module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Probability.Kernel.Disintegration.StandardBorel | {
"line": 269,
"column": 10
} | {
"line": 269,
"column": 36
} | {
"line": 269,
"column": 36
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nΩ : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmΩ : MeasurableSpace Ω\ninst✝³ : StandardBorelSpace Ω\ninst✝² : Nonempty Ω\nκ : Kernel α (β × Ω)\ninst✝¹ : IsSFiniteKernel κ\nη : Kernel (α × β) ℝ\ninst✝ : IsSFiniteKernel η\nhη : (κ.map (Prod.map id (embeddingRea... | [
"α : Type u_1\nβ : Type u_2\nΩ : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmΩ : MeasurableSpace Ω\ninst✝³ : StandardBorelSpace Ω\ninst✝² : Nonempty Ω\nκ : Kernel α (β × Ω)\ninst✝¹ : IsSFiniteKernel κ\nη : Kernel (α × β) ℝ\ninst✝ : IsSFiniteKernel η\nhη : (κ.map (Prod.map id (embeddingReal Ω))).fst ⊗... | map_apply' _ (by fun_prop) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Disintegration.StandardBorel | {
"line": 303,
"column": 53
} | {
"line": 303,
"column": 75
} | {
"line": 303,
"column": 75
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nΩ : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmΩ : MeasurableSpace Ω\ninst✝³ : StandardBorelSpace Ω\ninst✝² : Nonempty Ω\nκ : Kernel α (β × Ω)\ninst✝¹ : IsSFiniteKernel κ\nη : Kernel (α × β) ℝ\ninst✝ : IsSFiniteKernel η\nhη : (κ.map (Prod.map id (embeddingRea... | [
"α : Type u_1\nβ : Type u_2\nΩ : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmΩ : MeasurableSpace Ω\ninst✝³ : StandardBorelSpace Ω\ninst✝² : Nonempty Ω\nκ : Kernel α (β × Ω)\ninst✝¹ : IsSFiniteKernel κ\nη : Kernel (α × β) ℝ\ninst✝ : IsSFiniteKernel η\nhη : (κ.map (Prod.map id (embeddingReal Ω))).fst ⊗... | h_prod_embed.comap_map | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 463,
"column": 2
} | {
"line": 466,
"column": 74
} | {
"line": 468,
"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✝ : CountablyGenerated γ\nκ : Kernel α (γ × β)\nν : Kernel α γ\nhκν : κ.fst ≤ ν\na : α\nx : γ\ns s' : Set β\nh : s ⊆ s'\n⊢ κ.density ν a x s ≤ κ.density ν a x s'",
"ppTerm": "?m.32"... | [] | refine limsup_le_limsup ?_ ?_ ?_
· exact Eventually.of_forall (fun n ↦ densityProcess_mono_set hκν n a x h)
· exact isCoboundedUnder_le_of_le atTop (fun i ↦ densityProcess_nonneg _ _ _ _ _ _)
· exact isBoundedUnder_of ⟨1, fun n ↦ densityProcess_le_one hκν _ _ _ _⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 463,
"column": 2
} | {
"line": 466,
"column": 74
} | {
"line": 468,
"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✝ : CountablyGenerated γ\nκ : Kernel α (γ × β)\nν : Kernel α γ\nhκν : κ.fst ≤ ν\na : α\nx : γ\ns s' : Set β\nh : s ⊆ s'\n⊢ κ.density ν a x s ≤ κ.density ν a x s'",
"ppTerm": "?m.32"... | [] | refine limsup_le_limsup ?_ ?_ ?_
· exact Eventually.of_forall (fun n ↦ densityProcess_mono_set hκν n a x h)
· exact isCoboundedUnder_le_of_le atTop (fun i ↦ densityProcess_nonneg _ _ _ _ _ _)
· exact isBoundedUnder_of ⟨1, fun n ↦ densityProcess_le_one hκν _ _ _ _⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.CondDistrib | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 6
} | {
"line": 157,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nΩ : Type u_3\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : StandardBorelSpace Ω\ninst✝² : Nonempty Ω\nmα : MeasurableSpace α\nμ : Measure α\ninst✝¹ : IsFiniteMeasure μ\nX : α → β\nY : α → Ω\nmβ : MeasurableSpace β\nhX : Measurable X\nhY : Measurable Y\nκ : Kernel β Ω\ninst✝ : IsFini... | [
"α : Type u_1\nβ : Type u_2\nΩ : Type u_3\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : StandardBorelSpace Ω\ninst✝² : Nonempty Ω\nmα : MeasurableSpace α\nμ : Measure α\ninst✝¹ : IsFiniteMeasure μ\nX : α → β\nY : α → Ω\nmβ : MeasurableSpace β\nhX : Measurable X\nhY : Measurable Y\nκ : Kernel β Ω\ninst✝ : IsFiniteKernel κ\n... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Probability.Kernel.Disintegration.Unique | {
"line": 129,
"column": 53
} | {
"line": 129,
"column": 57
} | {
"line": 129,
"column": 57
} | [
{
"pp": "α : Type u_1\nΩ : Type u_3\nmα : MeasurableSpace α\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : StandardBorelSpace Ω\ninst✝² : Nonempty Ω\nμ : Measure α\ninst✝¹ : IsFiniteMeasure μ\nκ : Kernel α Ω\ninst✝ : IsMarkovKernel κ\nthis : ⇑κ =ᵐ[(μ ⊗ₘ κ).fst] ⇑(μ ⊗ₘ κ).condKernel\n⊢ ⇑(μ ⊗ₘ κ).condKernel =ᵐ[μ] ⇑κ",
... | [
"α : Type u_1\nΩ : Type u_3\nmα : MeasurableSpace α\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : StandardBorelSpace Ω\ninst✝² : Nonempty Ω\nμ : Measure α\ninst✝¹ : IsFiniteMeasure μ\nκ : Kernel α Ω\ninst✝ : IsMarkovKernel κ\nthis : ⇑κ =ᵐ[(μ ⊗ₘ κ).fst] ⇑(μ ⊗ₘ κ).condKernel\n⊢ ⇑κ =ᵐ[μ] ⇑(μ ⊗ₘ κ).condKernel"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 630,
"column": 8
} | {
"line": 630,
"column": 21
} | {
"line": 630,
"column": 21
} | [
{
"pp": "case refine_2\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 : Antitone seq\nhseq_iInter : ⋂ i... | [
"case refine_2\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 : Antitone seq\nhseq_iInter : ⋂ i, seq i = ∅\... | integral_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 714,
"column": 2
} | {
"line": 715,
"column": 35
} | {
"line": 716,
"column": 2
} | [
{
"pp": "case neg.refine_1\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝ : CountablyGenerated γ\nκ : Kernel α (γ × β)\nn : ℕ\na : α\nx : γ\nseq : ℕ → Set β\nhseq : Monotone seq\nhseq_iUnion : ⋃ i, seq i = univ\nh0 : ¬(κ.fst a) (countable... | [
"case neg.refine_2\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝ : CountablyGenerated γ\nκ : Kernel α (γ × β)\nn : ℕ\na : α\nx : γ\nseq : ℕ → Set β\nhseq : Monotone seq\nhseq_iUnion : ⋃ i, seq i = univ\nh0 : ¬(κ.fst a) (countablePartitionSet... | · convert! tendsto_measure_iUnion_atTop (monotone_const.set_prod hseq)
rw [← prod_iUnion, hseq_iUnion] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 730,
"column": 2
} | {
"line": 730,
"column": 20
} | {
"line": 732,
"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)\ninst✝ : IsFiniteKernel κ\na : α\nh : ∀ᵐ (a_1 : γ) ∂κ.fst a, ∀ (i : ℕ), κ.densityProcess κ.fst i a a_1 univ = 1\nx : γ\nhx : ∀ (i : ℕ), κ... | [] | simp [density, hx] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.ProductMeasure | {
"line": 438,
"column": 40
} | {
"line": 438,
"column": 64
} | {
"line": 438,
"column": 64
} | [
{
"pp": "ι✝ : Type u_1\nX✝ : ι✝ → Type u_2\nmX✝ : (i : ι✝) → MeasurableSpace (X✝ i)\nμ✝ : (i : ι✝) → Measure (X✝ i)\nι : 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)\ns : Set ι\nhs : Countable ↑s\nt : (i : ι) → Set (X i... | [
"ι✝ : Type u_1\nX✝ : ι✝ → Type u_2\nmX✝ : (i : ι✝) → MeasurableSpace (X✝ i)\nμ✝ : (i : ι✝) → Measure (X✝ i)\nι : 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)\ns : Set ι\nhs : Countable ↑s\nt : (i : ι) → Set (X i)\nmt : ∀ i ... | Set.iUnion_finset_eq_set | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 373,
"column": 2
} | {
"line": 373,
"column": 6
} | {
"line": 374,
"column": 2
} | [
{
"pp": "α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\n⊢ (κ.singularPart η) a ⟂ₘ η a",
"ppTerm": "?m.29",
"assigned": true,
"usedConsta... | [
"α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\n⊢ η a ⟂ₘ (κ.singularPart η) a"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Probability.Kernel.Composition.AbsolutelyContinuous | {
"line": 50,
"column": 2
} | {
"line": 50,
"column": 6
} | {
"line": 51,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ η : Kernel α β\ninst✝² : IsFiniteKernel κ\ninst✝¹ : IsFiniteKernel η\ninst✝ : MeasurableSpace.CountableOrCountablyGenerated α β\nμ ν : Measure α\nhκη : ∀ᵐ (a : α) ∂μ, κ a ⟂ₘ η a\nhμ : SFinite μ\nhν : SFinite ν\ns : ... | [
