module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj | {
"line": 232,
"column": 14
} | {
"line": 232,
"column": 57
} | {
"line": 232,
"column": 58
} | [
{
"pp": "case inr.succ\nX : ℕ → Type u_1\nmX : (n : ℕ) → MeasurableSpace (X n)\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\ninst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)\na b k : ℕ\nh : a ≤ k\nhk : partialTraj κ a k = (Kernel.id ×ₖ (partialTraj κ a k).map (restrict₂ ⋯)).map (_root_.IicProdIoc a k)\nthi... | [
"case inr.succ\nX : ℕ → Type u_1\nmX : (n : ℕ) → MeasurableSpace (X n)\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\ninst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)\na b k : ℕ\nh : a ≤ k\nhk : partialTraj κ a k = (Kernel.id ×ₖ (partialTraj κ a k).map (restrict₂ ⋯)).map (_root_.IicProdIoc a k)\nthis :\n _root... | ← partialTraj_comp_partialTraj h k.le_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 130,
"column": 2
} | {
"line": 133,
"column": 29
} | {
"line": 135,
"column": 0
} | [
{
"pp": "X : ℕ → Type u_1\na : ℕ\nx : (i : ↥(Iic a)) → X ↑i\nind : (n : ℕ) → ((i : ↥(Iic n)) → X ↑i) → X (n + 1)\n⊢ frestrictLe a (iterateInduction x ind) = x",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"iterateInduction._proof_5",
"congrArg",
"Finset"... | [] | ext i
simp only [frestrictLe_apply]
obtain ⟨(zero | j), hj⟩ := i <;> rw [iterateInduction]
rw [dif_pos (mem_Iic.1 hj)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 130,
"column": 2
} | {
"line": 133,
"column": 29
} | {
"line": 135,
"column": 0
} | [
{
"pp": "X : ℕ → Type u_1\na : ℕ\nx : (i : ↥(Iic a)) → X ↑i\nind : (n : ℕ) → ((i : ↥(Iic n)) → X ↑i) → X (n + 1)\n⊢ frestrictLe a (iterateInduction x ind) = x",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"iterateInduction._proof_5",
"congrArg",
"Finset"... | [] | ext i
simp only [frestrictLe_apply]
obtain ⟨(zero | j), hj⟩ := i <;> rw [iterateInduction]
rw [dif_pos (mem_Iic.1 hj)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.ProductMeasure | {
"line": 86,
"column": 2
} | {
"line": 93,
"column": 46
} | {
"line": 95,
"column": 0
} | [
{
"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)\ninst✝ : Fintype ι\ns : Set ((i : ι) → X i)\nhs : MeasurableSet s\n⊢ (piContent μ) s = (Measure.pi μ) s",
"ppTerm": "?m.20",
"assigned": true,
"usedC... | [] | let e : @Finset.univ ι _ ≃ ι :=
{ toFun i := i
invFun i := ⟨i, mem_univ i⟩ }
have : s = cylinder univ (MeasurableEquiv.piCongrLeft X e ⁻¹' s) := rfl
nth_rw 1 [this]
dsimp [e]
rw [piContent_cylinder _ (hs.preimage (by fun_prop)), ← Measure.pi_map_piCongrLeft e,
← Measure.map_apply (by fun_prop) hs]... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.ProductMeasure | {
"line": 86,
"column": 2
} | {
"line": 93,
"column": 46
} | {
"line": 95,
"column": 0
} | [
{
"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)\ninst✝ : Fintype ι\ns : Set ((i : ι) → X i)\nhs : MeasurableSet s\n⊢ (piContent μ) s = (Measure.pi μ) s",
"ppTerm": "?m.20",
"assigned": true,
"usedC... | [] | let e : @Finset.univ ι _ ≃ ι :=
{ toFun i := i
invFun i := ⟨i, mem_univ i⟩ }
have : s = cylinder univ (MeasurableEquiv.piCongrLeft X e ⁻¹' s) := rfl
nth_rw 1 [this]
dsimp [e]
rw [piContent_cylinder _ (hs.preimage (by fun_prop)), ← Measure.pi_map_piCongrLeft e,
