coe_eq] Ω : Type u_1 inst✝⁵ : MeasurableSpace Ω R : Type u_2 inst✝⁴ : SMul R ℝ≥0 inst✝³ : SMul R ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝ : TopologicalSpace Ω μ : FiniteMeasure Ω f g : Ω →ᵇ ℝ≥0 f_le_g : ↑f ≤ ↑g ⊢ ∫⁻ (ω : Ω), ↑(↑f ω) ∂↑μ ≤ ∫⁻ (ω : Ω), ↑(↑g ω) ∂↑μ exact lintegral_mono fun ω => ENNReal.coe_mono (f_le_g ω) ** Qed
MeasureTheory.FiniteMeasure.testAgainstNN_zero Ω : Type u_1 inst✝⁵ : MeasurableSpace Ω R : Type u_2 inst✝⁴ : SMul R ℝ≥0 inst✝³ : SMul R ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝ : TopologicalSpace Ω μ : FiniteMeasure Ω ⊢ testAgainstNN μ 0 = 0 simpa only [zero_mul] using μ.testAgainstNN_const 0 ** Qed
MeasureTheory.FiniteMeasure.testAgainstNN_one Ω : Type u_1 inst✝⁵ : MeasurableSpace Ω R : Type u_2 inst✝⁴ : SMul R ℝ≥0 inst✝³ : SMul R ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝ : TopologicalSpace Ω μ : FiniteMeasure Ω ⊢ testAgainstNN μ 1 = mass μ simp only [testAgainstNN, coe_one, Pi.one_apply, ENNReal.coe_one, lintegral_one] Ω : Type u_1 inst✝⁵ : MeasurableSpace Ω R : Type u_2 inst✝⁴ : SMul R ℝ≥0 inst✝³ : SMul R ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝ : TopologicalSpace Ω μ : FiniteMeasure Ω ⊢ ENNReal.toNNReal (↑↑↑μ univ) = mass μ rfl ** Qed
MeasureTheory.FiniteMeasure.zero_testAgainstNN Ω : Type u_1 inst✝⁵ : MeasurableSpace Ω R : Type u_2 inst✝⁴ : SMul R ℝ≥0 inst✝³ : SMul R ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝ : TopologicalSpace Ω ⊢ testAgainstNN 0 = 0 funext case h Ω : Type u_1 inst✝⁵ : MeasurableSpace Ω R : Type u_2 inst✝⁴ : SMul R ℝ≥0 inst✝³ : SMul R ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝ : TopologicalSpace Ω x✝ : Ω →ᵇ ℝ≥0 ⊢ testAgainstNN 0 x✝ = OfNat.ofNat 0 x✝ simp only [zero_testAgainstNN_apply, Pi.zero_apply] ** Qed
MeasureTheory.FiniteMeasure.testAgainstNN_lipschitz Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω μ : FiniteMeasure Ω ⊢ LipschitzWith (mass μ) fun f => testAgainstNN μ f rw [lipschitzWith_iff_dist_le_mul] ** Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω μ : FiniteMeasure Ω ⊢ ∀ (x y : Ω →ᵇ ℝ≥0), dist (testAgainstNN μ x) (testAgainstNN μ y) ≤ ↑(mass μ) * dist x y intro f₁ f₂ Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω μ : FiniteMeasure Ω f₁ f₂ : Ω →ᵇ ℝ≥0 ⊢ dist (testAgainstNN μ f₁) (testAgainstNN μ f₂) ≤ ↑(mass μ) * dist f₁ f₂ ** suffices abs (μ.testAgainstNN f₁ - μ.testAgainstNN f₂ : ℝ) ≤ μ.mass * dist f₁ f₂ by rwa [NNReal.dist_eq] ** Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω μ : FiniteMeasure Ω f₁ f₂ : Ω →ᵇ ℝ≥0 ⊢ \|↑(testAgainstNN μ f₁) - ↑(testAgainstNN μ f₂)\| ≤ ↑(mass μ) * dist f₁ f₂ apply abs_le.mpr Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω μ : FiniteMeasure Ω f₁ f₂ : Ω →ᵇ ℝ≥0 ⊢ -(↑(mass μ) * dist f₁ f₂) ≤ ↑(testAgainstNN μ f₁) - ↑(testAgainstNN μ f₂) ∧ ↑(testAgainstNN μ f₁) - ↑(testAgainstNN μ f₂) ≤ ↑(mass μ) * dist f₁ f₂ constructor Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω μ : FiniteMeasure Ω f₁ f₂ : Ω →ᵇ ℝ≥0 this : \|↑(testAgainstNN μ f₁) - ↑(testAgainstNN μ f₂)\| ≤ ↑(mass μ) * dist f₁ f₂ ⊢ dist (testAgainstNN μ f₁) (testAgainstNN μ f₂) ≤ ↑(mass μ) * dist f₁ f₂ rwa [NNReal.dist_eq] case left Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω μ : FiniteMeasure Ω f₁ f₂ : Ω →ᵇ ℝ≥0 ⊢ -(↑(mass μ) * dist f₁ f₂) ≤ ↑(testAgainstNN μ f₁) - ↑(testAgainstNN μ f₂) have key' := μ.testAgainstNN_lipschitz_estimate f₂ f₁ case left Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω μ : FiniteMeasure Ω f₁ f₂ : Ω →ᵇ ℝ≥0 key' : testAgainstNN μ f₂ ≤ testAgainstNN μ f₁ + nndist f₂ f₁ * mass μ ⊢ -(↑(mass μ) * dist f₁ f₂) ≤ ↑(testAgainstNN μ f₁) - ↑(testAgainstNN μ f₂) rw [mul_comm] at key' case left Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω μ : FiniteMeasure Ω f₁ f₂ : Ω →ᵇ ℝ≥0 key' : testAgainstNN μ f₂ ≤ testAgainstNN μ f₁ + mass μ * nndist f₂ f₁ ⊢ -(↑(mass μ) * dist f₁ f₂) ≤ ↑(testAgainstNN μ f₁) - ↑(testAgainstNN μ f₂) ** suffices ↑(μ.testAgainstNN f₂) ≤ ↑(μ.testAgainstNN f₁) + ↑μ.mass * dist f₁ f₂ by linarith ** case left Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω μ : FiniteMeasure Ω f₁ f₂ : Ω →ᵇ ℝ≥0 key' : testAgainstNN μ f₂ ≤ testAgainstNN μ f₁ + mass μ * nndist f₂ f₁ ⊢ ↑(testAgainstNN μ f₂) ≤ ↑(testAgainstNN μ f₁) + ↑(mass μ) * dist f₁ f₂ have key := NNReal.coe_mono key' case left Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω μ : FiniteMeasure Ω f₁ f₂ : Ω →ᵇ ℝ≥0 key' : testAgainstNN μ f₂ ≤ testAgainstNN μ f₁ + mass μ * nndist f₂ f₁ key : ↑(testAgainstNN μ f₂) ≤ ↑(testAgainstNN μ f₁ + mass μ * nndist f₂ f₁) ⊢ ↑(testAgainstNN μ f₂) ≤ ↑(testAgainstNN μ f₁) + ↑(mass μ) * dist f₁ f₂ rwa [NNReal.coe_add, NNReal.coe_mul, nndist_comm] at key Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω μ : FiniteMeasure Ω f₁ f₂ : Ω →ᵇ ℝ≥0 key' : testAgainstNN μ f₂ ≤ testAgainstNN μ f₁ + mass μ * nndist f₂ f₁ this : ↑(testAgainstNN μ f₂) ≤ ↑(testAgainstNN μ f₁) + ↑(mass μ) * dist f₁ f₂ ⊢ -(↑(mass μ) * dist f₁ f₂) ≤ ↑(testAgainstNN μ f₁) - ↑(testAgainstNN μ f₂) linarith case right Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω μ : FiniteMeasure Ω f₁ f₂ : Ω →ᵇ ℝ≥0 ⊢ ↑(testAgainstNN μ f₁) - ↑(testAgainstNN μ f₂) ≤ ↑(mass μ) * dist f₁ f₂ have key' := μ.testAgainstNN_lipschitz_estimate f₁ f₂ case right Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω μ : FiniteMeasure Ω f₁ f₂ : Ω →ᵇ ℝ≥0 key' : testAgainstNN μ f₁ ≤ testAgainstNN μ f₂ + nndist f₁ f₂ * mass μ ⊢ ↑(testAgainstNN μ f₁) - ↑(testAgainstNN μ f₂) ≤ ↑(mass μ) * dist f₁ f₂ rw [mul_comm] at key' case right Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω μ : FiniteMeasure Ω f₁ f₂ : Ω →ᵇ ℝ≥0 key' : testAgainstNN μ f₁ ≤ testAgainstNN μ f₂ + mass μ * nndist f₁ f₂ ⊢ ↑(testAgainstNN μ f₁) - ↑(testAgainstNN μ f₂) ≤ ↑(mass μ) * dist f₁ f₂ ** suffices ↑(μ.testAgainstNN f₁) ≤ ↑(μ.testAgainstNN f₂) + ↑μ.mass * dist f₁ f₂ by linarith ** case right Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω μ : FiniteMeasure Ω f₁ f₂ : Ω →ᵇ ℝ≥0 key' : testAgainstNN μ f₁ ≤ testAgainstNN μ f₂ + mass μ * nndist f₁ f₂ ⊢ ↑(testAgainstNN μ f₁) ≤ ↑(testAgainstNN μ f₂) + ↑(mass μ) * dist f₁ f₂ have key := NNReal.coe_mono key' case right Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω μ : FiniteMeasure Ω f₁ f₂ : Ω →ᵇ ℝ≥0 key' : testAgainstNN μ f₁ ≤ testAgainstNN μ f₂ + mass μ * nndist f₁ f₂ key : ↑(testAgainstNN μ f₁) ≤ ↑(testAgainstNN μ f₂ + mass μ * nndist f₁ f₂) ⊢ ↑(testAgainstNN μ f₁) ≤ ↑(testAgainstNN μ f₂) + ↑(mass μ) * dist f₁ f₂ rwa [NNReal.coe_add, NNReal.coe_mul] at key Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω μ : FiniteMeasure Ω f₁ f₂ : Ω →ᵇ ℝ≥0 key' : testAgainstNN μ f₁ ≤ testAgainstNN μ f₂ + mass μ * nndist f₁ f₂ this : ↑(testAgainstNN μ f₁) ≤ ↑(testAgainstNN μ f₂) + ↑(mass μ) * dist f₁ f₂ ⊢ ↑(testAgainstNN μ f₁) - ↑(testAgainstNN μ f₂) ≤ ↑(mass μ) * dist f₁ f₂ linarith Qed
MeasureTheory.FiniteMeasure.tendsto_iff_forall_testAgainstNN_tendsto Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_3 F : Filter γ μs : γ → FiniteMeasure Ω μ : FiniteMeasure Ω ⊢ Tendsto μs F (𝓝 μ) ↔ ∀ (f : Ω →ᵇ ℝ≥0), Tendsto (fun i => testAgainstNN (μs i) f) F (𝓝 (testAgainstNN μ f)) rw [FiniteMeasure.tendsto_iff_forall_toWeakDualBCNN_tendsto] Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_3 F : Filter γ μs : γ → FiniteMeasure Ω μ : FiniteMeasure Ω ⊢ (∀ (f : Ω →ᵇ ℝ≥0), Tendsto (fun i => ↑(toWeakDualBCNN (μs i)) f) F (𝓝 (↑(toWeakDualBCNN μ) f))) ↔ ∀ (f : Ω →ᵇ ℝ≥0), Tendsto (fun i => testAgainstNN (μs i) f) F (𝓝 (testAgainstNN μ f)) rfl ** Qed
MeasureTheory.FiniteMeasure.tendsto_zero_of_tendsto_zero_mass Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_3 F : Filter γ μs : γ → FiniteMeasure Ω mass_lim : Tendsto (fun i => mass (μs i)) F (𝓝 0) ⊢ Tendsto μs F (𝓝 0) rw [tendsto_iff_forall_testAgainstNN_tendsto] Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_3 F : Filter γ μs : γ → FiniteMeasure Ω mass_lim : Tendsto (fun i => mass (μs i)) F (𝓝 0) ⊢ ∀ (f : Ω →ᵇ ℝ≥0), Tendsto (fun i => testAgainstNN (μs i) f) F (𝓝 (testAgainstNN 0 f)) intro f Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_3 F : Filter γ μs : γ → FiniteMeasure Ω mass_lim : Tendsto (fun i => mass (μs i)) F (𝓝 0) f : Ω →ᵇ ℝ≥0 ⊢ Tendsto (fun i => testAgainstNN (μs i) f) F (𝓝 (testAgainstNN 0 f)) convert tendsto_zero_testAgainstNN_of_tendsto_zero_mass mass_lim f case h.e'_5.h.e'_3 Ω : Type u_1 inst✝⁶ : MeasurableSpace Ω R : Type u_2 inst✝⁵ : SMul R ℝ≥0 inst✝⁴ : SMul R ℝ≥0∞ inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞ inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞ inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_3 F : Filter γ μs : γ → FiniteMeasure Ω mass_lim : Tendsto (fun i => mass (μs i)) F (𝓝 0) f : Ω →ᵇ ℝ≥0 ⊢ testAgainstNN 0 f = 0 rw [zero_testAgainstNN_apply] ** Qed
MeasureTheory.FiniteMeasure.tendsto_testAgainstNN_filter_of_le_const Ω : Type u_1 inst✝³ : MeasurableSpace Ω inst✝² : TopologicalSpace Ω inst✝¹ : OpensMeasurableSpace Ω ι : Type u_2 L : Filter ι inst✝ : IsCountablyGenerated L μ : FiniteMeasure Ω fs : ι → Ω →ᵇ ℝ≥0 c : ℝ≥0 fs_le_const : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂↑μ, ↑(fs i) ω ≤ c f : Ω →ᵇ ℝ≥0 fs_lim : ∀ᵐ (ω : Ω) ∂↑μ, Tendsto (fun i => ↑(fs i) ω) L (𝓝 (↑f ω)) ⊢ Tendsto (fun i => testAgainstNN μ (fs i)) L (𝓝 (testAgainstNN μ f)) apply (ENNReal.tendsto_toNNReal (f.lintegral_lt_top_of_nnreal (μ : Measure Ω)).ne).comp Ω : Type u_1 inst✝³ : MeasurableSpace Ω inst✝² : TopologicalSpace Ω inst✝¹ : OpensMeasurableSpace Ω ι : Type u_2 L : Filter ι inst✝ : IsCountablyGenerated L μ : FiniteMeasure Ω fs : ι → Ω →ᵇ ℝ≥0 c : ℝ≥0 fs_le_const : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂↑μ, ↑(fs i) ω ≤ c f : Ω →ᵇ ℝ≥0 fs_lim : ∀ᵐ (ω : Ω) ∂↑μ, Tendsto (fun i => ↑(fs i) ω) L (𝓝 (↑f ω)) ⊢ Tendsto (fun i => ∫⁻ (ω : Ω), ↑(↑(fs i) ω) ∂↑μ) L (𝓝 (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ)) exact tendsto_lintegral_nn_filter_of_le_const μ fs_le_const fs_lim ** Qed
MeasureTheory.FiniteMeasure.tendsto_of_forall_integral_tendsto Ω : Type u_1 inst✝² : MeasurableSpace Ω inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_2 F : Filter γ μs : γ → FiniteMeasure Ω μ : FiniteMeasure Ω h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), ↑f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), ↑f x ∂↑μ)) ⊢ Tendsto μs F (𝓝 μ) apply (@tendsto_iff_forall_lintegral_tendsto Ω _ _ _ γ F μs μ).mpr Ω : Type u_1 inst✝² : MeasurableSpace Ω inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_2 F : Filter γ μs : γ → FiniteMeasure Ω μ : FiniteMeasure Ω h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), ↑f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), ↑f x ∂↑μ)) ⊢ ∀ (f : Ω →ᵇ ℝ≥0), Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ)) intro f Ω : Type u_1 inst✝² : MeasurableSpace Ω inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_2 F : Filter γ μs : γ → FiniteMeasure Ω μ : FiniteMeasure Ω h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), ↑f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), ↑f x ∂↑μ)) f : Ω →ᵇ ℝ≥0 ⊢ Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ)) have key := @ENNReal.tendsto_toReal_iff _ F _ (fun i => (f.lintegral_lt_top_of_nnreal (μs i)).ne) _ (f.lintegral_lt_top_of_nnreal μ).ne Ω : Type u_1 inst✝² : MeasurableSpace Ω inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_2 F : Filter γ μs : γ → FiniteMeasure Ω μ : FiniteMeasure Ω h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), ↑f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), ↑f x ∂↑μ)) f : Ω →ᵇ ℝ≥0 key : Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) ↔ Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ)) ⊢ Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ)) apply key.mp Ω : Type u_1 inst✝² : MeasurableSpace Ω inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_2 F : Filter γ μs : γ → FiniteMeasure Ω μ : FiniteMeasure Ω h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), ↑f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), ↑f x ∂↑μ)) f : Ω →ᵇ ℝ≥0 key : Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) ↔ Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ)) ⊢ Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) have lip : LipschitzWith 1 ((↑) : ℝ≥0 → ℝ) := isometry_subtype_coe.lipschitz Ω : Type u_1 inst✝² : MeasurableSpace Ω inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_2 F : Filter γ μs : γ → FiniteMeasure Ω μ : FiniteMeasure Ω h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), ↑f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), ↑f x ∂↑μ)) f : Ω →ᵇ ℝ≥0 key : Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) ↔ Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ)) lip : LipschitzWith 1 NNReal.toReal ⊢ Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) set f₀ := BoundedContinuousFunction.comp _ lip f with _def_f₀ Ω : Type u_1 inst✝² : MeasurableSpace Ω inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_2 F : Filter γ μs : γ → FiniteMeasure Ω μ : FiniteMeasure Ω h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), ↑f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), ↑f x ∂↑μ)) f : Ω →ᵇ ℝ≥0 key : Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) ↔ Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ)) lip : LipschitzWith 1 NNReal.toReal f₀ : Ω →ᵇ ℝ := comp NNReal.toReal lip f _def_f₀ : f₀ = comp NNReal.toReal lip f ⊢ Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) have f₀_eq : ⇑f₀ = ((↑) : ℝ≥0 → ℝ) ∘ ⇑f := rfl Ω : Type u_1 inst✝² : MeasurableSpace Ω inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_2 F : Filter γ μs : γ → FiniteMeasure Ω μ : FiniteMeasure Ω h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), ↑f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), ↑f x ∂↑μ)) f : Ω →ᵇ ℝ≥0 key : Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) ↔ Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ)) lip : LipschitzWith 1 NNReal.toReal f₀ : Ω →ᵇ ℝ := comp NNReal.toReal lip f _def_f₀ : f₀ = comp NNReal.toReal lip f f₀_eq : ↑f₀ = NNReal.toReal ∘ ↑f ⊢ Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) have f₀_nn : 0 ≤ ⇑f₀ := fun _ => by simp only [f₀_eq, Pi.zero_apply, Function.comp_apply, NNReal.zero_le_coe] Ω : Type u_1 inst✝² : MeasurableSpace Ω inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_2 F : Filter γ μs : γ → FiniteMeasure Ω μ : FiniteMeasure Ω h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), ↑f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), ↑f x ∂↑μ)) f : Ω →ᵇ ℝ≥0 key : Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) ↔ Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ)) lip : LipschitzWith 1 NNReal.toReal f₀ : Ω →ᵇ ℝ := comp NNReal.toReal lip f _def_f₀ : f₀ = comp NNReal.toReal lip f f₀_eq : ↑f₀ = NNReal.toReal ∘ ↑f f₀_nn : 0 ≤ ↑f₀ ⊢ Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) have f₀_ae_nn : 0 ≤ᵐ[(μ : Measure Ω)] ⇑f₀ := eventually_of_forall f₀_nn Ω : Type u_1 inst✝² : MeasurableSpace Ω inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_2 F : Filter γ μs : γ → FiniteMeasure Ω μ : FiniteMeasure Ω h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), ↑f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), ↑f x ∂↑μ)) f : Ω →ᵇ ℝ≥0 key : Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) ↔ Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ)) lip : LipschitzWith 1 NNReal.toReal f₀ : Ω →ᵇ ℝ := comp NNReal.toReal lip f _def_f₀ : f₀ = comp NNReal.toReal lip f f₀_eq : ↑f₀ = NNReal.toReal ∘ ↑f f₀_nn : 0 ≤ ↑f₀ f₀_ae_nn : 0 ≤ᵐ[↑μ] ↑f₀ ⊢ Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) have f₀_ae_nns : ∀ i, 0 ≤ᵐ[(μs i : Measure Ω)] ⇑f₀ := fun i => eventually_of_forall f₀_nn Ω : Type u_1 inst✝² : MeasurableSpace Ω inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_2 F : Filter γ μs : γ → FiniteMeasure Ω μ : FiniteMeasure Ω h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), ↑f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), ↑f x ∂↑μ)) f : Ω →ᵇ ℝ≥0 key : Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) ↔ Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ)) lip : LipschitzWith 1 NNReal.toReal f₀ : Ω →ᵇ ℝ := comp NNReal.toReal lip f _def_f₀ : f₀ = comp NNReal.toReal lip f f₀_eq : ↑f₀ = NNReal.toReal ∘ ↑f f₀_nn : 0 ≤ ↑f₀ f₀_ae_nn : 0 ≤ᵐ[↑μ] ↑f₀ f₀_ae_nns : ∀ (i : γ), 0 ≤ᵐ[↑(μs i)] ↑f₀ ⊢ Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) have aux := integral_eq_lintegral_of_nonneg_ae f₀_ae_nn f₀.continuous.measurable.aestronglyMeasurable Ω : Type u_1 inst✝² : MeasurableSpace Ω inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_2 F : Filter γ μs : γ → FiniteMeasure Ω μ : FiniteMeasure Ω h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), ↑f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), ↑f x ∂↑μ)) f : Ω →ᵇ ℝ≥0 key : Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) ↔ Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ)) lip : LipschitzWith 1 NNReal.toReal f₀ : Ω →ᵇ ℝ := comp NNReal.toReal lip f _def_f₀ : f₀ = comp NNReal.toReal lip f f₀_eq : ↑f₀ = NNReal.toReal ∘ ↑f f₀_nn : 0 ≤ ↑f₀ f₀_ae_nn : 0 ≤ᵐ[↑μ] ↑f₀ f₀_ae_nns : ∀ (i : γ), 0 ≤ᵐ[↑(μs i)] ↑f₀ aux : ∫ (a : Ω), ↑f₀ a ∂↑μ = ENNReal.toReal (∫⁻ (a : Ω), ENNReal.ofReal (↑f₀ a) ∂↑μ) ⊢ Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) have auxs := fun i => integral_eq_lintegral_of_nonneg_ae (f₀_ae_nns i) f₀.continuous.measurable.aestronglyMeasurable Ω : Type u_1 inst✝² : MeasurableSpace Ω inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_2 F : Filter γ μs : γ → FiniteMeasure Ω μ : FiniteMeasure Ω h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), ↑f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), ↑f x ∂↑μ)) f : Ω →ᵇ ℝ≥0 key : Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) ↔ Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ)) lip : LipschitzWith 1 NNReal.toReal f₀ : Ω →ᵇ ℝ := comp NNReal.toReal lip f _def_f₀ : f₀ = comp NNReal.toReal lip f f₀_eq : ↑f₀ = NNReal.toReal ∘ ↑f f₀_nn : 0 ≤ ↑f₀ f₀_ae_nn : 0 ≤ᵐ[↑μ] ↑f₀ f₀_ae_nns : ∀ (i : γ), 0 ≤ᵐ[↑(μs i)] ↑f₀ aux : ∫ (a : Ω), ↑f₀ a ∂↑μ = ENNReal.toReal (∫⁻ (a : Ω), ENNReal.ofReal (↑f₀ a) ∂↑μ) auxs : ∀ (i : γ), ∫ (a : Ω), ↑f₀ a ∂↑(μs i) = ENNReal.toReal (∫⁻ (a : Ω), ENNReal.ofReal (↑f₀ a) ∂↑(μs i)) ⊢ Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) simp_rw [f₀_eq, Function.comp_apply, ENNReal.ofReal_coe_nnreal] at aux auxs Ω : Type u_1 inst✝² : MeasurableSpace Ω inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_2 F : Filter γ μs : γ → FiniteMeasure Ω μ : FiniteMeasure Ω h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), ↑f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), ↑f x ∂↑μ)) f : Ω →ᵇ ℝ≥0 key : Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) ↔ Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ)) lip : LipschitzWith 1 NNReal.toReal f₀ : Ω →ᵇ ℝ := comp NNReal.toReal lip f _def_f₀ : f₀ = comp NNReal.toReal lip f f₀_eq : ↑f₀ = NNReal.toReal ∘ ↑f f₀_nn : 0 ≤ ↑f₀ f₀_ae_nn : 0 ≤ᵐ[↑μ] ↑f₀ f₀_ae_nns : ∀ (i : γ), 0 ≤ᵐ[↑(μs i)] ↑f₀ aux : ∫ (a : Ω), ↑(↑f a) ∂↑μ = ENNReal.toReal (∫⁻ (a : Ω), ↑(↑f a) ∂↑μ) auxs : ∀ (i : γ), ∫ (a : Ω), ↑(↑f a) ∂↑(μs i) = ENNReal.toReal (∫⁻ (a : Ω), ↑(↑f a) ∂↑(μs i)) ⊢ Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) simpa only [← aux, ← auxs] using h f₀ Ω : Type u_1 inst✝² : MeasurableSpace Ω inst✝¹ : TopologicalSpace Ω inst✝ : OpensMeasurableSpace Ω γ : Type u_2 F : Filter γ μs : γ → FiniteMeasure Ω μ : FiniteMeasure Ω h : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i => ∫ (x : Ω), ↑f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), ↑f x ∂↑μ)) f : Ω →ᵇ ℝ≥0 key : Tendsto (fun n => ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs n))) F (𝓝 (ENNReal.toReal (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ))) ↔ Tendsto (fun i => ∫⁻ (x : Ω), ↑(↑f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(↑f x) ∂↑μ)) lip : LipschitzWith 1 NNReal.toReal f₀ : Ω →ᵇ ℝ := comp NNReal.toReal lip f _def_f₀ : f₀ = comp NNReal.toReal lip f f₀_eq : ↑f₀ = NNReal.toReal ∘ ↑f x✝ : Ω ⊢ OfNat.ofNat 0 x✝ ≤ ↑f₀ x✝ simp only [f₀_eq, Pi.zero_apply, Function.comp_apply, NNReal.zero_le_coe] ** Qed
MeasureTheory.Memℒp.integrable_sq α : Type u_1 F : Type u_2 m : MeasurableSpace α μ : Measure α inst✝ : NormedAddCommGroup F f : α → ℝ h : Memℒp f 2 ⊢ Integrable fun x => f x ^ 2 simpa [← memℒp_one_iff_integrable] using h.norm_rpow two_ne_zero ENNReal.two_ne_top ** Qed
MeasureTheory.memℒp_two_iff_integrable_sq_norm α : Type u_1 F : Type u_2 m : MeasurableSpace α μ : Measure α inst✝ : NormedAddCommGroup F f : α → F hf : AEStronglyMeasurable f μ ⊢ Memℒp f 2 ↔ Integrable fun x => ‖f x‖ ^ 2 rw [← memℒp_one_iff_integrable] α : Type u_1 F : Type u_2 m : MeasurableSpace α μ : Measure α inst✝ : NormedAddCommGroup F f : α → F hf : AEStronglyMeasurable f μ ⊢ Memℒp f 2 ↔ Memℒp (fun x => ‖f x‖ ^ 2) 1 convert (memℒp_norm_rpow_iff hf two_ne_zero ENNReal.two_ne_top).symm case h.e'_2.h.e'_5.h α : Type u_1 F : Type u_2 m : MeasurableSpace α μ : Measure α inst✝ : NormedAddCommGroup F f : α → F hf : AEStronglyMeasurable f μ x✝ : α ⊢ ‖f x✝‖ ^ 2 = ‖f x✝‖ ^ ENNReal.toReal 2 simp case h.e'_2.h.e'_6 α : Type u_1 F : Type u_2 m : MeasurableSpace α μ : Measure α inst✝ : NormedAddCommGroup F f : α → F hf : AEStronglyMeasurable f μ ⊢ 1 = 2 / 2 rw [div_eq_mul_inv, ENNReal.mul_inv_cancel two_ne_zero ENNReal.two_ne_top] ** Qed
MeasureTheory.Memℒp.inner_const ** α : Type u_1 m : MeasurableSpace α p : ℝ≥0∞ μ : Measure α E : Type u_2 𝕜 : Type u_3 inst✝² : IsROrC 𝕜 inst✝¹ : NormedAddCommGroup E inst✝ : InnerProductSpace 𝕜 E f : α → E hf : Memℒp f p c : E x : α ⊢ ‖inner (f x) c‖ ≤ ?m.29864 hf c * ‖f x‖ rw [mul_comm] α : Type u_1 m : MeasurableSpace α p : ℝ≥0∞ μ : Measure α E : Type u_2 𝕜 : Type u_3 inst✝² : IsROrC 𝕜 inst✝¹ : NormedAddCommGroup E inst✝ : InnerProductSpace 𝕜 E f : α → E hf : Memℒp f p c : E x : α ⊢ ‖inner (f x) c‖ ≤ ‖f x‖ * ?m.29864 hf c α : Type u_1 m : MeasurableSpace α p : ℝ≥0∞ μ : Measure α E : Type u_2 𝕜 : Type u_3 inst✝² : IsROrC 𝕜 inst✝¹ : NormedAddCommGroup E inst✝ : InnerProductSpace 𝕜 E ⊢ {f : α → E} → Memℒp f p → E → ℝ exact norm_inner_le_norm _ _ Qed
MeasureTheory.Integrable.const_inner α : Type u_1 m : MeasurableSpace α p : ℝ≥0∞ μ : Measure α E : Type u_2 𝕜 : Type u_3 inst✝² : IsROrC 𝕜 inst✝¹ : NormedAddCommGroup E inst✝ : InnerProductSpace 𝕜 E f : α → E c : E hf : Integrable f ⊢ Integrable fun x => inner c (f x) rw [← memℒp_one_iff_integrable] at hf ⊢ α : Type u_1 m : MeasurableSpace α p : ℝ≥0∞ μ : Measure α E : Type u_2 𝕜 : Type u_3 inst✝² : IsROrC 𝕜 inst✝¹ : NormedAddCommGroup E inst✝ : InnerProductSpace 𝕜 E f : α → E c : E hf : Memℒp f 1 ⊢ Memℒp (fun x => inner c (f x)) 1 exact hf.const_inner c ** Qed
MeasureTheory.Integrable.inner_const α : Type u_1 m : MeasurableSpace α p : ℝ≥0∞ μ : Measure α E : Type u_2 𝕜 : Type u_3 inst✝² : IsROrC 𝕜 inst✝¹ : NormedAddCommGroup E inst✝ : InnerProductSpace 𝕜 E f : α → E hf : Integrable f c : E ⊢ Integrable fun x => inner (f x) c rw [← memℒp_one_iff_integrable] at hf ⊢ α : Type u_1 m : MeasurableSpace α p : ℝ≥0∞ μ : Measure α E : Type u_2 𝕜 : Type u_3 inst✝² : IsROrC 𝕜 inst✝¹ : NormedAddCommGroup E inst✝ : InnerProductSpace 𝕜 E f : α → E hf : Memℒp f 1 c : E ⊢ Memℒp (fun x => inner (f x) c) 1 exact hf.inner_const c ** Qed
integral_eq_zero_of_forall_integral_inner_eq_zero α : Type u_1 m : MeasurableSpace α p : ℝ≥0∞ μ : Measure α E : Type u_2 𝕜 : Type u_3 inst✝⁴ : IsROrC 𝕜 inst✝³ : NormedAddCommGroup E inst✝² : InnerProductSpace 𝕜 E f✝ : α → E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f : α → E hf : Integrable f hf_int : ∀ (c : E), ∫ (x : α), inner c (f x) ∂μ = 0 ⊢ ∫ (x : α), f x ∂μ = 0 specialize hf_int (∫ x, f x ∂μ) α : Type u_1 m : MeasurableSpace α p : ℝ≥0∞ μ : Measure α E : Type u_2 𝕜 : Type u_3 inst✝⁴ : IsROrC 𝕜 inst✝³ : NormedAddCommGroup E inst✝² : InnerProductSpace 𝕜 E f✝ : α → E inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E f : α → E hf : Integrable f hf_int : ∫ (x : α), inner (∫ (x : α), f x ∂μ) (f x) ∂μ = 0 ⊢ ∫ (x : α), f x ∂μ = 0 rwa [integral_inner hf, inner_self_eq_zero] at hf_int ** Qed
MeasureTheory.L2.snorm_rpow_two_norm_lt_top α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f : { x // x ∈ Lp F 2 } ⊢ snorm (fun x => ‖↑↑f x‖ ^ 2) 1 μ < ⊤ have h_two : ENNReal.ofReal (2 : ℝ) = 2 := by simp [zero_le_one] α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f : { x // x ∈ Lp F 2 } h_two : ENNReal.ofReal 2 = 2 ⊢ snorm (fun x => ‖↑↑f x‖ ^ 2) 1 μ < ⊤ rw [snorm_norm_rpow f zero_lt_two, one_mul, h_two] α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f : { x // x ∈ Lp F 2 } h_two : ENNReal.ofReal 2 = 2 ⊢ snorm (↑↑f) 2 μ ^ 2 < ⊤ exact ENNReal.rpow_lt_top_of_nonneg zero_le_two (Lp.snorm_ne_top f) α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f : { x // x ∈ Lp F 2 } ⊢ ENNReal.ofReal 2 = 2 simp [zero_le_one] ** Qed
MeasureTheory.L2.integral_inner_eq_sq_snorm α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f : { x // x ∈ Lp E 2 } ⊢ ∫ (a : α), inner (↑↑f a) (↑↑f a) ∂μ = ↑(ENNReal.toReal (∫⁻ (a : α), ↑‖↑↑f a‖₊ ^ 2 ∂μ)) simp_rw [inner_self_eq_norm_sq_to_K] α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f : { x // x ∈ Lp E 2 } ⊢ ∫ (a : α), ↑‖↑↑f a‖ ^ 2 ∂μ = ↑(ENNReal.toReal (∫⁻ (a : α), ↑‖↑↑f a‖₊ ^ 2 ∂μ)) norm_cast α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f : { x // x ∈ Lp E 2 } ⊢ ∫ (a : α), ‖↑↑f a‖ ^ 2 ∂μ = ENNReal.toReal (∫⁻ (a : α), ↑(‖↑↑f a‖₊ ^ 2) ∂μ) rw [integral_eq_lintegral_of_nonneg_ae] α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f : { x // x ∈ Lp E 2 } ⊢ ENNReal.toReal (∫⁻ (a : α), ENNReal.ofReal (‖↑↑f a‖ ^ 2) ∂μ) = ENNReal.toReal (∫⁻ (a : α), ↑(‖↑↑f a‖₊ ^ 2) ∂μ) case hf α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f : { x // x ∈ Lp E 2 } ⊢ 0 ≤ᵐ[μ] fun a => ‖↑↑f a‖ ^ 2 case hfm α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f : { x // x ∈ Lp E 2 } ⊢ AEStronglyMeasurable (fun a => ‖↑↑f a‖ ^ 2) μ rotate_left α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f : { x // x ∈ Lp E 2 } ⊢ ENNReal.toReal (∫⁻ (a : α), ENNReal.ofReal (‖↑↑f a‖ ^ 2) ∂μ) = ENNReal.toReal (∫⁻ (a : α), ↑(‖↑↑f a‖₊ ^ 2) ∂μ) congr case e_a.e_f α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f : { x // x ∈ Lp E 2 } ⊢ (fun a => ENNReal.ofReal (‖↑↑f a‖ ^ 2)) = fun a => ↑(‖↑↑f a‖₊ ^ 2) ext1 x case e_a.e_f.h α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f : { x // x ∈ Lp E 2 } x : α ⊢ ENNReal.ofReal (‖↑↑f x‖ ^ 2) = ↑(‖↑↑f x‖₊ ^ 2) have h_two : (2 : ℝ) = ((2 : ℕ) : ℝ) := by simp case e_a.e_f.h α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f : { x // x ∈ Lp E 2 } x : α h_two : 2 = ↑2 ⊢ ENNReal.ofReal (‖↑↑f x‖ ^ 2) = ↑(‖↑↑f x‖₊ ^ 2) rw [← Real.rpow_nat_cast _ 2, ← h_two, ← ENNReal.ofReal_rpow_of_nonneg (norm_nonneg _) zero_le_two, ofReal_norm_eq_coe_nnnorm] case e_a.e_f.h α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f : { x // x ∈ Lp E 2 } x : α h_two : 2 = ↑2 ⊢ ↑‖↑↑f x‖₊ ^ 2 = ↑(‖↑↑f x‖₊ ^ 2) norm_cast case hf α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f : { x // x ∈ Lp E 2 } ⊢ 0 ≤ᵐ[μ] fun a => ‖↑↑f a‖ ^ 2 exact Filter.eventually_of_forall fun x => sq_nonneg _ case hfm α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f : { x // x ∈ Lp E 2 } ⊢ AEStronglyMeasurable (fun a => ‖↑↑f a‖ ^ 2) μ exact ((Lp.aestronglyMeasurable f).norm.aemeasurable.pow_const _).aestronglyMeasurable α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f : { x // x ∈ Lp E 2 } x : α ⊢ 2 = ↑2 simp ** Qed
MeasureTheory.L2.add_left' α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f f' g : { x // x ∈ Lp E 2 } ⊢ inner (f + f') g = inner f g + inner f' g simp_rw [inner_def, ← integral_add (integrable_inner f g) (integrable_inner f' g), ← inner_add_left] α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f f' g : { x // x ∈ Lp E 2 } ⊢ ∫ (a : α), inner (↑↑(f + f') a) (↑↑g a) ∂μ = ∫ (a : α), inner (↑↑f a + ↑↑f' a) (↑↑g a) ∂μ refine' integral_congr_ae ((coeFn_add f f').mono fun x hx => _) α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f f' g : { x // x ∈ Lp E 2 } x : α hx : ↑↑(f + f') x = (↑↑f + ↑↑f') x ⊢ (fun a => inner (↑↑(f + f') a) (↑↑g a)) x = (fun a => inner (↑↑f a + ↑↑f' a) (↑↑g a)) x simp only α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f f' g : { x // x ∈ Lp E 2 } x : α hx : ↑↑(f + f') x = (↑↑f + ↑↑f') x ⊢ inner (↑↑(f + f') x) (↑↑g x) = inner (↑↑f x + ↑↑f' x) (↑↑g x) rw [hx, Pi.add_apply] ** Qed
MeasureTheory.L2.smul_left' ** α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f g : { x // x ∈ Lp E 2 } r : 𝕜 ⊢ inner (r • f) g = ↑(starRingEnd 𝕜) r * inner f g rw [inner_def, inner_def, ← smul_eq_mul, ← integral_smul] α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f g : { x // x ∈ Lp E 2 } r : 𝕜 ⊢ ∫ (a : α), inner (↑↑(r • f) a) (↑↑g a) ∂μ = ∫ (a : α), ↑(starRingEnd 𝕜) r • inner (↑↑f a) (↑↑g a) ∂μ refine' integral_congr_ae ((coeFn_smul r f).mono fun x hx => _) α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f g : { x // x ∈ Lp E 2 } r : 𝕜 x : α hx : ↑↑(r • f) x = (r • ↑↑f) x ⊢ (fun a => inner (↑↑(r • f) a) (↑↑g a)) x = (fun a => ↑(starRingEnd 𝕜) r • inner (↑↑f a) (↑↑g a)) x simp only α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁴ : IsROrC 𝕜 inst✝³ : MeasurableSpace α μ : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : NormedAddCommGroup F f g : { x // x ∈ Lp E 2 } r : 𝕜 x : α hx : ↑↑(r • f) x = (r • ↑↑f) x ⊢ inner (↑↑(r • f) x) (↑↑g x) = ↑(starRingEnd 𝕜) r • inner (↑↑f x) (↑↑g x) rw [smul_eq_mul, ← inner_smul_left, hx, Pi.smul_apply] Qed