"case pos\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ η : Kernel α β\ninst✝² : IsFiniteKernel κ\ninst✝¹ : IsFiniteKernel η\ninst✝ : MeasurableSpace.CountableOrCountablyGenerated α β\nμ ν : Measure α\nhκη : ∀ᵐ (a : α) ∂μ, κ a ⟂ₘ η a\nhμ : SFinite μ\nhν : SFinite ν\ns : Set (α × β) ... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Probability.Kernel.Posterior | {
"line": 169,
"column": 2
} | {
"line": 169,
"column": 55
} | {
"line": 170,
"column": 2
} | [
{
"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\nν : Measure 𝓧\ninst✝ : SFinite ν\nh_ac : ∀ᵐ (ω : Ω) ∂μ, κ ω ≪ ν\n⊢ μ ⊗ₘ κ ≪ μ ⊗ₘ Ker... | [
"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\nν : Measure 𝓧\ninst✝ : SFinite ν\nh_ac : ∀ᵐ (ω : Ω) ∂μ, κ ω ≪ ν\n⊢ ∀ᵐ (a : Ω) ∂μ, κ a ≪ (Kernel.... | refine Measure.AbsolutelyContinuous.compProd_right ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 506,
"column": 4
} | {
"line": 507,
"column": 81
} | {
"line": 508,
"column": 2
} | [
{
"pp": "case compl\nX : 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)\na : ℕ\nt : Set ((n : ℕ) → X n)\nmt : MeasurableSet t\nht : Measurable fun x₀ ↦ (trajFun κ a x₀) t\n⊢ Measurable fun x₀ ↦ (trajFun κ ... | [] | have := isProbabilityMeasure_trajFun κ a
simpa [measure_compl mt (measure_ne_top _ _)] using Measurable.const_sub ht _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 506,
"column": 4
} | {
"line": 507,
"column": 81
} | {
"line": 508,
"column": 2
} | [
{
"pp": "case compl\nX : 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)\na : ℕ\nt : Set ((n : ℕ) → X n)\nmt : MeasurableSet t\nht : Measurable fun x₀ ↦ (trajFun κ a x₀) t\n⊢ Measurable fun x₀ ↦ (trajFun κ ... | [] | have := isProbabilityMeasure_trajFun κ a
simpa [measure_compl mt (measure_ne_top _ _)] using Measurable.const_sub ht _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Posterior | {
"line": 221,
"column": 2
} | {
"line": 221,
"column": 55
} | {
"line": 222,
"column": 2
} | [
{
"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\ninst✝ : MeasurableSpace.CountableOrCountablyGenerated Ω 𝓧\nh_ac : ∀ᵐ (b : 𝓧) ∂⇑κ ∘ₘ... | [
"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\ninst✝ : MeasurableSpace.CountableOrCountablyGenerated Ω 𝓧\nh_ac : ∀ᵐ (b : 𝓧) ∂⇑κ ∘ₘ μ, (κ†μ) b ... | refine Measure.AbsolutelyContinuous.compProd_right ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Probability.Decision.BayesEstimator | {
"line": 84,
"column": 6
} | {
"line": 84,
"column": 24
} | {
"line": 84,
"column": 25
} | [
{
"pp": "case e_f\nΘ : 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∞\ninst✝⁴ : StandardBorelSpace Θ\ninst✝³ : Nonempty Θ\nhl : Measurable (Function.uncurry ℓ)\nP : Kernel Θ 𝓧\ninst✝² : IsFiniteKernel P\nκ : Kernel 𝓧 𝓨\nin... | [
"case e_f\nΘ : 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∞\ninst✝⁴ : StandardBorelSpace Θ\ninst✝³ : Nonempty Θ\nhl : Measurable (Function.uncurry ℓ)\nP : Kernel Θ 𝓧\ninst✝² : IsFiniteKernel P\nκ : Kernel 𝓧 𝓨\ninst✝¹ : IsSFi... | Kernel.prod_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Decision.Risk.Basic | {
"line": 238,
"column": 48
} | {
"line": 242,
"column": 67
} | {
"line": 244,
"column": 0
} | [
{
"pp": "Θ : Type u_1\n𝓧 : Type u_2\n𝓧' : Type u_3\n𝓨 : Type u_4\nmΘ : MeasurableSpace Θ\nm𝓧 : MeasurableSpace 𝓧\nm𝓧' : MeasurableSpace 𝓧'\nm𝓨 : MeasurableSpace 𝓨\nℓ : Θ → 𝓨 → ℝ≥0∞\nP : Kernel Θ 𝓧\nπ : Measure Θ\nη : Kernel 𝓧 𝓧'\ninst✝ : IsMarkovKernel η\n⊢ bayesRisk ℓ P π ≤ bayesRisk ℓ (η ∘ₖ P) π"... | [] | by
simp only [bayesRisk, avgRisk, le_iInf_iff]
intro κ hκ
rw [← κ.comp_assoc η]
exact iInf_le_of_le (κ ∘ₖ η) (iInf_le_of_le inferInstance le_rfl) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Decision.Risk.Basic | {
"line": 248,
"column": 49
} | {
"line": 250,
"column": 43
} | {
"line": 252,
"column": 0
} | [
{
"pp": "Θ : Type u_1\n𝓧 : Type u_2\n𝓧' : Type u_3\n𝓨 : Type u_4\nmΘ : MeasurableSpace Θ\nm𝓧 : MeasurableSpace 𝓧\nm𝓧' : MeasurableSpace 𝓧'\nm𝓨 : MeasurableSpace 𝓨\nℓ : Θ → 𝓨 → ℝ≥0∞\nP : Kernel Θ 𝓧\nπ : Measure Θ\nf : 𝓧 → 𝓧'\nhf : Measurable f\n⊢ bayesRisk ℓ P π ≤ bayesRisk ℓ (P.map f) π",
"ppTe... | [] | by
rw [← Kernel.deterministic_comp_eq_map hf]
exact bayesRisk_le_bayesRisk_comp _ _ _ _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Distributions.Exponential | {
"line": 57,
"column": 65
} | {
"line": 58,
"column": 42
} | {
"line": 60,
"column": 0
} | [
{
"pp": "r x : ℝ\nhx : 0 ≤ x\n⊢ exponentialPDF r x = ENNReal.ofReal (r * rexp (-(r * x)))",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Real.instLE",
"Real",
"HMul.hMul",
"Real.instZero",
"ENNReal.ofReal",
"congrArg",
"ProbabilityTheory.exponent... | [] | by
simp only [exponentialPDF_eq, if_pos hx] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Distributions.Exponential | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 32
} | {
"line": 123,
"column": 2
} | [
{
"pp": "b x : ℝ\nhb : 0 < b\n⊢ IntegrableOn (fun x ↦ rexp (-(b * x))) (Ioc 0 x) volume",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Ioc",
"NormedCommRing.toSeminormedCommRing",
"Real",
"MeasureTheory.Measure",
"NonUnitalCommRing.toNonU... | [
"b x : ℝ\nhb : 0 < b\n⊢ IntegrableOn (fun x ↦ rexp (-b * x)) (Ioc 0 x) volume"
] | simp only [neg_mul_eq_neg_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.ProbabilityMassFunction.Basic | {
"line": 100,
"column": 2
} | {
"line": 118,
"column": 88
} | {
"line": 120,
"column": 0
} | [
{
"pp": "α : Type u_1\np : PMF α\na : α\n⊢ p a = 1 ↔ p.support = {a}",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"PMF.tsum_coe",
"Set.Subset.antisymm",
"Iff.mpr",
"Eq.mpr",
"ENNReal.instAdd",
"Trans.trans",
... | [] | refine ⟨fun h => Set.Subset.antisymm (fun a' ha' => by_contra fun ha => ?_)
fun a' ha' => ha'.symm ▸ (p.mem_support_iff a).2 fun ha => zero_ne_one <| ha.symm.trans h,
fun h => _root_.trans (symm <| tsum_eq_single a
fun a' ha' => (p.apply_eq_zero_iff a').2 (h.symm ▸ ha')) p.tsum_coe⟩
suffices 1 < ∑' a, p... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.ProbabilityMassFunction.Basic | {
"line": 100,
"column": 2
} | {
"line": 118,
"column": 88
} | {
"line": 120,
"column": 0
} | [
{
"pp": "α : Type u_1\np : PMF α\na : α\n⊢ p a = 1 ↔ p.support = {a}",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"PMF.tsum_coe",
"Set.Subset.antisymm",
"Iff.mpr",
"Eq.mpr",
"ENNReal.instAdd",
"Trans.trans",
... | [] | refine ⟨fun h => Set.Subset.antisymm (fun a' ha' => by_contra fun ha => ?_)
fun a' ha' => ha'.symm ▸ (p.mem_support_iff a).2 fun ha => zero_ne_one <| ha.symm.trans h,
fun h => _root_.trans (symm <| tsum_eq_single a
fun a' ha' => (p.apply_eq_zero_iff a').2 (h.symm ▸ ha')) p.tsum_coe⟩
suffices 1 < ∑' a, p... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.ProbabilityMassFunction.Basic | {
"line": 234,
"column": 6
} | {
"line": 234,
"column": 51
} | {
"line": 234,
"column": 52
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\np : PMF α\ns : Set α\nhs : MeasurableSet s\n⊢ p.toMeasure s = 0 ↔ Disjoint p.support s",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"ChainCompletePartialOrder.instOfCompleteLattice",
... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\np : PMF α\ns : Set α\nhs : MeasurableSet s\n⊢ p.toOuterMeasure s = 0 ↔ Disjoint p.support s"
] | p.toMeasure_apply_eq_toOuterMeasure_apply hs, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.ProbabilityMassFunction.Basic | {
"line": 276,
"column": 4
} | {
"line": 276,
"column": 49
} | {
"line": 276,
"column": 50
