module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 375,
"column": 58
} | {
"line": 380,
"column": 26
} | {
"line": 382,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝¹ : CountablyGenerated γ\nκ : Kernel α (γ × β)\nν : Kernel α γ\nhκν : κ.fst ≤ ν\ninst✝ : IsFiniteKernel ν\na : α\ns : Set β\nhs : MeasurableSet s\n⊢ ∀ᵐ (x : γ) ∂ν a,\n Tendsto (fun ... | [] | by
refine Submartingale.ae_tendsto_limitProcess (martingale_densityProcess hκν a hs).submartingale
(R := (ν a univ).toNNReal) (fun n ↦ ?_)
refine (eLpNorm_densityProcess_le hκν n a s).trans_eq ?_
rw [ENNReal.coe_toNNReal]
exact measure_ne_top _ _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.CondDistrib | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 56
} | {
"line": 124,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nΩ : Type u_3\nF : Type u_4\ninst✝⁵ : MeasurableSpace Ω\ninst✝⁴ : StandardBorelSpace Ω\ninst✝³ : Nonempty Ω\ninst✝² : NormedAddCommGroup F\nmα : MeasurableSpace α\nμ : Measure α\ninst✝¹ : IsFiniteMeasure μ\nX : α → β\nY : α → Ω\nmβ : MeasurableSpace β\nf : β × Ω → F\ninst✝ : ... | [] | rw [condDistrib]; exact hf.integral_kernel_prod_right' | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.CondDistrib | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 56
} | {
"line": 124,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nΩ : Type u_3\nF : Type u_4\ninst✝⁵ : MeasurableSpace Ω\ninst✝⁴ : StandardBorelSpace Ω\ninst✝³ : Nonempty Ω\ninst✝² : NormedAddCommGroup F\nmα : MeasurableSpace α\nμ : Measure α\ninst✝¹ : IsFiniteMeasure μ\nX : α → β\nY : α → Ω\nmβ : MeasurableSpace β\nf : β × Ω → F\ninst✝ : ... | [] | rw [condDistrib]; exact hf.integral_kernel_prod_right' | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Disintegration.Unique | {
"line": 110,
"column": 72
} | {
"line": 110,
"column": 75
} | {
"line": 111,
"column": 6
} | [
{
"pp": "α : Type u_1\nΩ : Type u_3\nmα : MeasurableSpace α\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : StandardBorelSpace Ω\ninst✝² : Nonempty Ω\nρ : Measure (α × Ω)\ninst✝¹ : IsFiniteMeasure ρ\nκ : Kernel α Ω\ninst✝ : IsFiniteKernel κ\nhκ : ρ = ρ.fst ⊗ₘ κ\nf : Ω → ℝ := embeddingReal Ω\nhf : MeasurableEmbedding (emb... | [
"α : Type u_1\nΩ : Type u_3\nmα : MeasurableSpace α\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : StandardBorelSpace Ω\ninst✝² : Nonempty Ω\nρ : Measure (α × Ω)\ninst✝¹ : IsFiniteMeasure ρ\nκ : Kernel α Ω\ninst✝ : IsFiniteKernel κ\nhκ : ρ = ρ.fst ⊗ₘ κ\nf : Ω → ℝ := embeddingReal Ω\nhf : MeasurableEmbedding (embeddingReal Ω... | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 605,
"column": 21
} | {
"line": 605,
"column": 33
} | {
"line": 605,
"column": 34
} | [
{
"pp": "case e'_5\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝¹ : CountablyGenerated γ\nκ : Kernel α (γ × β)\nν : Kernel α γ\nhκν : κ.fst ≤ ν\ninst✝ : IsFiniteKernel ν\na : α\nseq : ℕ → Set β\nhseq : Monotone seq\nhseq_iUnion : ⋃ i, se... | [
"case e'_5\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝¹ : CountablyGenerated γ\nκ : Kernel α (γ × β)\nν : Kernel α γ\nhκν : κ.fst ≤ ν\ninst✝ : IsFiniteKernel ν\na : α\nseq : ℕ → Set β\nhseq : Monotone seq\nhseq_iUnion : ⋃ i, seq i = univ\n... | hseq_iUnion, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj | {
"line": 201,
"column": 33
} | {
"line": 201,
"column": 80
} | {
"line": 202,
"column": 4
} | [
{
"pp": "X : Type u_2\nY : Type u_3\nZ : Type u_4\nmX : MeasurableSpace X\nmY : MeasurableSpace Y\nmZ : MeasurableSpace Z\nκ : Kernel X Y\ninst✝¹ : IsSFiniteKernel κ\nη : Kernel (X × Y) Z\ninst✝ : IsSFiniteKernel η\nx : X\ns : Set (X × Z)\nms : MeasurableSet s\n⊢ ∫⁻ (b : X × Y), ((deterministic Prod.fst ⋯ ×ₖ η)... | [
"X : Type u_2\nY : Type u_3\nZ : Type u_4\nmX : MeasurableSpace X\nmY : MeasurableSpace Y\nmZ : MeasurableSpace Z\nκ : Kernel X Y\ninst✝¹ : IsSFiniteKernel κ\nη : Kernel (X × Y) Z\ninst✝ : IsSFiniteKernel η\nx : X\ns : Set (X × Z)\nms : MeasurableSet s\n⊢ ∫⁻ (b : Y), ((deterministic Prod.fst ⋯ ×ₖ η) (x, b)) s ∂κ x ... | lintegral_id_prod (Kernel.measurable_coe _ ms), | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Probability.Kernel.CondDistrib | {
"line": 352,
"column": 10
} | {
"line": 352,
"column": 38
} | {
"line": 352,
"column": 38
} | [
{
"pp": "case refine_2\nα : Type u_1\nβ : Type u_2\nΩ : Type u_3\nF : Type u_4\ninst✝⁶ : MeasurableSpace Ω\ninst✝⁵ : StandardBorelSpace Ω\ninst✝⁴ : Nonempty Ω\ninst✝³ : NormedAddCommGroup F\nmα : MeasurableSpace α\nμ : Measure α\ninst✝² : IsFiniteMeasure μ\nX : α → β\nY : α → Ω\nmβ : MeasurableSpace β\nf : β × ... | [
"case refine_2\nα : Type u_1\nβ : Type u_2\nΩ : Type u_3\nF : Type u_4\ninst✝⁶ : MeasurableSpace Ω\ninst✝⁵ : StandardBorelSpace Ω\ninst✝⁴ : Nonempty Ω\ninst✝³ : NormedAddCommGroup F\nmα : MeasurableSpace α\nμ : Measure α\ninst✝² : IsFiniteMeasure μ\nX : α → β\nY : α → Ω\nmβ : MeasurableSpace β\nf : β × Ω → F\ninst✝... | ← Measure.restrict_map hX ht | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 667,
"column": 2
} | {
"line": 672,
"column": 36
} | {
"line": 673,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝¹ : CountablyGenerated γ\nκ : Kernel α (γ × β)\ninst✝ : IsFiniteKernel κ\nn : ℕ\na : α\nthis : {x | ¬κ.densityProcess κ.fst n a x univ = 1} ⊆ {x | (κ.fst a) (countablePartitionSet n x)... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝¹ : CountablyGenerated γ\nκ : Kernel α (γ × β)\ninst✝ : IsFiniteKernel κ\nn : ℕ\na : α\nthis✝ : {x | ¬κ.densityProcess κ.fst n a x univ = 1} ⊆ {x | (κ.fst a) (countablePartitionSet n x) = 0}\nthis... | have : {x | fst κ a (countablePartitionSet n x) = 0}
⊆ ⋃ (u) (_ : u ∈ countablePartition γ n) (_ : fst κ a u = 0), u := by
intro t ht
simp only [mem_setOf_eq, mem_iUnion, exists_prop] at ht ⊢
exact ⟨countablePartitionSet n t, countablePartitionSet_mem _ _, ht,
mem_countablePartitionSet _ _⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 248,
"column": 61
} | {
"line": 248,
"column": 81
} | {
"line": 248,
"column": 81
} | [
{
"pp": "X : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → MeasurableSpace (X n)\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\ninst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)\nb : ℕ\nS : Set ((i : ↥(Iic b)) → X ↑i)\nmS : MeasurableSet S\nx₀ : (n : ℕ) → X n\na : ℕ\n⊢ ∫⁻ (a : (i : ↥(Iic b)) → X ↑i), S.indicator 1 a ∂(par... | [
"X : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → MeasurableSpace (X n)\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\ninst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)\nb : ℕ\nS : Set ((i : ↥(Iic b)) → X ↑i)\nmS : MeasurableSet S\nx₀ : (n : ℕ) → X n\na : ℕ\n⊢ ∫⁻ (a : (i : ↥(Iic b)) → X ↑i), S.indicator 1 a ∂(partialTraj κ a... | lmarginalPartialTraj | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 291,
"column": 2
} | {
"line": 299,
"column": 19
} | {
"line": 301,
"column": 2
} | [
{
"pp": "X : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → MeasurableSpace (X n)\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\ninst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)\nf : ℕ → ((n : ℕ) → X n) → ℝ≥0∞\na : ℕ → ℕ\nhcte : ∀ (n : ℕ), DependsOn (f n) ↑(Iic (a n))\nmf : ∀ (n : ℕ), Measurable (f n)\nbound : ℝ≥0∞\nfin_b... | [
"X : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → MeasurableSpace (X n)\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\ninst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)\nf : ℕ → ((n : ℕ) → X n) → ℝ≥0∞\na : ℕ → ℕ\nhcte : ∀ (n : ℕ), DependsOn (f n) ↑(Iic (a n))\nmf : ∀ (n : ℕ), Measurable (f n)\nbound : ℝ≥0∞\nfin_bound : bound... | have f_eq x n : lmarginalPartialTraj κ k (a n) (f n) x =
lmarginalPartialTraj κ k (k + 1) (F n) x := by
simp_rw [F]
obtain h | h | h := lt_trichotomy (k + 1) (a n)
· rw [← lmarginalPartialTraj_self k.le_succ h.le (mf n)]
· rw [← h, lmarginalPartialTraj_le _ le_rfl (mf n)]
· rw [lmarginalPartia... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 307,
"column": 19
} | {
"line": 307,
"column": 39
} | {
"line": 307,
"column": 39
} | [
{
"pp": "X : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → MeasurableSpace (X n)\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\ninst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)\nf : ℕ → ((n : ℕ) → X n) → ℝ≥0∞\na : ℕ → ℕ\nhcte : ∀ (n : ℕ), DependsOn (f n) ↑(Iic (a n))\nmf : ∀ (n : ℕ), Measurable (f n)\nbound : ℝ≥0∞\nfin_b... | [