← Measure.map_apply (by fun_prop) hs]... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Decision.Risk.Defs | {
"line": 97,
"column": 62
} | {
"line": 98,
"column": 30
} | {
"line": 100,
"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\nκ : Kernel 𝓧 𝓨\nπ : Measure Θ\ninst✝ : IsEmpty 𝓧\n⊢ avgRisk ℓ P κ π = 0",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
... | [] | by
simp [Subsingleton.elim P 0] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.ProductMeasure | {
"line": 176,
"column": 2
} | {
"line": 188,
"column": 19
} | {
"line": 189,
"column": 2
} | [
{
"pp": "case inl\nX : ℕ → Type u_1\nmX : (n : ℕ) → MeasurableSpace (X n)\nμ : (n : ℕ) → Measure (X n)\nhμ : ∀ (n : ℕ), IsProbabilityMeasure (μ n)\na b : ℕ\nhab : a < b\n⊢ (partialTraj (fun n ↦ const ((i : ↥(Iic n)) → X ↑i) (μ (n + 1))) a b).map (restrict₂ ⋯) =\n const ((i : ↥(Iic a)) → X ↑i) (Measure.pi fun... | [
"case inr\nX : ℕ → Type u_1\nmX : (n : ℕ) → MeasurableSpace (X n)\nμ : (n : ℕ) → Measure (X n)\nhμ : ∀ (n : ℕ), IsProbabilityMeasure (μ n)\na b : ℕ\nhba : b ≤ a\n⊢ (partialTraj (fun n ↦ const ((i : ↥(Iic n)) → X ↑i) (μ (n + 1))) a b).map (restrict₂ ⋯) =\n const ((i : ↥(Iic a)) → X ↑i) (Measure.pi fun i ↦ μ ↑i)"
... | · refine Nat.le_induction ?_ (fun n hn hind ↦ ?_) b (Nat.succ_le_of_lt hab) <;> ext1 x₀
· rw [partialTraj_succ_self, ← map_comp_right, map_apply, prod_apply, map_apply, const_apply,
const_apply, Measure.map_piSingleton, restrict₂_comp_IicProdIoc, Measure.map_snd_prod,
measure_univ, one_smul]
a... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj | {
"line": 239,
"column": 6
} | {
"line": 239,
"column": 49
} | {
"line": 239,
"column": 50
} | [
{
"pp": "case inr.succ\nX : ℕ → Type u_1\nmX : (n : ℕ) → MeasurableSpace (X n)\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\ninst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)\na b k : ℕ\nh : a ≤ k\nhk : partialTraj κ a k = (Kernel.id ×ₖ (partialTraj κ a k).map (restrict₂ ⋯)).map (_root_.IicProdIoc a k)\nthi... | [
"case inr.succ\nX : ℕ → Type u_1\nmX : (n : ℕ) → MeasurableSpace (X n)\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\ninst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)\na b k : ℕ\nh : a ≤ k\nhk : partialTraj κ a k = (Kernel.id ×ₖ (partialTraj κ a k).map (restrict₂ ⋯)).map (_root_.IicProdIoc a k)\nthis :\n _root... | ← partialTraj_comp_partialTraj h k.le_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 99,
"column": 4
} | {
"line": 99,
"column": 31
} | {
"line": 100,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ η : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nhκη : κ ≤ η\na : α\nx : γ\nhα : Countable α\n⊢ 0 ≤ ((∂κ a/∂η a) x).toReal",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": ... | [] | exact ENNReal.toReal_nonneg | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 99,
"column": 4
} | {
"line": 99,
"column": 31
} | {
"line": 100,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ η : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nhκη : κ ≤ η\na : α\nx : γ\nhα : Countable α\n⊢ 0 ≤ ((∂κ a/∂η a) x).toReal",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": ... | [] | exact ENNReal.toReal_nonneg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 99,