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MeasurableSpace α\np : PMF α\ninst✝ : MeasurableSingletonClass α\ns : Set α\nhs : MeasurableSet (s ∩ p.support)\n⊢ p.toMeasure (s ∩ p.support) = p.toOuterMeasure s",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",... | [
"α : Type u_1\ninst✝¹ : MeasurableSpace α\np : PMF α\ninst✝ : MeasurableSingletonClass α\ns : Set α\nhs : MeasurableSet (s ∩ p.support)\n⊢ p.toOuterMeasure (s ∩ p.support) = p.toOuterMeasure s"
] | p.toMeasure_apply_eq_toOuterMeasure_apply hs, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.ProbabilityMassFunction.Monad | {
"line": 131,
"column": 37
} | {
"line": 131,
"column": 80
} | {
"line": 131,
"column": 80
} | [
{
"pp": "α : Type u_1\np : PMF α\nx y : α\nhy : y ≠ x\n⊢ p y * (pure y) x = 0",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"ENNReal.instAddCommMonoid",
"MulZeroClass.toMul",
"congrArg",
"PMF",
"CommSemiring.toSemiring",
... | [] | rw [pure_apply_of_ne _ _ hy.symm, mul_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.ProbabilityMassFunction.Monad | {
"line": 131,
"column": 37
} | {
"line": 131,
"column": 80
} | {
"line": 131,
"column": 80
} | [
{
"pp": "α : Type u_1\np : PMF α\nx y : α\nhy : y ≠ x\n⊢ p y * (pure y) x = 0",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"ENNReal.instAddCommMonoid",
"MulZeroClass.toMul",
"congrArg",
"PMF",
"CommSemiring.toSemiring",
... | [] | rw [pure_apply_of_ne _ _ hy.symm, mul_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.ProbabilityMassFunction.Monad | {
"line": 131,
"column": 37
} | {
"line": 131,
"column": 80
} | {
"line": 131,
"column": 80
} | [
{
"pp": "α : Type u_1\np : PMF α\nx y : α\nhy : y ≠ x\n⊢ p y * (pure y) x = 0",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"ENNReal.instAddCommMonoid",
"MulZeroClass.toMul",
"congrArg",
"PMF",
"CommSemiring.toSemiring",
... | [] | rw [pure_apply_of_ne _ _ hy.symm, mul_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.ProbabilityMassFunction.Constructions | {
"line": 314,
"column": 39
} | {
"line": 314,
"column": 56
} | {
"line": 314,
"column": 57
} | [
{
"pp": "case false\np : ℝ≥0\nh : p ≤ 1\n⊢ ¬1 - ↑p = 0 ↔ false ∈ {b | if b = true then ¬p = 0 else ¬p = 1}",
"ppTerm": "?false",
"assigned": true,
"usedConstants": [
"ENNReal.ofNNReal",
"setOf",
"HSub.hSub",
"Membership.mem",
"id",
"instDecidableEqBool",
"NN... | [
"case false\np : ℝ≥0\nh : p ≤ 1\n⊢ ¬1 - ↑p = 0 ↔ if false = true then ¬p = 0 else ¬p = 1"
] | Set.mem_setOf_eq, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Probability.Independence.InfinitePi | {
"line": 133,
"column": 6
} | {
"line": 133,
"column": 32
} | {
"line": 133,
"column": 33
} | [
{
"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)\ni : ι\n⊢ map (X i) (P i) = map (fun ω ... | [
"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)\ni : ι\n⊢ map (X i) (map (fun x ↦ x i) (infinitePi ... | ← infinitePi_map_eval P i, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.IdentDistribIndep | {
"line": 63,
"column": 12
} | {
"line": 69,
"column": 21
} | {
"line": 71,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nι : Type u_3\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nμ : Measure Ω\nν : Measure Ω'\ninst✝ : Countable ι\nE : ι → Type u_6\nmE : (i : ι) → MeasurableSpace (E i)\nX : (i : ι) → Ω → E i\nY : (i : ι) → Ω' → E i\nh : ∀ (i : ι), IdentDistrib (X i) (Y i) μ ν\nhX_ind : i... | [] | by
have : IsProbabilityMeasure μ := hX_ind.isProbabilityMeasure
have : IsProbabilityMeasure ν := hY_ind.isProbabilityMeasure
rw [(iIndepFun_iff_map_fun_eq_infinitePi_map₀' (fun i ↦ (h i).aemeasurable_fst)).mp hX_ind,
(iIndepFun_iff_map_fun_eq_infinitePi_map₀' (fun i ↦ (h i).aemeasurable_snd)).mp hY_in... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Independence.ZeroOne | {
"line": 185,
"column": 9
} | {
"line": 185,
"column": 19
} | {
"line": 185,
"column": 19
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\ns : ι → MeasurableSpace Ω\nm0 : MeasurableSpace Ω\nκ : Kernel α Ω\nμα : Measure α\nβ : Type u_4\np : Set ι → Prop\nf : Filter ι\nns : β → Set ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s κ μα\nhf : ∀ (t : Set ι), p t → tᶜ ∈ f\nhns ... | [
"α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\ns : ι → MeasurableSpace Ω\nm0 : MeasurableSpace Ω\nκ : Kernel α Ω\nμα : Measure α\nβ : Type u_4\np : Set ι → Prop\nf : Filter ι\nns : β → Set ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s κ μα\nhf : ∀ (t : Set ι), p t → tᶜ ∈ f\nhns : Directed (... | iSup_const | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 180,
"column": 6
} | {
"line": 180,
"column": 19
} | {
"line": 181,
"column": 4
} | [
{
"pp": "case refine_1\nX Y : SFinKer\n⊢ { hom := Kernel.swap X.carrier Y.carrier, property := ⋯ } ≫\n { hom := Kernel.swap Y.carrier X.carrier, property := ⋯ } =\n 𝟙 (X ⊗ Y)",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"SFinKer.instMonoidalCategory",
"SFinKer.h... | [] | ext : 1; simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 180,
"column": 6
} | {
"line": 180,
"column": 19
} | {
"line": 181,
"column": 4
} | [
{
"pp": "case refine_1\nX Y : SFinKer\n⊢ { hom := Kernel.swap X.carrier Y.carrier, property := ⋯ } ≫\n { hom := Kernel.swap Y.carrier X.carrier, property := ⋯ } =\n 𝟙 (X ⊗ Y)",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"SFinKer.instMonoidalCategory",
"SFinKer.h... | [] | ext : 1; simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 181,
"column": 6
} | {
"line": 181,
"column": 19
} | {
"line": 182,
"column": 2
} | [
{
"pp": "case refine_2\nX Y : SFinKer\n⊢ { hom := Kernel.swap Y.carrier X.carrier, property := ⋯ } ≫\n { hom := Kernel.swap X.carrier Y.carrier, property := ⋯ } =\n 𝟙 (Y ⊗ X)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"SFinKer.instMonoidalCategory",
"SFinKer.h... | [] | ext : 1; simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 181,
"column": 6
} | {
"line": 181,
"column": 19
} | {
"line": 182,
"column": 2
} | [
{
"pp": "case refine_2\nX Y : SFinKer\n⊢ { hom := Kernel.swap Y.carrier X.carrier, property := ⋯ } ≫\n { hom := Kernel.swap X.carrier Y.carrier, property := ⋯ } =\n 𝟙 (Y ⊗ X)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"SFinKer.instMonoidalCategory",
"SFinKer.h... | [] | ext : 1; simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 201,
"column": 4
} | {
"line": 201,
"column": 17
} | {
"line": 203,
"column": 0
} | [
{
"pp": "X Y : SFinKer\n⊢ { hom := { hom := Kernel.swap X.carrier Y.carrier, property := ⋯ },\n inv := { hom := Kernel.swap Y.carrier X.carrier, property := ⋯ }, hom_inv_id := ⋯, inv_hom_id := ⋯ }.hom ≫\n { hom := { hom := Kernel.swap Y.carrier X.carrier, property := ⋯ },\n inv := { hom :... | [] | ext : 1; simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 201,
"column": 4
} | {
"line": 201,
"column": 17
} | {
"line": 203,
"column": 0
} | [
{
"pp": "X Y : SFinKer\n⊢ { hom := { hom := Kernel.swap X.carrier Y.carrier, property := ⋯ },\n inv := { hom := Kernel.swap Y.carrier X.carrier, property := ⋯ }, hom_inv_id := ⋯, inv_hom_id := ⋯ }.hom ≫\n { hom := { hom := Kernel.swap Y.carrier X.carrier, property := ⋯ },\n inv := { hom :... | [] | ext : 1; simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Independence.Conditional | {
"line": 790,
"column": 83
} | {
"line": 791,
"column": 98
} | {
"line": 793,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nm' mΩ : MeasurableSpace Ω\ninst✝² : StandardBorelSpace Ω\nhm' : m' ≤ mΩ\nμ : Measure Ω\ninst✝¹ : IsFiniteMeasure μ\nf : Ω → β\ng : Ω → β'\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\ninst✝ : CountableOrCountablyGenerated Ω (β × β')\nhf : Measurable f\nhg... | [] | by