MeasureTheory.L2.inner_indicatorConstLp_eq_inner_set_integral α : Type u_1 E : Type u_2 F : Type u_3 𝕜 : Type u_4 inst✝⁶ : IsROrC 𝕜 inst✝⁵ : MeasurableSpace α μ : Measure α inst✝⁴ : NormedAddCommGroup E inst✝³ : InnerProductSpace 𝕜 E inst✝² : NormedAddCommGroup F s : Set α inst✝¹ : CompleteSpace E inst✝ : NormedSpace ℝ E hs : MeasurableSet s hμs : ↑↑μ s ≠ ⊤ c : E f : { x // x ∈ Lp E 2 } ⊢ inner (indicatorConstLp 2 hs hμs c) f = inner c (∫ (x : α) in s, ↑↑f x ∂μ) rw [← integral_inner (integrableOn_Lp_of_measure_ne_top f fact_one_le_two_ennreal.elim hμs), L2.inner_indicatorConstLp_eq_set_integral_inner] ** Qed
MeasureTheory.BoundedContinuousFunction.inner_toLp ** α : Type u_1 inst✝⁴ : TopologicalSpace α inst✝³ : MeasureSpace α inst✝² : BorelSpace α 𝕜 : Type u_2 inst✝¹ : IsROrC 𝕜 μ : Measure α inst✝ : IsFiniteMeasure μ f g : α →ᵇ 𝕜 ⊢ inner (↑(BoundedContinuousFunction.toLp 2 μ 𝕜) f) (↑(BoundedContinuousFunction.toLp 2 μ 𝕜) g) = ∫ (x : α), ↑(starRingEnd ((fun a => 𝕜) x)) (↑f x) * ↑g x ∂μ apply integral_congr_ae case h α : Type u_1 inst✝⁴ : TopologicalSpace α inst✝³ : MeasureSpace α inst✝² : BorelSpace α 𝕜 : Type u_2 inst✝¹ : IsROrC 𝕜 μ : Measure α inst✝ : IsFiniteMeasure μ f g : α →ᵇ 𝕜 ⊢ (fun a => inner (↑↑(↑(BoundedContinuousFunction.toLp 2 μ 𝕜) f) a) (↑↑(↑(BoundedContinuousFunction.toLp 2 μ 𝕜) g) a)) =ᵐ[μ] fun a => ↑(starRingEnd ((fun a => 𝕜) a)) (↑f a) * ↑g a have hf_ae := f.coeFn_toLp 2 μ 𝕜 case h α : Type u_1 inst✝⁴ : TopologicalSpace α inst✝³ : MeasureSpace α inst✝² : BorelSpace α 𝕜 : Type u_2 inst✝¹ : IsROrC 𝕜 μ : Measure α inst✝ : IsFiniteMeasure μ f g : α →ᵇ 𝕜 hf_ae : ↑↑(↑(BoundedContinuousFunction.toLp 2 μ 𝕜) f) =ᵐ[μ] ↑f ⊢ (fun a => inner (↑↑(↑(BoundedContinuousFunction.toLp 2 μ 𝕜) f) a) (↑↑(↑(BoundedContinuousFunction.toLp 2 μ 𝕜) g) a)) =ᵐ[μ] fun a => ↑(starRingEnd ((fun a => 𝕜) a)) (↑f a) * ↑g a have hg_ae := g.coeFn_toLp 2 μ 𝕜 case h α : Type u_1 inst✝⁴ : TopologicalSpace α inst✝³ : MeasureSpace α inst✝² : BorelSpace α 𝕜 : Type u_2 inst✝¹ : IsROrC 𝕜 μ : Measure α inst✝ : IsFiniteMeasure μ f g : α →ᵇ 𝕜 hf_ae : ↑↑(↑(BoundedContinuousFunction.toLp 2 μ 𝕜) f) =ᵐ[μ] ↑f hg_ae : ↑↑(↑(BoundedContinuousFunction.toLp 2 μ 𝕜) g) =ᵐ[μ] ↑g ⊢ (fun a => inner (↑↑(↑(BoundedContinuousFunction.toLp 2 μ 𝕜) f) a) (↑↑(↑(BoundedContinuousFunction.toLp 2 μ 𝕜) g) a)) =ᵐ[μ] fun a => ↑(starRingEnd ((fun a => 𝕜) a)) (↑f a) * ↑g a filter_upwards [hf_ae, hg_ae] with _ hf hg case h α : Type u_1 inst✝⁴ : TopologicalSpace α inst✝³ : MeasureSpace α inst✝² : BorelSpace α 𝕜 : Type u_2 inst✝¹ : IsROrC 𝕜 μ : Measure α inst✝ : IsFiniteMeasure μ f g : α →ᵇ 𝕜 hf_ae : ↑↑(↑(BoundedContinuousFunction.toLp 2 μ 𝕜) f) =ᵐ[μ] ↑f hg_ae : ↑↑(↑(BoundedContinuousFunction.toLp 2 μ 𝕜) g) =ᵐ[μ] ↑g a✝ : α hf : ↑↑(↑(BoundedContinuousFunction.toLp 2 μ 𝕜) f) a✝ = ↑f a✝ hg : ↑↑(↑(BoundedContinuousFunction.toLp 2 μ 𝕜) g) a✝ = ↑g a✝ ⊢ inner (↑↑(↑(BoundedContinuousFunction.toLp 2 μ 𝕜) f) a✝) (↑↑(↑(BoundedContinuousFunction.toLp 2 μ 𝕜) g) a✝) = ↑(starRingEnd 𝕜) (↑f a✝) * ↑g a✝ rw [hf, hg] case h α : Type u_1 inst✝⁴ : TopologicalSpace α inst✝³ : MeasureSpace α inst✝² : BorelSpace α 𝕜 : Type u_2 inst✝¹ : IsROrC 𝕜 μ : Measure α inst✝ : IsFiniteMeasure μ f g : α →ᵇ 𝕜 hf_ae : ↑↑(↑(BoundedContinuousFunction.toLp 2 μ 𝕜) f) =ᵐ[μ] ↑f hg_ae : ↑↑(↑(BoundedContinuousFunction.toLp 2 μ 𝕜) g) =ᵐ[μ] ↑g a✝ : α hf : ↑↑(↑(BoundedContinuousFunction.toLp 2 μ 𝕜) f) a✝ = ↑f a✝ hg : ↑↑(↑(BoundedContinuousFunction.toLp 2 μ 𝕜) g) a✝ = ↑g a✝ ⊢ inner (↑f a✝) (↑g a✝) = ↑(starRingEnd 𝕜) (↑f a✝) * ↑g a✝ simp Qed
MeasureTheory.ContinuousMap.inner_toLp ** α : Type u_1 inst✝⁵ : TopologicalSpace α inst✝⁴ : MeasureSpace α inst✝³ : BorelSpace α 𝕜 : Type u_2 inst✝² : IsROrC 𝕜 μ : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : CompactSpace α f g : C(α, 𝕜) ⊢ inner (↑(ContinuousMap.toLp 2 μ 𝕜) f) (↑(ContinuousMap.toLp 2 μ 𝕜) g) = ∫ (x : α), ↑(starRingEnd ((fun x => 𝕜) x)) (↑f x) * ↑g x ∂μ apply integral_congr_ae case h α : Type u_1 inst✝⁵ : TopologicalSpace α inst✝⁴ : MeasureSpace α inst✝³ : BorelSpace α 𝕜 : Type u_2 inst✝² : IsROrC 𝕜 μ : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : CompactSpace α f g : C(α, 𝕜) ⊢ (fun a => inner (↑↑(↑(ContinuousMap.toLp 2 μ 𝕜) f) a) (↑↑(↑(ContinuousMap.toLp 2 μ 𝕜) g) a)) =ᵐ[μ] fun a => ↑(starRingEnd ((fun x => 𝕜) a)) (↑f a) * ↑g a have hf_ae := f.coeFn_toLp (p := 2) (𝕜 := 𝕜) μ case h α : Type u_1 inst✝⁵ : TopologicalSpace α inst✝⁴ : MeasureSpace α inst✝³ : BorelSpace α 𝕜 : Type u_2 inst✝² : IsROrC 𝕜 μ : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : CompactSpace α f g : C(α, 𝕜) hf_ae : ↑↑(↑(ContinuousMap.toLp 2 μ 𝕜) f) =ᵐ[μ] ↑f ⊢ (fun a => inner (↑↑(↑(ContinuousMap.toLp 2 μ 𝕜) f) a) (↑↑(↑(ContinuousMap.toLp 2 μ 𝕜) g) a)) =ᵐ[μ] fun a => ↑(starRingEnd ((fun x => 𝕜) a)) (↑f a) * ↑g a have hg_ae := g.coeFn_toLp (p := 2) (𝕜 := 𝕜) μ case h α : Type u_1 inst✝⁵ : TopologicalSpace α inst✝⁴ : MeasureSpace α inst✝³ : BorelSpace α 𝕜 : Type u_2 inst✝² : IsROrC 𝕜 μ : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : CompactSpace α f g : C(α, 𝕜) hf_ae : ↑↑(↑(ContinuousMap.toLp 2 μ 𝕜) f) =ᵐ[μ] ↑f hg_ae : ↑↑(↑(ContinuousMap.toLp 2 μ 𝕜) g) =ᵐ[μ] ↑g ⊢ (fun a => inner (↑↑(↑(ContinuousMap.toLp 2 μ 𝕜) f) a) (↑↑(↑(ContinuousMap.toLp 2 μ 𝕜) g) a)) =ᵐ[μ] fun a => ↑(starRingEnd ((fun x => 𝕜) a)) (↑f a) * ↑g a filter_upwards [hf_ae, hg_ae] with _ hf hg case h α : Type u_1 inst✝⁵ : TopologicalSpace α inst✝⁴ : MeasureSpace α inst✝³ : BorelSpace α 𝕜 : Type u_2 inst✝² : IsROrC 𝕜 μ : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : CompactSpace α f g : C(α, 𝕜) hf_ae : ↑↑(↑(ContinuousMap.toLp 2 μ 𝕜) f) =ᵐ[μ] ↑f hg_ae : ↑↑(↑(ContinuousMap.toLp 2 μ 𝕜) g) =ᵐ[μ] ↑g a✝ : α hf : ↑↑(↑(ContinuousMap.toLp 2 μ 𝕜) f) a✝ = ↑f a✝ hg : ↑↑(↑(ContinuousMap.toLp 2 μ 𝕜) g) a✝ = ↑g a✝ ⊢ inner (↑↑(↑(ContinuousMap.toLp 2 μ 𝕜) f) a✝) (↑↑(↑(ContinuousMap.toLp 2 μ 𝕜) g) a✝) = ↑(starRingEnd 𝕜) (↑f a✝) * ↑g a✝ rw [hf, hg] case h α : Type u_1 inst✝⁵ : TopologicalSpace α inst✝⁴ : MeasureSpace α inst✝³ : BorelSpace α 𝕜 : Type u_2 inst✝² : IsROrC 𝕜 μ : Measure α inst✝¹ : IsFiniteMeasure μ inst✝ : CompactSpace α f g : C(α, 𝕜) hf_ae : ↑↑(↑(ContinuousMap.toLp 2 μ 𝕜) f) =ᵐ[μ] ↑f hg_ae : ↑↑(↑(ContinuousMap.toLp 2 μ 𝕜) g) =ᵐ[μ] ↑g a✝ : α hf : ↑↑(↑(ContinuousMap.toLp 2 μ 𝕜) f) a✝ = ↑f a✝ hg : ↑↑(↑(ContinuousMap.toLp 2 μ 𝕜) g) a✝ = ↑g a✝ ⊢ inner (↑f a✝) (↑g a✝) = ↑(starRingEnd 𝕜) (↑f a✝) * ↑g a✝ simp Qed
Turing.ToPartrec.Code.zero'_eval ⊢ eval zero' = fun v => pure (0 :: v) simp [eval] ** Qed
Turing.ToPartrec.Code.succ_eval ⊢ eval succ = fun v => pure [Nat.succ (List.headI v)] simp [eval] ** Qed
Turing.ToPartrec.Code.tail_eval ⊢ eval tail = fun v => pure (List.tail v) simp [eval] ** Qed
Turing.ToPartrec.Code.cons_eval f fs : Code ⊢ eval (cons f fs) = fun v => do let n ← eval f v let ns ← eval fs v pure (List.headI n :: ns) simp [eval] ** Qed
Turing.ToPartrec.Code.comp_eval f g : Code ⊢ eval (comp f g) = fun v => eval g v >>= eval f simp [eval] ** Qed
Turing.ToPartrec.Code.case_eval f g : Code ⊢ eval (case f g) = fun v => Nat.rec (eval f (List.tail v)) (fun y x => eval g (y :: List.tail v)) (List.headI v) simp [eval] ** Qed
Turing.ToPartrec.Code.fix_eval f : Code ⊢ eval (fix f) = PFun.fix fun v => Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v)) (eval f v) simp [eval] ** Qed
Turing.ToPartrec.Code.nil_eval v : List ℕ ⊢ eval nil v = pure [] simp [nil] ** Qed
Turing.ToPartrec.Code.head_eval v : List ℕ ⊢ eval head v = pure [List.headI v] simp [head] ** Qed
Turing.ToPartrec.Code.exists_code.comp m n : ℕ f : Vector ℕ n →. ℕ g : Fin n → Vector ℕ m →. ℕ hf : ∃ c, ∀ (v : Vector ℕ n), eval c ↑v = pure <$> f v hg : ∀ (i : Fin n), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g i v ⊢ ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> ((Vector.mOfFn fun i => g i v) >>= f) rsuffices ⟨cg, hg⟩ : ∃ c : Code, ∀ v : Vector ℕ m, c.eval v.1 = Subtype.val <$> Vector.mOfFn fun i => g i v m n : ℕ f : Vector ℕ n →. ℕ g : Fin n → Vector ℕ m →. ℕ hf : ∃ c, ∀ (v : Vector ℕ n), eval c ↑v = pure <$> f v hg : ∀ (i : Fin n), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g i v ⊢ ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = Subtype.val <$> Vector.mOfFn fun i => g i v clear hf f m n : ℕ g : Fin n → Vector ℕ m →. ℕ hg : ∀ (i : Fin n), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g i v ⊢ ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = Subtype.val <$> Vector.mOfFn fun i => g i v induction' n with n IH case intro m n : ℕ f : Vector ℕ n →. ℕ g : Fin n → Vector ℕ m →. ℕ hf : ∃ c, ∀ (v : Vector ℕ n), eval c ↑v = pure <$> f v hg✝ : ∀ (i : Fin n), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g i v cg : Code hg : ∀ (v : Vector ℕ m), eval cg ↑v = Subtype.val <$> Vector.mOfFn fun i => g i v ⊢ ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> ((Vector.mOfFn fun i => g i v) >>= f) obtain ⟨cf, hf⟩ := hf case intro.intro m n : ℕ f : Vector ℕ n →. ℕ g : Fin n → Vector ℕ m →. ℕ hg✝ : ∀ (i : Fin n), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g i v cg : Code hg : ∀ (v : Vector ℕ m), eval cg ↑v = Subtype.val <$> Vector.mOfFn fun i => g i v cf : Code hf : ∀ (v : Vector ℕ n), eval cf ↑v = pure <$> f v ⊢ ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> ((Vector.mOfFn fun i => g i v) >>= f) exact ⟨cf.comp cg, fun v => by simp [hg, hf, map_bind, seq_bind_eq, Function.comp] rfl⟩ m n : ℕ f : Vector ℕ n →. ℕ g : Fin n → Vector ℕ m →. ℕ hg✝ : ∀ (i : Fin n), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g i v cg : Code hg : ∀ (v : Vector ℕ m), eval cg ↑v = Subtype.val <$> Vector.mOfFn fun i => g i v cf : Code hf : ∀ (v : Vector ℕ n), eval cf ↑v = pure <$> f v v : Vector ℕ m ⊢ eval (Code.comp cf cg) ↑v = pure <$> ((Vector.mOfFn fun i => g i v) >>= f) simp [hg, hf, map_bind, seq_bind_eq, Function.comp] m n : ℕ f : Vector ℕ n →. ℕ g : Fin n → Vector ℕ m →. ℕ hg✝ : ∀ (i : Fin n), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g i v cg : Code hg : ∀ (v : Vector ℕ m), eval cg ↑v = Subtype.val <$> Vector.mOfFn fun i => g i v cf : Code hf : ∀ (v : Vector ℕ n), eval cf ↑v = pure <$> f v v : Vector ℕ m ⊢ (do let x ← Vector.mOfFn fun i => g i v pure <$> f x) = do let a ← Vector.mOfFn fun i => g i v pure <$> f a rfl case zero m n : ℕ g✝ : Fin n → Vector ℕ m →. ℕ hg✝ : ∀ (i : Fin n), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g✝ i v g : Fin Nat.zero → Vector ℕ m →. ℕ hg : ∀ (i : Fin Nat.zero), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g i v ⊢ ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = Subtype.val <$> Vector.mOfFn fun i => g i v exact ⟨nil, fun v => by simp [Vector.mOfFn, Bind.bind]; rfl⟩ m n : ℕ g✝ : Fin n → Vector ℕ m →. ℕ hg✝ : ∀ (i : Fin n), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g✝ i v g : Fin Nat.zero → Vector ℕ m →. ℕ hg : ∀ (i : Fin Nat.zero), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g i v v : Vector ℕ m ⊢ eval nil ↑v = Subtype.val <$> Vector.mOfFn fun i => g i v simp [Vector.mOfFn, Bind.bind] m n : ℕ g✝ : Fin n → Vector ℕ m →. ℕ hg✝ : ∀ (i : Fin n), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g✝ i v g : Fin Nat.zero → Vector ℕ m →. ℕ hg : ∀ (i : Fin Nat.zero), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g i v v : Vector ℕ m ⊢ Part.some [] = Subtype.val <$> Part.some Vector.nil rfl case succ m n✝ : ℕ g✝ : Fin n✝ → Vector ℕ m →. ℕ hg✝ : ∀ (i : Fin n✝), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g✝ i v n : ℕ IH : ∀ {g : Fin n → Vector ℕ m →. ℕ}, (∀ (i : Fin n), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g i v) → ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = Subtype.val <$> Vector.mOfFn fun i => g i v g : Fin (Nat.succ n) → Vector ℕ m →. ℕ hg : ∀ (i : Fin (Nat.succ n)), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g i v ⊢ ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = Subtype.val <$> Vector.mOfFn fun i => g i v obtain ⟨cg, hg₁⟩ := hg 0 case succ.intro m n✝ : ℕ g✝ : Fin n✝ → Vector ℕ m →. ℕ hg✝ : ∀ (i : Fin n✝), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g✝ i v n : ℕ IH : ∀ {g : Fin n → Vector ℕ m →. ℕ}, (∀ (i : Fin n), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g i v) → ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = Subtype.val <$> Vector.mOfFn fun i => g i v g : Fin (Nat.succ n) → Vector ℕ m →. ℕ hg : ∀ (i : Fin (Nat.succ n)), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g i v cg : Code hg₁ : ∀ (v : Vector ℕ m), eval cg ↑v = pure <$> g 0 v ⊢ ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = Subtype.val <$> Vector.mOfFn fun i => g i v obtain ⟨cl, hl⟩ := IH fun i => hg i.succ m n✝ : ℕ g✝ : Fin n✝ → Vector ℕ m →. ℕ hg✝ : ∀ (i : Fin n✝), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g✝ i v n : ℕ IH : ∀ {g : Fin n → Vector ℕ m →. ℕ}, (∀ (i : Fin n), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g i v) → ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = Subtype.val <$> Vector.mOfFn fun i => g i v g : Fin (Nat.succ n) → Vector ℕ m →. ℕ hg : ∀ (i : Fin (Nat.succ n)), ∃ c, ∀ (v : Vector ℕ m), eval c ↑v = pure <$> g i v cg : Code hg₁ : ∀ (v : Vector ℕ m), eval cg ↑v = pure <$> g 0 v cl : Code hl : ∀ (v : Vector ℕ m), eval cl ↑v = Subtype.val <$> Vector.mOfFn fun i => g (Fin.succ i) v v : Vector ℕ m ⊢ (do let x ← g 0 v let x_1 ← Vector.mOfFn fun i => g (Fin.succ i) v Part.some (List.headI (pure x) :: ↑x_1)) = do let a ← g 0 v let a_1 ← Vector.mOfFn fun i => g (Fin.succ i) v Subtype.val <$> Part.some (a ::ᵥ a_1) rfl ** Qed
Turing.ToPartrec.Cont.then_eval k k' : Cont v : List ℕ ⊢ eval (then k k') v = eval k v >>= eval k' induction' k with _ _ _ _ _ _ _ _ _ k_ih _ _ k_ih generalizing v <;> simp only [Cont.eval, Cont.then, bind_assoc, pure_bind, ] * case comp k' : Cont v✝ : List ℕ a✝¹ : Code a✝ : Cont k_ih : ∀ {v : List ℕ}, eval (then a✝ k') v = eval a✝ v >>= eval k' v : List ℕ ⊢ Code.eval a✝¹ v >>= eval (then a✝ k') = Code.eval a✝¹ v >>= fun x => eval a✝ x >>= eval k' simp only [← k_ih] case fix k' : Cont v✝ : List ℕ a✝¹ : Code a✝ : Cont k_ih : ∀ {v : List ℕ}, eval (then a✝ k') v = eval a✝ v >>= eval k' v : List ℕ ⊢ (if List.headI v = 0 then eval a✝ (List.tail v) >>= eval k' else Code.eval (Code.fix a✝¹) (List.tail v) >>= eval (then a✝ k')) = (if List.headI v = 0 then eval a✝ (List.tail v) else Code.eval (Code.fix a✝¹) (List.tail v) >>= eval a✝) >>= eval k' split_ifs <;> [rfl; simp only [← k_ih, bind_assoc]] Qed