"X : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → MeasurableSpace (X n)\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\ninst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)\nf : ℕ → ((n : ℕ) → X n) → ℝ≥0∞\na : ℕ → ℕ\nhcte : ∀ (n : ℕ), DependsOn (f n) ↑(Iic (a n))\nmf : ∀ (n : ℕ), Measurable (f n)\nbound : ℝ≥0∞\nfin_bound : bound... | lmarginalPartialTraj | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Probability.ProductMeasure | {
"line": 255,
"column": 55
} | {
"line": 255,
"column": 70
} | {
"line": 255,
"column": 71
} | [
{
"pp": "ι : Type u_1\nX : ι → Type u_2\nmX : (i : ι) → MeasurableSpace (X i)\nμ : (i : ι) → Measure (X i)\nhμ : ∀ (i : ι), IsProbabilityMeasure (μ i)\ne : ℕ ≃ ι\nI : Finset ι\nS : Set ((i : ↥I) → X ↑i)\nhS : MeasurableSet S\nhs : cylinder I S ∈ measurableCylinders X\n⊢ (Measure.pi fun i ↦ μ (e ↑i)) (⇑(Equiv.pi... | [
"ι : Type u_1\nX : ι → Type u_2\nmX : (i : ι) → MeasurableSpace (X i)\nμ : (i : ι) → Measure (X i)\nhμ : ∀ (i : ι), IsProbabilityMeasure (μ i)\ne : ℕ ≃ ι\nI : Finset ι\nS : Set ((i : ↥I) → X ↑i)\nhS : MeasurableSet S\nhs : cylinder I S ∈ measurableCylinders X\n⊢ (Measure.pi fun i ↦ μ (e ↑i)) (⇑(Equiv.piCongrLeft (f... | map_apply _ hS, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.ProductMeasure | {
"line": 276,
"column": 2
} | {
"line": 276,
"column": 29
} | {
"line": 278,
"column": 2
} | [
{
"pp": "ι : Type u_1\nX : ι → Type u_2\nmX : (i : ι) → MeasurableSpace (X i)\nμ : (i : ι) → Measure (X i)\nhμ : ∀ (i : ι), IsProbabilityMeasure (μ i)\nA : ℕ → Set ((i : ι) → X i)\nA_mem : ∀ (n : ℕ), A n ∈ measurableCylinders X\nA_anti : Antitone A\nA_inter : ⋂ n, A n = ∅\nthis : ∀ (i : ι), Nonempty (X i)\ns : ... | [
"ι : Type u_1\nX : ι → Type u_2\nmX : (i : ι) → MeasurableSpace (X i)\nμ : (i : ι) → Measure (X i)\nhμ : ∀ (i : ι), IsProbabilityMeasure (μ i)\nA : ℕ → Set ((i : ι) → X i)\nA_mem : ∀ (n : ℕ), A n ∈ measurableCylinders X\nA_anti : Antitone A\nA_inter : ⋂ n, A n = ∅\nthis : ∀ (i : ι), Nonempty (X i)\ns : ℕ → Finset ι... | let u := ⋃ n, (s n : Set ι) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 622,
"column": 11
} | {
"line": 622,
"column": 34
} | {
"line": 622,
"column": 35
} | [
{
"pp": "α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\ninst✝¹ : IsFiniteKernel κ\nf : α → γ → ℝ≥0∞\ninst✝ : IsFiniteKernel (κ.withDensity f)\nhf : Measurable (Function.uncurry f)\na : α\nh_ae : (κ.withDensity ... | [
"α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\ninst✝¹ : IsFiniteKernel κ\nf : α → γ → ℝ≥0∞\ninst✝ : IsFiniteKernel (κ.withDensity f)\nhf : Measurable (Function.uncurry f)\na : α\nh_ae : (κ.withDensity f).rnDeriv κ... | κ.withDensity_apply hf, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Posterior | {
"line": 122,
"column": 39
} | {
"line": 122,
"column": 64
} | {
"line": 124,
"column": 0
} | [
{
"pp": "Ω : Type u_1\n𝓧 : Type u_2\nmΩ : MeasurableSpace Ω\nm𝓧 : MeasurableSpace 𝓧\nκ : Kernel Ω 𝓧\nμ : Measure Ω\ninst✝⁴ : IsFiniteMeasure μ\ninst✝³ : IsFiniteKernel κ\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\nη : Kernel 𝓧 Ω\ninst✝ : IsFiniteKernel η\nh : (⇑κ ∘ₘ μ) ⊗ₘ η = ⇑(Kernel.swap Ω 𝓧) ∘... | [] | rw [h, Measure.swap_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Kernel.Posterior | {
"line": 122,
"column": 39
} | {
"line": 122,
"column": 64
} | {
"line": 124,
"column": 0
} | [
{
"pp": "Ω : Type u_1\n𝓧 : Type u_2\nmΩ : MeasurableSpace Ω\nm𝓧 : MeasurableSpace 𝓧\nκ : Kernel Ω 𝓧\nμ : Measure Ω\ninst✝⁴ : IsFiniteMeasure μ\ninst✝³ : IsFiniteKernel κ\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\nη : Kernel 𝓧 Ω\ninst✝ : IsFiniteKernel η\nh : (⇑κ ∘ₘ μ) ⊗ₘ η = ⇑(Kernel.swap Ω 𝓧) ∘... | [] | rw [h, Measure.swap_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Posterior | {
"line": 122,
"column": 39
} | {
"line": 122,
"column": 64
} | {
"line": 124,
"column": 0
} | [
{
"pp": "Ω : Type u_1\n𝓧 : Type u_2\nmΩ : MeasurableSpace Ω\nm𝓧 : MeasurableSpace 𝓧\nκ : Kernel Ω 𝓧\nμ : Measure Ω\ninst✝⁴ : IsFiniteMeasure μ\ninst✝³ : IsFiniteKernel κ\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\nη : Kernel 𝓧 Ω\ninst✝ : IsFiniteKernel η\nh : (⇑κ ∘ₘ μ) ⊗ₘ η = ⇑(Kernel.swap Ω 𝓧) ∘... | [] | rw [h, Measure.swap_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Decision.Risk.Basic | {
"line": 119,
"column": 2
} | {
"line": 119,
"column": 26
} | {
"line": 120,
"column": 2
} | [
{
"pp": "case h\nΘ : Type u_1\n𝓧 : Type u_2\n𝓨 : Type u_4\nmΘ : MeasurableSpace Θ\nm𝓧 : MeasurableSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\nℓ : Θ → 𝓨 → ℝ≥0∞\nhl : Measurable (uncurry ℓ)\nμ : Measure 𝓧\ninst✝¹ : SFinite μ\nπ : Measure Θ\ninst✝ : SFinite π\nhl_pos : μ Set.univ = ∞ → ⨅ y, ∫⁻ (θ : Θ), ℓ θ y ∂π = 0 →... | [
"case h.hf\nΘ : Type u_1\n𝓧 : Type u_2\n𝓨 : Type u_4\nmΘ : MeasurableSpace Θ\nm𝓧 : MeasurableSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\nℓ : Θ → 𝓨 → ℝ≥0∞\nhl : Measurable (uncurry ℓ)\nμ : Measure 𝓧\ninst✝¹ : SFinite μ\nπ : Measure Θ\ninst✝ : SFinite π\nhl_pos : μ Set.univ = ∞ → ⨅ y, ∫⁻ (θ : Θ), ℓ θ y ∂π = 0 → ∃ y, ∫⁻ ... | rw [lintegral_mul_const] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Kernel.Posterior | {
"line": 307,
"column": 11
} | {
"line": 307,
"column": 14
} | {
"line": 307,
"column": 15
} | [
{
"pp": "𝓧 : Type u_2\nm𝓧 : MeasurableSpace 𝓧\nΩ : Type u_5\ninst✝⁵ : Countable Ω\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : Nonempty Ω\ninst✝² : StandardBorelSpace Ω\nκ : Kernel Ω 𝓧\ninst✝¹ : IsFiniteKernel κ\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nh_rnDeriv : ∀ᵐ (a : 𝓧) ∂⇑κ ∘ₘ μ, ∀ (i : Ω), κ.rnDeriv (Kern... | [
"𝓧 : Type u_2\nm𝓧 : MeasurableSpace 𝓧\nΩ : Type u_5\ninst✝⁵ : Countable Ω\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : Nonempty Ω\ninst✝² : StandardBorelSpace Ω\nκ : Kernel Ω 𝓧\ninst✝¹ : IsFiniteKernel κ\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nh_rnDeriv : ∀ᵐ (a : 𝓧) ∂⇑κ ∘ₘ μ, ∀ (i : Ω), κ.rnDeriv (Kernel.const Ω (... | hx, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Probability.Decision.BayesEstimator | {
"line": 126,
"column": 75
} | {
"line": 130,
"column": 55
} | {
"line": 132,
"column": 0
} | [
{
"pp": "Θ : Type u_1\n𝓧 : Type u_2\n𝓨 : Type u_3\nmΘ : MeasurableSpace Θ\nm𝓧 : MeasurableSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\nℓ : Θ → 𝓨 → ℝ≥0∞\nP : Kernel Θ 𝓧\nπ : Measure Θ\ninst✝³ : StandardBorelSpace Θ\ninst✝² : Nonempty Θ\nf : 𝓧 → 𝓨\ninst✝¹ : IsFiniteKernel P\ninst✝ : IsFiniteMeasure π\nhf : IsArgmin... | [] | by
rw [avgRisk_eq_lintegral_lintegral_lintegral hl]
refine lintegral_congr_ae ?_
filter_upwards [hf.property] with x hx
rwa [Kernel.lintegral_deterministic' _ (by fun_prop)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Distributions.Gamma | {
"line": 114,
"column": 19
} | {
"line": 114,
"column": 33
} | {
"line": 114,
"column": 34
} | [
{
"pp": "a r : ℝ\nha : 0 < a\nhr : 0 < r\nleftSide : ∫⁻ (x : ℝ) in Iio 0, gammaPDF a r x = 0\nrightSide :\n ∫⁻ (x : ℝ) in Ici 0, gammaPDF a r x =\n ∫⁻ (x : ℝ) in Ici 0, ENNReal.ofReal (r ^ a / Gamma a * x ^ (a - 1) * rexp (-(r * x)))\n⊢ r ^ a * (1 / r) ^ a * Gamma a / Gamma a = 1",
"ppTerm": "?m.199",
... | [
"a r : ℝ\nha : 0 < a\nhr : 0 < r\nleftSide : ∫⁻ (x : ℝ) in Iio 0, gammaPDF a r x = 0\nrightSide :\n ∫⁻ (x : ℝ) in Ici 0, gammaPDF a r x =\n ∫⁻ (x : ℝ) in Ici 0, ENNReal.ofReal (r ^ a / Gamma a * x ^ (a - 1) * rexp (-(r * x)))\n⊢ r ^ a * (1 / r) ^ a * (Gamma a / Gamma a) = 1"
] | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.ProbabilityMassFunction.Basic | {
"line": 105,
"column": 2
} | {
"line": 118,
"column": 88
} | {
"line": 120,
"column": 0
} | [
{
"pp": "α : Type u_1\np : PMF α\na : α\nh : p a = 1\na' : α\nha' : a' ∈ p.support\nha : a' ∉ {a}\n⊢ 1 < ∑' (a : α), p a",
"ppTerm": "?m.87",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"Iff.mpr",
"Eq.mpr",
"ENNReal.instAdd",
"Trans.trans",... | [] | classical
have : 0 < ∑' b, ite (b = a) 0 (p b) := by
rw [pos_iff_ne_zero, ENNReal.summable.tsum_ne_zero_iff]
exact ⟨a', ite_ne_left_iff.2 ⟨ha, Ne.symm <| (p.mem_support_iff a').2 ha'⟩⟩
calc
1 = 1 + 0 := (add_zero 1).symm
_ < p a + ∑' b, ite (b = a) 0 (p b) :=
(ENNReal.add_lt_add_of_le_of_lt EN... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Probability.Distributions.Gamma | {
"line": 103,
"column": 32
} | {
"line": 120,
"column": 54
} | {
"line": 122,
"column": 0
} | [
{
"pp": "a r : ℝ\nha : 0 < a\nhr : 0 < r\n⊢ ∫⁻ (x : ℝ), gammaPDF a r x = 1",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Iff.mpr",