rw [condIndepFun_iff_compProd_map_prod_eq_compProd_prod_map_map hf hg, ← Kernel.compProd_eq_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.ProbabilityMassFunction.Binomial | {
"line": 75,
"column": 77
} | {
"line": 75,
"column": 81
} | {
"line": 75,
"column": 82
} | [
{
"pp": "k b : ℕ\nhb : k ≤ b\nx : ℝ≥0\nh : x ≤ 1\neq0 : k % (b + 1) = k\neq1 : 1 - ↑x = ENNReal.ofReal (1 - ↑x)\nthis : 1 - ↑x ≥ 0\n⊢ ENNReal.ofReal (↑(b.choose k) * ↑x ^ k * (1 - ↑x) ^ (b - k)) =\n ↑x ^ (k % (b + 1)) * (1 - ↑x) ^ (b - k % (b + 1)) * ↑(b.choose (k % (b + 1)))",
"ppTerm": "?m.122",
"a... | [
"k b : ℕ\nhb : k ≤ b\nx : ℝ≥0\nh : x ≤ 1\neq0 : k % (b + 1) = k\neq1 : 1 - ↑x = ENNReal.ofReal (1 - ↑x)\nthis : 1 - ↑x ≥ 0\n⊢ ENNReal.ofReal (↑(b.choose k) * ↑x ^ k * (1 - ↑x) ^ (b - k)) = ↑x ^ k * (1 - ↑x) ^ (b - k) * ↑(b.choose k)"
] | eq0, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 266,
"column": 63
} | {
"line": 266,
"column": 88
} | {
"line": 268,
"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\nω' : Ω'\nhm : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)\nt : ℝ\n⊢ mgf (-X) (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)",
"ppTerm": "?m.72",... | [] | simpa [mgf] using hm (-t) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Probability.Moments.SubGaussian | {
"line": 266,
"column": 63
} | {
"line": 266,
"column": 88
} | {
"line": 268,
"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\nω' : Ω'\nhm : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)\nt : ℝ\n⊢ mgf (-X) (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)",
"ppTerm": "?m.72",... | [] | simpa [mgf] using hm (-t) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Moments.SubGaussian | {
"line": 266,
"column": 63
} | {
"line": 266,
"column": 88
} | {
"line": 268,
"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\nω' : Ω'\nhm : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)\nt : ℝ\n⊢ mgf (-X) (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)",
"ppTerm": "?m.72",... | [] | simpa [mgf] using hm (-t) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Process.LocalProperty | {
"line": 251,
"column": 4
} | {
"line": 251,
"column": 21
} | {
"line": 252,
"column": 4
} | [
{
"pp": "ι : Type u_1\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝⁴ : ConditionallyCompleteLinearOrderBot ι\ninst✝³ : TopologicalSpace ι\ninst✝² : OrderTopology ι\n𝓕 : Filtration ι mΩ\ninst✝¹ : SecondCountableTopology ι\ninst✝ : IsFiniteMeasure P\nτ : ℕ → Ω → WithTop ι\nσ : ℕ → ℕ → Ω → WithTop ι... | [
"ι : Type u_1\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝⁴ : ConditionallyCompleteLinearOrderBot ι\ninst✝³ : TopologicalSpace ι\ninst✝² : OrderTopology ι\n𝓕 : Filtration ι mΩ\ninst✝¹ : SecondCountableTopology ι\ninst✝ : IsFiniteMeasure P\nτ : ℕ → Ω → WithTop ι\nσ : ℕ → ℕ → Ω → WithTop ι\nhτ : IsLoc... | specialize hω hij | Lean.Elab.Tactic.evalSpecialize | Lean.Parser.Tactic.specialize |
Mathlib.Probability.Moments.SubGaussian | {
"line": 449,
"column": 8
} | {
"line": 450,
"column": 47
} | {
"line": 450,
"column": 48
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF X cX κ ν\nhY : HasSubgaussianMGF Y cY κ ν\nhX0 : ¬cX = 0\nhY0 : ¬cY = 0\np : ℝ≥0 := (NNReal.sqrt cX + NNReal.sqrt cY) / NNReal.sqrt cX\nq : ℝ≥... | [] | linear_combination t ^ 2 * (-√↑cY * Real.sq_sqrt cX.coe_nonneg
-√↑cX * Real.sq_sqrt cY.coe_nonneg) | Mathlib.Tactic.LinearCombination._aux_Mathlib_Tactic_LinearCombination___elabRules_Mathlib_Tactic_LinearCombination_linearCombination_1 | Mathlib.Tactic.LinearCombination.linearCombination |
Mathlib.Probability.Moments.SubGaussian | {
"line": 481,
"column": 15
} | {
"line": 481,
"column": 68
} | {
"line": 481,
"column": 68
} | [
{
"pp": "case pos.inr.refine_1\nΩ : 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\nΩ'' : Type u_3\nmΩ'' : MeasurableSpace Ω''\nY : Ω'' → ℝ\ncY : ℝ≥0\ninst✝¹ : SFinite ν\nη : Kernel (Ω' × Ω) Ω''\ninst✝ : IsZeroOrMarkovKernel η\nhX :... | [
"case pos.inr.refine_1\nΩ : 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\nΩ'' : Type u_3\nmΩ'' : MeasurableSpace Ω''\nY : Ω'' → ℝ\ncY : ℝ≥0\ninst✝¹ : SFinite ν\nη : Kernel (Ω' × Ω) Ω''\ninst✝ : IsZeroOrMarkovKernel η\nhX : HasSubgauss... | memLp_map_measure_iff h.1 measurable_fst.aemeasurable | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RepresentationTheory.Action | {
"line": 85,
"column": 32
} | {
"line": 85,
"column": 57
} | {
"line": 87,
"column": 0
} | [
{
"pp": "k : Type u\nG : Type v\nV : Type u'\nW : Type v'\ninst✝⁵ : Monoid G\ninst✝⁴ : Semiring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AddCommGroup W\ninst✝ : Module k W\nσ : Representation k G V\nρ : Representation k G W\nX Y Z : Action (Type w) G\ng : G\n⊢ (↑(MonoidAlgebra.uniqueLinearEquiv... | [] | simp [linearize_single _] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RepresentationTheory.Action | {
"line": 97,
"column": 31
} | {
"line": 97,
"column": 56
} | {
"line": 99,
"column": 0
} | [
{
"pp": "k : Type u\nG : Type v\nV : Type u'\nW : Type v'\ninst✝⁵ : Monoid G\ninst✝⁴ : Semiring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AddCommGroup W\ninst✝ : Module k W\nσ : Representation k G V\nρ : Representation k G W\nX Y Z : Action (Type w) G\ng : G\nx✝ : (𝟙_ (Action (Type u) G)).V\n⊢ ... | [] | simp [linearize_single _] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RepresentationTheory.Action | {
"line": 120,
"column": 31
} | {
"line": 120,
"column": 56
} | {
"line": 120,
"column": 56
} | [
{
"pp": "k✝ : Type u\nG : Type v\nV : Type u'\nW : Type v'\ninst✝⁸ : Monoid G\ninst✝⁷ : Semiring k✝\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : Module k✝ V\ninst✝⁴ : AddCommGroup W\ninst✝³ : Module k✝ W\nσ✝ : Representation k✝ G V\nρ✝ : Representation k✝ G W\nX Y Z : Action (Type w) G\nk : Type u\ninst✝² : CommSemiring ... | [
"k✝ : Type u\nG : Type v\nV : Type u'\nW : Type v'\ninst✝⁸ : Monoid G\ninst✝⁷ : Semiring k✝\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : Module k✝ V\ninst✝⁴ : AddCommGroup W\ninst✝³ : Module k✝ W\nσ✝ : Representation k✝ G V\nρ✝ : Representation k✝ G W\nX Y Z : Action (Type w) G\nk : Type u\ninst✝² : CommSemiring k\ninst✝¹ : ... | simp [linearize_single _] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.Moments.SubGaussian | {
"line": 773,
"column": 4
} | {
"line": 773,
"column": 49
} | {
"line": 774,
"column": 4
} | [
{
"pp": "case h_indep\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nι : Type u_2\nX : ι → Ω → ℝ\nh_indep : iIndepFun X μ\nc : ι → ℝ≥0\ns : Finset ι\nh_subG : ∀ i ∈ s, HasSubgaussianMGF (X i) (c i) μ\n⊢ iIndepFun (fun i ↦ X ↑i) μ",
"ppTerm": "?h_indep",
"assigned": true,
"usedConstants": [
... | [
"case h_meas\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nι : Type u_2\nX : ι → Ω → ℝ\nh_indep : iIndepFun X μ\nc : ι → ℝ≥0\ns : Finset ι\nh_subG : ∀ i ∈ s, HasSubgaussianMGF (X i) (c i) μ\n⊢ ∀ (i : ↥s), AEMeasurable (X ↑i) μ",
"case h_subG\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nι : Type u... | · exact h_indep.precomp Subtype.val_injective | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.StrongLaw | {
"line": 404,
"column": 8
} | {
"line": 406,
"column": 58
} | {
"line": 407,
"column": 6
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : ℕ → Ω → ℝ\nhint : Integrable (X 0) ℙ\nhindep : Pairwise ((fun f g ↦ f ⟂ᵢ g) on X)\nhident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ\nhnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω\nc : ℝ\nc_one : 1 < c\nε : ℝ\nεpos : 0 < ε\nc_pos : 0 < ... | [] | gcongr with j
rw [(hident j).truncation.variance_eq]