Turing.ToPartrec.stepNormal_then c : Code k k' : Cont v : List ℕ ⊢ stepNormal c (Cont.then k k') v = Cfg.then (stepNormal c k v) k' induction c generalizing k v <;> simp only [Cont.then, stepNormal, ] <;> try { simp only [Cfg.then]; done } * case cons k' : Cont a✝¹ a✝ : Code a_ih✝¹ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝¹ (Cont.then k k') v = Cfg.then (stepNormal a✝¹ k v) k' a_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝ (Cont.then k k') v = Cfg.then (stepNormal a✝ k v) k' k : Cont v : List ℕ ⊢ stepNormal a✝¹ (Cont.cons₁ a✝ v (Cont.then k k')) v = Cfg.then (stepNormal a✝¹ (Cont.cons₁ a✝ v k) v) k' case comp k' : Cont a✝¹ a✝ : Code a_ih✝¹ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝¹ (Cont.then k k') v = Cfg.then (stepNormal a✝¹ k v) k' a_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝ (Cont.then k k') v = Cfg.then (stepNormal a✝ k v) k' k : Cont v : List ℕ ⊢ stepNormal a✝ (Cont.comp a✝¹ (Cont.then k k')) v = Cfg.then (stepNormal a✝ (Cont.comp a✝¹ k) v) k' case case k' : Cont a✝¹ a✝ : Code a_ih✝¹ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝¹ (Cont.then k k') v = Cfg.then (stepNormal a✝¹ k v) k' a_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝ (Cont.then k k') v = Cfg.then (stepNormal a✝ k v) k' k : Cont v : List ℕ ⊢ Nat.rec (Cfg.then (stepNormal a✝¹ k (List.tail v)) k') (fun y x => Cfg.then (stepNormal a✝ k (y :: List.tail v)) k') (List.headI v) = Cfg.then (Nat.rec (stepNormal a✝¹ k (List.tail v)) (fun y x => stepNormal a✝ k (y :: List.tail v)) (List.headI v)) k' case fix k' : Cont a✝ : Code a_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝ (Cont.then k k') v = Cfg.then (stepNormal a✝ k v) k' k : Cont v : List ℕ ⊢ stepNormal a✝ (Cont.fix a✝ (Cont.then k k')) v = Cfg.then (stepNormal a✝ (Cont.fix a✝ k) v) k' case cons c c' ih _ => rw [← ih, Cont.then] case comp k' : Cont a✝¹ a✝ : Code a_ih✝¹ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝¹ (Cont.then k k') v = Cfg.then (stepNormal a✝¹ k v) k' a_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝ (Cont.then k k') v = Cfg.then (stepNormal a✝ k v) k' k : Cont v : List ℕ ⊢ stepNormal a✝ (Cont.comp a✝¹ (Cont.then k k')) v = Cfg.then (stepNormal a✝ (Cont.comp a✝¹ k) v) k' case case k' : Cont a✝¹ a✝ : Code a_ih✝¹ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝¹ (Cont.then k k') v = Cfg.then (stepNormal a✝¹ k v) k' a_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝ (Cont.then k k') v = Cfg.then (stepNormal a✝ k v) k' k : Cont v : List ℕ ⊢ Nat.rec (Cfg.then (stepNormal a✝¹ k (List.tail v)) k') (fun y x => Cfg.then (stepNormal a✝ k (y :: List.tail v)) k') (List.headI v) = Cfg.then (Nat.rec (stepNormal a✝¹ k (List.tail v)) (fun y x => stepNormal a✝ k (y :: List.tail v)) (List.headI v)) k' case fix k' : Cont a✝ : Code a_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝ (Cont.then k k') v = Cfg.then (stepNormal a✝ k v) k' k : Cont v : List ℕ ⊢ stepNormal a✝ (Cont.fix a✝ (Cont.then k k')) v = Cfg.then (stepNormal a✝ (Cont.fix a✝ k) v) k' case comp c c' _ ih' => rw [← ih', Cont.then] case fix k' : Cont a✝ : Code a_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝ (Cont.then k k') v = Cfg.then (stepNormal a✝ k v) k' k : Cont v : List ℕ ⊢ stepNormal a✝ (Cont.fix a✝ (Cont.then k k')) v = Cfg.then (stepNormal a✝ (Cont.fix a✝ k) v) k' case fix c ih => rw [← ih, Cont.then] k' : Cont c c' : Code ih : ∀ (k : Cont) (v : List ℕ), stepNormal c (Cont.then k k') v = Cfg.then (stepNormal c k v) k' a_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal c' (Cont.then k k') v = Cfg.then (stepNormal c' k v) k' k : Cont v : List ℕ ⊢ stepNormal c (Cont.cons₁ c' v (Cont.then k k')) v = Cfg.then (stepNormal c (Cont.cons₁ c' v k) v) k' rw [← ih, Cont.then] k' : Cont c c' : Code a_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal c (Cont.then k k') v = Cfg.then (stepNormal c k v) k' ih' : ∀ (k : Cont) (v : List ℕ), stepNormal c' (Cont.then k k') v = Cfg.then (stepNormal c' k v) k' k : Cont v : List ℕ ⊢ stepNormal c' (Cont.comp c (Cont.then k k')) v = Cfg.then (stepNormal c' (Cont.comp c k) v) k' rw [← ih', Cont.then] case case k' : Cont a✝¹ a✝ : Code a_ih✝¹ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝¹ (Cont.then k k') v = Cfg.then (stepNormal a✝¹ k v) k' a_ih✝ : ∀ (k : Cont) (v : List ℕ), stepNormal a✝ (Cont.then k k') v = Cfg.then (stepNormal a✝ k v) k' k : Cont v : List ℕ ⊢ Nat.rec (Cfg.then (stepNormal a✝¹ k (List.tail v)) k') (fun y x => Cfg.then (stepNormal a✝ k (y :: List.tail v)) k') (List.headI v) = Cfg.then (Nat.rec (stepNormal a✝¹ k (List.tail v)) (fun y x => stepNormal a✝ k (y :: List.tail v)) (List.headI v)) k' cases v.headI <;> simp only [Nat.rec] k' : Cont c : Code ih : ∀ (k : Cont) (v : List ℕ), stepNormal c (Cont.then k k') v = Cfg.then (stepNormal c k v) k' k : Cont v : List ℕ ⊢ stepNormal c (Cont.fix c (Cont.then k k')) v = Cfg.then (stepNormal c (Cont.fix c k) v) k' rw [← ih, Cont.then] Qed
Turing.ToPartrec.stepRet_then k k' : Cont v : List ℕ ⊢ stepRet (Cont.then k k') v = Cfg.then (stepRet k v) k' induction k generalizing v <;> simp only [Cont.then, stepRet, ] <;> try { simp only [Cfg.then]; done } * case cons₁ k' : Cont a✝² : Code a✝¹ : List ℕ a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, stepRet (Cont.then a✝ k') v = Cfg.then (stepRet a✝ v) k' v : List ℕ ⊢ stepNormal a✝² (Cont.cons₂ v (Cont.then a✝ k')) a✝¹ = Cfg.then (stepNormal a✝² (Cont.cons₂ v a✝) a✝¹) k' case comp k' : Cont a✝¹ : Code a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, stepRet (Cont.then a✝ k') v = Cfg.then (stepRet a✝ v) k' v : List ℕ ⊢ stepNormal a✝¹ (Cont.then a✝ k') v = Cfg.then (stepNormal a✝¹ a✝ v) k' case fix k' : Cont a✝¹ : Code a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, stepRet (Cont.then a✝ k') v = Cfg.then (stepRet a✝ v) k' v : List ℕ ⊢ (if List.headI v = 0 then Cfg.then (stepRet a✝ (List.tail v)) k' else stepNormal a✝¹ (Cont.fix a✝¹ (Cont.then a✝ k')) (List.tail v)) = Cfg.then (if List.headI v = 0 then stepRet a✝ (List.tail v) else stepNormal a✝¹ (Cont.fix a✝¹ a✝) (List.tail v)) k' case cons₁ => rw [← stepNormal_then] rfl case comp k' : Cont a✝¹ : Code a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, stepRet (Cont.then a✝ k') v = Cfg.then (stepRet a✝ v) k' v : List ℕ ⊢ stepNormal a✝¹ (Cont.then a✝ k') v = Cfg.then (stepNormal a✝¹ a✝ v) k' case fix k' : Cont a✝¹ : Code a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, stepRet (Cont.then a✝ k') v = Cfg.then (stepRet a✝ v) k' v : List ℕ ⊢ (if List.headI v = 0 then Cfg.then (stepRet a✝ (List.tail v)) k' else stepNormal a✝¹ (Cont.fix a✝¹ (Cont.then a✝ k')) (List.tail v)) = Cfg.then (if List.headI v = 0 then stepRet a✝ (List.tail v) else stepNormal a✝¹ (Cont.fix a✝¹ a✝) (List.tail v)) k' case comp => rw [← stepNormal_then] k' : Cont a✝² : Code a✝¹ : List ℕ a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, stepRet (Cont.then a✝ k') v = Cfg.then (stepRet a✝ v) k' v : List ℕ ⊢ stepNormal a✝² (Cont.cons₂ v (Cont.then a✝ k')) a✝¹ = Cfg.then (stepNormal a✝² (Cont.cons₂ v a✝) a✝¹) k' rw [← stepNormal_then] k' : Cont a✝² : Code a✝¹ : List ℕ a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, stepRet (Cont.then a✝ k') v = Cfg.then (stepRet a✝ v) k' v : List ℕ ⊢ stepNormal a✝² (Cont.cons₂ v (Cont.then a✝ k')) a✝¹ = stepNormal a✝² (Cont.then (Cont.cons₂ v a✝) k') a✝¹ rfl k' : Cont a✝¹ : Code a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, stepRet (Cont.then a✝ k') v = Cfg.then (stepRet a✝ v) k' v : List ℕ ⊢ stepNormal a✝¹ (Cont.then a✝ k') v = Cfg.then (stepNormal a✝¹ a✝ v) k' rw [← stepNormal_then] k' : Cont a✝¹ : Code a✝ : Cont k_ih : ∀ {v : List ℕ}, stepRet (Cont.then a✝ k') v = Cfg.then (stepRet a✝ v) k' v : List ℕ ⊢ (if List.headI v = 0 then Cfg.then (stepRet a✝ (List.tail v)) k' else stepNormal a✝¹ (Cont.fix a✝¹ (Cont.then a✝ k')) (List.tail v)) = Cfg.then (if List.headI v = 0 then stepRet a✝ (List.tail v) else stepNormal a✝¹ (Cont.fix a✝¹ a✝) (List.tail v)) k' split_ifs case pos k' : Cont a✝¹ : Code a✝ : Cont k_ih : ∀ {v : List ℕ}, stepRet (Cont.then a✝ k') v = Cfg.then (stepRet a✝ v) k' v : List ℕ h✝ : List.headI v = 0 ⊢ Cfg.then (stepRet a✝ (List.tail v)) k' = Cfg.then (stepRet a✝ (List.tail v)) k' rw [← k_ih] case neg k' : Cont a✝¹ : Code a✝ : Cont k_ih : ∀ {v : List ℕ}, stepRet (Cont.then a✝ k') v = Cfg.then (stepRet a✝ v) k' v : List ℕ h✝ : ¬List.headI v = 0 ⊢ stepNormal a✝¹ (Cont.fix a✝¹ (Cont.then a✝ k')) (List.tail v) = Cfg.then (stepNormal a✝¹ (Cont.fix a✝¹ a✝) (List.tail v)) k' rw [← stepNormal_then] case neg k' : Cont a✝¹ : Code a✝ : Cont k_ih : ∀ {v : List ℕ}, stepRet (Cont.then a✝ k') v = Cfg.then (stepRet a✝ v) k' v : List ℕ h✝ : ¬List.headI v = 0 ⊢ stepNormal a✝¹ (Cont.fix a✝¹ (Cont.then a✝ k')) (List.tail v) = stepNormal a✝¹ (Cont.then (Cont.fix a✝¹ a✝) k') (List.tail v) rfl Qed
Turing.ToPartrec.Code.Ok.zero c : Code h : Ok c v : List ℕ ⊢ Turing.eval step (stepNormal c Cont.halt v) = Cfg.halt <$> eval c v rw [h, ← bind_pure_comp] c : Code h : Ok c v : List ℕ ⊢ (do let v ← eval c v Turing.eval step (Cfg.ret Cont.halt v)) = do let a ← eval c v pure (Cfg.halt a) congr case e_a c : Code h : Ok c v : List ℕ ⊢ (fun v => Turing.eval step (Cfg.ret Cont.halt v)) = fun a => pure (Cfg.halt a) funext v case e_a.h c : Code h : Ok c v✝ v : List ℕ ⊢ Turing.eval step (Cfg.ret Cont.halt v) = pure (Cfg.halt v) exact Part.eq_some_iff.2 (mem_eval.2 ⟨ReflTransGen.single rfl, rfl⟩) ** Qed
Turing.ToPartrec.stepNormal.is_ret c : Code k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal c k v = Cfg.ret k' v' induction c generalizing k v case zero' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal Code.zero' k v = Cfg.ret k' v' case succ k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal Code.succ k v = Cfg.ret k' v' case tail k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal Code.tail k v = Cfg.ret k' v' case cons a✝¹ a✝ : Code a_ih✝¹ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝¹ k v = Cfg.ret k' v' a_ih✝ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝ k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.cons a✝¹ a✝) k v = Cfg.ret k' v' case comp a✝¹ a✝ : Code a_ih✝¹ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝¹ k v = Cfg.ret k' v' a_ih✝ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝ k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.comp a✝¹ a✝) k v = Cfg.ret k' v' case case a✝¹ a✝ : Code a_ih✝¹ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝¹ k v = Cfg.ret k' v' a_ih✝ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝ k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.case a✝¹ a✝) k v = Cfg.ret k' v' case fix a✝ : Code a_ih✝ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝ k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.fix a✝) k v = Cfg.ret k' v' iterate 3 exact ⟨_, _, rfl⟩ case cons a✝¹ a✝ : Code a_ih✝¹ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝¹ k v = Cfg.ret k' v' a_ih✝ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝ k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.cons a✝¹ a✝) k v = Cfg.ret k' v' case comp a✝¹ a✝ : Code a_ih✝¹ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝¹ k v = Cfg.ret k' v' a_ih✝ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝ k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.comp a✝¹ a✝) k v = Cfg.ret k' v' case case a✝¹ a✝ : Code a_ih✝¹ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝¹ k v = Cfg.ret k' v' a_ih✝ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝ k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.case a✝¹ a✝) k v = Cfg.ret k' v' case fix a✝ : Code a_ih✝ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝ k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.fix a✝) k v = Cfg.ret k' v' case cons _f fs IHf _IHfs => apply IHf case comp a✝¹ a✝ : Code a_ih✝¹ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝¹ k v = Cfg.ret k' v' a_ih✝ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝ k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.comp a✝¹ a✝) k v = Cfg.ret k' v' case case a✝¹ a✝ : Code a_ih✝¹ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝¹ k v = Cfg.ret k' v' a_ih✝ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝ k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.case a✝¹ a✝) k v = Cfg.ret k' v' case fix a✝ : Code a_ih✝ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝ k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.fix a✝) k v = Cfg.ret k' v' case comp f _g _IHf IHg => apply IHg case case a✝¹ a✝ : Code a_ih✝¹ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝¹ k v = Cfg.ret k' v' a_ih✝ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝ k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.case a✝¹ a✝) k v = Cfg.ret k' v' case fix a✝ : Code a_ih✝ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝ k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.fix a✝) k v = Cfg.ret k' v' case case f g IHf IHg => rw [stepNormal] simp only [] cases v.headI <;> simp only [] <;> [apply IHf; apply IHg] case fix a✝ : Code a_ih✝ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝ k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.fix a✝) k v = Cfg.ret k' v' case fix f IHf => apply IHf case tail k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal Code.tail k v = Cfg.ret k' v' case cons a✝¹ a✝ : Code a_ih✝¹ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝¹ k v = Cfg.ret k' v' a_ih✝ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝ k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.cons a✝¹ a✝) k v = Cfg.ret k' v' case comp a✝¹ a✝ : Code a_ih✝¹ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝¹ k v = Cfg.ret k' v' a_ih✝ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝ k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.comp a✝¹ a✝) k v = Cfg.ret k' v' case case a✝¹ a✝ : Code a_ih✝¹ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝¹ k v = Cfg.ret k' v' a_ih✝ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝ k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.case a✝¹ a✝) k v = Cfg.ret k' v' case fix a✝ : Code a_ih✝ : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal a✝ k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.fix a✝) k v = Cfg.ret k' v' exact ⟨_, _, rfl⟩ _f fs : Code IHf : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal _f k v = Cfg.ret k' v' _IHfs : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal fs k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.cons _f fs) k v = Cfg.ret k' v' apply IHf f _g : Code _IHf : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal f k v = Cfg.ret k' v' IHg : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal _g k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.comp f _g) k v = Cfg.ret k' v' apply IHg f g : Code IHf : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal f k v = Cfg.ret k' v' IHg : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal g k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.case f g) k v = Cfg.ret k' v' rw [stepNormal] f g : Code IHf : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal f k v = Cfg.ret k' v' IHg : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal g k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', (fun k v => Nat.rec (stepNormal f k (List.tail v)) (fun y x => stepNormal g k (y :: List.tail v)) (List.headI v)) k v = Cfg.ret k' v' simp only [] f g : Code IHf : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal f k v = Cfg.ret k' v' IHg : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal g k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', Nat.rec (stepNormal f k (List.tail v)) (fun y x => stepNormal g k (y :: List.tail v)) (List.headI v) = Cfg.ret k' v' cases v.headI <;> simp only [] <;> [apply IHf; apply IHg] f : Code IHf : ∀ (k : Cont) (v : List ℕ), ∃ k' v', stepNormal f k v = Cfg.ret k' v' k : Cont v : List ℕ ⊢ ∃ k' v', stepNormal (Code.fix f) k v = Cfg.ret k' v' apply IHf ** Qed