"MeasureTheory.AEStronglyMeasurable.congr",
"ProbabilityTheory.gammaPDF_of_nonneg",
"Real.instIsOrderedRin... | [] | by
have leftSide : ∫⁻ x in Iio 0, gammaPDF a r x = 0 := by
rw [setLIntegral_congr_fun measurableSet_Iio
(fun x (hx : x < 0) ↦ gammaPDF_of_neg hx), lintegral_zero]
have rightSide : ∫⁻ x in Ici 0, gammaPDF a r x =
∫⁻ x in Ici 0, ENNReal.ofReal (r ^ a / Gamma a * x ^ (a - 1) * exp (-(r * x))) :=
se... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Distributions.Poisson.PoissonLimitThm | {
"line": 73,
"column": 6
} | {
"line": 73,
"column": 20
} | {
"line": 73,
"column": 21
} | [
{
"pp": "p : ℕ → ℝ\nr : ℝ\nk : ℕ\nhr : Tendsto (fun n ↦ ↑n * p n) atTop (𝓝 r)\n⊢ Tendsto (fun n ↦ ↑(n.choose k) * p n ^ k * (1 - p n) ^ (n - k)) atTop (𝓝 (Real.exp (-r) * r ^ k / ↑k.factorial))",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Nat.choos... | [
"p : ℕ → ℝ\nr : ℝ\nk : ℕ\nhr : Tendsto (fun n ↦ ↑n * p n) atTop (𝓝 r)\n⊢ Tendsto (fun n ↦ ↑(n.choose k) * p n ^ k * (1 - p n) ^ (n - k)) atTop (𝓝 (Real.exp (-r) * (r ^ k / ↑k.factorial)))"
] | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Independence.InfinitePi | {
"line": 131,
"column": 61
} | {
"line": 131,
"column": 83
} | {
"line": 131,
"column": 83
} | [
{
"pp": "ι : Type u_1\n𝓧 : ι → Type u_3\nm𝓧 : (i : ι) → MeasurableSpace (𝓧 i)\nΩ : ι → Type u_4\nmΩ : (i : ι) → MeasurableSpace (Ω i)\nP : (i : ι) → Measure (Ω i)\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (P i)\nX : (i : ι) → Ω i → 𝓧 i\nmX : ∀ (i : ι), Measurable (X i)\n⊢ map (fun ω i ↦ X i (ω i)) (infiniteP... | [
"ι : Type u_1\n𝓧 : ι → Type u_3\nm𝓧 : (i : ι) → MeasurableSpace (𝓧 i)\nΩ : ι → Type u_4\nmΩ : (i : ι) → MeasurableSpace (Ω i)\nP : (i : ι) → Measure (Ω i)\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (P i)\nX : (i : ι) → Ω i → 𝓧 i\nmX : ∀ (i : ι), Measurable (X i)\n⊢ (infinitePi fun i ↦ map (X i) (P i)) = infiniteP... | infinitePi_map_pi _ mX | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Independence.InfinitePi | {
"line": 180,
"column": 6
} | {
"line": 180,
"column": 70
} | {
"line": 180,
"column": 70
} | [
{
"pp": "ι : Type u_1\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nκ : ι → Type u_4\n𝓧 : (i : ι) → κ i → Type u_5\nm𝓧 : (i : ι) → (j : κ i) → MeasurableSpace (𝓧 i j)\nX : (i : ι) → (j : κ i) → Ω → 𝓧 i j\nmX : ∀ (i : ι) (j : κ i), Measurable (X i j)\nh1 : iIndepFun (fun i ω x ↦ X i x ω) P\nh2 : ∀ (i... | [
"ι : Type u_1\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nκ : ι → Type u_4\n𝓧 : (i : ι) → κ i → Type u_5\nm𝓧 : (i : ι) → (j : κ i) → MeasurableSpace (𝓧 i j)\nX : (i : ι) → (j : κ i) → Ω → 𝓧 i j\nmX : ∀ (i : ι) (j : κ i), Measurable (X i j)\nh1 : iIndepFun (fun i ω x ↦ X i x ω) P\nh2 : ∀ (i : ι), iInde... | (iIndepFun_iff_map_fun_eq_infinitePi_map (by fun_prop)).1 (h2 i) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Distributions.Uniform | {
"line": 155,
"column": 4
} | {
"line": 156,
"column": 75
} | {
"line": 157,
"column": 2
} | [
{
"pp": "case neg.left\nΩ : Type u_2\nx✝ : MeasurableSpace Ω\nℙ : Measure Ω\nX : Ω → ℝ\ns : Set ℝ\nhcs : IsCompact s\nhuX : IsUniform X s ℙ volume\nhnt : ¬volume s = 0 ∧ ¬volume s = ∞\nthis : IsProbabilityMeasure ℙ\n⊢ AEStronglyMeasurable (fun x ↦ x * (pdf X ℙ volume x).toReal) volume",
"ppTerm": "?neg.left... | [] | exact aestronglyMeasurable_id.mul
(measurable_pdf X ℙ).aemeasurable.ennreal_toReal.aestronglyMeasurable | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Distributions.Uniform | {
"line": 155,
"column": 4
} | {
"line": 156,
"column": 75
} | {
"line": 157,
"column": 2
} | [
{
"pp": "case neg.left\nΩ : Type u_2\nx✝ : MeasurableSpace Ω\nℙ : Measure Ω\nX : Ω → ℝ\ns : Set ℝ\nhcs : IsCompact s\nhuX : IsUniform X s ℙ volume\nhnt : ¬volume s = 0 ∧ ¬volume s = ∞\nthis : IsProbabilityMeasure ℙ\n⊢ AEStronglyMeasurable (fun x ↦ x * (pdf X ℙ volume x).toReal) volume",
"ppTerm": "?neg.left... | [] | exact aestronglyMeasurable_id.mul
(measurable_pdf X ℙ).aemeasurable.ennreal_toReal.aestronglyMeasurable | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Distributions.Uniform | {
"line": 155,
"column": 4
} | {
"line": 156,
"column": 75
} | {
"line": 157,
"column": 2
} | [
{
"pp": "case neg.left\nΩ : Type u_2\nx✝ : MeasurableSpace Ω\nℙ : Measure Ω\nX : Ω → ℝ\ns : Set ℝ\nhcs : IsCompact s\nhuX : IsUniform X s ℙ volume\nhnt : ¬volume s = 0 ∧ ¬volume s = ∞\nthis : IsProbabilityMeasure ℙ\n⊢ AEStronglyMeasurable (fun x ↦ x * (pdf X ℙ volume x).toReal) volume",
"ppTerm": "?neg.left... | [] | exact aestronglyMeasurable_id.mul
(measurable_pdf X ℙ).aemeasurable.ennreal_toReal.aestronglyMeasurable | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Distributions.Uniform | {
"line": 171,
"column": 6
} | {
"line": 171,
"column": 20
} | {
"line": 171,
"column": 21
} | [
{
"pp": "Ω : Type u_2\nx✝ : MeasurableSpace Ω\nℙ : Measure Ω\nX : Ω → ℝ\ns : Set ℝ\nhuX : IsUniform X s ℙ volume\n⊢ ∫ (x : Ω), X x ∂ℙ = (volume s)⁻¹.toReal * ∫ (x : ℝ) in s, x",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Re... | [
"Ω : Type u_2\nx✝ : MeasurableSpace Ω\nℙ : Measure Ω\nX : Ω → ℝ\ns : Set ℝ\nhuX : IsUniform X s ℙ volume\n⊢ ∫ (x : Ω), X x ∂ℙ = (volume s)⁻¹.toReal • ∫ (x : ℝ) in s, x"
] | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Distributions.Uniform | {
"line": 219,
"column": 4
} | {
"line": 223,
"column": 70
} | {
"line": 224,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\ns : Finset α\nhs : s.Nonempty\n⊢ (∑ a ∈ s, if a ∈ s then (↑(#s))⁻¹ else 0) = 1",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"PMF.uniformOfFinset._simp_2",
"Finset.sum_ite_mem",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidW... | [] | simp only [Finset.sum_ite_mem, Finset.inter_self, Finset.sum_const, nsmul_eq_mul]
have : (s.card : ℝ≥0∞) ≠ 0 := by
simpa only [Ne, Nat.cast_eq_zero, Finset.card_eq_zero] using
Finset.nonempty_iff_ne_empty.1 hs
exact ENNReal.mul_inv_cancel this <| ENNReal.natCast_ne_top s.card | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Distributions.Uniform | {
"line": 219,
"column": 4
} | {
"line": 223,
"column": 70
} | {
"line": 224,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\ns : Finset α\nhs : s.Nonempty\n⊢ (∑ a ∈ s, if a ∈ s then (↑(#s))⁻¹ else 0) = 1",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"PMF.uniformOfFinset._simp_2",
"Finset.sum_ite_mem",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidW... | [] | simp only [Finset.sum_ite_mem, Finset.inter_self, Finset.sum_const, nsmul_eq_mul]
have : (s.card : ℝ≥0∞) ≠ 0 := by
simpa only [Ne, Nat.cast_eq_zero, Finset.card_eq_zero] using
Finset.nonempty_iff_ne_empty.1 hs
exact ENNReal.mul_inv_cancel this <| ENNReal.natCast_ne_top s.card | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Condexp | {
"line": 305,
"column": 6
} | {
"line": 305,
"column": 53
} | {
"line": 306,
"column": 4
} | [
{
"pp": "case neg.refine_1\nΩ : Type u_1\nF : Type u_2\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝³ : IsFiniteMeasure μ\ns : Set Ω\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nhs : MeasurableSet s\nf : Ω → F\nhf : Integrable f μ\nhμs : ¬μ s = 0\nt : Set Ω\n⊢ ‖∫ (x : Ω), f... | [] | exact Or.inr <| measure_lt_top (μ.restrict s) t | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Kernel.Category.Stoch | {
"line": 111,
"column": 4
} | {
"line": 111,
"column": 59
} | {
"line": 112,
"column": 4
} | [
{
"pp": "X Y : Stoch\nκ✝ : X ⟶ Y\nX✝ Y✝ Z✝ : Stoch\nκ : X✝ ⟶ Y✝\nη : Y✝ ⟶ Z✝\nx✝ : Deterministic (κ ≫ η)\n⊢ η.hom.hom ∥ₖ Kernel.id ∘ₖ copy Y✝.obj.carrier ∘ₖ κ.hom.hom =\n η.hom.hom ∘ₖ κ.hom.hom ∥ₖ κ.hom.hom ∘ₖ copy X✝.obj.carrier",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Ca... | [
"X Y : Stoch\nκ✝ : X ⟶ Y\nX✝ Y✝ Z✝ : Stoch\nκ : X✝ ⟶ Y✝\nη : Y✝ ⟶ Z✝\nx✝ : Deterministic (κ ≫ η)\nthis : IsDeterministic (κ ≫ η).hom.hom\n⊢ η.hom.hom ∥ₖ Kernel.id ∘ₖ copy Y✝.obj.carrier ∘ₖ κ.hom.hom =\n η.hom.hom ∘ₖ κ.hom.hom ∥ₖ κ.hom.hom ∘ₖ copy X✝.obj.carrier"
] | have : IsDeterministic (κ ≫ η).hom.hom := inferInstance | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.Independence.BoundedContinuousFunction | {
"line": 135,
"column": 2
} | {
"line": 135,
"column": 76
} | {
"line": 136,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nT : Type u_3\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nF : T → Type u_5\nG : Type u_6\ninst✝⁹ : (t : T) → TopologicalSpace (F t)\ninst✝⁸ : (t : T) → MeasurableSpace (F t)\ninst✝⁷ : ∀ (t : T), BorelSpace (F t)\ninst✝⁶ : ∀ (t : T), HasOuterApproxClosed (F t)\ninst✝⁵ : TopologicalSpace G\ninst... | [