exact variance_le_expectation_sq (hX 0).truncation | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.StrongLaw | {
"line": 404,
"column": 8
} | {
"line": 406,
"column": 58
} | {
"line": 407,
"column": 6
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : ℕ → Ω → ℝ\nhint : Integrable (X 0) ℙ\nhindep : Pairwise ((fun f g ↦ f ⟂ᵢ g) on X)\nhident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ\nhnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω\nc : ℝ\nc_one : 1 < c\nε : ℝ\nεpos : 0 < ε\nc_pos : 0 < ... | [] | gcongr with j
rw [(hident j).truncation.variance_eq]
exact variance_le_expectation_sq (hX 0).truncation | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RepresentationTheory.Invariants | {
"line": 95,
"column": 36
} | {
"line": 95,
"column": 49
} | {
"line": 95,
"column": 50
} | [
{
"pp": "case succ\nk : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝³ : CommRing k\ninst✝² : Group G\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nρ : Representation k G V\ng : G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g\nx : V\nhx : (ρ g) x = x\ni : ℕ\na✝ : (ρ ((fun x ↦ g ^ x) ↑i)) x = x\n⊢ (ρ ((fun x ↦ g ^ x) (↑... | [] | | succ i _ => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RepresentationTheory.Invariants | {
"line": 96,
"column": 4
} | {
"line": 96,
"column": 82
} | {
"line": 96,
"column": 82
} | [
{
"pp": "case pred\nk : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝³ : CommRing k\ninst✝² : Group G\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nρ : Representation k G V\ng : G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g\nx : V\nhx : (ρ g) x = x\ni : ℕ\nh : (ρ ((fun x ↦ g ^ x) (-↑i))) x = x\n⊢ (ρ ((fun x ↦ g ^ x) ... | [] | simpa [neg_sub_comm _ (1 : ℤ), zpow_sub] using congr(ρ g⁻¹ $(h.trans hx.symm)) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RepresentationTheory.Invariants | {
"line": 189,
"column": 66
} | {
"line": 198,
"column": 46
} | {
"line": 200,
"column": 0
} | [
{
"pp": "k : Type u\ninst✝¹ : CommRing k\nG : Type v\ninst✝ : Group G\nX Y : Rep k G\nf : ↑X →ₗ[k] ↑Y\ng : G\n⊢ ((X.ρ.linHom Y.ρ) g) f = f ↔ f ∘ₗ X.ρ g = Y.ρ g ∘ₗ f",
"ppTerm": "?m.92",
"assigned": true,
"usedConstants": [
"LinearMap.id",
"Eq.mpr",
"Rep.V",
"MonoidHom.instMon... | [] | by
dsimp
constructor
· intro h
nth_rw 1 [← h]
rw [LinearMap.comp_assoc, LinearMap.comp_assoc, ← Rep.ρ_mul, inv_mul_cancel, map_one,
Module.End.one_eq_id, LinearMap.comp_id]
· intro h
rw [← LinearMap.comp_assoc, ← h, LinearMap.comp_assoc, ← Rep.ρ_mul, mul_inv_cancel, map_one,
Module.End.o... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RepresentationTheory.Character | {
"line": 172,
"column": 78
} | {
"line": 175,
"column": 36
} | {
"line": 177,
"column": 0
} | [
{
"pp": "k : Type u\ninst✝¹ : Field k\nG : Type v\ninst✝ : Monoid G\nV W : FDRep k G\ni : V ≅ W\n⊢ V.character = W.character",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"LinearMap.trace",
"Eq.mpr",
"MonoidHom.instFunLike",
"Semiring.toModule",
"MonoidHom",... | [] | by
ext g
simp only [character, FDRep.Iso.conj_ρ i]
exact (trace_conj' (V.ρ g) _).symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RepresentationTheory.Coinvariants | {
"line": 131,
"column": 31
} | {
"line": 131,
"column": 84
} | {
"line": 133,
"column": 0
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nX : Type u_5\ninst✝⁷ : CommRing k\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AddCommGroup W\ninst✝² : Module k W\ninst✝¹ : AddCommGroup X\ninst✝ : Module k X\nρ : Representation k G V\nτ : Representation k G W\nυ : ... | [] | by simpa using congr($((f.isIntertwining' g).symm) x) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.StrongLaw | {
"line": 754,
"column": 2
} | {
"line": 755,
"column": 36
} | {
"line": 756,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁵ : IsProbabilityMeasure μ\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nX : ℕ → Ω → E\nhint : Integrable (X 0) μ\nh' : StronglyMeasurable (X 0... | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁵ : IsProbabilityMeasure μ\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nX : ℕ → Ω → E\nhint : Integrable (X 0) μ\nh' : StronglyMeasurable (X 0)\nhindep : ... | have I : ∀ᶠ n in atTop, (∑ i ∈ range n, ‖(X i - Y k i) ω‖) / n < δ :=
(tendsto_order.1 (h'ω k)).2 δ hk | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RepresentationTheory.FinGroupCharZero | {
"line": 51,
"column": 6
} | {
"line": 51,
"column": 61
} | {
"line": 52,
"column": 4
} | [
{
"pp": "k : Type u\ninst✝³ : Field k\nG : Type u\ninst✝² : Finite G\ninst✝¹ : Group G\ninst✝ : NeZero ↑(Nat.card G)\nV : Rep k G\n⊢ Injective V",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Injective",
"CategoryTheory.Equivalence.map_injective_... | [
"k : Type u\ninst✝³ : Field k\nG : Type u\ninst✝² : Finite G\ninst✝¹ : Group G\ninst✝ : NeZero ↑(Nat.card G)\nV : Rep k G\n⊢ Injective (equivalenceModuleMonoidAlgebra.functor.obj V)"
] | ← Rep.equivalenceModuleMonoidAlgebra.map_injective_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RepresentationTheory.FinGroupCharZero | {
"line": 125,
"column": 4
} | {
"line": 125,
"column": 8
} | {
"line": 125,
"column": 8
} | [
{
"pp": "case mp\nk : Type u\ninst✝⁴ : Field k\nG : Type u\ninst✝³ : Group G\ninst✝² : IsAlgClosed k\ninst✝¹ : CharZero k\ninst✝ : Fintype G\nV : FDRep k G\nthis : Invertible ↑(Nat.card G)\nh : Simple V\n⊢ ∑ g, V.character g * V.character g⁻¹ = ↑(Nat.card G)",
"ppTerm": "?mp",
"assigned": true,
"use... | [
"case mp\nk : Type u\ninst✝⁴ : Field k\nG : Type u\ninst✝³ : Group G\ninst✝² : IsAlgClosed k\ninst✝¹ : CharZero k\ninst✝ : Fintype G\nV : FDRep k G\nthis : Invertible ↑(Nat.card G)\nh : Simple V\n⊢ ↑(Nat.card G) = ∑ g, V.character g * V.character g⁻¹"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 15
} | {
"line": 116,
"column": 2
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\n⊢ ModuleCat.ofHom A.ρ.invariants.subtype ≫ d₀₁ A = 0",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Pi.Function.module",
"Submodule",
"Rep.V",
"CategoryTheory.CategoryStruct.toQuiver",
... | [
"k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nx : ↑A\nhx : x ∈ A.ρ.invariants\ng : G\n⊢ (ModuleCat.Hom.hom (ModuleCat.ofHom A.ρ.invariants.subtype ≫ d₀₁ A)) ⟨x, hx⟩ g = (ModuleCat.Hom.hom 0) ⟨x, hx⟩ g"
] | ext ⟨x, hx⟩ g | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree | {
"line": 380,
"column": 2
} | {
"line": 380,
"column": 74
} | {
"line": 382,
"column": 0
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nf : ↥(cocycles₂ A)\ng : G\nthis : (A.ρ 1) (f (1, g)) + f (1, 1 * g) = f (1 * 1, g) + f (1, 1)\n⊢ f (1, g) = f (1, 1)",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Pi.Function.module",
"Submodule",
... | [] | simpa only [map_one, Module.End.one_apply, one_mul, add_right_inj, this] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RepresentationTheory.Homological.GroupHomology.LongExactSequence | {
"line": 156,
"column": 86
} | {
"line": 158,
"column": 27
} | {
"line": 160,
"column": 0
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nX : ShortComplex (Rep k G)\nhX : X.ShortExact\ny : G × G →₀ ↑X.X₂\nx : G →₀ ↑X.X₁\nhx : (mapRange.linearMap (Rep.Hom.hom X.f).toLinearMap) x = (ConcreteCategory.hom (d₂₁ X.X₂)) y\n⊢ (Rep.Hom.hom X.f) ((ModuleCat.Hom.hom (d₁₀ X.X₁)) x) = (Rep.Hom.hom X... | [] | by
have := congr($((mapShortComplexH1 (MonoidHom.id G) X.f).comm₂₃.symm) x)
simp_all [shortComplexH1] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RepresentationTheory.Homological.TateCohomology.Basic | {
"line": 81,
"column": 2
} | {
"line": 82,
"column": 21
} | {
"line": 84,
"column": 0
} | [