Turing.ToPartrec.code_is_ok c : Code ⊢ Code.Ok c induction c <;> intro k v <;> rw [stepNormal] case zero' k : Cont v : List ℕ ⊢ eval step ((fun k v => Cfg.ret k (0 :: v)) k v) = do let v ← Code.eval Code.zero' v eval step (Cfg.ret k v) case succ k : Cont v : List ℕ ⊢ eval step ((fun k v => Cfg.ret k [Nat.succ (List.headI v)]) k v) = do let v ← Code.eval Code.succ v eval step (Cfg.ret k v) case tail k : Cont v : List ℕ ⊢ eval step ((fun k v => Cfg.ret k (List.tail v)) k v) = do let v ← Code.eval Code.tail v eval step (Cfg.ret k v) case cons a✝¹ a✝ : Code a_ih✝¹ : Code.Ok a✝¹ a_ih✝ : Code.Ok a✝ k : Cont v : List ℕ ⊢ eval step ((fun k v => stepNormal a✝¹ (Cont.cons₁ a✝ v k) v) k v) = do let v ← Code.eval (Code.cons a✝¹ a✝) v eval step (Cfg.ret k v) case comp a✝¹ a✝ : Code a_ih✝¹ : Code.Ok a✝¹ a_ih✝ : Code.Ok a✝ k : Cont v : List ℕ ⊢ eval step ((fun k v => stepNormal a✝ (Cont.comp a✝¹ k) v) k v) = do let v ← Code.eval (Code.comp a✝¹ a✝) v eval step (Cfg.ret k v) case case a✝¹ a✝ : Code a_ih✝¹ : Code.Ok a✝¹ a_ih✝ : Code.Ok a✝ k : Cont v : List ℕ ⊢ eval step ((fun k v => Nat.rec (stepNormal a✝¹ k (List.tail v)) (fun y x => stepNormal a✝ k (y :: List.tail v)) (List.headI v)) k v) = do let v ← Code.eval (Code.case a✝¹ a✝) v eval step (Cfg.ret k v) case fix a✝ : Code a_ih✝ : Code.Ok a✝ k : Cont v : List ℕ ⊢ eval step ((fun k v => stepNormal a✝ (Cont.fix a✝ k) v) k v) = do let v ← Code.eval (Code.fix a✝) v eval step (Cfg.ret k v) iterate 3 simp only [Code.eval, pure_bind] case cons a✝¹ a✝ : Code a_ih✝¹ : Code.Ok a✝¹ a_ih✝ : Code.Ok a✝ k : Cont v : List ℕ ⊢ eval step ((fun k v => stepNormal a✝¹ (Cont.cons₁ a✝ v k) v) k v) = do let v ← Code.eval (Code.cons a✝¹ a✝) v eval step (Cfg.ret k v) case comp a✝¹ a✝ : Code a_ih✝¹ : Code.Ok a✝¹ a_ih✝ : Code.Ok a✝ k : Cont v : List ℕ ⊢ eval step ((fun k v => stepNormal a✝ (Cont.comp a✝¹ k) v) k v) = do let v ← Code.eval (Code.comp a✝¹ a✝) v eval step (Cfg.ret k v) case case a✝¹ a✝ : Code a_ih✝¹ : Code.Ok a✝¹ a_ih✝ : Code.Ok a✝ k : Cont v : List ℕ ⊢ eval step ((fun k v => Nat.rec (stepNormal a✝¹ k (List.tail v)) (fun y x => stepNormal a✝ k (y :: List.tail v)) (List.headI v)) k v) = do let v ← Code.eval (Code.case a✝¹ a✝) v eval step (Cfg.ret k v) case fix a✝ : Code a_ih✝ : Code.Ok a✝ k : Cont v : List ℕ ⊢ eval step ((fun k v => stepNormal a✝ (Cont.fix a✝ k) v) k v) = do let v ← Code.eval (Code.fix a✝) v eval step (Cfg.ret k v) case cons f fs IHf IHfs => rw [Code.eval, IHf] simp only [bind_assoc, Cont.eval, pure_bind]; congr; funext v rw [reaches_eval]; swap; exact ReflTransGen.single rfl rw [stepRet, IHfs]; congr; funext v' refine' Eq.trans _ (Eq.symm _) <;> try exact reaches_eval (ReflTransGen.single rfl) case comp a✝¹ a✝ : Code a_ih✝¹ : Code.Ok a✝¹ a_ih✝ : Code.Ok a✝ k : Cont v : List ℕ ⊢ eval step ((fun k v => stepNormal a✝ (Cont.comp a✝¹ k) v) k v) = do let v ← Code.eval (Code.comp a✝¹ a✝) v eval step (Cfg.ret k v) case case a✝¹ a✝ : Code a_ih✝¹ : Code.Ok a✝¹ a_ih✝ : Code.Ok a✝ k : Cont v : List ℕ ⊢ eval step ((fun k v => Nat.rec (stepNormal a✝¹ k (List.tail v)) (fun y x => stepNormal a✝ k (y :: List.tail v)) (List.headI v)) k v) = do let v ← Code.eval (Code.case a✝¹ a✝) v eval step (Cfg.ret k v) case fix a✝ : Code a_ih✝ : Code.Ok a✝ k : Cont v : List ℕ ⊢ eval step ((fun k v => stepNormal a✝ (Cont.fix a✝ k) v) k v) = do let v ← Code.eval (Code.fix a✝) v eval step (Cfg.ret k v) case comp f g IHf IHg => rw [Code.eval, IHg] simp only [bind_assoc, Cont.eval, pure_bind]; congr; funext v rw [reaches_eval]; swap; exact ReflTransGen.single rfl rw [stepRet, IHf] case case a✝¹ a✝ : Code a_ih✝¹ : Code.Ok a✝¹ a_ih✝ : Code.Ok a✝ k : Cont v : List ℕ ⊢ eval step ((fun k v => Nat.rec (stepNormal a✝¹ k (List.tail v)) (fun y x => stepNormal a✝ k (y :: List.tail v)) (List.headI v)) k v) = do let v ← Code.eval (Code.case a✝¹ a✝) v eval step (Cfg.ret k v) case fix a✝ : Code a_ih✝ : Code.Ok a✝ k : Cont v : List ℕ ⊢ eval step ((fun k v => stepNormal a✝ (Cont.fix a✝ k) v) k v) = do let v ← Code.eval (Code.fix a✝) v eval step (Cfg.ret k v) case case f g IHf IHg => simp only [Code.eval] cases v.headI <;> simp only [Code.eval] <;> [apply IHf; apply IHg] case fix a✝ : Code a_ih✝ : Code.Ok a✝ k : Cont v : List ℕ ⊢ eval step ((fun k v => stepNormal a✝ (Cont.fix a✝ k) v) k v) = do let v ← Code.eval (Code.fix a✝) v eval step (Cfg.ret k v) case fix f IHf => rw [cont_eval_fix IHf] case tail k : Cont v : List ℕ ⊢ eval step ((fun k v => Cfg.ret k (List.tail v)) k v) = do let v ← Code.eval Code.tail v eval step (Cfg.ret k v) case cons a✝¹ a✝ : Code a_ih✝¹ : Code.Ok a✝¹ a_ih✝ : Code.Ok a✝ k : Cont v : List ℕ ⊢ eval step ((fun k v => stepNormal a✝¹ (Cont.cons₁ a✝ v k) v) k v) = do let v ← Code.eval (Code.cons a✝¹ a✝) v eval step (Cfg.ret k v) case comp a✝¹ a✝ : Code a_ih✝¹ : Code.Ok a✝¹ a_ih✝ : Code.Ok a✝ k : Cont v : List ℕ ⊢ eval step ((fun k v => stepNormal a✝ (Cont.comp a✝¹ k) v) k v) = do let v ← Code.eval (Code.comp a✝¹ a✝) v eval step (Cfg.ret k v) case case a✝¹ a✝ : Code a_ih✝¹ : Code.Ok a✝¹ a_ih✝ : Code.Ok a✝ k : Cont v : List ℕ ⊢ eval step ((fun k v => Nat.rec (stepNormal a✝¹ k (List.tail v)) (fun y x => stepNormal a✝ k (y :: List.tail v)) (List.headI v)) k v) = do let v ← Code.eval (Code.case a✝¹ a✝) v eval step (Cfg.ret k v) case fix a✝ : Code a_ih✝ : Code.Ok a✝ k : Cont v : List ℕ ⊢ eval step ((fun k v => stepNormal a✝ (Cont.fix a✝ k) v) k v) = do let v ← Code.eval (Code.fix a✝) v eval step (Cfg.ret k v) simp only [Code.eval, pure_bind] f fs : Code IHf : Code.Ok f IHfs : Code.Ok fs k : Cont v : List ℕ ⊢ eval step ((fun k v => stepNormal f (Cont.cons₁ fs v k) v) k v) = do let v ← Code.eval (Code.cons f fs) v eval step (Cfg.ret k v) rw [Code.eval, IHf] f fs : Code IHf : Code.Ok f IHfs : Code.Ok fs k : Cont v : List ℕ ⊢ (do let v_1 ← Code.eval f v eval step (Cfg.ret (Cont.cons₁ fs v k) v_1)) = do let v ← (fun v => do let n ← Code.eval f v let ns ← Code.eval fs v pure (List.headI n :: ns)) v eval step (Cfg.ret k v) simp only [bind_assoc, Cont.eval, pure_bind] f fs : Code IHf : Code.Ok f IHfs : Code.Ok fs k : Cont v : List ℕ ⊢ (do let v_1 ← Code.eval f v eval step (Cfg.ret (Cont.cons₁ fs v k) v_1)) = do let x ← Code.eval f v let x_1 ← Code.eval fs v eval step (Cfg.ret k (List.headI x :: x_1)) congr case e_a f fs : Code IHf : Code.Ok f IHfs : Code.Ok fs k : Cont v : List ℕ ⊢ (fun v_1 => eval step (Cfg.ret (Cont.cons₁ fs v k) v_1)) = fun x => do let x_1 ← Code.eval fs v eval step (Cfg.ret k (List.headI x :: x_1)) funext v case e_a.h f fs : Code IHf : Code.Ok f IHfs : Code.Ok fs k : Cont v✝ v : List ℕ ⊢ eval step (Cfg.ret (Cont.cons₁ fs v✝ k) v) = do let x ← Code.eval fs v✝ eval step (Cfg.ret k (List.headI v :: x)) rw [reaches_eval] case e_a.h f fs : Code IHf : Code.Ok f IHfs : Code.Ok fs k : Cont v✝ v : List ℕ ⊢ eval step ?m.200354 = do let x ← Code.eval fs v✝ eval step (Cfg.ret k (List.headI v :: x)) case e_a.h f fs : Code IHf : Code.Ok f IHfs : Code.Ok fs k : Cont v✝ v : List ℕ ⊢ Reaches step (Cfg.ret (Cont.cons₁ fs v✝ k) v) ?m.200354 f fs : Code IHf : Code.Ok f IHfs : Code.Ok fs k : Cont v✝ v : List ℕ ⊢ Cfg swap case e_a.h f fs : Code IHf : Code.Ok f IHfs : Code.Ok fs k : Cont v✝ v : List ℕ ⊢ Reaches step (Cfg.ret (Cont.cons₁ fs v✝ k) v) ?m.200354 case e_a.h f fs : Code IHf : Code.Ok f IHfs : Code.Ok fs k : Cont v✝ v : List ℕ ⊢ eval step ?m.200354 = do let x ← Code.eval fs v✝ eval step (Cfg.ret k (List.headI v :: x)) f fs : Code IHf : Code.Ok f IHfs : Code.Ok fs k : Cont v✝ v : List ℕ ⊢ Cfg exact ReflTransGen.single rfl case e_a.h f fs : Code IHf : Code.Ok f IHfs : Code.Ok fs k : Cont v✝ v : List ℕ ⊢ eval step (stepRet (Cont.cons₁ fs v✝ k) v) = do let x ← Code.eval fs v✝ eval step (Cfg.ret k (List.headI v :: x)) rw [stepRet, IHfs] case e_a.h f fs : Code IHf : Code.Ok f IHfs : Code.Ok fs k : Cont v✝ v : List ℕ ⊢ (do let v_1 ← Code.eval fs v✝ eval step (Cfg.ret (Cont.cons₂ v k) v_1)) = do let x ← Code.eval fs v✝ eval step (Cfg.ret k (List.headI v :: x)) congr case e_a.h.e_a f fs : Code IHf : Code.Ok f IHfs : Code.Ok fs k : Cont v✝ v : List ℕ ⊢ (fun v_1 => eval step (Cfg.ret (Cont.cons₂ v k) v_1)) = fun x => eval step (Cfg.ret k (List.headI v :: x)) funext v' case e_a.h.e_a.h f fs : Code IHf : Code.Ok f IHfs : Code.Ok fs k : Cont v✝ v v' : List ℕ ⊢ eval step (Cfg.ret (Cont.cons₂ v k) v') = eval step (Cfg.ret k (List.headI v :: v')) refine' Eq.trans _ (Eq.symm _) <;> try exact reaches_eval (ReflTransGen.single rfl) case e_a.h.e_a.h.refine'_3 f fs : Code IHf : Code.Ok f IHfs : Code.Ok fs k : Cont v✝ v v' : List ℕ ⊢ eval step (Cfg.ret k (List.headI v :: v')) = eval step (stepRet (Cont.cons₂ v k) v') exact reaches_eval (ReflTransGen.single rfl) f g : Code IHf : Code.Ok f IHg : Code.Ok g k : Cont v : List ℕ ⊢ eval step ((fun k v => stepNormal g (Cont.comp f k) v) k v) = do let v ← Code.eval (Code.comp f g) v eval step (Cfg.ret k v) rw [Code.eval, IHg] f g : Code IHf : Code.Ok f IHg : Code.Ok g k : Cont v : List ℕ ⊢ (do let v ← Code.eval g v eval step (Cfg.ret (Cont.comp f k) v)) = do let v ← (fun v => Code.eval g v >>= Code.eval f) v eval step (Cfg.ret k v) simp only [bind_assoc, Cont.eval, pure_bind] f g : Code IHf : Code.Ok f IHg : Code.Ok g k : Cont v : List ℕ ⊢ (do let v ← Code.eval g v eval step (Cfg.ret (Cont.comp f k) v)) = do let x ← Code.eval g v let v ← Code.eval f x eval step (Cfg.ret k v) congr case e_a f g : Code IHf : Code.Ok f IHg : Code.Ok g k : Cont v : List ℕ ⊢ (fun v => eval step (Cfg.ret (Cont.comp f k) v)) = fun x => do let v ← Code.eval f x eval step (Cfg.ret k v) funext v case e_a.h f g : Code IHf : Code.Ok f IHg : Code.Ok g k : Cont v✝ v : List ℕ ⊢ eval step (Cfg.ret (Cont.comp f k) v) = do let v ← Code.eval f v eval step (Cfg.ret k v) rw [reaches_eval] case e_a.h f g : Code IHf : Code.Ok f IHg : Code.Ok g k : Cont v✝ v : List ℕ ⊢ eval step ?m.201128 = do let v ← Code.eval f v eval step (Cfg.ret k v) case e_a.h f g : Code IHf : Code.Ok f IHg : Code.Ok g k : Cont v✝ v : List ℕ ⊢ Reaches step (Cfg.ret (Cont.comp f k) v) ?m.201128 f g : Code IHf : Code.Ok f IHg : Code.Ok g k : Cont v✝ v : List ℕ ⊢ Cfg swap case e_a.h f g : Code IHf : Code.Ok f IHg : Code.Ok g k : Cont v✝ v : List ℕ ⊢ Reaches step (Cfg.ret (Cont.comp f k) v) ?m.201128 case e_a.h f g : Code IHf : Code.Ok f IHg : Code.Ok g k : Cont v✝ v : List ℕ ⊢ eval step ?m.201128 = do let v ← Code.eval f v eval step (Cfg.ret k v) f g : Code IHf : Code.Ok f IHg : Code.Ok g k : Cont v✝ v : List ℕ ⊢ Cfg exact ReflTransGen.single rfl case e_a.h f g : Code IHf : Code.Ok f IHg : Code.Ok g k : Cont v✝ v : List ℕ ⊢ eval step (stepRet (Cont.comp f k) v) = do let v ← Code.eval f v eval step (Cfg.ret k v) rw [stepRet, IHf] f g : Code IHf : Code.Ok f IHg : Code.Ok g k : Cont v : List ℕ ⊢ eval step ((fun k v => Nat.rec (stepNormal f k (List.tail v)) (fun y x => stepNormal g k (y :: List.tail v)) (List.headI v)) k v) = do let v ← Code.eval (Code.case f g) v eval step (Cfg.ret k v) simp only [Code.eval] f g : Code IHf : Code.Ok f IHg : Code.Ok g k : Cont v : List ℕ ⊢ eval step (Nat.rec (stepNormal f k (List.tail v)) (fun y x => stepNormal g k (y :: List.tail v)) (List.headI v)) = do let v ← Nat.rec (Code.eval f (List.tail v)) (fun y x => Code.eval g (y :: List.tail v)) (List.headI v) eval step (Cfg.ret k v) cases v.headI <;> simp only [Code.eval] <;> [apply IHf; apply IHg] f : Code IHf : Code.Ok f k : Cont v : List ℕ ⊢ eval step ((fun k v => stepNormal f (Cont.fix f k) v) k v) = do let v ← Code.eval (Code.fix f) v eval step (Cfg.ret k v) rw [cont_eval_fix IHf] ** Qed