"Ω : Type u_1\nT : Type u_3\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nF : T → Type u_5\nG : Type u_6\ninst✝⁹ : (t : T) → TopologicalSpace (F t)\ninst✝⁸ : (t : T) → MeasurableSpace (F t)\ninst✝⁷ : ∀ (t : T), BorelSpace (F t)\ninst✝⁶ : ∀ (t : T), HasOuterApproxClosed (F t)\ninst✝⁵ : TopologicalSpace G\ninst✝⁴ : Measura... | refine eq_prod_of_integral_mul_prod_boundedContinuousFunction fun f g ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Probability.Kernel.Proper | {
"line": 101,
"column": 50
} | {
"line": 101,
"column": 97
} | {
"line": 102,
"column": 6
} | [
{
"pp": "case refine_1\nX : Type u_1\n𝓑 𝓧 : MeasurableSpace X\nπ : Kernel X X\nB : Set X\nf : X → ℝ≥0∞\nx₀ : X\nhπ : π.IsProper\nh𝓑𝓧 : 𝓑 ≤ 𝓧\nhf : Measurable f\nhB : MeasurableSet B\nc : ℝ≥0∞\nA : Set X\nhA : MeasurableSet A\n⊢ c * ∫⁻ (a : X), B.indicator 1 a * A.indicator 1 a ∂π x₀ = B.indicator 1 x₀ * (... | [
"case refine_1\nX : Type u_1\n𝓑 𝓧 : MeasurableSpace X\nπ : Kernel X X\nB : Set X\nf : X → ℝ≥0∞\nx₀ : X\nhπ : π.IsProper\nh𝓑𝓧 : 𝓑 ≤ 𝓧\nhf : Measurable f\nhB : MeasurableSet B\nc : ℝ≥0∞\nA : Set X\nhA : MeasurableSet A\n⊢ c * (B.indicator 1 x₀ * ∫⁻ (x : X), A.indicator 1 x ∂π x₀) =\n B.indicator 1 x₀ * (c * ... | hπ.lintegral_indicator_mul_indicator h𝓑𝓧 hA hB, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Independence.Conditional | {
"line": 257,
"column": 6
} | {
"line": 257,
"column": 37
} | {
"line": 257,
"column": 37
} | [
{
"pp": "Ω : Type u_1\nm' mΩ : MeasurableSpace Ω\ninst✝¹ : StandardBorelSpace Ω\nhm' : m' ≤ mΩ\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\ns t : Set Ω\nhs : MeasurableSet s\nht : MeasurableSet t\n⊢ CondIndepSets m' hm' {s} {t} μ ↔\n μ[(s ∩ t).indicator fun ω ↦ 1 | m'] =ᵐ[μ] μ[s.indicator fun ω ↦ 1 | m'] * μ[t... | [
"Ω : Type u_1\nm' mΩ : MeasurableSpace Ω\ninst✝¹ : StandardBorelSpace Ω\nhm' : m' ≤ mΩ\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\ns t : Set Ω\nhs : MeasurableSet s\nht : MeasurableSet t\n⊢ (∀ (t1 t2 : Set Ω),\n t1 ∈ {s} →\n t2 ∈ {t} →\n μ[(t1 ∩ t2).indicator fun ω ↦ 1 | m'] =ᵐ[μ] μ[t1.indicator... | condIndepSets_iff _ _ _ _ ?_ ?_ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Independence.Conditional | {
"line": 298,
"column": 4
} | {
"line": 298,
"column": 13
} | {
"line": 299,
"column": 2
} | [
{
"pp": "case hs1\nΩ : Type u_1\nm' m₁ m₂ mΩ : MeasurableSpace Ω\ninst✝¹ : StandardBorelSpace Ω\nhm' : m' ≤ mΩ\nhm₁ : m₁ ≤ mΩ\nhm₂ : m₂ ≤ mΩ\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\n⊢ ∀ s ∈ {s | MeasurableSet s}, MeasurableSet s",
"ppTerm": "?hs1",
"assigned": true,
"usedConstants": [],
"usedF... | [] | exact hm₁ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Independence.Conditional | {
"line": 298,
"column": 4
} | {
"line": 298,
"column": 13
} | {
"line": 299,
"column": 2
} | [
{
"pp": "case hs1\nΩ : Type u_1\nm' m₁ m₂ mΩ : MeasurableSpace Ω\ninst✝¹ : StandardBorelSpace Ω\nhm' : m' ≤ mΩ\nhm₁ : m₁ ≤ mΩ\nhm₂ : m₂ ≤ mΩ\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\n⊢ ∀ s ∈ {s | MeasurableSet s}, MeasurableSet s",
"ppTerm": "?hs1",
"assigned": true,
"usedConstants": [],
"usedF... | [] | exact hm₁ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Independence.Conditional | {
"line": 298,
"column": 4
} | {
"line": 298,
"column": 13
} | {
"line": 299,
"column": 2
} | [
{
"pp": "case hs1\nΩ : Type u_1\nm' m₁ m₂ mΩ : MeasurableSpace Ω\ninst✝¹ : StandardBorelSpace Ω\nhm' : m' ≤ mΩ\nhm₁ : m₁ ≤ mΩ\nhm₂ : m₂ ≤ mΩ\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\n⊢ ∀ s ∈ {s | MeasurableSet s}, MeasurableSet s",
"ppTerm": "?hs1",
"assigned": true,
"usedConstants": [],
"usedF... | [] | exact hm₁ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Representation | {
"line": 95,
"column": 6
} | {
"line": 99,
"column": 11
} | {
"line": 100,
"column": 2
} | [
{
"pp": "case neg.refine_2\nX : Type u_1\nmX : MeasurableSpace X\nκ : Kernel X ↑I\ninst✝ : IsMarkovKernel κ\nf : X → ↑I → ↑I := fun s t ↦ sSup {x | (κ s).real (Icc 0 x) < ↑t}\nmeasurable_f : Measurable (uncurry f)\na : X\nx : ↑I\nIic_to_Icc : Iic x = Icc 0 x\nκ_in_I : (κ a).real (Icc 0 x) ∈ I\nξ : ↑I\nhξ : f a ... | [] | refine eventually_nhdsWithin_of_forall fun y hy ↦ ?_
by_contra! h
simp only [sSup_le_iff, f] at hξ
specialize hξ y h
grind | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Representation | {
"line": 95,
"column": 6
} | {
"line": 99,
"column": 11
} | {
"line": 100,
"column": 2
} | [
{
"pp": "case neg.refine_2\nX : Type u_1\nmX : MeasurableSpace X\nκ : Kernel X ↑I\ninst✝ : IsMarkovKernel κ\nf : X → ↑I → ↑I := fun s t ↦ sSup {x | (κ s).real (Icc 0 x) < ↑t}\nmeasurable_f : Measurable (uncurry f)\na : X\nx : ↑I\nIic_to_Icc : Iic x = Icc 0 x\nκ_in_I : (κ a).real (Icc 0 x) ∈ I\nξ : ↑I\nhξ : f a ... | [] | refine eventually_nhdsWithin_of_forall fun y hy ↦ ?_
by_contra! h
simp only [sSup_le_iff, f] at hξ
specialize hξ y h
grind | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Martingale.OptionalSampling | {
"line": 184,
"column": 6
} | {
"line": 184,
"column": 83
} | {
"line": 185,
"column": 6
} | [
{
"pp": "case refine_2\nΩ : Type u_1\nE : Type u_2\nm : MeasurableSpace Ω\nμ : Measure Ω\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace ℝ E\ninst✝¹¹ : CompleteSpace E\nι : Type u_3\ninst✝¹⁰ : LinearOrder ι\ninst✝⁹ : LocallyFiniteOrder ι\ninst✝⁸ : OrderBot ι\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : Discret... | [
"case refine_2\nΩ : Type u_1\nE : Type u_2\nm : MeasurableSpace Ω\nμ : Measure Ω\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace ℝ E\ninst✝¹¹ : CompleteSpace E\nι : Type u_3\ninst✝¹⁰ : LinearOrder ι\ninst✝⁹ : LocallyFiniteOrder ι\ninst✝⁸ : OrderBot ι\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : DiscreteTopology ι\... | refine StronglyMeasurable.indicator ?_ (hτ.measurableSet_le_stopping_time hσ) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Probability.Process.Kolmogorov | {
"line": 115,
"column": 69
} | {
"line": 115,
"column": 87
} | {
"line": 117,
"column": 0
} | [
{
"pp": "T : Type u_1\nΩ : Type u_2\nE : Type u_3\ninst✝¹ : PseudoEMetricSpace T\nmΩ : MeasurableSpace Ω\ninst✝ : PseudoEMetricSpace E\np q : ℝ\nM : ℝ≥0\nP : Measure Ω\nX : T → Ω → E\nhX : IsAEKolmogorovProcess X P p q M\ns t : T\nω : Ω\nhω₁ : X s ω = IsAEKolmogorovProcess.mk X hX s ω\nhω₂ : X t ω = IsAEKolmogo... | [] | by simp [hω₁, hω₂] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Moments.SubGaussian | {
"line": 346,
"column": 33
} | {
"line": 346,
"column": 48
} | {
"line": 348,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nh✝ : HasSubgaussianMGF X c κ ν\nε : ℝ\nhε : 0 ≤ ε\nhc0 : ¬c = 0\nω' : Ω'\nh : ∀ (t : ℝ), 0 ≤ t → (κ ω').real {ω | ε ≤ X ω} ≤ rexp (-t * ε + ↑c * t ^ 2 / 2)\n⊢ rexp (-(ε / ... | [] | by congr; field | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Process.LocalProperty | {
"line": 178,
"column": 6
} | {
"line": 178,
"column": 38
} | {
"line": 178,
"column": 39
} | [
{
"pp": "ι : Type u_1\nΩ : Type u_2\nE : Type u_3\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝⁴ : LinearOrder ι\n𝓕 : Filtration ι mΩ\nX : ι → Ω → E\np✝ q✝ : (ι → Ω → E) → Prop\ninst✝³ : OrderBot ι\ninst✝² : Zero E\ninst✝¹ : TopologicalSpace ι\ninst✝ : OrderTopology ι\nhp✝ : IsStable 𝓕 p✝\nhq✝ : IsStable 𝓕 q... | [
"ι : Type u_1\nΩ : Type u_2\nE : Type u_3\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝⁴ : LinearOrder ι\n𝓕 : Filtration ι mΩ\nX : ι → Ω → E\np✝ q✝ : (ι → Ω → E) → Prop\ninst✝³ : OrderBot ι\ninst✝² : Zero E\ninst✝¹ : TopologicalSpace ι\ninst✝ : OrderTopology ι\nhp✝ : IsStable 𝓕 p✝\nhq✝ : IsStable 𝓕 q✝\nx✝ : Loca... | ← stoppedProcess_stoppedProcess, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Process.LocalProperty | {
"line": 213,
"column": 23
} | {
"line": 223,
"column": 94
} | {
"line": 225,
"column": 0
} | [
{
"pp": "ι : Type u_1\nΩ : Type u_2\nE : Type u_3\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝⁷ : ConditionallyCompleteLinearOrderBot ι\ninst✝⁶ : TopologicalSpace ι\ninst✝⁵ : OrderTopology ι\n𝓕 : Filtration ι mΩ\nX : ι → Ω → E\np : (ι → Ω → E) → Prop\ninst✝⁴ : Zero E\ninst✝³ : DenselyOrdered ι\ninst✝² : First... | [] | by
refine ⟨_, hτ.isLocalizingSequence_biInf, fun n ↦ ?_⟩
rw [stoppedProcess_indicator_comm', ← stoppedProcess_stoppedProcess_of_le_right
(τ := fun ω ↦ τ n ω) (fun _ ↦ (iInf_le _ n).trans <| iInf_le _ le_rfl),
← stoppedProcess_indicator_comm']
convert!