"column": 4
} | {
"line": 99,
"column": 31
} | {
"line": 100,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ η : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nhκη : κ ≤ η\na : α\nx : γ\nhα : Countable α\n⊢ 0 ≤ ((∂κ a/∂η a) x).toReal",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": ... | [] | exact ENNReal.toReal_nonneg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.ProductMeasure | {
"line": 419,
"column": 2
} | {
"line": 421,
"column": 50
} | {
"line": 423,
"column": 0
} | [
{
"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)\nI : Set ι\ns : Finset ↑I\nt : (i : ↑I) → Set (X ↑i)\nht : ∀ (i : ↑I), MeasurableSet (t i)\n⊢ (map I.restrict (infinitePi μ)) ((↑s).pi t) = ∏ i ∈ s, (μ ↑i) (t i)... | [] | rw [map_apply (by fun_prop), restrict_preimage, infinitePi_pi _ (by measurability)]
· simp
· exact .pi s.countable_toSet (by measurability) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.ProductMeasure | {
"line": 419,
"column": 2
} | {
"line": 421,
"column": 50
} | {
"line": 423,
"column": 0
} | [
{
"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)\nI : Set ι\ns : Finset ↑I\nt : (i : ↑I) → Set (X ↑i)\nht : ∀ (i : ↑I), MeasurableSet (t i)\n⊢ (map I.restrict (infinitePi μ)) ((↑s).pi t) = ∏ i ∈ s, (μ ↑i) (t i)... | [] | rw [map_apply (by fun_prop), restrict_preimage, infinitePi_pi _ (by measurability)]
· simp
· exact .pi s.countable_toSet (by measurability) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 415,
"column": 42
} | {
"line": 422,
"column": 41
} | {
"line": 424,
"column": 0
} | [
{
"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 = 0 ↔ κ a ≪ η a",
"ppTerm": "?m.33",
"assigned": true,
"u... | [] | by
conv_rhs => rw [← rnDeriv_add_singularPart κ η, coe_add, Pi.add_apply]
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
· rw [h, add_zero]
exact withDensity_absolutelyContinuous _ _
rw [Measure.AbsolutelyContinuous.add_left_iff] at h
exact Measure.eq_zero_of_absolutelyContinuous_of_mutuallySingular h.2
(mutuallySin... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.ProductMeasure | {
"line": 535,
"column": 4
} | {
"line": 536,
"column": 83
} | {
"line": 538,
"column": 0
} | [
{
"pp": "case mt\nι : Type u_3\nκ : ι → Type u_4\nX : (i : ι) → κ i → Type u_5\nmX : (i : ι) → (j : κ i) → MeasurableSpace (X i j)\nμ : (i : ι) → (j : κ i) → Measure (X i j)\nhμ : ∀ (i : ι) (j : κ i), IsProbabilityMeasure (μ i j)\ns : Finset ((i : ι) × κ i)\nt : (i : (i : ι) × κ i) → Set (X i.fst i.snd)\nht : ∀... | [] | simp only [mem_image, Sigma.exists, exists_and_right, exists_eq_right, forall_exists_index]
exact fun i j hij ↦ MeasurableSet.pi (countable_toSet _) fun k hk ↦ by simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.ProductMeasure | {
"line": 535,
"column": 4
} | {
"line": 536,
"column": 83
} | {
"line": 538,
"column": 0
} | [
{
"pp": "case mt\nι : Type u_3\nκ : ι → Type u_4\nX : (i : ι) → κ i → Type u_5\nmX : (i : ι) → (j : κ i) → MeasurableSpace (X i j)\nμ : (i : ι) → (j : κ i) → Measure (X i j)\nhμ : ∀ (i : ι) (j : κ i), IsProbabilityMeasure (μ i j)\ns : Finset ((i : ι) × κ i)\nt : (i : (i : ι) × κ i) → Set (X i.fst i.snd)\nht : ∀... | [] | simp only [mem_image, Sigma.exists, exists_and_right, exists_eq_right, forall_exists_index]