Turing.ToPartrec.stepRet_eval k : Cont v : List ℕ ⊢ eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v induction k generalizing v case halt v : List ℕ ⊢ eval step (stepRet Cont.halt v) = Cfg.halt <$> Cont.eval Cont.halt v case cons₁ a✝² : Code a✝¹ : List ℕ a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, eval step (stepRet a✝ v) = Cfg.halt <$> Cont.eval a✝ v v : List ℕ ⊢ eval step (stepRet (Cont.cons₁ a✝² a✝¹ a✝) v) = Cfg.halt <$> Cont.eval (Cont.cons₁ a✝² a✝¹ a✝) v case cons₂ a✝¹ : List ℕ a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, eval step (stepRet a✝ v) = Cfg.halt <$> Cont.eval a✝ v v : List ℕ ⊢ eval step (stepRet (Cont.cons₂ a✝¹ a✝) v) = Cfg.halt <$> Cont.eval (Cont.cons₂ a✝¹ a✝) v case comp a✝¹ : Code a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, eval step (stepRet a✝ v) = Cfg.halt <$> Cont.eval a✝ v v : List ℕ ⊢ eval step (stepRet (Cont.comp a✝¹ a✝) v) = Cfg.halt <$> Cont.eval (Cont.comp a✝¹ a✝) v case fix a✝¹ : Code a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, eval step (stepRet a✝ v) = Cfg.halt <$> Cont.eval a✝ v v : List ℕ ⊢ eval step (stepRet (Cont.fix a✝¹ a✝) v) = Cfg.halt <$> Cont.eval (Cont.fix a✝¹ a✝) v case halt => simp only [mem_eval, Cont.eval, map_pure] exact Part.eq_some_iff.2 (mem_eval.2 ⟨ReflTransGen.refl, rfl⟩) case cons₁ a✝² : Code a✝¹ : List ℕ a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, eval step (stepRet a✝ v) = Cfg.halt <$> Cont.eval a✝ v v : List ℕ ⊢ eval step (stepRet (Cont.cons₁ a✝² a✝¹ a✝) v) = Cfg.halt <$> Cont.eval (Cont.cons₁ a✝² a✝¹ a✝) v case cons₂ a✝¹ : List ℕ a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, eval step (stepRet a✝ v) = Cfg.halt <$> Cont.eval a✝ v v : List ℕ ⊢ eval step (stepRet (Cont.cons₂ a✝¹ a✝) v) = Cfg.halt <$> Cont.eval (Cont.cons₂ a✝¹ a✝) v case comp a✝¹ : Code a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, eval step (stepRet a✝ v) = Cfg.halt <$> Cont.eval a✝ v v : List ℕ ⊢ eval step (stepRet (Cont.comp a✝¹ a✝) v) = Cfg.halt <$> Cont.eval (Cont.comp a✝¹ a✝) v case fix a✝¹ : Code a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, eval step (stepRet a✝ v) = Cfg.halt <$> Cont.eval a✝ v v : List ℕ ⊢ eval step (stepRet (Cont.fix a✝¹ a✝) v) = Cfg.halt <$> Cont.eval (Cont.fix a✝¹ a✝) v case cons₁ fs as k IH => rw [Cont.eval, stepRet, code_is_ok] simp only [← bind_pure_comp, bind_assoc]; congr; funext v' rw [reaches_eval]; swap; exact ReflTransGen.single rfl rw [stepRet, IH, bind_pure_comp] case cons₂ a✝¹ : List ℕ a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, eval step (stepRet a✝ v) = Cfg.halt <$> Cont.eval a✝ v v : List ℕ ⊢ eval step (stepRet (Cont.cons₂ a✝¹ a✝) v) = Cfg.halt <$> Cont.eval (Cont.cons₂ a✝¹ a✝) v case comp a✝¹ : Code a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, eval step (stepRet a✝ v) = Cfg.halt <$> Cont.eval a✝ v v : List ℕ ⊢ eval step (stepRet (Cont.comp a✝¹ a✝) v) = Cfg.halt <$> Cont.eval (Cont.comp a✝¹ a✝) v case fix a✝¹ : Code a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, eval step (stepRet a✝ v) = Cfg.halt <$> Cont.eval a✝ v v : List ℕ ⊢ eval step (stepRet (Cont.fix a✝¹ a✝) v) = Cfg.halt <$> Cont.eval (Cont.fix a✝¹ a✝) v case cons₂ ns k IH => rw [Cont.eval, stepRet]; exact IH case comp a✝¹ : Code a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, eval step (stepRet a✝ v) = Cfg.halt <$> Cont.eval a✝ v v : List ℕ ⊢ eval step (stepRet (Cont.comp a✝¹ a✝) v) = Cfg.halt <$> Cont.eval (Cont.comp a✝¹ a✝) v case fix a✝¹ : Code a✝ : Cont a_ih✝ : ∀ {v : List ℕ}, eval step (stepRet a✝ v) = Cfg.halt <$> Cont.eval a✝ v v : List ℕ ⊢ eval step (stepRet (Cont.fix a✝¹ a✝) v) = Cfg.halt <$> Cont.eval (Cont.fix a✝¹ a✝) v case comp f k IH => rw [Cont.eval, stepRet, code_is_ok] simp only [← bind_pure_comp, bind_assoc]; congr; funext v' rw [reaches_eval]; swap; exact ReflTransGen.single rfl rw [IH, bind_pure_comp] v : List ℕ ⊢ eval step (stepRet Cont.halt v) = Cfg.halt <$> Cont.eval Cont.halt v simp only [mem_eval, Cont.eval, map_pure] v : List ℕ ⊢ eval step (stepRet Cont.halt v) = pure (Cfg.halt v) exact Part.eq_some_iff.2 (mem_eval.2 ⟨ReflTransGen.refl, rfl⟩) fs : Code as : List ℕ k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v : List ℕ ⊢ eval step (stepRet (Cont.cons₁ fs as k) v) = Cfg.halt <$> Cont.eval (Cont.cons₁ fs as k) v rw [Cont.eval, stepRet, code_is_ok] fs : Code as : List ℕ k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v : List ℕ ⊢ (do let v_1 ← Code.eval fs as eval step (Cfg.ret (Cont.cons₂ v k) v_1)) = Cfg.halt <$> (fun v => do let ns ← Code.eval fs as Cont.eval k (List.headI v :: ns)) v simp only [← bind_pure_comp, bind_assoc] fs : Code as : List ℕ k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v : List ℕ ⊢ (do let v_1 ← Code.eval fs as eval step (Cfg.ret (Cont.cons₂ v k) v_1)) = do let x ← Code.eval fs as let a ← Cont.eval k (List.headI v :: x) pure (Cfg.halt a) congr case e_a fs : Code as : List ℕ k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v : List ℕ ⊢ (fun v_1 => eval step (Cfg.ret (Cont.cons₂ v k) v_1)) = fun x => do let a ← Cont.eval k (List.headI v :: x) pure (Cfg.halt a) funext v' case e_a.h fs : Code as : List ℕ k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v v' : List ℕ ⊢ eval step (Cfg.ret (Cont.cons₂ v k) v') = do let a ← Cont.eval k (List.headI v :: v') pure (Cfg.halt a) rw [reaches_eval] case e_a.h fs : Code as : List ℕ k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v v' : List ℕ ⊢ eval step ?m.202867 = do let a ← Cont.eval k (List.headI v :: v') pure (Cfg.halt a) case e_a.h fs : Code as : List ℕ k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v v' : List ℕ ⊢ Reaches step (Cfg.ret (Cont.cons₂ v k) v') ?m.202867 fs : Code as : List ℕ k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v v' : List ℕ ⊢ Cfg swap case e_a.h fs : Code as : List ℕ k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v v' : List ℕ ⊢ Reaches step (Cfg.ret (Cont.cons₂ v k) v') ?m.202867 case e_a.h fs : Code as : List ℕ k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v v' : List ℕ ⊢ eval step ?m.202867 = do let a ← Cont.eval k (List.headI v :: v') pure (Cfg.halt a) fs : Code as : List ℕ k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v v' : List ℕ ⊢ Cfg exact ReflTransGen.single rfl case e_a.h fs : Code as : List ℕ k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v v' : List ℕ ⊢ eval step (stepRet (Cont.cons₂ v k) v') = do let a ← Cont.eval k (List.headI v :: v') pure (Cfg.halt a) rw [stepRet, IH, bind_pure_comp] ns : List ℕ k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v : List ℕ ⊢ eval step (stepRet (Cont.cons₂ ns k) v) = Cfg.halt <$> Cont.eval (Cont.cons₂ ns k) v rw [Cont.eval, stepRet] ns : List ℕ k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v : List ℕ ⊢ eval step (stepRet k (List.headI ns :: v)) = Cfg.halt <$> (fun v => Cont.eval k (List.headI ns :: v)) v exact IH f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v : List ℕ ⊢ eval step (stepRet (Cont.comp f k) v) = Cfg.halt <$> Cont.eval (Cont.comp f k) v rw [Cont.eval, stepRet, code_is_ok] f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v : List ℕ ⊢ (do let v ← Code.eval f v eval step (Cfg.ret k v)) = Cfg.halt <$> (fun v => Code.eval f v >>= Cont.eval k) v simp only [← bind_pure_comp, bind_assoc] f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v : List ℕ ⊢ (do let v ← Code.eval f v eval step (Cfg.ret k v)) = do let x ← Code.eval f v let a ← Cont.eval k x pure (Cfg.halt a) congr case e_a f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v : List ℕ ⊢ (fun v => eval step (Cfg.ret k v)) = fun x => do let a ← Cont.eval k x pure (Cfg.halt a) funext v' case e_a.h f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v v' : List ℕ ⊢ eval step (Cfg.ret k v') = do let a ← Cont.eval k v' pure (Cfg.halt a) rw [reaches_eval] case e_a.h f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v v' : List ℕ ⊢ eval step ?m.203559 = do let a ← Cont.eval k v' pure (Cfg.halt a) case e_a.h f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v v' : List ℕ ⊢ Reaches step (Cfg.ret k v') ?m.203559 f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v v' : List ℕ ⊢ Cfg swap case e_a.h f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v v' : List ℕ ⊢ Reaches step (Cfg.ret k v') ?m.203559 case e_a.h f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v v' : List ℕ ⊢ eval step ?m.203559 = do let a ← Cont.eval k v' pure (Cfg.halt a) f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v v' : List ℕ ⊢ Cfg exact ReflTransGen.single rfl case e_a.h f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v v' : List ℕ ⊢ eval step (stepRet k v') = do let a ← Cont.eval k v' pure (Cfg.halt a) rw [IH, bind_pure_comp] f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v : List ℕ ⊢ eval step (stepRet (Cont.fix f k) v) = Cfg.halt <$> Cont.eval (Cont.fix f k) v rw [Cont.eval, stepRet] f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v : List ℕ ⊢ eval step (if List.headI v = 0 then stepRet k (List.tail v) else stepNormal f (Cont.fix f k) (List.tail v)) = Cfg.halt <$> (fun v => if List.headI v = 0 then Cont.eval k (List.tail v) else Code.eval (Code.fix f) (List.tail v) >>= Cont.eval k) v simp only [bind_pure_comp] f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v : List ℕ ⊢ eval step (if List.headI v = 0 then stepRet k (List.tail v) else stepNormal f (Cont.fix f k) (List.tail v)) = Cfg.halt <$> if List.headI v = 0 then Cont.eval k (List.tail v) else Code.eval (Code.fix f) (List.tail v) >>= Cont.eval k split_ifs case neg f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v : List ℕ h✝ : ¬List.headI v = 0 ⊢ eval step (stepNormal f (Cont.fix f k) (List.tail v)) = Cfg.halt <$> (Code.eval (Code.fix f) (List.tail v) >>= Cont.eval k) simp only [← bind_pure_comp, bind_assoc, cont_eval_fix (code_is_ok _)] case neg f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v : List ℕ h✝ : ¬List.headI v = 0 ⊢ (do let v ← Code.eval (Code.fix f) (List.tail v) eval step (Cfg.ret k v)) = do let x ← Code.eval (Code.fix f) (List.tail v) let a ← Cont.eval k x pure (Cfg.halt a) congr case neg.e_a f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v : List ℕ h✝ : ¬List.headI v = 0 ⊢ (fun v => eval step (Cfg.ret k v)) = fun x => do let a ← Cont.eval k x pure (Cfg.halt a) funext case neg.e_a.h f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v : List ℕ h✝ : ¬List.headI v = 0 x✝ : List ℕ ⊢ eval step (Cfg.ret k x✝) = do let a ← Cont.eval k x✝ pure (Cfg.halt a) rw [bind_pure_comp, ← IH] case neg.e_a.h f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v : List ℕ h✝ : ¬List.headI v = 0 x✝ : List ℕ ⊢ eval step (Cfg.ret k x✝) = eval step (stepRet k x✝) exact reaches_eval (ReflTransGen.single rfl) case pos f : Code k : Cont IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> Cont.eval k v v : List ℕ h✝ : List.headI v = 0 ⊢ eval step (stepRet k (List.tail v)) = Cfg.halt <$> Cont.eval k (List.tail v) exact IH ** Qed
Turing.PartrecToTM2.trNat_zero ⊢ trNat 0 = [] rw [trNat, Nat.cast_zero] ⊢ trNum 0 = [] rfl ** Qed
Turing.PartrecToTM2.K'.elim_update_main a b c d a' : List Γ' ⊢ update (elim a b c d) main a' = elim a' b c d funext x case h a b c d a' : List Γ' x : K' ⊢ update (elim a b c d) main a' x = elim a' b c d x cases x <;> rfl ** Qed
Turing.PartrecToTM2.K'.elim_update_rev a b c d b' : List Γ' ⊢ update (elim a b c d) rev b' = elim a b' c d funext x case h a b c d b' : List Γ' x : K' ⊢ update (elim a b c d) rev b' x = elim a b' c d x cases x <;> rfl ** Qed
Turing.PartrecToTM2.K'.elim_update_aux a b c d c' : List Γ' ⊢ update (elim a b c d) aux c' = elim a b c' d funext x case h a b c d c' : List Γ' x : K' ⊢ update (elim a b c d) aux c' x = elim a b c' d x cases x <;> rfl ** Qed
Turing.PartrecToTM2.K'.elim_update_stack a b c d d' : List Γ' ⊢ update (elim a b c d) stack d' = elim a b c d' funext x case h a b c d d' : List Γ' x : K' ⊢ update (elim a b c d) stack d' x = elim a b c d' x cases x <;> rfl ** Qed
Turing.PartrecToTM2.splitAtPred_eq α : Type u_1 p : α → Bool l₁ : List α o : α l₂ : List α x✝ : ∀ (x : α), x ∈ l₁ → p x = false left✝ : p o = true h₃ : [] = l₁ ++ o :: l₂ ⊢ splitAtPred p [] = (l₁, some o, l₂) simp at h₃ α : Type u_1 p : α → Bool a : α L l₁ : List α o : Option α l₂ : List α h₁ : ∀ (x : α), x ∈ l₁ → p x = false h₂ : Option.elim' (a :: L = l₁ ∧ l₂ = []) (fun a_1 => p a_1 = true ∧ a :: L = l₁ ++ a_1 :: l₂) o ⊢ splitAtPred p (a :: L) = (l₁, o, l₂) rw [splitAtPred] α : Type u_1 p : α → Bool a : α L l₁ : List α o : Option α l₂ : List α h₁ : ∀ (x : α), x ∈ l₁ → p x = false h₂ : Option.elim' (a :: L = l₁ ∧ l₂ = []) (fun a_1 => p a_1 = true ∧ a :: L = l₁ ++ a_1 :: l₂) o ⊢ (bif p a then ([], some a, L) else match splitAtPred p L with \| (l₁, o, l₂) => (a :: l₁, o, l₂)) = (l₁, o, l₂) have IH := splitAtPred_eq p L α : Type u_1 p : α → Bool a : α L l₁ : List α o : Option α l₂ : List α h₁ : ∀ (x : α), x ∈ l₁ → p x = false h₂ : Option.elim' (a :: L = l₁ ∧ l₂ = []) (fun a_1 => p a_1 = true ∧ a :: L = l₁ ++ a_1 :: l₂) o IH : ∀ (l₁ : List α) (o : Option α) (l₂ : List α), (∀ (x : α), x ∈ l₁ → p x = false) → Option.elim' (L = l₁ ∧ l₂ = []) (fun a => p a = true ∧ L = l₁ ++ a :: l₂) o → splitAtPred p L = (l₁, o, l₂) ⊢ (bif p a then ([], some a, L) else match splitAtPred p L with \| (l₁, o, l₂) => (a :: l₁, o, l₂)) = (l₁, o, l₂) cases' o with o case none α : Type u_1 p : α → Bool a : α L l₁ l₂ : List α h₁ : ∀ (x : α), x ∈ l₁ → p x = false IH : ∀ (l₁ : List α) (o : Option α) (l₂ : List α), (∀ (x : α), x ∈ l₁ → p x = false) → Option.elim' (L = l₁ ∧ l₂ = []) (fun a => p a = true ∧ L = l₁ ++ a :: l₂) o → splitAtPred p L = (l₁, o, l₂) h₂ : Option.elim' (a :: L = l₁ ∧ l₂ = []) (fun a_1 => p a_1 = true ∧ a :: L = l₁ ++ a_1 :: l₂) none ⊢ (bif p a then ([], some a, L) else match splitAtPred p L with \| (l₁, o, l₂) => (a :: l₁, o, l₂)) = (l₁, none, l₂) cases' l₁ with a' l₁ <;> rcases h₂ with ⟨⟨⟩, rfl⟩ case none.cons.intro.refl α : Type u_1 p : α → Bool a : α L : List α IH : ∀ (l₁ : List α) (o : Option α) (l₂ : List α), (∀ (x : α), x ∈ l₁ → p x = false) → Option.elim' (L = l₁ ∧ l₂ = []) (fun a => p a = true ∧ L = l₁ ++ a :: l₂) o → splitAtPred p L = (l₁, o, l₂) h₁ : ∀ (x : α), x ∈ a :: L → p x = false ⊢ (bif p a then ([], some a, L) else match splitAtPred p L with \| (l₁, o, l₂) => (a :: l₁, o, l₂)) = (a :: L, none, []) rw [h₁ a (List.Mem.head _), cond, IH L none [] _ ⟨rfl, rfl⟩] α : Type u_1 p : α → Bool a : α L : List α IH : ∀ (l₁ : List α) (o : Option α) (l₂ : List α), (∀ (x : α), x ∈ l₁ → p x = false) → Option.elim' (L = l₁ ∧ l₂ = []) (fun a => p a = true ∧ L = l₁ ++ a :: l₂) o → splitAtPred p L = (l₁, o, l₂) h₁ : ∀ (x : α), x ∈ a :: L → p x = false ⊢ ∀ (x : α), x ∈ L → p x = false exact fun x h => h₁ x (List.Mem.tail _ h) case some α : Type u_1 p : α → Bool a : α L l₁ l₂ : List α h₁ : ∀ (x : α), x ∈ l₁ → p x = false IH : ∀ (l₁ : List α) (o : Option α) (l₂ : List α), (∀ (x : α), x ∈ l₁ → p x = false) → Option.elim' (L = l₁ ∧ l₂ = []) (fun a => p a = true ∧ L = l₁ ++ a :: l₂) o → splitAtPred p L = (l₁, o, l₂) o : α h₂ : Option.elim' (a :: L = l₁ ∧ l₂ = []) (fun a_1 => p a_1 = true ∧ a :: L = l₁ ++ a_1 :: l₂) (some o) ⊢ (bif p a then ([], some a, L) else match splitAtPred p L with \| (l₁, o, l₂) => (a :: l₁, o, l₂)) = (l₁, some o, l₂) cases' l₁ with a' l₁ <;> rcases h₂ with ⟨h₂, ⟨⟩⟩ case some.cons.intro.refl α : Type u_1 p : α → Bool a : α l₂ : List α o : α l₁ : List α h₂ : p o = true h₁ : ∀ (x : α), x ∈ a :: l₁ → p x = false IH : ∀ (l₁_1 : List α) (o_1 : Option α) (l₂_1 : List α), (∀ (x : α), x ∈ l₁_1 → p x = false) → Option.elim' (List.append l₁ (o :: l₂) = l₁_1 ∧ l₂_1 = []) (fun a => p a = true ∧ List.append l₁ (o :: l₂) = l₁_1 ++ a :: l₂_1) o_1 → splitAtPred p (List.append l₁ (o :: l₂)) = (l₁_1, o_1, l₂_1) ⊢ (bif p a then ([], some a, List.append l₁ (o :: l₂)) else match splitAtPred p (List.append l₁ (o :: l₂)) with \| (l₁, o, l₂) => (a :: l₁, o, l₂)) = (a :: l₁, some o, l₂) rw [h₁ a (List.Mem.head _), cond, IH l₁ (some o) l₂ _ ⟨h₂, _⟩] <;> try rfl α : Type u_1 p : α → Bool a : α l₂ : List α o : α l₁ : List α h₂ : p o = true h₁ : ∀ (x : α), x ∈ a :: l₁ → p x = false IH : ∀ (l₁_1 : List α) (o_1 : Option α) (l₂_1 : List α), (∀ (x : α), x ∈ l₁_1 → p x = false) → Option.elim' (List.append l₁ (o :: l₂) = l₁_1 ∧ l₂_1 = []) (fun a => p a = true ∧ List.append l₁ (o :: l₂) = l₁_1 ++ a :: l₂_1) o_1 → splitAtPred p (List.append l₁ (o :: l₂)) = (l₁_1, o_1, l₂_1) ⊢ ∀ (x : α), x ∈ l₁ → p x = false exact fun x h => h₁ x (List.Mem.tail _ h) case some.nil.intro.refl α : Type u_1 p : α → Bool a : α L : List α IH : ∀ (l₁ : List α) (o : Option α) (l₂ : List α), (∀ (x : α), x ∈ l₁ → p x = false) → Option.elim' (L = l₁ ∧ l₂ = []) (fun a => p a = true ∧ L = l₁ ++ a :: l₂) o → splitAtPred p L = (l₁, o, l₂) h₁ : ∀ (x : α), x ∈ [] → p x = false h₂ : p a = true ⊢ (bif p a then ([], some a, L) else match splitAtPred p L with \| (l₁, o, l₂) => (a :: l₁, o, l₂)) = ([], some a, L) rw [h₂, cond] α : Type u_1 p : α → Bool a : α l₂ : List α o : α l₁ : List α h₂ : p o = true h₁ : ∀ (x : α), x ∈ a :: l₁ → p x = false IH : ∀ (l₁_1 : List α) (o_1 : Option α) (l₂_1 : List α), (∀ (x : α), x ∈ l₁_1 → p x = false) → Option.elim' (List.append l₁ (o :: l₂) = l₁_1 ∧ l₂_1 = []) (fun a => p a = true ∧ List.append l₁ (o :: l₂) = l₁_1 ++ a :: l₂_1) o_1 → splitAtPred p (List.append l₁ (o :: l₂)) = (l₁_1, o_1, l₂_1) ⊢ List.append l₁ (o :: l₂) = l₁ ++ o :: l₂ rfl ** Qed