{
"pp": "R G : Type u\ninst✝² : CommRing R\ninst✝¹ : Group G\ninst✝ : Fintype G\nM : Rep R G\n⊢ d₁₀ M ≫ Hom.toModuleCatHom M.norm = 0",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Rep.V",
"Representation",
"MonoidHom.instFunLike",
"LinearMap.comp.congr_simp",
... | [] | ext
simp [d₁₀_single M] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RepresentationTheory.Homological.TateCohomology.Basic | {
"line": 81,
"column": 2
} | {
"line": 82,
"column": 21
} | {
"line": 84,
"column": 0
} | [
{
"pp": "R G : Type u\ninst✝² : CommRing R\ninst✝¹ : Group G\ninst✝ : Fintype G\nM : Rep R G\n⊢ d₁₀ M ≫ Hom.toModuleCatHom M.norm = 0",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Rep.V",
"Representation",
"MonoidHom.instFunLike",
"LinearMap.comp.congr_simp",
... | [] | ext
simp [d₁₀_single M] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RepresentationTheory.Homological.TateCohomology.Basic | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 24
} | {
"line": 88,
"column": 0
} | [
{
"pp": "R G : Type u\ninst✝² : CommRing R\ninst✝¹ : Group G\ninst✝ : Fintype G\nM : Rep R G\n⊢ (ChainComplex.of.d (fun n ↦ ModuleCat.of R ((Fin n → G) →₀ ↑M)) (fun n ↦ inhomogeneousChains.d M n) 1 0 ≫\n (chainsIso₀ M).hom) ≫\n Hom.toModuleCatHom M.norm =\n 0",
"ppTerm": "?m.34",
"assigne... | [] | simp [← comp_d₁₀_eq _] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality | {
"line": 425,
"column": 4
} | {
"line": 427,
"column": 72
} | {
"line": 428,
"column": 2
} | [
{
"pp": "case h.refine_3.H\nk G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\nA : Rep k G\nS : Subgroup G\ninst✝ : S.Normal\nx : ↥(cocycles₁ A)\nhx :\n (ConcreteCategory.hom (H1π (res S.subtype A)))\n ((ConcreteCategory.hom (mapCocycles₁ S.subtype (𝟙 (res S.subtype A)))) x) =\n 0\ny : ↑A\nhy :\n (... | [] | induction h using QuotientGroup.induction_on with | @H h =>
apply Subtype.ext
simp [← QuotientGroup.mk_mul, h1 g h, sub_add_eq_add_sub, add_assoc] | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality | {
"line": 425,
"column": 4
} | {
"line": 427,
"column": 72
} | {
"line": 428,
"column": 2
} | [
{
"pp": "case h.refine_3.H\nk G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\nA : Rep k G\nS : Subgroup G\ninst✝ : S.Normal\nx : ↥(cocycles₁ A)\nhx :\n (ConcreteCategory.hom (H1π (res S.subtype A)))\n ((ConcreteCategory.hom (mapCocycles₁ S.subtype (𝟙 (res S.subtype A)))) x) =\n 0\ny : ↑A\nhy :\n (... | [] | induction h using QuotientGroup.induction_on with | @H h =>
apply Subtype.ext
simp [← QuotientGroup.mk_mul, h1 g h, sub_add_eq_add_sub, add_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality | {
"line": 425,
"column": 4
} | {
"line": 427,
"column": 72
} | {
"line": 428,
"column": 2
} | [
{
"pp": "case h.refine_3.H\nk G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\nA : Rep k G\nS : Subgroup G\ninst✝ : S.Normal\nx : ↥(cocycles₁ A)\nhx :\n (ConcreteCategory.hom (H1π (res S.subtype A)))\n ((ConcreteCategory.hom (mapCocycles₁ S.subtype (𝟙 (res S.subtype A)))) x) =\n 0\ny : ↑A\nhy :\n (... | [] | induction h using QuotientGroup.induction_on with | @H h =>
apply Subtype.ext
simp [← QuotientGroup.mk_mul, h1 g h, sub_add_eq_add_sub, add_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality | {
"line": 428,
"column": 4
} | {
"line": 428,
"column": 8
} | {
"line": 429,
"column": 4
} | [
{
"pp": "case h.refine_4\nk G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\nA : Rep k G\nS : Subgroup G\ninst✝ : S.Normal\nx : ↥(cocycles₁ A)\nhx :\n (ConcreteCategory.hom (H1π (res S.subtype A)))\n ((ConcreteCategory.hom (mapCocycles₁ S.subtype (𝟙 (res S.subtype A)))) x) =\n 0\ny : ↑A\nhy :\n (Mo... | [
"case h.refine_4\nk G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\nA : Rep k G\nS : Subgroup G\ninst✝ : S.Normal\nx : ↥(cocycles₁ A)\nhx :\n (ConcreteCategory.hom (H1π (res S.subtype A)))\n ((ConcreteCategory.hom (mapCocycles₁ S.subtype (𝟙 (res S.subtype A)))) x) =\n 0\ny : ↑A\nhy :\n (ModuleCat.Hom.... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.RepresentationTheory.Tannaka | {
"line": 195,
"column": 4
} | {
"line": 195,
"column": 12
} | {
"line": 196,
"column": 4
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Finite G\ns x✝ : G\nf : ↑rightFDRep.V.obj\n⊢ (Hom.hom (rightFDRep.ρ x✝ ≫ InducedCategory.homMk (↟(leftRegular s))).hom) f =\n (Hom.hom (InducedCategory.homMk (↟(leftRegular s)) ≫ rightFDRep.ρ x✝).hom) f",
"ppTerm": "?m.60",
"assign... | [
"k G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Finite G\ns x✝¹ : G\nf : ↑rightFDRep.V.obj\nx✝ : G\n⊢ (Hom.hom (rightFDRep.ρ x✝¹ ≫ InducedCategory.homMk (↟(leftRegular s))).hom) f x✝ =\n (Hom.hom (InducedCategory.homMk (↟(leftRegular s)) ≫ rightFDRep.ρ x✝¹).hom) f x✝"
] | funext _ | _aux_Init_NotationExtra___macroRules_tacticFunext____1 | tacticFunext___ |
Mathlib.RingTheory.MvPolynomial.Ideal | {
"line": 189,
"column": 2
} | {
"line": 189,
"column": 42
} | {
"line": 190,
"column": 2
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nk : Type u_3\ninst✝ : Field k\nB : Set (MvPolynomial σ k)\n⊢ span (m.leadingTerm '' B) = span ((fun p ↦ (MvPolynomial.monomial (m.degree p)) 1) '' (B \\ {0}))",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"MonomialOrder.span_leadingTerm_e... | [
"σ : Type u_1\nm : MonomialOrder σ\nk : Type u_3\ninst✝ : Field k\nB : Set (MvPolynomial σ k)\n⊢ ∀ p ∈ B, IsUnit (m.leadingCoeff p) ∨ p = 0"
] | apply span_leadingTerm_eq_span_monomial₀ | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.MvPowerSeries.Equiv | {
"line": 104,
"column": 2
} | {
"line": 104,
"column": 6
} | {
"line": 104,
"column": 6
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : Finite σ\np : MvPolynomial σ R\n⊢ (toAdicCompletion σ R) ↑p = (AdicCompletion.of (MvPolynomial.idealOfVars σ R) (MvPolynomial σ R)) p",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"Semir... | [
"σ : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : Finite σ\np : MvPolynomial σ R\n⊢ (AdicCompletion.of (MvPolynomial.idealOfVars σ R) (MvPolynomial σ R)) p = (toAdicCompletion σ R) ↑p"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 6
} | {
"line": 106,
"column": 2
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\n⊢ (ModuleCat.Hom.hom (d₁₀ A)).range = Coinvariants.ker A.ρ",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Submodule",
"Representation.Coinvariants.ker",
"Rep.V",
"RingHomSurjective.ids",
... | [
"k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\n⊢ Coinvariants.ker A.ρ = (ModuleCat.Hom.hom (d₁₀ A)).range"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.RingTheory.MvPowerSeries.Rename | {
"line": 312,
"column": 4
} | {
"line": 312,
"column": 75
} | {
"line": 314,
"column": 0
} | [
{
"pp": "case refine_3\nσ : Type u_1\nτ : Type u_2\nf : σ → τ\ninst✝¹ : TendstoCofinite f\nR : Type u_6\ninst✝ : CommRing R\np : MvPowerSeries σ R\nn : τ →₀ ℕ\nthis : ∀ (d : σ →₀ ℕ), (coeff d) p * (coeff n) (d.prod fun s e ↦ X (f s) ^ e) ≠ 0 → mapDomain f d = n\nx : σ →₀ ℕ\nhx : x ∈ Function.support fun d ↦ (co... | [] | simp [← this _ hx, ← monomial_mapDomain_apply_one, coeff_monomial_same] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.AdicCompletion.LocalRing | {
"line": 104,
"column": 2
} | {
"line": 108,
"column": 21
} | {
"line": 110,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsLocalRing R\nx : AdicCompletion (maximalIdeal R) R\n⊢ x ∈ maximalIdeal (AdicCompletion (maximalIdeal R) R) ↔ ↑x 1 = 0",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Ideal.fg_of_isNoetherianRing",
... | [] | have : (AdicCompletion.eval (maximalIdeal R) R 1).ker =