exact fun i j hij ↦ MeasurableSet.pi (countable_toSet _) fun k hk ↦ by simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Posterior | {
"line": 167,
"column": 4
} | {
"line": 167,
"column": 34
} | {
"line": 168,
"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\nthis : μ ⊗ₘ κ ≪ μ.p... | [] | exact this.map measurable_swap | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Kernel.Posterior | {
"line": 219,
"column": 4
} | {
"line": 219,
"column": 34
} | {
"line": 220,
"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... | [] | exact this.map measurable_swap | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 560,
"column": 2
} | {
"line": 561,
"column": 77
} | {
"line": 562,
"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)\na n : ℕ\nη : Kernel ((i : ↥(Iic a)) → X ↑i) ((n : ℕ) → X n)\nhη : ∀ b ≥ n, η.map (frestrictLe b) = partialTraj κ a b\nx : (i : ↥(Iic a)) → X ↑i\... | [
"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)\na n : ℕ\nη : Kernel ((i : ↥(Iic a)) → X ↑i) ((n : ℕ) → X n)\nhη : ∀ b ≥ n, η.map (frestrictLe b) = partialTraj κ a b\nx : (i : ↥(Iic a)) → X ↑i\n⊢ IsProject... | · intro k hk
rw [inducedFamily_Iic, ← map_apply _ (measurable_frestrictLe k), hη k hk] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Decision.Risk.Basic | {
"line": 204,
"column": 2
} | {
"line": 204,
"column": 21
} | {
"line": 205,
"column": 2
} | [
{
"pp": "case inr\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∞\nP : Kernel Θ 𝓧\nπ : Measure Θ\ninst✝¹ : Subsingleton 𝓧\ninst✝ : Nonempty 𝓨\nhX : Nonempty 𝓧\n⊢ bayesRisk ℓ P π = ⨅ μ, ⨅ (_ : IsProbabilityMeasure μ), ... | [
"case inr\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∞\nP : Kernel Θ 𝓧\nπ : Measure Θ\ninst✝¹ : Subsingleton 𝓧\ninst✝ : Nonempty 𝓨\nhX : Nonempty 𝓧\nx : 𝓧\n⊢ bayesRisk ℓ P π = ⨅ μ, ⨅ (_ : IsProbabilityMeasure μ), avgR... | obtain x := hX.some | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 603,
"column": 15
} | {
"line": 603,
"column": 24
} | {
"line": 603,
"column": 25
} | [
{
"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)\nn : ℕ\nx : (i : ↥(Iic n)) → X ↑i\nthis : (fun x_1 ↦ updateFinset x_1 (Iic n) x) = ⇑(IicProdIoi n) ∘ Prod.mk x ∘ (Set.Ioi n).restrict\n⊢ Measure.... | [
"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)\nn : ℕ\nx : (i : ↥(Iic n)) → X ↑i\nthis : (fun x_1 ↦ updateFinset x_1 (Iic n) x) = ⇑(IicProdIoi n) ∘ Prod.mk x ∘ (Set.Ioi n).restrict\n⊢ Measure.map (⇑(IicPr... | id_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Distributions.Exponential | {
"line": 157,
"column": 6
} | {
"line": 157,
"column": 72
} | {
"line": 158,
"column": 6
} | [
{
"pp": "r : ℝ\nhr : 0 < r\nx : ℝ\nh : 0 ≤ x\n⊢ AEStronglyMeasurable (fun x ↦ r * rexp (-(r * x))) (volume.restrict (Icc 0 x))",
"ppTerm": "?m.167",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"MeasureTheory.Integrable.aestronglyMeasurable",
"Real",
"H... | [
"r : ℝ\nhr : 0 < r\nx : ℝ\nh : 0 ≤ x\n⊢ Integrable (fun x ↦ rexp (-(r * x))) (volume.restrict (Icc 0 x))"
] | refine Integrable.aestronglyMeasurable (Integrable.const_mul ?_ _) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 690,
"column": 47
} | {
"line": 690,
"column": 69
} | {
"line": 690,
"column": 69
} | [
{