Turing.PartrecToTM2.unrev_ok q : Λ' s : Option Γ' S : K' → List Γ' ⊢ rev ≠ main decide ** Qed
Turing.PartrecToTM2.move₂_ok p : Γ' → Bool k₁ k₂ : K' q : Λ' s : Option Γ' L₁ : List Γ' o : Option Γ' L₂ : List Γ' S : K' → List Γ' h₁ : k₁ ≠ rev ∧ k₂ ≠ rev ∧ k₁ ≠ k₂ h₂ : S rev = [] e : splitAtPred p (S k₁) = (L₁, o, L₂) ⊢ Reaches₁ (TM2.step tr) { l := some (move₂ p k₁ k₂ q), var := s, stk := S } { l := some q, var := none, stk := update (update S k₁ (Option.elim o id List.cons L₂)) k₂ (L₁ ++ S k₂) } refine' (move_ok h₁.1 e).trans (TransGen.head rfl _) p : Γ' → Bool k₁ k₂ : K' q : Λ' s : Option Γ' L₁ : List Γ' o : Option Γ' L₂ : List Γ' S : K' → List Γ' h₁ : k₁ ≠ rev ∧ k₂ ≠ rev ∧ k₁ ≠ k₂ h₂ : S rev = [] e : splitAtPred p (S k₁) = (L₁, o, L₂) ⊢ TransGen (fun a b => b ∈ TM2.step tr a) (TM2.stepAux (tr (Λ'.push k₁ id (Λ'.move (fun x => false) rev k₂ q))) o (update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev)))) { l := some q, var := none, stk := update (update S k₁ (Option.elim o id List.cons L₂)) k₂ (L₁ ++ S k₂) } simp only [TM2.step, Option.mem_def, TM2.stepAux, id_eq, ne_eq, Option.elim] p : Γ' → Bool k₁ k₂ : K' q : Λ' s : Option Γ' L₁ : List Γ' o : Option Γ' L₂ : List Γ' S : K' → List Γ' h₁ : k₁ ≠ rev ∧ k₂ ≠ rev ∧ k₁ ≠ k₂ h₂ : S rev = [] e : splitAtPred p (S k₁) = (L₁, o, L₂) ⊢ TransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif Option.isSome o then { l := some (Λ'.move (fun x => false) rev k₂ q), var := o, stk := update (update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev))) k₁ (Option.iget o :: update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev)) k₁) } else { l := some (Λ'.move (fun x => false) rev k₂ q), var := o, stk := update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev)) }) { l := some q, var := none, stk := update (update S k₁ ((match (motive := Option Γ' → (List Γ' → List Γ') → (Γ' → List Γ' → List Γ') → List Γ' → List Γ') o, id, List.cons with \| some x, x_1, f => f x \| none, y, x => y) L₂)) k₂ (L₁ ++ S k₂) } cases o <;> simp only [Option.elim, id.def] case none p : Γ' → Bool k₁ k₂ : K' q : Λ' s : Option Γ' L₁ L₂ : List Γ' S : K' → List Γ' h₁ : k₁ ≠ rev ∧ k₂ ≠ rev ∧ k₁ ≠ k₂ h₂ : S rev = [] e : splitAtPred p (S k₁) = (L₁, none, L₂) ⊢ TransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif Option.isSome none then { l := some (Λ'.move (fun x => false) rev k₂ q), var := none, stk := update (update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev))) k₁ (Option.iget none :: update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev)) k₁) } else { l := some (Λ'.move (fun x => false) rev k₂ q), var := none, stk := update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev)) }) { l := some q, var := none, stk := update (update S k₁ L₂) k₂ (L₁ ++ S k₂) } simp only [TM2.stepAux, Option.isSome, cond_false] case none p : Γ' → Bool k₁ k₂ : K' q : Λ' s : Option Γ' L₁ L₂ : List Γ' S : K' → List Γ' h₁ : k₁ ≠ rev ∧ k₂ ≠ rev ∧ k₁ ≠ k₂ h₂ : S rev = [] e : splitAtPred p (S k₁) = (L₁, none, L₂) ⊢ TransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) { l := some (Λ'.move (fun x => false) rev k₂ q), var := none, stk := update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev)) } { l := some q, var := none, stk := update (update S k₁ L₂) k₂ (L₁ ++ S k₂) } convert move_ok h₁.2.1.symm (splitAtPred_false _) using 2 case h.e'_2.h.e'_7 p : Γ' → Bool k₁ k₂ : K' q : Λ' s : Option Γ' L₁ L₂ : List Γ' S : K' → List Γ' h₁ : k₁ ≠ rev ∧ k₂ ≠ rev ∧ k₁ ≠ k₂ h₂ : S rev = [] e : splitAtPred p (S k₁) = (L₁, none, L₂) ⊢ update (update S k₁ L₂) k₂ (L₁ ++ S k₂) = update (update (update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev))) rev []) k₂ (List.reverseAux (update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev)) rev) (update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev)) k₂)) simp only [Function.update_comm h₁.1, Function.update_idem] case h.e'_2.h.e'_7 p : Γ' → Bool k₁ k₂ : K' q : Λ' s : Option Γ' L₁ L₂ : List Γ' S : K' → List Γ' h₁ : k₁ ≠ rev ∧ k₂ ≠ rev ∧ k₁ ≠ k₂ h₂ : S rev = [] e : splitAtPred p (S k₁) = (L₁, none, L₂) ⊢ update (update S k₁ L₂) k₂ (L₁ ++ S k₂) = update (update (update S rev []) k₁ L₂) k₂ (List.reverseAux (update (update S rev (List.reverseAux L₁ (S rev))) k₁ L₂ rev) (update (update S rev (List.reverseAux L₁ (S rev))) k₁ L₂ k₂)) rw [show update S rev [] = S by rw [← h₂, Function.update_eq_self]] case h.e'_2.h.e'_7 p : Γ' → Bool k₁ k₂ : K' q : Λ' s : Option Γ' L₁ L₂ : List Γ' S : K' → List Γ' h₁ : k₁ ≠ rev ∧ k₂ ≠ rev ∧ k₁ ≠ k₂ h₂ : S rev = [] e : splitAtPred p (S k₁) = (L₁, none, L₂) ⊢ update (update S k₁ L₂) k₂ (L₁ ++ S k₂) = update (update S k₁ L₂) k₂ (List.reverseAux (update (update S rev (List.reverseAux L₁ (S rev))) k₁ L₂ rev) (update (update S rev (List.reverseAux L₁ (S rev))) k₁ L₂ k₂)) simp only [Function.update_noteq h₁.2.2.symm, Function.update_noteq h₁.2.1, Function.update_noteq h₁.1.symm, List.reverseAux_eq, h₂, Function.update_same, List.append_nil, List.reverse_reverse] p : Γ' → Bool k₁ k₂ : K' q : Λ' s : Option Γ' L₁ L₂ : List Γ' S : K' → List Γ' h₁ : k₁ ≠ rev ∧ k₂ ≠ rev ∧ k₁ ≠ k₂ h₂ : S rev = [] e : splitAtPred p (S k₁) = (L₁, none, L₂) ⊢ update S rev [] = S rw [← h₂, Function.update_eq_self] case some p : Γ' → Bool k₁ k₂ : K' q : Λ' s : Option Γ' L₁ L₂ : List Γ' S : K' → List Γ' h₁ : k₁ ≠ rev ∧ k₂ ≠ rev ∧ k₁ ≠ k₂ h₂ : S rev = [] val✝ : Γ' e : splitAtPred p (S k₁) = (L₁, some val✝, L₂) ⊢ TransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif Option.isSome (some val✝) then { l := some (Λ'.move (fun x => false) rev k₂ q), var := some val✝, stk := update (update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev))) k₁ (Option.iget (some val✝) :: update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev)) k₁) } else { l := some (Λ'.move (fun x => false) rev k₂ q), var := some val✝, stk := update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev)) }) { l := some q, var := none, stk := update (update S k₁ (val✝ :: L₂)) k₂ (L₁ ++ S k₂) } simp only [TM2.stepAux, Option.isSome, cond_true] case some p : Γ' → Bool k₁ k₂ : K' q : Λ' s : Option Γ' L₁ L₂ : List Γ' S : K' → List Γ' h₁ : k₁ ≠ rev ∧ k₂ ≠ rev ∧ k₁ ≠ k₂ h₂ : S rev = [] val✝ : Γ' e : splitAtPred p (S k₁) = (L₁, some val✝, L₂) ⊢ TransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) { l := some (Λ'.move (fun x => false) rev k₂ q), var := some val✝, stk := update (update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev))) k₁ (Option.iget (some val✝) :: update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev)) k₁) } { l := some q, var := none, stk := update (update S k₁ (val✝ :: L₂)) k₂ (L₁ ++ S k₂) } convert move_ok h₁.2.1.symm (splitAtPred_false _) using 2 case h.e'_2.h.e'_7 p : Γ' → Bool k₁ k₂ : K' q : Λ' s : Option Γ' L₁ L₂ : List Γ' S : K' → List Γ' h₁ : k₁ ≠ rev ∧ k₂ ≠ rev ∧ k₁ ≠ k₂ h₂ : S rev = [] val✝ : Γ' e : splitAtPred p (S k₁) = (L₁, some val✝, L₂) ⊢ update (update S k₁ (val✝ :: L₂)) k₂ (L₁ ++ S k₂) = update (update (update (update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev))) k₁ (Option.iget (some val✝) :: update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev)) k₁)) rev []) k₂ (List.reverseAux (update (update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev))) k₁ (Option.iget (some val✝) :: update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev)) k₁) rev) (update (update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev))) k₁ (Option.iget (some val✝) :: update (update S k₁ L₂) rev (List.reverseAux L₁ (S rev)) k₁) k₂)) simp only [h₂, Function.update_comm h₁.1, List.reverseAux_eq, Function.update_same, List.append_nil, Function.update_idem] case h.e'_2.h.e'_7 p : Γ' → Bool k₁ k₂ : K' q : Λ' s : Option Γ' L₁ L₂ : List Γ' S : K' → List Γ' h₁ : k₁ ≠ rev ∧ k₂ ≠ rev ∧ k₁ ≠ k₂ h₂ : S rev = [] val✝ : Γ' e : splitAtPred p (S k₁) = (L₁, some val✝, L₂) ⊢ update (update S k₁ (val✝ :: L₂)) k₂ (L₁ ++ S k₂) = update (update (update S rev []) k₁ (Option.iget (some val✝) :: L₂)) k₂ (List.reverse (update (update S rev (List.reverse L₁)) k₁ (Option.iget (some val✝) :: L₂) rev) ++ update (update S rev (List.reverse L₁)) k₁ (Option.iget (some val✝) :: L₂) k₂) rw [show update S rev [] = S by rw [← h₂, Function.update_eq_self]] case h.e'_2.h.e'_7 p : Γ' → Bool k₁ k₂ : K' q : Λ' s : Option Γ' L₁ L₂ : List Γ' S : K' → List Γ' h₁ : k₁ ≠ rev ∧ k₂ ≠ rev ∧ k₁ ≠ k₂ h₂ : S rev = [] val✝ : Γ' e : splitAtPred p (S k₁) = (L₁, some val✝, L₂) ⊢ update (update S k₁ (val✝ :: L₂)) k₂ (L₁ ++ S k₂) = update (update S k₁ (Option.iget (some val✝) :: L₂)) k₂ (List.reverse (update (update S rev (List.reverse L₁)) k₁ (Option.iget (some val✝) :: L₂) rev) ++ update (update S rev (List.reverse L₁)) k₁ (Option.iget (some val✝) :: L₂) k₂) simp only [Function.update_noteq h₁.1.symm, Function.update_noteq h₁.2.2.symm, Function.update_noteq h₁.2.1, Function.update_same, List.reverse_reverse] p : Γ' → Bool k₁ k₂ : K' q : Λ' s : Option Γ' L₁ L₂ : List Γ' S : K' → List Γ' h₁ : k₁ ≠ rev ∧ k₂ ≠ rev ∧ k₁ ≠ k₂ h₂ : S rev = [] val✝ : Γ' e : splitAtPred p (S k₁) = (L₁, some val✝, L₂) ⊢ update S rev [] = S rw [← h₂, Function.update_eq_self] ** Qed
Turing.PartrecToTM2.clear_ok p : Γ' → Bool k : K' q : Λ' s : Option Γ' L₁ : List Γ' o : Option Γ' L₂ : List Γ' S : K' → List Γ' e : splitAtPred p (S k) = (L₁, o, L₂) ⊢ Reaches₁ (TM2.step tr) { l := some (Λ'.clear p k q), var := s, stk := S } { l := some q, var := o, stk := update S k L₂ } induction' L₁ with a L₁ IH generalizing S s case nil p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e✝ : splitAtPred p (S✝ k) = (L₁, o, L₂) s : Option Γ' S : K' → List Γ' e : splitAtPred p (S k) = ([], o, L₂) ⊢ Reaches₁ (TM2.step tr) { l := some (Λ'.clear p k q), var := s, stk := S } { l := some q, var := o, stk := update S k L₂ } refine' TransGen.head' rfl _ case nil p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e✝ : splitAtPred p (S✝ k) = (L₁, o, L₂) s : Option Γ' S : K' → List Γ' e : splitAtPred p (S k) = ([], o, L₂) ⊢ ReflTransGen (fun a b => b ∈ TM2.step tr a) (TM2.stepAux (tr (Λ'.clear p k q)) s S) { l := some q, var := o, stk := update S k L₂ } simp only [TM2.step, Option.mem_def, TM2.stepAux, Option.elim] case nil p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e✝ : splitAtPred p (S✝ k) = (L₁, o, L₂) s : Option Γ' S : K' → List Γ' e : splitAtPred p (S k) = ([], o, L₂) ⊢ ReflTransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif match List.head? (S k), true, p with \| some x, x_1, f => f x \| none, y, x => y then { l := some q, var := List.head? (S k), stk := update S k (List.tail (S k)) } else { l := some (Λ'.clear p k q), var := List.head? (S k), stk := update S k (List.tail (S k)) }) { l := some q, var := o, stk := update S k L₂ } revert e case nil p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e : splitAtPred p (S✝ k) = (L₁, o, L₂) s : Option Γ' S : K' → List Γ' ⊢ splitAtPred p (S k) = ([], o, L₂) → ReflTransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif match List.head? (S k), true, p with \| some x, x_1, f => f x \| none, y, x => y then { l := some q, var := List.head? (S k), stk := update S k (List.tail (S k)) } else { l := some (Λ'.clear p k q), var := List.head? (S k), stk := update S k (List.tail (S k)) }) { l := some q, var := o, stk := update S k L₂ } cases' S k with a Sk <;> intro e case nil.cons p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e✝ : splitAtPred p (S✝ k) = (L₁, o, L₂) s : Option Γ' S : K' → List Γ' a : Γ' Sk : List Γ' e : splitAtPred p (a :: Sk) = ([], o, L₂) ⊢ ReflTransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif match List.head? (a :: Sk), true, p with \| some x, x_1, f => f x \| none, y, x => y then { l := some q, var := List.head? (a :: Sk), stk := update S k (List.tail (a :: Sk)) } else { l := some (Λ'.clear p k q), var := List.head? (a :: Sk), stk := update S k (List.tail (a :: Sk)) }) { l := some q, var := o, stk := update S k L₂ } simp only [splitAtPred, Option.elim, List.head?, List.tail_cons] at e ⊢ case nil.cons p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e✝ : splitAtPred p (S✝ k) = (L₁, o, L₂) s : Option Γ' S : K' → List Γ' a : Γ' Sk : List Γ' e : (bif p a then ([], some a, Sk) else (a :: (splitAtPred p Sk).1, (splitAtPred p Sk).2.1, (splitAtPred p Sk).2.2)) = ([], o, L₂) ⊢ ReflTransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif p a then { l := some q, var := some a, stk := update S k Sk } else { l := some (Λ'.clear p k q), var := some a, stk := update S k Sk }) { l := some q, var := o, stk := update S k L₂ } revert e case nil.cons p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e : splitAtPred p (S✝ k) = (L₁, o, L₂) s : Option Γ' S : K' → List Γ' a : Γ' Sk : List Γ' ⊢ (bif p a then ([], some a, Sk) else (a :: (splitAtPred p Sk).1, (splitAtPred p Sk).2.1, (splitAtPred p