hp _ (hpτ n) (fun ω ↦ ⨅ j ≥ n, τ j ω) <| hτ.isLoca... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RepresentationTheory.Subrepresentation | {
"line": 64,
"column": 6
} | {
"line": 65,
"column": 93
} | {
"line": 67,
"column": 0
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nW : Type u_3\nM : Type u_4\ninst✝⁵ : Semiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid W\ninst✝² : Module A W\nρ : Representation A G W\ninst✝¹ : AddCommMonoid M\ninst✝ : Module A[G] M\nρ₁ ρ₂ : Subrepresentation ρ\ng : G\nx₁ : W\nhx₁ : x₁ ∈ ρ₁.toSubmodule\nx₂ : W\nhx₂ : ... | [] | exact Submodule.mem_sup.mpr
⟨ρ g x₁, ρ₁.apply_mem_toSubmodule g hx₁, ρ g x₂, ρ₂.apply_mem_toSubmodule g hx₂, rfl⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RepresentationTheory.Subrepresentation | {
"line": 129,
"column": 23
} | {
"line": 129,
"column": 37
} | {
"line": 129,
"column": 38
} | [
{
"pp": "case single\nA : Type u_1\nG : Type u_2\nW : Type u_3\nM : Type u_4\ninst✝⁵ : CommSemiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid W\ninst✝² : Module A W\nρ : Representation A G W\ninst✝¹ : AddCommMonoid M\ninst✝ : Module A[G] M\nσ : Subrepresentation (Representation.ofModule M)\nm : M\nhm : m ∈ σ... | [
"case single\nA : Type u_1\nG : Type u_2\nW : Type u_3\nM : Type u_4\ninst✝⁵ : CommSemiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid W\ninst✝² : Module A W\nρ : Representation A G W\ninst✝¹ : AddCommMonoid M\ninst✝ : Module A[G] M\nσ : Subrepresentation (Representation.ofModule M)\nm : M\nhm : m ∈ σ.toSubmodule... | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.StrongLaw | {
"line": 159,
"column": 6
} | {
"line": 168,
"column": 30
} | {
"line": 169,
"column": 4
} | [
{
"pp": "case inr\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : AEStronglyMeasurable f μ\nA : ℝ\nn : ℕ\nhn : n ≠ 0\nh'f : 0 ≤ f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : A < 0\n⊢ ∫ (y : ℝ), (Set.Ioc 0 A).indicator id y ^ n ∂Measure.map f μ =\n -∫ (x : ℝ),... | [] | simp only [Set.Ioc_eq_empty_of_le hA.le, zero_pow hn, Set.indicator_empty, integral_zero,
zero_eq_neg]
apply integral_eq_zero_of_ae
have : ∀ᵐ x ∂Measure.map f μ, (0 : ℝ) ≤ x :=
(ae_map_iff hf.aemeasurable measurableSet_Ici).2 (Eventually.of_forall h'f)
filter_upwards [this] with x hx
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.StrongLaw | {
"line": 159,
"column": 6
} | {
"line": 168,
"column": 30
} | {
"line": 169,
"column": 4
} | [
{
"pp": "case inr\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : AEStronglyMeasurable f μ\nA : ℝ\nn : ℕ\nhn : n ≠ 0\nh'f : 0 ≤ f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : A < 0\n⊢ ∫ (y : ℝ), (Set.Ioc 0 A).indicator id y ^ n ∂Measure.map f μ =\n -∫ (x : ℝ),... | [] | simp only [Set.Ioc_eq_empty_of_le hA.le, zero_pow hn, Set.indicator_empty, integral_zero,
zero_eq_neg]
apply integral_eq_zero_of_ae
have : ∀ᵐ x ∂Measure.map f μ, (0 : ℝ) ≤ x :=
(ae_map_iff hf.aemeasurable measurableSet_Ici).2 (Eventually.of_forall h'f)
filter_upwards [this] with x hx
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RepresentationTheory.Rep.Res | {
"line": 125,
"column": 4
} | {
"line": 125,
"column": 18
} | {
"line": 126,
"column": 4
} | [
{
"pp": "case mp\nG : Type v1\nH : Type v2\ninst✝² : Monoid G\ninst✝¹ : Monoid H\nk : Type u\ninst✝ : CommRing k\nφ : H →* G\nS : ShortComplex (Rep k G)\nh : (S.map (resFunctor φ)).ShortExact\n⊢ S.ShortExact",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Rep.instAdditiveResFunctor",
... | [
"case mp\nG : Type v1\nH : Type v2\ninst✝² : Monoid G\ninst✝¹ : Monoid H\nk : Type u\ninst✝ : CommRing k\nφ : H →* G\nS : ShortComplex (Rep k G)\nh : (S.map (resFunctor φ)).ShortExact\nh₁ : (S.map (resFunctor φ)).Exact\n⊢ S.ShortExact"
] | have h₁ := h.1 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.StrongLaw | {
"line": 618,
"column": 2
} | {
"line": 618,
"column": 67
} | {
"line": 619,
"column": 2
} | [
{
"pp": "case neg\nΩ : Type u_2\nm : MeasurableSpace Ω\nμ : Measure Ω\nX : ℕ → Ω → ℝ\nhint : Integrable (X 0) μ\nhindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on X)\nhident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ\nmΩ : MeasureSpace Ω := { toMeasurableSpace := m, volume := μ }\nh : ¬∀ᵐ (ω : Ω), X 0 ω = 0\nthis ... | [
"case neg\nΩ : Type u_2\nm : MeasurableSpace Ω\nμ : Measure Ω\nX : ℕ → Ω → ℝ\nhint : Integrable (X 0) μ\nhindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on X)\nhident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ\nmΩ : MeasureSpace Ω := { toMeasurableSpace := m, volume := μ }\nh : ¬∀ᵐ (ω : Ω), X 0 ω = 0\nthis : IsProbabil... | have posm : Measurable pos := measurable_id'.max measurable_const | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RepresentationTheory.Rep.Basic | {
"line": 983,
"column": 45
} | {
"line": 983,
"column": 84
} | {
"line": 984,
"column": 2
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝¹ : CommRing k\ninst✝ : Monoid G\nX Y Z : Action (Type u) G\n⊢ Hom.hom\n (ofHom (δ (X ⊗ Y) Z) ≫\n ofHom (δ X Y) ▷ (linearization k G).obj Z ≫\n (α_ ((linearization k G).obj X) ((linearization k G).obj Y) ((linearization k G).obj Z)).hom) =\n Hom.hom ... | [] | simpa using assoc_comp_δ X Y Z (k := k) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RepresentationTheory.Rep.Basic | {
"line": 983,
"column": 45
} | {
"line": 983,
"column": 84
} | {
"line": 984,
"column": 2
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝¹ : CommRing k\ninst✝ : Monoid G\nX Y Z : Action (Type u) G\n⊢ Hom.hom\n (ofHom (δ (X ⊗ Y) Z) ≫\n ofHom (δ X Y) ▷ (linearization k G).obj Z ≫\n (α_ ((linearization k G).obj X) ((linearization k G).obj Y) ((linearization k G).obj Z)).hom) =\n Hom.hom ... | [] | simpa using assoc_comp_δ X Y Z (k := k) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RepresentationTheory.Rep.Basic | {
"line": 983,
"column": 45
} | {
"line": 983,
"column": 84
} | {
"line": 984,
"column": 2
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝¹ : CommRing k\ninst✝ : Monoid G\nX Y Z : Action (Type u) G\n⊢ Hom.hom\n (ofHom (δ (X ⊗ Y) Z) ≫\n ofHom (δ X Y) ▷ (linearization k G).obj Z ≫\n (α_ ((linearization k G).obj X) ((linearization k G).obj Y) ((linearization k G).obj Z)).hom) =\n Hom.hom ... | [] | simpa using assoc_comp_δ X Y Z (k := k) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RepresentationTheory.Homological.Resolution | {
"line": 142,
"column": 2
} | {
"line": 147,
"column": 80
} | {
"line": 149,
"column": 0
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\nn : ℕ\ninst✝ : Monoid G\n⊢ (compForgetAugmented G).ExtraDegeneracy",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"CategoryTheory.Comma.right",
"Opposite",
"SimplexCategory.instFintypeToTypeOrderHomFinHAddNatLenOfNat",
"Ca... | [] | refine
ExtraDegeneracy.ofIso (?_ : (Arrow.mk <| terminal.from G).augmentedCechNerve ≅ _)
(extraDegeneracyAugmentedCechNerve G)
exact
Comma.isoMk (CechNerveTerminalFrom.iso G ≪≫ cechNerveTerminalFromIsoCompForget G)
(Iso.refl _) (by ext : 1; exact IsTerminal.hom_ext terminalIsTerminal _ _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RepresentationTheory.Homological.Resolution | {
"line": 142,
"column": 2
} | {
"line": 147,
"column": 80
} | {
"line": 149,
"column": 0
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\nn : ℕ\ninst✝ : Monoid G\n⊢ (compForgetAugmented G).ExtraDegeneracy",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"CategoryTheory.Comma.right",
"Opposite",
"SimplexCategory.instFintypeToTypeOrderHomFinHAddNatLenOfNat",
"Ca... | [] | refine
ExtraDegeneracy.ofIso (?_ : (Arrow.mk <| terminal.from G).augmentedCechNerve ≅ _)
(extraDegeneracyAugmentedCechNerve G)
exact
Comma.isoMk (CechNerveTerminalFrom.iso G ≪≫ cechNerveTerminalFromIsoCompForget G)
(Iso.refl _) (by ext : 1; exact IsTerminal.hom_ext terminalIsTerminal _ _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RepresentationTheory.Homological.Resolution | {
"line": 196,
"column": 19
} | {
"line": 196,
"column": 33
} | {
"line": 196,
"column": 34
} | [
{
"pp": "k G : Type u\ninst✝ : CommRing k\nn : ℕ\nc : Fin (n + 1) → G\nr : k\n⊢ (d k G n) (MonoidAlgebra.single c (r * 1)) = ∑ p, MonoidAlgebra.single (c ∘ p.succAbove) (r * 1 * (-1) ^ ↑p)",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Fin.succAbove",
"Eq.mpr",
"NegZero... | [
"k G : Type u\ninst✝ : CommRing k\nn : ℕ\nc : Fin (n + 1) → G\nr : k\n⊢ (d k G n) (MonoidAlgebra.single c (r • 1)) = ∑ p, MonoidAlgebra.single (c ∘ p.succAbove) (r • 1 * (-1) ^ ↑p)"
] | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPowerSeries.Equiv | {
"line": 131,
"column": 6
} | {
"line": 131,
"column": 20
} | {
"line": 131,
"column": 21
} | [
{
"pp": "case a\nσ : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : Finite σ\nn : ℕ\nf : AdicCompletion (MvPolynomial.idealOfVars σ R) (MvPolynomial σ R)\nx : σ →₀ ℕ\nh : degree x < n\n⊢ Quotient.out (↑f (degree x + 1)) - Quotient.out (↑f n) ∈ MvPolynomial.idealOfVars σ R ^ (degree x + 1) * ⊤",