"pp": "X : ℕ → Type u_1\ninst✝³ : (n : ℕ) → MeasurableSpace (X n)\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\ninst✝² : ∀ (n : ℕ), IsMarkovKernel (κ n)\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b : ℕ\nhab : a ≤ b\nx₀ : (i : ↥(Iic a)) → X ↑i\nf : ((n : ℕ) → X n) → E... | [
"X : ℕ → Type u_1\ninst✝³ : (n : ℕ) → MeasurableSpace (X n)\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\ninst✝² : ∀ (n : ℕ), IsMarkovKernel (κ n)\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b : ℕ\nhab : a ≤ b\nx₀ : (i : ↥(Iic a)) → X ↑i\nf : ((n : ℕ) → X n) → E\nhf : Integ... | ← Measure.snd_compProd | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Distributions.Exponential | {
"line": 167,
"column": 2
} | {
"line": 172,
"column": 16
} | {
"line": 174,
"column": 0
} | [
{
"pp": "r : ℝ\nhr : 0 < r\nx : ℝ\n⊢ ↑(cdf (expMeasure r)) x = if 0 ≤ x then 1 - rexp (-(r * x)) else 0",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Real.instIsOrderedRing",
"Eq.mpr",
"mul_nonneg",
"Real.partialOrder",
... | [] | rw [cdf_expMeasure_eq_lintegral hr, lintegral_exponentialPDF_eq_antiDeriv hr x,
ENNReal.toReal_ofReal_eq_iff]
split_ifs with h
· simp only [sub_nonneg, exp_le_one_iff, Left.neg_nonpos_iff]
exact mul_nonneg hr.le h
· exact le_rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Distributions.Exponential | {
"line": 167,
"column": 2
} | {
"line": 172,
"column": 16
} | {
"line": 174,
"column": 0
} | [
{
"pp": "r : ℝ\nhr : 0 < r\nx : ℝ\n⊢ ↑(cdf (expMeasure r)) x = if 0 ≤ x then 1 - rexp (-(r * x)) else 0",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Real.instIsOrderedRing",
"Eq.mpr",
"mul_nonneg",
"Real.partialOrder",
... | [] | rw [cdf_expMeasure_eq_lintegral hr, lintegral_exponentialPDF_eq_antiDeriv hr x,
ENNReal.toReal_ofReal_eq_iff]
split_ifs with h
· simp only [sub_nonneg, exp_le_one_iff, Left.neg_nonpos_iff]
exact mul_nonneg hr.le h
· exact le_rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Distributions.Pareto | {
"line": 110,
"column": 35
} | {
"line": 110,
"column": 71
} | {
"line": 111,
"column": 2
} | [
{
"pp": "t r : ℝ\nht : 0 < t\nhr : 0 < r\nleftSide : ∫⁻ (x : ℝ) in Iio t, paretoPDF t r x = 0\nrightSide : ∫⁻ (x : ℝ) in Ici t, paretoPDF t r x = ∫⁻ (x : ℝ) in Ici t, ENNReal.ofReal (r * t ^ r * x ^ (-(r + 1)))\nx : ℝ\nhx : x ∈ Ici t\n⊢ 0 x ≤ r * t ^ r * x ^ (-(r + 1))",
"ppTerm": "?m.191",
"assigned": ... | [] | by positivity [lt_of_lt_of_le ht hx] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 712,
"column": 47
} | {
"line": 712,
"column": 69
} | {
"line": 712,
"column": 69
} | [
{
"pp": "X : ℕ → Type u_1\ninst✝³ : (n : ℕ) → MeasurableSpace (X n)\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\ninst✝² : ∀ (n : ℕ), IsMarkovKernel (κ n)\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b : ℕ\nhab : a ≤ b\nx₀ : (i : ↥(Iic a)) → X ↑i\nf : ((n : ℕ) → X n) → E... | [
"X : ℕ → Type u_1\ninst✝³ : (n : ℕ) → MeasurableSpace (X n)\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\ninst✝² : ∀ (n : ℕ), IsMarkovKernel (κ n)\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b : ℕ\nhab : a ≤ b\nx₀ : (i : ↥(Iic a)) → X ↑i\nf : ((n : ℕ) → X n) → E\nhf : Integ... | ← Measure.snd_compProd | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 723,
"column": 2
} | {