Sk).2.2)) = ([], o, L₂) → ReflTransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif p a then { l := some q, var := some a, stk := update S k Sk } else { l := some (Λ'.clear p k q), var := some a, stk := update S k Sk }) { l := some q, var := o, stk := update S k L₂ } cases p a <;> intro e <;> simp only [cond_false, cond_true, Prod.mk.injEq, true_and, false_and] at e ⊢ case nil.nil p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e✝ : splitAtPred p (S✝ k) = (L₁, o, L₂) s : Option Γ' S : K' → List Γ' e : splitAtPred p [] = ([], o, L₂) ⊢ ReflTransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif match List.head? [], true, p with \| some x, x_1, f => f x \| none, y, x => y then { l := some q, var := List.head? [], stk := update S k (List.tail []) } else { l := some (Λ'.clear p k q), var := List.head? [], stk := update S k (List.tail []) }) { l := some q, var := o, stk := update S k L₂ } cases e case nil.nil.refl p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁ : List Γ' S✝ : K' → List Γ' s : Option Γ' S : K' → List Γ' e : splitAtPred p (S✝ k) = (L₁, none, []) ⊢ ReflTransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif match List.head? [], true, p with \| some x, x_1, f => f x \| none, y, x => y then { l := some q, var := List.head? [], stk := update S k (List.tail []) } else { l := some (Λ'.clear p k q), var := List.head? [], stk := update S k (List.tail []) }) { l := some q, var := none, stk := update S k [] } rfl case nil.cons.true p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e✝ : splitAtPred p (S✝ k) = (L₁, o, L₂) s : Option Γ' S : K' → List Γ' a : Γ' Sk : List Γ' e : some a = o ∧ Sk = L₂ ⊢ ReflTransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) { l := some q, var := some a, stk := update S k Sk } { l := some q, var := o, stk := update S k L₂ } rcases e with ⟨e₁, e₂⟩ case nil.cons.true.intro p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e : splitAtPred p (S✝ k) = (L₁, o, L₂) s : Option Γ' S : K' → List Γ' a : Γ' Sk : List Γ' e₁ : some a = o e₂ : Sk = L₂ ⊢ ReflTransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) { l := some q, var := some a, stk := update S k Sk } { l := some q, var := o, stk := update S k L₂ } rw [e₁, e₂] case cons p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁✝ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e✝ : splitAtPred p (S✝ k) = (L₁✝, o, L₂) a : Γ' L₁ : List Γ' IH : ∀ {s : Option Γ'} {S : K' → List Γ'}, splitAtPred p (S k) = (L₁, o, L₂) → Reaches₁ (TM2.step tr) { l := some (Λ'.clear p k q), var := s, stk := S } { l := some q, var := o, stk := update S k L₂ } s : Option Γ' S : K' → List Γ' e : splitAtPred p (S k) = (a :: L₁, o, L₂) ⊢ Reaches₁ (TM2.step tr) { l := some (Λ'.clear p k q), var := s, stk := S } { l := some q, var := o, stk := update S k L₂ } refine' TransGen.head rfl _ case cons p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁✝ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e✝ : splitAtPred p (S✝ k) = (L₁✝, o, L₂) a : Γ' L₁ : List Γ' IH : ∀ {s : Option Γ'} {S : K' → List Γ'}, splitAtPred p (S k) = (L₁, o, L₂) → Reaches₁ (TM2.step tr) { l := some (Λ'.clear p k q), var := s, stk := S } { l := some q, var := o, stk := update S k L₂ } s : Option Γ' S : K' → List Γ' e : splitAtPred p (S k) = (a :: L₁, o, L₂) ⊢ TransGen (fun a b => b ∈ TM2.step tr a) (TM2.stepAux (tr (Λ'.clear p k q)) s S) { l := some q, var := o, stk := update S k L₂ } simp only [TM2.step, Option.mem_def, TM2.stepAux, Option.elim] case cons p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁✝ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e✝ : splitAtPred p (S✝ k) = (L₁✝, o, L₂) a : Γ' L₁ : List Γ' IH : ∀ {s : Option Γ'} {S : K' → List Γ'}, splitAtPred p (S k) = (L₁, o, L₂) → Reaches₁ (TM2.step tr) { l := some (Λ'.clear p k q), var := s, stk := S } { l := some q, var := o, stk := update S k L₂ } s : Option Γ' S : K' → List Γ' e : splitAtPred p (S k) = (a :: L₁, o, L₂) ⊢ TransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif match List.head? (S k), true, p with \| some x, x_1, f => f x \| none, y, x => y then { l := some q, var := List.head? (S k), stk := update S k (List.tail (S k)) } else { l := some (Λ'.clear p k q), var := List.head? (S k), stk := update S k (List.tail (S k)) }) { l := some q, var := o, stk := update S k L₂ } cases' e₁ : S k with a' Sk <;> rw [e₁, splitAtPred] at e case cons.cons p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁✝ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e✝ : splitAtPred p (S✝ k) = (L₁✝, o, L₂) a : Γ' L₁ : List Γ' IH : ∀ {s : Option Γ'} {S : K' → List Γ'}, splitAtPred p (S k) = (L₁, o, L₂) → Reaches₁ (TM2.step tr) { l := some (Λ'.clear p k q), var := s, stk := S } { l := some q, var := o, stk := update S k L₂ } s : Option Γ' S : K' → List Γ' a' : Γ' Sk : List Γ' e : (bif p a' then ([], some a', Sk) else match splitAtPred p Sk with \| (l₁, o, l₂) => (a' :: l₁, o, l₂)) = (a :: L₁, o, L₂) e₁ : S k = a' :: Sk ⊢ TransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif match List.head? (a' :: Sk), true, p with \| some x, x_1, f => f x \| none, y, x => y then { l := some q, var := List.head? (a' :: Sk), stk := update S k (List.tail (a' :: Sk)) } else { l := some (Λ'.clear p k q), var := List.head? (a' :: Sk), stk := update S k (List.tail (a' :: Sk)) }) { l := some q, var := o, stk := update S k L₂ } cases e₂ : p a' <;> simp only [e₂, cond] at e case cons.cons.false p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁✝ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e✝ : splitAtPred p (S✝ k) = (L₁✝, o, L₂) a : Γ' L₁ : List Γ' IH : ∀ {s : Option Γ'} {S : K' → List Γ'}, splitAtPred p (S k) = (L₁, o, L₂) → Reaches₁ (TM2.step tr) { l := some (Λ'.clear p k q), var := s, stk := S } { l := some q, var := o, stk := update S k L₂ } s : Option Γ' S : K' → List Γ' a' : Γ' Sk : List Γ' e₁ : S k = a' :: Sk e₂ : p a' = false e : (a' :: (splitAtPred p Sk).1, (splitAtPred p Sk).2.1, (splitAtPred p Sk).2.2) = (a :: L₁, o, L₂) ⊢ TransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif match List.head? (a' :: Sk), true, p with \| some x, x_1, f => f x \| none, y, x => y then { l := some q, var := List.head? (a' :: Sk), stk := update S k (List.tail (a' :: Sk)) } else { l := some (Λ'.clear p k q), var := List.head? (a' :: Sk), stk := update S k (List.tail (a' :: Sk)) }) { l := some q, var := o, stk := update S k L₂ } case cons.cons.true p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁✝ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e✝ : splitAtPred p (S✝ k) = (L₁✝, o, L₂) a : Γ' L₁ : List Γ' IH : ∀ {s : Option Γ'} {S : K' → List Γ'}, splitAtPred p (S k) = (L₁, o, L₂) → Reaches₁ (TM2.step tr) { l := some (Λ'.clear p k q), var := s, stk := S } { l := some q, var := o, stk := update S k L₂ } s : Option Γ' S : K' → List Γ' a' : Γ' Sk : List Γ' e₁ : S k = a' :: Sk e₂ : p a' = true e : ([], some a', Sk) = (a :: L₁, o, L₂) ⊢ TransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif match List.head? (a' :: Sk), true, p with \| some x, x_1, f => f x \| none, y, x => y then { l := some q, var := List.head? (a' :: Sk), stk := update S k (List.tail (a' :: Sk)) } else { l := some (Λ'.clear p k q), var := List.head? (a' :: Sk), stk := update S k (List.tail (a' :: Sk)) }) { l := some q, var := o, stk := update S k L₂ } swap case cons.cons.false p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁✝ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e✝ : splitAtPred p (S✝ k) = (L₁✝, o, L₂) a : Γ' L₁ : List Γ' IH : ∀ {s : Option Γ'} {S : K' → List Γ'}, splitAtPred p (S k) = (L₁, o, L₂) → Reaches₁ (TM2.step tr) { l := some (Λ'.clear p k q), var := s, stk := S } { l := some q, var := o, stk := update S k L₂ } s : Option Γ' S : K' → List Γ' a' : Γ' Sk : List Γ' e₁ : S k = a' :: Sk e₂ : p a' = false e : (a' :: (splitAtPred p Sk).1, (splitAtPred p Sk).2.1, (splitAtPred p Sk).2.2) = (a :: L₁, o, L₂) ⊢ TransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif match List.head? (a' :: Sk), true, p with \| some x, x_1, f => f x \| none, y, x => y then { l := some q, var := List.head? (a' :: Sk), stk := update S k (List.tail (a' :: Sk)) } else { l := some (Λ'.clear p k q), var := List.head? (a' :: Sk), stk := update S k (List.tail (a' :: Sk)) }) { l := some q, var := o, stk := update S k L₂ } rcases e₃ : splitAtPred p Sk with ⟨_, _, _⟩ case cons.cons.false.mk.mk p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁✝ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e✝ : splitAtPred p (S✝ k) = (L₁✝, o, L₂) a : Γ' L₁ : List Γ' IH : ∀ {s : Option Γ'} {S : K' → List Γ'}, splitAtPred p (S k) = (L₁, o, L₂) → Reaches₁ (TM2.step tr) { l := some (Λ'.clear p k q), var := s, stk := S } { l := some q, var := o, stk := update S k L₂ } s : Option Γ' S : K' → List Γ' a' : Γ' Sk : List Γ' e₁ : S k = a' :: Sk e₂ : p a' = false e : (a' :: (splitAtPred p Sk).1, (splitAtPred p Sk).2.1, (splitAtPred p Sk).2.2) = (a :: L₁, o, L₂) fst✝¹ : List Γ' fst✝ : Option Γ' snd✝ : List Γ' e₃ : splitAtPred p Sk = (fst✝¹, fst✝, snd✝) ⊢ TransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif match List.head? (a' :: Sk), true, p with \| some x, x_1, f => f x \| none, y, x => y then { l := some q, var := List.head? (a' :: Sk), stk := update S k (List.tail (a' :: Sk)) } else { l := some (Λ'.clear p k q), var := List.head? (a' :: Sk), stk := update S k (List.tail (a' :: Sk)) }) { l := some q, var := o, stk := update S k L₂ } rw [e₃] at e case cons.cons.false.mk.mk p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁✝ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e✝ : splitAtPred p (S✝ k) = (L₁✝, o, L₂) a : Γ' L₁ : List Γ' IH : ∀ {s : Option Γ'} {S : K' → List Γ'}, splitAtPred p (S k) = (L₁, o, L₂) → Reaches₁ (TM2.step tr) { l := some (Λ'.clear p k q), var := s, stk := S } { l := some q, var := o, stk := update S k L₂ } s : Option Γ' S : K' → List Γ' a' : Γ' Sk : List Γ' e₁ : S k = a' :: Sk e₂ : p a' = false fst✝¹ : List Γ' fst✝ : Option Γ' snd✝ : List Γ' e : (a' :: (fst✝¹, fst✝, snd✝).1, (fst✝¹, fst✝, snd✝).2.1, (fst✝¹, fst✝, snd✝).2.2) = (a :: L₁, o, L₂) e₃ : splitAtPred p Sk = (fst✝¹, fst✝, snd✝) ⊢ TransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif match List.head? (a' :: Sk), true, p with \| some x, x_1, f => f x \| none, y, x => y then { l := some q, var := List.head? (a' :: Sk), stk := update S k (List.tail (a' :: Sk)) } else { l := some (Λ'.clear p k q), var := List.head? (a' :: Sk), stk := update S k (List.tail (a' :: Sk)) }) { l := some q, var := o, stk := update S k L₂ } cases e case cons.cons.false.mk.mk.refl p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁ : List Γ' S✝ : K' → List Γ' a : Γ' s : Option Γ' S : K' → List Γ' Sk fst✝¹ : List Γ' fst✝ : Option Γ' snd✝ : List Γ' e₃ : splitAtPred p Sk = (fst✝¹, fst✝, snd✝) e₁ : S k = a :: Sk e₂ : p a = false e : splitAtPred p (S✝ k) = (L₁, (fst✝¹, fst✝, snd✝).2.1, (fst✝¹, fst✝, snd✝).2.2) IH : ∀ {s : Option Γ'} {S : K' → List Γ'}, splitAtPred p (S k) = ((fst✝¹, fst✝, snd✝).1, (fst✝¹, fst✝, snd✝).2.1, (fst✝¹, fst✝, snd✝).2.2) → Reaches₁ (TM2.step tr) { l := some (Λ'.clear p k q), var := s, stk := S } { l := some q, var := (fst✝¹, fst✝, snd✝).2.1, stk := update S k (fst✝¹, fst✝, snd✝).2.2 } ⊢ TransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif match List.head? (a :: Sk), true, p with \| some x, x_1, f => f x \| none, y, x => y then { l := some q, var := List.head? (a :: Sk), stk := update S k (List.tail (a :: Sk)) } else { l := some (Λ'.clear p k q), var := List.head? (a :: Sk), stk := update S k (List.tail (a :: Sk)) }) { l := some q, var := (fst✝¹, fst✝, snd✝).2.1, stk := update S k (fst✝¹, fst✝, snd✝).2.2 } simp only [List.head?_cons, e₂, List.tail_cons, cond_false] case cons.cons.false.mk.mk.refl p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁ : List Γ' S✝ : K' → List Γ' a : Γ' s : Option Γ' S : K' → List Γ' Sk fst✝¹ : List Γ' fst✝ : Option Γ' snd✝ : List Γ' e₃ : splitAtPred p Sk = (fst✝¹, fst✝, snd✝) e₁ : S k = a :: Sk e₂ : p a = false e : splitAtPred p (S✝ k) = (L₁, (fst✝¹, fst✝, snd✝).2.1, (fst✝¹, fst✝, snd✝).2.2) IH : ∀ {s : Option Γ'} {S : K' → List Γ'}, splitAtPred p (S k) = ((fst✝¹, fst✝, snd✝).1, (fst✝¹, fst✝, snd✝).2.1, (fst✝¹, fst✝, snd✝).2.2) → Reaches₁ (TM2.step tr) { l := some (Λ'.clear p k q), var := s, stk := S } { l := some q, var := (fst✝¹, fst✝, snd✝).2.1, stk := update S k (fst✝¹, fst✝, snd✝).2.2 } ⊢ TransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) { l := some (Λ'.clear p k q), var := some a, stk := update S k Sk } { l := some q, var := fst✝, stk := update S k snd✝ } convert @IH _ (update S k Sk) _ using 2 <;> simp [e₃] case cons.nil p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁✝ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e✝ : splitAtPred p (S✝ k) = (L₁✝, o, L₂) a : Γ' L₁ : List Γ' IH : ∀ {s : Option Γ'} {S : K' → List Γ'}, splitAtPred p (S k) = (L₁, o, L₂) → Reaches₁ (TM2.step tr) { l := some (Λ'.clear p k q), var := s, stk := S } { l := some q, var := o, stk := update S k L₂ } s : Option Γ' S : K' → List Γ' e : ([], none, []) = (a :: L₁, o, L₂) e₁ : S k = [] ⊢ TransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif match List.head? [], true, p with \| some x, x_1, f => f x \| none, y, x => y then { l := some q, var := List.head? [], stk := update S k (List.tail []) } else { l := some (Λ'.clear p k q), var := List.head? [], stk := update S k (List.tail []) }) { l := some q, var := o, stk := update S k L₂ } cases e case cons.cons.true p : Γ' → Bool k : K' q : Λ' s✝ : Option Γ' L₁✝ : List Γ' o : Option Γ' L₂ : List Γ' S✝ : K' → List Γ' e✝ : splitAtPred p (S✝ k) = (L₁✝, o, L₂) a : Γ' L₁ : List Γ' IH : ∀ {s : Option Γ'} {S : K' → List Γ'}, splitAtPred p (S k) = (L₁, o, L₂) → Reaches₁ (TM2.step tr) { l := some (Λ'.clear p k q), var := s, stk := S } { l := some q, var := o, stk := update S k L₂ } s : Option Γ' S : K' → List Γ' a' : Γ' Sk : List Γ' e₁ : S k = a' :: Sk e₂ : p a' = true e : ([], some a', Sk) = (a :: L₁, o, L₂) ⊢ TransGen (fun a b => (match a with \| { l := none, var := var, stk := stk } => none \| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif match List.head? (a' :: Sk), true, p with \| some x, x_1, f => f x \| none, y, x => y then { l := some q, var := List.head? (a' :: Sk), stk := update S k (List.tail (a' :: Sk)) } else { l := some (Λ'.clear p k q), var := List.head? (a' :: Sk), stk := update S k (List.tail (a' :: Sk)) }) { l := some q, var := o, stk := update S k L₂ } cases e ** Qed