"ppTer... | [
"case a\nσ : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : Finite σ\nn : ℕ\nf : AdicCompletion (MvPolynomial.idealOfVars σ R) (MvPolynomial σ R)\nx : σ →₀ ℕ\nh : degree x < n\n⊢ Quotient.out (↑f (degree x + 1)) - Quotient.out (↑f n) ∈ MvPolynomial.idealOfVars σ R ^ (degree x + 1) • ⊤"
] | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 486,
"column": 2
} | {
"line": 486,
"column": 24
} | {
"line": 487,
"column": 2
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\ng h : G\na : ↑A\n⊢ single (h, 1) ((A.ρ g⁻¹) a) - single (g * h, 1) a ∈ boundaries₂ A",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"groupHomology.d₃₂",
"Rep.V",
"Representation",
"MonoidHom.... | [
"case h\nk G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\ng h : G\na : ↑A\n⊢ (ModuleCat.Hom.hom (d₃₂ A)) (single (g, h, 1) a) = single (h, 1) ((A.ρ g⁻¹) a) - single (g * h, 1) a"
] | use single (g, h, 1) a | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.RingTheory.AdicCompletion.Topology | {
"line": 72,
"column": 4
} | {
"line": 72,
"column": 82
} | {
"line": 74,
"column": 0
} | [
{
"pp": "case mpr\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : UniformSpace R\ninst✝ : IsUniformAddGroup R\nI : Ideal R\nhI : IsAdic I\nthis✝ : (𝓝 0).IsCountablyGenerated\nthis : (𝓤 R).IsCountablyGenerated\nH : CompleteSpace R\nf : ℕ → R\nhf : ∀ {m n : ℕ}, m ≤ n → f m - f n ∈ I ^ m\nL : R\nhL : Filter.map f F... | [] | simpa using Ideal.add_mem _ (hN (max i N) le_sup_right) (hf (le_max_left i N)) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.AdicCompletion.RingHom | {
"line": 177,
"column": 61
} | {
"line": 179,
"column": 71
} | {
"line": 181,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : NonAssocSemiring R\ninst✝¹ : CommRing S\nI : Ideal S\na : ℕ → ℕ\nha : StrictMono a\nf : (n : ℕ) → R →+* S ⧸ I ^ a n\nhf : ∀ {m : ℕ}, (factorPow I ⋯).comp (f (m + 1)) = f m\ninst✝ : IsAdicComplete I S\nn : ℕ\nx : R\n⊢ (Ideal.Quotient.mk (I ^ a n)) ((liftRingHom I ha ... | [] | by
simp [liftRingHom, IsAdicComplete.liftRingHom,
factorPow_comp_eq_of_factorPow_comp_succ_eq' ha f hf ha.le_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Coalgebra.GroupLike | {
"line": 106,
"column": 6
} | {
"line": 106,
"column": 44
} | {
"line": 106,
"column": 44
} | [
{
"pp": "R : Type u_2\nA : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : AddCommGroup A\ninst✝² : Module R A\ninst✝¹ : Coalgebra R A\ninst✝ : IsTorsionFree R A\n⊢ LinearIndepOn R id {a | IsGroupLikeElem R a}",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"R : Type u_2\nA : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : AddCommGroup A\ninst✝² : Module R A\ninst✝¹ : Coalgebra R A\ninst✝ : IsTorsionFree R A\n⊢ ∀ (t : Finset A), ↑t ⊆ {a | IsGroupLikeElem R a} → LinearIndepOn R id ↑t"
] | linearIndepOn_iff_linearIndepOn_finset | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.SelmerGroup | {
"line": 202,
"column": 2
} | {
"line": 225,
"column": 77
} | {
"line": 227,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type v\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nn : ℕ\nhn : Fact (0 < n)\n⊢ fromUnit.ker = (powMonoidHom n).range",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"I... | [] | ext ⟨_, _, _, _⟩
constructor
· intro hx
rcases (QuotientGroup.eq_one_iff _).mp (Subtype.mk.inj hx) with ⟨⟨v, i, vi, iv⟩, hx⟩
have hv : ↑(_ ^ n : Kˣ) = algebraMap R K _ := congr_arg Units.val hx
have hi : ↑(_ ^ n : Kˣ)⁻¹ = algebraMap R K _ := congr_arg Units.inv hx
rw [Units.val_pow_eq_pow_val] at hv... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DedekindDomain.SelmerGroup | {
"line": 202,
"column": 2
} | {
"line": 225,
"column": 77
} | {
"line": 227,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type v\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nn : ℕ\nhn : Fact (0 < n)\n⊢ fromUnit.ker = (powMonoidHom n).range",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"I... | [] | ext ⟨_, _, _, _⟩
constructor
· intro hx
rcases (QuotientGroup.eq_one_iff _).mp (Subtype.mk.inj hx) with ⟨⟨v, i, vi, iv⟩, hx⟩
have hv : ↑(_ ^ n : Kˣ) = algebraMap R K _ := congr_arg Units.val hx
have hi : ↑(_ ^ n : Kˣ)⁻¹ = algebraMap R K _ := congr_arg Units.inv hx
rw [Units.val_pow_eq_pow_val] at hv... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Regular.LinearMap | {
"line": 56,
"column": 8
} | {
"line": 57,
"column": 31
} | {
"line": 58,
"column": 6
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R M\ninst✝³ : Module R N\ninst✝² : IsNoetherianRing R\ninst✝¹ : Module.Finite R M\ninst✝ : Module.Finite R N\nhom0 : Subsingleton (N →ₗ[R] M)\nh✝ : Nontrivial M\nh : ∀ r ∈ an... | [] | rw [biUnion_associatedPrimes_eq_compl_regular R M]
exact fun r hr ↦ h r hr | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Regular.LinearMap | {
"line": 56,
"column": 8
} | {
"line": 57,
"column": 31
} | {
"line": 58,
"column": 6
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R M\ninst✝³ : Module R N\ninst✝² : IsNoetherianRing R\ninst✝¹ : Module.Finite R M\ninst✝ : Module.Finite R N\nhom0 : Subsingleton (N →ₗ[R] M)\nh✝ : Nontrivial M\nh : ∀ r ∈ an... | [] | rw [biUnion_associatedPrimes_eq_compl_regular R M]
exact fun r hr ↦ h r hr | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.AssociatedPrime.Localization | {
"line": 132,
"column": 69
} | {
"line": 148,
"column": 85
} | {
"line": 150,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsNoetherianRing R\ninst✝ : Module.Finite R M\n⊢ (annihilator R M).minimalPrimes ⊆ associatedPrimes R M",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Nontrivial",
... | [] | by
intro p hp
have prime := hp.isPrime
let Rₚ := Localization.AtPrime p
have : Nontrivial (LocalizedModule p.primeCompl M) := by
simpa [← Module.mem_support_iff (p := ⟨p, prime⟩), Module.support_eq_zeroLocus] using! hp.1.2
rcases associatedPrimes.nonempty Rₚ (LocalizedModule p.primeCompl M) with ⟨q, hq⟩
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Regular.RegularSequence | {
"line": 250,
"column": 40
} | {
"line": 250,
"column": 49
} | {
"line": 250,
"column": 49
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nrs : List R\n⊢ (IsSMulRegular M r ∧ IsWeaklyRegular (QuotSMulTop r M) rs) ∧ ¬⊤ = Ideal.ofList rs • ⊤ ↔\n IsSMulRegular M r ∧ IsWeaklyRegular (QuotSMulTop r M) rs ∧ ⊤ ≠ Ideal.ofList rs • ⊤",
"ppTe... | [
"R : Type u_1\nM : Type u_3\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nrs : List R\n⊢ IsSMulRegular M r ∧ IsWeaklyRegular (QuotSMulTop r M) rs ∧ ¬⊤ = Ideal.ofList rs • ⊤ ↔\n IsSMulRegular M r ∧ IsWeaklyRegular (QuotSMulTop r M) rs ∧ ⊤ ≠ Ideal.ofList rs • ⊤"
] | and_assoc | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DividedPowers.Basic | {
"line": 321,
"column": 6
} | {
"line": 321,
"column": 65
} | {
"line": 322,
"column": 4
} | [
{
"pp": "A : Type u_1\ninst✝² : CommSemiring A\nM : Type u_2\ninst✝¹ : AddCommMonoid M\nI : AddSubmonoid M\ndpow : ℕ → M → A\ndpow_zero : ∀ {x : M}, x ∈ I → dpow 0 x = 1\ndpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0\nι : Type u_3\ninst✝ : DecidableEq ι\nx : ι → M\ndpow_add : ∀ {n : ℕ} {x y : M}, x ∈ I → y ∈... | [] | exact Sym.count_coe_fill_of_ne (ne_of_mem_of_not_mem hi ha) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Regular.RegularSequence | {
"line": 259,
"column": 54
} | {
"line": 259,
"column": 63
} | {
"line": 259,
"column": 63
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nrs : List R\n⊢ (IsSMulRegular M r ∧ IsWeaklyRegular (QuotSMulTop r M) (List.map (⇑(algebraMap R (R ⧸ Ideal.span {r}))) rs)) ∧\n ¬⊤ = Ideal.ofList rs • ⊤ ↔\n IsSMulRegular M r ∧\n IsWeaklyRe... | [
"R : Type u_1\nM : Type u_3\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nrs : List R\n⊢ IsSMulRegular M r ∧\n IsWeaklyRegular (QuotSMulTop r M) (List.map (⇑(algebraMap R (R ⧸ Ideal.span {r}))) rs) ∧\n ¬⊤ = Ideal.ofList rs • ⊤ ↔\n IsSMulRegular M r ∧\n IsWeaklyRegular... | and_assoc | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Regular.RegularSequence | {
"line": 619,
"column": 4
} | {
"line": 619,
"column": 45
} | {
"line": 620,
"column": 2
} | [
{
"pp": "case swap\nR : Type u_1\nM : Type u_3\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nrs : List R\nh✝ : IsWeaklyRegular M rs\nrs' : List R\nh'' : rs ~ rs'\nh' :\n ∀ (a b : R) (rs' : List R),\n a :: b :: rs' <+~ rs →\n let K := torsionBy R (M ⧸ Ideal.ofList rs' • ⊤) b;\n ... | [] | exact h.imp_left (swap · (h' b a rs₀ H₁)) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 353,
"column": 28
} | {
"line": 353,
"column": 42
} | {
"line": 353,
"column": 43
} | [
{
"pp": "α : Type u_1\nR : Type u_2\nM : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nA : Type u_4\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Ideal A\nhI : DividedPowers I\ng : M →ₗ[R] A\nhg : ∀ (m : M), g m ∈ I\nn : ℕ\nr : R\nm : M\n⊢ (algebraMap R A) r ^ n * hI.dpo... | [