(maximalIdeal R) • (⊤ : Submodule R (AdicCompletion (maximalIdeal R) R)) := by
simp [← pow_smul_top_eq_ker_eval (maximalIdeal R).fg_of_isNoetherianRing]
rw [maximalIdeal_eq_map, ← Submodule.restrictScalars_mem R, ← Ideal.smul_top_eq_map]
simp [← this, ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.AdicCompletion.LocalRing | {
"line": 104,
"column": 2
} | {
"line": 108,
"column": 21
} | {
"line": 110,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsLocalRing R\nx : AdicCompletion (maximalIdeal R) R\n⊢ x ∈ maximalIdeal (AdicCompletion (maximalIdeal R) R) ↔ ↑x 1 = 0",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Ideal.fg_of_isNoetherianRing",
... | [] | have : (AdicCompletion.eval (maximalIdeal R) R 1).ker =
(maximalIdeal R) • (⊤ : Submodule R (AdicCompletion (maximalIdeal R) R)) := by
simp [← pow_smul_top_eq_ker_eval (maximalIdeal R).fg_of_isNoetherianRing]
rw [maximalIdeal_eq_map, ← Submodule.restrictScalars_mem R, ← Ideal.smul_top_eq_map]
simp [← this, ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.AdicCompletion.RingHom | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 6
} | {
"line": 155,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹ : NonAssocSemiring R\ninst✝ : 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\nm n : ℕ\nhle : m ≤ n\nx : R\n⊢ ((factorPow I ⋯).comp (f n)) x = (f m) x",
"ppTerm": "?m.70... | [
"R : Type u_1\nS : Type u_2\ninst✝¹ : NonAssocSemiring R\ninst✝ : 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\nm n : ℕ\nhle : m ≤ n\nx : R\n⊢ (f m) x = ((factorPow I ⋯).comp (f n)) x"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.RingTheory.AdicCompletion.LocalRing | {
"line": 165,
"column": 4
} | {
"line": 165,
"column": 14
} | {
"line": 166,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsLocalRing R\nfg : (maximalIdeal R).FG\ncomapeq :\n comap (algebraMap R (AdicCompletion (maximalIdeal R) R)) (maximalIdeal (AdicCompletion (maximalIdeal R) R)) =\n maximalIdeal R\nf : (maximalIdeal R).Cotangent →ₗ[R] (maximalI... | [
"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsLocalRing R\nfg : (maximalIdeal R).FG\ncomapeq :\n comap (algebraMap R (AdicCompletion (maximalIdeal R) R)) (maximalIdeal (AdicCompletion (maximalIdeal R) R)) =\n maximalIdeal R\nf : (maximalIdeal R).Cotangent →ₗ[R] (maximalIdeal (AdicCo... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.Congruence.Star | {
"line": 28,
"column": 17
} | {
"line": 30,
"column": 39
} | {
"line": 31,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NonUnitalNonAssocSemiring R\ninst✝ : StarRing R\nr : R → R → Prop\nhr : ∀ (a b : R), r a b → r (Star.star a) (Star.star b)\na b w✝ x✝ y✝ z✝ : R\nh1 : Rel r w✝ x✝\nh2 : Rel r y✝ z✝\n⊢ Rel r (Star.star (w✝ * y✝)) (Star.star (x✝ * z✝))",
"ppTerm": "?m.110",
"assigned": true,... | [] | by
rw [star_mul, star_mul]
exact (h2.star hr).mul (h1.star hr) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
"line": 565,
"column": 2
} | {
"line": 566,
"column": 90
} | {
"line": 567,
"column": 2
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\nA : Rep k G\nS : Subgroup G\ninst✝ : S.Normal\nx : ↑(shortComplexH1 (A.coinvariantsShortComplex S).X₁).X₂\na✝ : x ∈ ⊤\nX : G →₀ ↥S →₀ ↑A\nhX : mapRange ⇑(ConcreteCategory.hom (chains₁ToCoinvariantsKer (res S.subtype A))) ⋯ X = x\n⊢ x ∈\n Submodule... | [
"k G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\nA : Rep k G\nS : Subgroup G\ninst✝ : S.Normal\nx : ↑(shortComplexH1 (A.coinvariantsShortComplex S).X₁).X₂\na✝ : x ∈ ⊤\nX : G →₀ ↥S →₀ ↑A\nhX : mapRange ⇑(ConcreteCategory.hom (chains₁ToCoinvariantsKer (res S.subtype A))) ⋯ X = x\nY : ↥S →₀ ↑A :=\n X.sum fun g f... | let Y : S →₀ A := X.sum fun g f =>
mapRange.linearMap (A.ρ g⁻¹) (lmapDomain _ k (fun s => MulAut.conjNormal g⁻¹ s) f) - f | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
"line": 626,
"column": 2
} | {
"line": 627,
"column": 51
} | {
"line": 630,
"column": 2
} | [
{
"pp": "case h\nk G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\nA : Rep k G\nS : Subgroup G\ninst✝ : S.Normal\nx : ↥(cycles₁ A)\nhx : (ConcreteCategory.hom (H1π A)) x ∈ (ModuleCat.Hom.hom (H1CoresCoinf A S).g).ker\n⊢ (ConcreteCategory.hom (H1π A)) x ∈ (ModuleCat.Hom.hom (H1CoresCoinf A S).f).range",
"... | [
"case h\nk G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\nA : Rep k G\nS : Subgroup G\ninst✝ : S.Normal\nx : ↥(cycles₁ A)\nhx :\n (ConcreteCategory.hom (H1π (A.quotientToCoinvariants S)))\n ((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.toCoinvariantsMkQ S))) x) =\n 0\n⊢ (ConcreteCategory.... | simp only [H1CoresCoinf_X₂, H1CoresCoinf_X₃, LinearMap.mem_ker, H1CoresCoinf_g,
H1π_comp_map_apply (QuotientGroup.mk' S)] at hx | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Polynomial.ContentIdeal | {
"line": 101,
"column": 2
} | {
"line": 104,
"column": 81
} | {
"line": 106,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\np q : R[X]\n⊢ (p * q).contentIdeal ≤ p.contentIdeal * q.contentIdeal",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Ideal.span_le",
"Polynomial.contentIdeal",
"Eq.mpr",
"Submodule",
"SetLike.mem_coe._simp_1",
... | [] | rw [contentIdeal_def, span_le]
simp only [Set.subset_def, Finset.mem_coe, mem_coeffs_iff]
rintro r ⟨n, _, rfl⟩
simp [coeff_mul, _root_.sum_mem, Submodule.mul_mem_mul, coeff_mem_contentIdeal] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.ContentIdeal | {
"line": 101,
"column": 2
} | {
"line": 104,
"column": 81
} | {
"line": 106,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\np q : R[X]\n⊢ (p * q).contentIdeal ≤ p.contentIdeal * q.contentIdeal",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Ideal.span_le",
"Polynomial.contentIdeal",
"Eq.mpr",
"Submodule",
"SetLike.mem_coe._simp_1",
... | [] | rw [contentIdeal_def, span_le]
simp only [Set.subset_def, Finset.mem_coe, mem_coeffs_iff]
rintro r ⟨n, _, rfl⟩
simp [coeff_mul, _root_.sum_mem, Submodule.mul_mem_mul, coeff_mem_contentIdeal] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DividedPowers.Basic | {
"line": 115,
"column": 6
} | {
"line": 115,
"column": 10
} | {
"line": 116,
"column": 6
} | [
{
"pp": "case neg\nA : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nn✝ : ℕ\nx✝ y✝ : A\nha : x✝ ∈ ⊥\nhb : y✝ ∈ ⊥\nh : ¬n✝ = 0\n⊢ 0 = ∑ x ∈ antidiagonal n✝, if x.2 = 0 then if x.1 = 0 then 1 else 0 else 0",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMono... | [
"case neg\nA : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nn✝ : ℕ\nx✝ y✝ : A\nha : x✝ ∈ ⊥\nhb : y✝ ∈ ⊥\nh : ¬n✝ = 0\n⊢ (∑ x ∈ antidiagonal n✝, if x.2 = 0 then if x.1 = 0 then 1 else 0 else 0) = 0"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.RingTheory.Depth.Rees | {
"line": 75,
"column": 8
} | {
"line": 75,
"column": 36
} | {
"line": 75,
"column": 37
} | [
{
"pp": "case succ\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : Small.{v, u} R\ninst✝² : IsNoetherianRing R\nI : Ideal R\nN : ModuleCat R\ninst✝¹ : Module.Finite R ↑N\nh_supp : Module.support R ↑N = PrimeSpectrum.zeroLocus ↑I\nn : ℕ\nih :\n ∀ (M : ModuleCat R) [Module.Finite R ↑M],\n I • ⊤ < ⊤ →\n (∀ i <... | [
"case succ\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : Small.{v, u} R\ninst✝² : IsNoetherianRing R\nI : Ideal R\nN : ModuleCat R\ninst✝¹ : Module.Finite R ↑N\nh_supp : PrimeSpectrum.zeroLocus ↑(Module.annihilator R ↑N) = PrimeSpectrum.zeroLocus ↑I\nn : ℕ\nih :\n ∀ (M : ModuleCat R) [Module.Finite R ↑M],\n I • ⊤ ... | Module.support_eq_zeroLocus, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Depth.Rees | {
"line": 113,
"column": 8
} | {
"line": 113,
"column": 36
} | {
"line": 113,
"column": 37
} | [
{