"α : Type u_1\nR : Type u_2\nM : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nA : Type u_4\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Ideal A\nhI : DividedPowers I\ng : M →ₗ[R] A\nhg : ∀ (m : M), g m ∈ I\nn : ℕ\nr : R\nm : M\n⊢ (algebraMap R A) r ^ n • hI.dpow n (g m) = ... | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DividedPowers.Padic | {
"line": 165,
"column": 4
} | {
"line": 166,
"column": 63
} | {
"line": 167,
"column": 2
} | [
{
"pp": "case pos\np : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\nx : ℤ_[p]\nhx : x ∈ Ideal.span {↑p}\nhinj : Injective ⇑Coe.ringHom\nheq : Coe.ringHom ⟨dpow' p n ↑x, ⋯⟩ = (↑n !)⁻¹ʳ * Coe.ringHom x ^ n\n⊢ ⋯.choose = ⟨dpow' p n ↑x, ⋯⟩",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Norm.norm... | [] | simpa only [← hinj.eq_iff, (Exists.choose_spec (_ : ∃ a, ∃ _, Coe.ringHom a = _)).2,
RatAlgebra.dpow_apply, Submodule.mem_top] using! heq.symm | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 291,
"column": 58
} | {
"line": 291,
"column": 72
} | {
"line": 291,
"column": 73
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommSemiring R\nI : Ideal R\ninst✝ : Algebra ℚ R\nhI : DividedPowers I\nn : ℕ\nx : R\nhx : x ∈ I\n⊢ hI.dpow n x = (↑n !)⁻¹ • (↑n ! * hI.dpow n x)",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",... | [
"R : Type u_1\ninst✝¹ : CommSemiring R\nI : Ideal R\ninst✝ : Algebra ℚ R\nhI : DividedPowers I\nn : ℕ\nx : R\nhx : x ∈ I\n⊢ hI.dpow n x = (↑n !)⁻¹ • ↑n ! • hI.dpow n x"
] | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 317,
"column": 31
} | {
"line": 319,
"column": 39
} | {
"line": 320,
"column": 6
} | [
{
"pp": "A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nS : Set hI.SubDPIdeal\nx : A\nhx : x ∈ ⨅ s ∈ insert ⊤ S, s.carrier\n⊢ x ∈ I",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Submodule",
"DividedPowers.SubDPIdeal",
"iInf",
"Semiring.t... | [] | by
simp only [mem_iInf] at hx
exact hx ⊤ (Set.mem_insert ⊤ S) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 407,
"column": 8
} | {
"line": 407,
"column": 53
} | {
"line": 408,
"column": 4
} | [
{
"pp": "case neg\nA : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nS : Set A\nhS : S ⊆ ↑I\nk : ℕ\nhk : k ≠ 0\nz : A\nhz : z ∈ span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x}\nJ : Ideal A := span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x}\nhSI : span... | [] | exact (span _).mul_mem_right _ (hx_pow m hm0) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 407,
"column": 8
} | {
"line": 407,
"column": 53
} | {
"line": 408,
"column": 4
} | [
{
"pp": "case neg\nA : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nS : Set A\nhS : S ⊆ ↑I\nk : ℕ\nhk : k ≠ 0\nz : A\nhz : z ∈ span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x}\nJ : Ideal A := span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x}\nhSI : span... | [] | exact (span _).mul_mem_right _ (hx_pow m hm0) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 407,
"column": 8
} | {
"line": 407,
"column": 53
} | {
"line": 408,
"column": 4
} | [
{
"pp": "case neg\nA : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nS : Set A\nhS : S ⊆ ↑I\nk : ℕ\nhk : k ≠ 0\nz : A\nhz : z ∈ span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x}\nJ : Ideal A := span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x}\nhSI : span... | [] | exact (span _).mul_mem_right _ (hx_pow m hm0) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Frobenius | {
"line": 205,
"column": 2
} | {
"line": 208,
"column": 76
} | {
"line": 209,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nG : Type u_3\ninst✝² : Group G\ninst✝¹ : MulSemiringAction G S\ninst✝ : SMulCommClass G R S\nQ : Ideal S\nσ : G\nH : IsArithFrobAt R σ Q\nτ : G\nx : S\n⊢ (MulSemiringAction.toAlgHom R S (τ * σ * τ⁻¹)) x - x ^ Na... | [
"R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nG : Type u_3\ninst✝² : Group G\ninst✝¹ : MulSemiringAction G S\ninst✝ : SMulCommClass G R S\nQ : Ideal S\nσ : G\nH : IsArithFrobAt R σ Q\nτ : G\nx : S\nthis : Ideal.under R (Ideal.map ((MulSemiringAction.toRingEquiv G S) τ)... | have : (Q.map (MulSemiringAction.toRingEquiv G S τ)).under R = Q.under R := by
rw [← Ideal.comap_symm, ← Ideal.comap_coe, Ideal.under, Ideal.comap_comap]
congr 1
exact (MulSemiringAction.toAlgEquiv R S τ).symm.toAlgHom.comp_algebraMap | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Grassmannian | {
"line": 183,
"column": 10
} | {
"line": 183,
"column": 13
} | {
"line": 183,
"column": 14
} | [
{
"pp": "case add\nR : Type u\ninst✝⁸ : CommRing R\nM : Type v\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nk : ℕ\nA : Type w\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra R A\nB : Type w\ninst✝³ : CommRing B\ninst✝² : Algebra R B\nf : A →ₐ[R] B\nC : Type w\ninst✝¹ : CommRing C\ninst✝ : Algebra R C\ng : B →ₐ[R] C\nN... | [
"case add\nR : Type u\ninst✝⁸ : CommRing R\nM : Type v\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nk : ℕ\nA : Type w\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra R A\nB : Type w\ninst✝³ : CommRing B\ninst✝² : Algebra R B\nf : A →ₐ[R] B\nC : Type w\ninst✝¹ : CommRing C\ninst✝ : Algebra R C\ng : B →ₐ[R] C\nN : G(k, A ⊗[... | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 311,
"column": 6
} | {
"line": 311,
"column": 14
} | {
"line": 311,
"column": 14
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\nα : Type u_5\ninst✝¹ : PartialOrder Γ\ninst✝ : AddCommMonoid R\ns : SummableFamily Γ R α\ng : Γ\na : α\nha : ↑g = (s a).orderTop\nhg : ∀ (b : α), ∀ g' ∈ (s b).support, g ≤ g'\nhna : ∀ (b : α), b ≠ a → (s b).coeff g = 0\nthis : s.hsum.orderTop = ↑g\n⊢ s.hsum.leadingCoeff = (s... | [
"Γ : Type u_1\nR : Type u_3\nα : Type u_5\ninst✝¹ : PartialOrder Γ\ninst✝ : AddCommMonoid R\ns : SummableFamily Γ R α\ng : Γ\na : α\nha : ↑g = (s a).orderTop\nhg : ∀ (b : α), ∀ g' ∈ (s b).support, g ≤ g'\nhna : ∀ (b : α), b ≠ a → (s b).coeff g = 0\nthis : (if h : s.hsum = 0 then ⊤ else ↑(⋯.min ⋯)) = ↑g\n⊢ s.hsum.le... | orderTop | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 406,
"column": 53
} | {
"line": 406,
"column": 56
} | {
"line": 406,
"column": 57
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝⁶ : PartialOrder Γ\ninst✝⁵ : PartialOrder Γ'\ninst✝⁴ : AddCommMonoid V\ninst✝³ : AddCommMonoid R\ninst✝² : SMulWithZero R V\ninst✝¹ : VAdd Γ Γ'\ninst✝ : IsOrderedCancelVAdd Γ Γ'\ns : α → R⟦Γ⟧\nt : β → V⟦Γ'⟧\nhs : ... | [
"Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝⁶ : PartialOrder Γ\ninst✝⁵ : PartialOrder Γ'\ninst✝⁴ : AddCommMonoid V\ninst✝³ : AddCommMonoid R\ninst✝² : SMulWithZero R V\ninst✝¹ : VAdd Γ Γ'\ninst✝ : IsOrderedCancelVAdd Γ Γ'\ns : α → R⟦Γ⟧\nt : β → V⟦Γ'⟧\nhs : (⋃ a, (s a).... | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 406,
"column": 4
} | {
"line": 406,
"column": 67
} | {
"line": 407,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝⁶ : PartialOrder Γ\ninst✝⁵ : PartialOrder Γ'\ninst✝⁴ : AddCommMonoid V\ninst✝³ : AddCommMonoid R\ninst✝² : SMulWithZero R V\ninst✝¹ : VAdd Γ Γ'\ninst✝ : IsOrderedCancelVAdd Γ Γ'\ns : α → R⟦Γ⟧\nt : β → V⟦Γ'⟧\nhs : ... | [] | rw [mem_support, not_not, HahnModule.coeff_smul, hx, sum_empty] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 561,
"column": 11
} | {
"line": 561,
"column": 25
} | {
"line": 561,
"column": 26
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\nα : Type u_5\nβ : Type u_6\ninst✝³ : AddCommMonoid Γ\ninst✝² : PartialOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : Semiring R\ns : SummableFamily Γ R α\nt : SummableFamily Γ R β\ng : Γ\n⊢ (s.mul t).hsum.coeff g = ∑ gh ∈ antidiagonal ⋯ ⋯ g, s.hsum.coeff gh.1 * t.hsum... | [
"Γ : Type u_1\nR : Type u_3\nα : Type u_5\nβ : Type u_6\ninst✝³ : AddCommMonoid Γ\ninst✝² : PartialOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : Semiring R\ns : SummableFamily Γ R α\nt : SummableFamily Γ R β\ng : Γ\n⊢ (s.mul t).hsum.coeff g = ∑ x ∈ antidiagonal ⋯ ⋯ g, s.hsum.coeff x.1 • t.hsum.coeff x.2"
] | ← smul_eq_mul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 566,
"column": 6
} | {
"line": 566,
"column": 20
} | {
"line": 566,
"column": 21
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\nα : Type u_5\nβ : Type u_6\ninst✝³ : AddCommMonoid Γ\ninst✝² : PartialOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : Semiring R\ns : SummableFamily Γ R α\nt : SummableFamily Γ R β\n⊢ (s.mul t).hsum = s.hsum * t.hsum",
"ppTerm": "?m.39",
"assigned": true,
"... | [
"Γ : Type u_1\nR : Type u_3\nα : Type u_5\nβ : Type u_6\ninst✝³ : AddCommMonoid Γ\ninst✝² : PartialOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : Semiring R\ns : SummableFamily Γ R α\nt : SummableFamily Γ R β\n⊢ (s.mul t).hsum = s.hsum • t.hsum"
] | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Henselian | {
"line": 271,
"column": 66
} | {
"line": 273,
"column": 9
} | {
"line": 274,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsLocalRing R\nf : R[X]\na b : R\nha : Polynomial.eval a f = 0\nhb : Polynomial.eval b f = 0\nh : ¬IsUnit (a - b)\nh' : IsUnit (Polynomial.eval a (derivative f))\nc : R\nh✝ :\n c * (b - a) ^ 2 + Polynomial.eval a (derivative f) * (b - a) + Polynomial.eval a f... | [] | by
rw [notMem_maximalIdeal, isUnit_iff_exists] at this
grind | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Ideal.Pure | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 45
} | {
"line": 135,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nI J : Ideal R\ninst✝¹ : I.Pure\ninst✝ : J.Pure\nh : zeroLocus ↑I = zeroLocus ↑J\ns : Set (PrimeSpectrum R)\nhs : zeroLocus ↑I = s\nt : Set (PrimeSpectrum R)\nht : zeroLocus ↑J = t\n⊢ RingHom.ker (RingHom.pi fun p ↦ algebraMap R (Localization.AtPrime (↑p).asIdeal)) =\n... | [
"R : Type u_1\ninst✝² : CommRing R\nI J : Ideal R\ninst✝¹ : I.Pure\ninst✝ : J.Pure\nh : zeroLocus ↑I = zeroLocus ↑J\ns : Set (PrimeSpectrum R)\nhs : zeroLocus ↑I = s\nht : zeroLocus ↑J = s\n⊢ RingHom.ker (RingHom.pi fun p ↦ algebraMap R (Localization.AtPrime (↑p).asIdeal)) =\n RingHom.ker (RingHom.pi fun p ↦ alg... | obtain rfl : s = t := by rw [← hs, ← ht, h] | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Spectrum.Prime.LTSeries | {
"line": 51,
"column": 2
} | {
"line": 51,
"column": 16
} | {
"line": 53,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsLocalRing R\np₁ p₀ : PrimeSpectrum R\nh₀ : p₀ < p₁\nh₁ : p₁ < closedPoint R\nx : R\nhx : x ∈ 𝔪\nhn : x ∉ p₀.asIdeal\ne : ↑(zeroLocus ↑p₀.asIdeal) ≃o PrimeSpectrum (R ⧸ p₀.asIdeal) := p₀.asIdeal.primeSpectrumQuotientOrd... | [] | simpa using h₀ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 483,
"column": 15
} | {
"line": 483,
"column": 23
} | {
"line": 483,
"column": 24
} | [
{
"pp": "R : Type u_1\ninst✝⁸ : CommRing R\ninst✝⁷ : IsNoetherianRing R\nS : Type u_2\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : IsNoetherianRing S\ninst✝³ : Algebra.HasGoingDown R S\np : Ideal R\ninst✝² : p.IsPrime\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : P.LiesOver p\nlp : LTSeries (PrimeSpectrum R... | [
"R : Type u_1\ninst✝⁸ : CommRing R\ninst✝⁷ : IsNoetherianRing R\nS : Type u_2\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : IsNoetherianRing S\ninst✝³ : Algebra.HasGoingDown R S\np : Ideal R\ninst✝² : p.IsPrime\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : P.LiesOver p\nlp : LTSeries (PrimeSpectrum R)\nhlp : Rel... | ← hlenq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.LittleWedderburn | {
"line": 95,
"column": 6
} | {
"line": 97,
"column": 96
} | {
"line": 98,
"column": 4
} | [
{
"pp": "case refine_2\nD : Type u_1\ninst✝¹ : DivisionRing D\ninst✝ : Finite D\nhD : InductionHyp D\nval✝ : Fintype D\nZ : Subring D := Subring.center D\nhZ : Z ≠ ⊤\nthis✝ : Field ↥Z := hD.field ⋯\nq : ℕ := card ↥Z\ncard_Z : q = card ↥Z\nhq : 1 < q\nn : ℕ := finrank (↥Z) D\ncard_D : card D = q ^ n\nh1qn : 1 ≤ ... | [] | convert! Int.natAbs_dvd_natAbs.mpr contra
clear_value q
simp only [eq_comm, Int.natAbs_eq_iff, Nat.cast_sub hq.le, Nat.cast_one, neg_sub, true_or] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.LittleWedderburn | {
"line": 95,
"column": 6
} | {
"line": 97,
"column": 96
} | {
"line": 98,
"column": 4
} | [
{
"pp": "case refine_2\nD : Type u_1\ninst✝¹ : DivisionRing D\ninst✝ : Finite D\nhD : InductionHyp D\nval✝ : Fintype D\nZ : Subring D := Subring.center D\nhZ : Z ≠ ⊤\nthis✝ : Field ↥Z := hD.field ⋯\nq : ℕ := card ↥Z\ncard_Z : q = card ↥Z\nhq : 1 < q\nn : ℕ := finrank (↥Z) D\ncard_D : card D = q ^ n\nh1qn : 1 ≤ ... | [] | convert! Int.natAbs_dvd_natAbs.mpr contra
clear_value q
simp only [eq_comm, Int.natAbs_eq_iff, Nat.cast_sub hq.le, Nat.cast_one, neg_sub, true_or] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Lasker | {
"line": 185,
"column": 6
} | {
"line": 185,
"column": 19
} | {
"line": 186,
"column": 6
} | [
{
"pp": "R : Type u_3\nM : Type u_4\ninst✝² : CommRing R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\ns₀ : Finset ↑N.associatedPrimes\nhs₀ : IsLowerSet ↑s₀\nq : Submodule R M\nhqp : q.IsPrimary\np : ↑N.associatedPrimes\nhq : (q.colon Set.univ).radical = ↑p\nS : Submonoid R := ⨅ q ∈ s₀, (↑q)... | [
"R : Type u_3\nM : Type u_4\ninst✝² : CommRing R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\ns₀ : Finset ↑N.associatedPrimes\nhs₀ : IsLowerSet ↑s₀\nq : Submodule R M\nhqp : q.IsPrimary\np : ↑N.associatedPrimes\nhq : (q.colon Set.univ).radical = ↑p\nS : Submonoid R := ⨅ q ∈ s₀, (↑q).primeCompl\... | contrapose hp | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose_1 | Mathlib.Tactic.Contrapose.contrapose |
Mathlib.RingTheory.LocalProperties.InjectiveDimension | {
"line": 109,
"column": 10
} | {
"line": 109,
"column": 37
} | {
"line": 109,
"column": 37
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : Small.{v, u} R\ninst✝ : IsNoetherianRing R\nM : ModuleCat R\nh : ∀ (m : MaximalSpectrum R), Injective (M.localizedModule m.asIdeal.primeCompl)\np : Ideal R\nhp : p.IsMaximal\nthis✝ : Small.{v, u} (Localization.AtPrime p) := small_of_surjective Localization.mkHo... | [
"R : Type u\ninst✝² : CommRing R\ninst✝¹ : Small.{v, u} R\ninst✝ : IsNoetherianRing R\nM : ModuleCat R\nh : ∀ (m : MaximalSpectrum R), Injective (M.localizedModule m.asIdeal.primeCompl)\np : Ideal R\nhp : p.IsMaximal\nthis✝ : Small.{v, u} (Localization.AtPrime p) := small_of_surjective Localization.mkHom_surjective... | ← Module.Baer.iff_injective | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Invariant.Profinite | {
"line": 122,
"column": 2
} | {
"line": 123,
"column": 72
} | {
"line": 125,
"column": 0
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝⁶ : CommRing A\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra A B\nG : Type u\ninst✝³ : Group G\ninst✝² : MulSemiringAction G B\ninst✝¹ : SMulCommClass G A B\ninst✝ : TopologicalSpace G\nQ : Ideal B\nN N' : OpenNormalSubgroup G\ne : N ≤ N'\nh : FixedPoints.subalgebra A B ↥↑N'.t... | [] | simpa only [Ideal.pointwise_smul_eq_comap,
← Ideal.comap_coe (F := RingEquiv _ _), Ideal.comap_comap] using! hx | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.RingTheory.Invariant.Profinite | {
"line": 168,
"column": 2
} | {
"line": 168,
"column": 16
} | {
"line": 169,
"column": 2
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹⁵ : CommRing A\ninst✝¹⁴ : CommRing B\ninst✝¹³ : Algebra A B\nG : Type u\ninst✝¹² : Group G\ninst✝¹¹ : MulSemiringAction G B\ninst✝¹⁰ : SMulCommClass G A B\nP✝ : Ideal A\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : CompactSpace G\ninst✝⁷ : TotallyDisconnectedSpace G\ninst✝⁶ :... | [
"A : Type u_1\nB : Type u_2\ninst✝¹⁵ : CommRing A\ninst✝¹⁴ : CommRing B\ninst✝¹³ : Algebra A B\nG : Type u\ninst✝¹² : Group G\ninst✝¹¹ : MulSemiringAction G B\ninst✝¹⁰ : SMulCommClass G A B\nP✝ : Ideal A\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : CompactSpace G\ninst✝⁷ : TotallyDisconnectedSpace G\ninst✝⁶ : IsTopologic... | refine ⟨τ, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.LaurentSeries | {
"line": 602,
"column": 4
} | {
"line": 602,
"column": 12
} | {
"line": 603,
"column": 4
} | [
{
"pp": "case neg\nK : Type u_2\ninst✝ : Field K\nd n : ℤ\nf g : K⸨X⸩\nH : Valued.v (g - f) ≤ exp (-d)\ntriv : ¬g = f\n⊢ n < d → g.coeff n = f.coeff n",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Int",
"Int.instLTInt",
"LT.lt"
],
"usedFVars": [
"n",
... | [
"case neg\nK : Type u_2\ninst✝ : Field K\nd n : ℤ\nf g : K⸨X⸩\nH : Valued.v (g - f) ≤ exp (-d)\ntriv : ¬g = f\nhn : n < d\n⊢ g.coeff n = f.coeff n"
] | intro hn | Lean.Elab.Tactic.evalIntro | null |
Mathlib.RingTheory.LaurentSeries | {
"line": 602,
"column": 4
} | {
"line": 602,
"column": 12
} | {
"line": 603,
"column": 4
} | [
{
"pp": "case neg\nK : Type u_2\ninst✝ : Field K\nd n : ℤ\nf g : K⸨X⸩\nH : Valued.v (g - f) ≤ exp (-d)\ntriv : ¬g = f\n⊢ n < d → g.coeff n = f.coeff n",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Int",
"Int.instLTInt",
"LT.lt"
],
"usedFVars": [
"n",
... | [
"case neg\nK : Type u_2\ninst✝ : Field K\nd n : ℤ\nf g : K⸨X⸩\nH : Valued.v (g - f) ≤ exp (-d)\ntriv : ¬g = f\nhn : n < d\n⊢ g.coeff n = f.coeff n"
] | intro hn | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem | {
"line": 83,
"column": 2
} | {
"line": 83,
"column": 43
} | {
"line": 85,
"column": 0
} | [
{
"pp": "case convert_7\ni n m : ℕ\nhin : i < n\nhim : i + 1 < m\nt : Fin n → ℕ\n⊢ ⟨i, hin⟩ ∈ {x | i ≤ ↑x}",
"ppTerm": "?convert_7",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Finset.mem_univ",
"le_rfl",
"Finset.univ",
"Finset",
"Membership.mem",
"Fin.... | [] | · exact mem_filter.2 ⟨mem_univ _, le_rfl⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
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