"pp": "case succ\nR : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsNoetherianRing R\nI : Ideal R\nN : ModuleCat R\nNfin : Module.Finite R ↑N\nNsupp : Module.support R ↑N ⊆ PrimeSpectrum.zeroLocus ↑I\nn : ℕ\nih :\n ∀ (M : ModuleCat R) [Module.Finite R ↑M],\n I • ⊤ < ⊤ →\n ∀ (rs : L... | [
"case succ\nR : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsNoetherianRing R\nI : Ideal R\nN : ModuleCat R\nNfin : Module.Finite R ↑N\nNsupp : Module.support R ↑N ⊆ PrimeSpectrum.zeroLocus ↑I\nn : ℕ\nih :\n ∀ (M : ModuleCat R) [Module.Finite R ↑M],\n I • ⊤ < ⊤ →\n ∀ (rs : List R),\n ... | Module.support_eq_zeroLocus, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DividedPowers.Basic | {
"line": 287,
"column": 6
} | {
"line": 289,
"column": 52
} | {
"line": 290,
"column": 4
} | [
{
"pp": "case pos\nA : 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... | [] | rw [hn]
rw [dpow_zero I.zero_mem]
simp only [sym_zero, card_singleton, cast_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DividedPowers.Basic | {
"line": 287,
"column": 6
} | {
"line": 289,
"column": 52
} | {
"line": 290,
"column": 4
} | [
{
"pp": "case pos\nA : 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... | [] | rw [hn]
rw [dpow_zero I.zero_mem]
simp only [sym_zero, card_singleton, cast_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DividedPowers.Basic | {
"line": 310,
"column": 6
} | {
"line": 310,
"column": 16
} | {
"line": 311,
"column": 6
} | [
{
"pp": "case insert.right_neg\nA : 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}... | [
"case insert.right_neg\nA : 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}, ... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.Depth.Rees | {
"line": 173,
"column": 45
} | {
"line": 173,
"column": 73
} | {
"line": 173,
"column": 74
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsNoetherianRing R\nI : Ideal R\nn : ℕ\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\nsmul_lt : I • ⊤ < ⊤\nntrQ : Nontrivial (R ⧸ I)\n⊢ Module.support R (R ⧸ I) = PrimeSpectrum.zeroLocus ↑I",
"ppTerm": "?m.136",
"assigned": true,... | [
"R : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsNoetherianRing R\nI : Ideal R\nn : ℕ\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\nsmul_lt : I • ⊤ < ⊤\nntrQ : Nontrivial (R ⧸ I)\n⊢ PrimeSpectrum.zeroLocus ↑(Module.annihilator R (R ⧸ I)) = PrimeSpectrum.zeroLocus ↑I"
] | Module.support_eq_zeroLocus, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DividedPowers.Basic | {
"line": 313,
"column": 6
} | {
"line": 313,
"column": 16
} | {
"line": 314,
"column": 6
} | [
{
"pp": "case insert.h\nA : 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 : ... | [
"case insert.h\nA : 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 → ... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.DividedPowers.Basic | {
"line": 322,
"column": 6
} | {
"line": 322,
"column": 16
} | {
"line": 323,
"column": 6
} | [
{
"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 ∈... | [
"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 ∈ I → dpow n ... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.DividedPowers.Basic | {
"line": 411,
"column": 4
} | {
"line": 411,
"column": 25
} | {
"line": 412,
"column": 4
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : CommSemiring B\nJ : Ideal B\ne : A ≃+* B\nh : Ideal.map e I = J\nhI : DividedPowers I\nn✝ : ℕ\na✝ x✝ : B\nhx : x✝ ∈ J\n⊢ e (e.symm a✝ ^ n✝ * hI.dpow n✝ (e.symm x✝)) = a✝ ^ n✝ * e (hI.dpow n✝ (e.symm x✝))",
"ppTerm": "?m.219",... | [
"A : Type u_1\nB : Type u_2\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : CommSemiring B\nJ : Ideal B\ne : A ≃+* B\nh : Ideal.map e I = J\nhI : DividedPowers I\nn✝ : ℕ\na✝ x✝ : B\nhx : x✝ ∈ J\n⊢ e (e.symm a✝) ^ n✝ * e (hI.dpow n✝ (e.symm x✝)) = a✝ ^ n✝ * e (hI.dpow n✝ (e.symm x✝))"
] | rw [map_mul, map_pow] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 100,
"column": 4
} | {
"line": 114,
"column": 77
} | {
"line": 116,
"column": 0
} | [
{
"pp": "case neg\nA : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : DecidablePred fun x ↦ x ∈ I\nn : ℕ\nhn_fac : IsUnit ↑(n - 1)!\nhnI : I ^ n = 0\nm : ℕ\nx : A\nhx : x ∈ I\ny : A\nhy : y ∈ I\nhmn : n ≤ m\n⊢ dpow I m (x + y) = ∑ k ∈ Finset.antidiagonal m, dpow I k.1 x * dpow I k.2 y",
"ppTerm": "... | [] | have h_sub : I ^ m ≤ I ^ n := Ideal.pow_le_pow_right hmn
rw [dpow_eq_of_mem (Ideal.add_mem I hx hy)]
simp only [dpow]
have hxy : (x + y) ^ m = 0 := by
rw [← Ideal.mem_bot, ← Ideal.zero_eq_bot, ← hnI]
exact Set.mem_of_subset_of_mem h_sub (Ideal.pow_mem_pow (Ideal.add_mem I hx hy) m)
rw [hxy, ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 100,
"column": 4
} | {
"line": 114,
"column": 77
} | {
"line": 116,
"column": 0
} | [
{
"pp": "case neg\nA : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : DecidablePred fun x ↦ x ∈ I\nn : ℕ\nhn_fac : IsUnit ↑(n - 1)!\nhnI : I ^ n = 0\nm : ℕ\nx : A\nhx : x ∈ I\ny : A\nhy : y ∈ I\nhmn : n ≤ m\n⊢ dpow I m (x + y) = ∑ k ∈ Finset.antidiagonal m, dpow I k.1 x * dpow I k.2 y",
"ppTerm": "... | [] | have h_sub : I ^ m ≤ I ^ n := Ideal.pow_le_pow_right hmn
rw [dpow_eq_of_mem (Ideal.add_mem I hx hy)]
simp only [dpow]
have hxy : (x + y) ^ m = 0 := by
rw [← Ideal.mem_bot, ← Ideal.zero_eq_bot, ← hnI]
exact Set.mem_of_subset_of_mem h_sub (Ideal.pow_mem_pow (Ideal.add_mem I hx hy) m)
rw [hxy, ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 290,
"column": 6
} | {
"line": 290,
"column": 49
} | {
"line": 292,
"column": 0
} | [
{
"pp": "R : Type u_4\nM : Type u_5\nι : Type u_6\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nv : ι → M\nhv : Submodule.span R (Set.range v) = ⊤\nm✝ m n : M\nhm_mem : m ∈ Submodule.span R (Set.range v)\nhn_mem : n ∈ Submodule.span R (Set.range v)\nhm :\n ∀ (x : DividedPowerAlgebra R M) (... | [] | exact hn (x * dp R c.1 m) c.2 (hm x c.1 hx) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.DualNumber | {
"line": 33,
"column": 41
} | {
"line": 42,
"column": 17
} | {
"line": 44,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : TrivSqZeroExt R M\n⊢ IsNilpotent x ↔ IsNilpotent x.fst",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | by
constructor <;> rintro ⟨n, hn⟩
· refine ⟨n, ?_⟩
rw [← fst_pow, hn, fst_zero]
· refine ⟨n * 2, ?_⟩
rw [pow_mul]
ext
· rw [fst_pow, fst_pow, hn, zero_pow two_ne_zero, fst_zero]
· rw [pow_two, snd_mul, fst_pow, hn, MulOpposite.op_zero, zero_smul, zero_smul, zero_add,
snd_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 200,
"column": 4
} | {
"line": 200,
"column": 30
} | {
"line": 202,
"column": 0
} | [
{
"pp": "case inl\nA : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nι : Type u_3\nJ : ι → Ideal A\nhJ : ∀ (i : ι), hI.IsSubDPIdeal (J i)\nh✝ : IsEmpty ι\n⊢ hI.IsSubDPIdeal I",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"DividedPowers.IsSubDPIdeal.self"
]... | [] | exact IsSubDPIdeal.self hI | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 445,
"column": 58
} | {
"line": 445,
"column": 69
} | {
"line": 446,
"column": 4
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nN : Type u_5\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nf : M →ₗ[R] N\nhf : Function.Surjective ⇑f\n⊢ Subalgebra.map (map R f) ⊤ = ⊤",
"ppTerm": "?m.79",
"assigned": true,
"usedConstants"... | [
"R : Type u_2\nM : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nN : Type u_5\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nf : M →ₗ[R] N\nhf : Function.Surjective ⇑f\n⊢ ⊤ ≤ Subalgebra.map (map R f) ⊤"
] | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
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