"line": 736,
"column": 78
} | {
"line": 738,
"column": 0
} | [
{
"pp": "X : ℕ → Type u_1\ninst✝⁴ : (n : ℕ) → MeasurableSpace (X n)\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\ninst✝³ : ∀ (n : ℕ), IsMarkovKernel (κ n)\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\na b : ℕ\nhab : a ≤ b\nx₀ : (i : ↥(Iic a)) → X ↑... | [] | have i_f' : Integrable (fun x ↦ ∫ y, f y ∂(traj κ b) x)
(((traj κ a) x₀).map (frestrictLe b)) := by
rw [← map_apply _ (measurable_frestrictLe _), traj_map_frestrictLe _ _]
rw [← traj_comp_partialTraj hab] at i_f
exact i_f.integral_comp
refine ae_eq_condExp_of_forall_setIntegral_eq (piLE.le _) i_f
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 723,
"column": 2
} | {
"line": 736,
"column": 78
} | {
"line": 738,
"column": 0
} | [
{
"pp": "X : ℕ → Type u_1\ninst✝⁴ : (n : ℕ) → MeasurableSpace (X n)\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\ninst✝³ : ∀ (n : ℕ), IsMarkovKernel (κ n)\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\na b : ℕ\nhab : a ≤ b\nx₀ : (i : ↥(Iic a)) → X ↑... | [] | have i_f' : Integrable (fun x ↦ ∫ y, f y ∂(traj κ b) x)
(((traj κ a) x₀).map (frestrictLe b)) := by
rw [← map_apply _ (measurable_frestrictLe _), traj_map_frestrictLe _ _]
rw [← traj_comp_partialTraj hab] at i_f
exact i_f.integral_comp
refine ae_eq_condExp_of_forall_setIntegral_eq (piLE.le _) i_f
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.ProbabilityMassFunction.Constructions | {
"line": 313,
"column": 4
} | {
"line": 314,
"column": 100
} | {
"line": 315,
"column": 4
} | [
{
"pp": "case false\np : ℝ≥0\nh : p ≤ 1\n⊢ false ∈ (bernoulli p h).support ↔ false ∈ {b | bif b then p ≠ 0 else p ≠ 1}",
"ppTerm": "?false",
"assigned": true,
"usedConstants": [
"cond",
"Eq.mpr",
"False",
"ENNReal.ofNNReal",
"congrArg",
"PMF",
"_private.Math... | [
"case false\np : ℝ≥0\nh : p ≤ 1\n⊢ 1 - ↑p = 0 ↔ p = 1"
] | simp_rw [mem_support_iff, bernoulli_apply, Bool.cond_false, Ne, ENNReal.coe_sub,
ENNReal.coe_one, Bool.cond_prop, Set.mem_setOf_eq, Bool.false_eq_true, ite_false, not_iff_not] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Probability.Distributions.TwoValued | {
"line": 80,
"column": 6
} | {
"line": 80,
"column": 38
} | {
"line": 81,
"column": 6
} | [
{
"pp": "Ω✝ : Type u_1\nm✝ : MeasurableSpace Ω✝\nX✝ : Ω✝ → ℝ\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nm₀ : MeasurableSpace Ω\nhm : m ≤ m₀\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nhXmeas✝ : AEMeasurable X μ\nhX : ∀ᵐ (ω : Ω) ∂μ, X ω = 0 ∨ X ω = 1\nhXmeas : Measurable X\n⊢ μ[X | m] - μ[X | m] ^ 2 =ᵐ[μ] μ... | [
"Ω✝ : Type u_1\nm✝ : MeasurableSpace Ω✝\nX✝ : Ω✝ → ℝ\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nm₀ : MeasurableSpace Ω\nhm : m ≤ m₀\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nhXmeas✝ : AEMeasurable X μ\nhX : ∀ᵐ (ω : Ω) ∂μ, X ω = 0 ∨ X ω = 1\nhXmeas : Measurable X\n⊢ μ[X | m] * (1 - μ[X | m]) =ᵐ[μ] μ[X | m] * ... | rw [sq, ← one_sub_mul, mul_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Independence.Conditional | {
"line": 229,
"column": 2
} | {
"line": 231,
"column": 73
} | {
"line": 232,
"column": 2
} | [
{
"pp": "case refine_1\nΩ : Type u_1\nm' mΩ : MeasurableSpace Ω\ninst✝¹ : StandardBorelSpace Ω\nhm' : m' ≤ mΩ\ns1 s2 : Set (Set Ω)\nhs1 : ∀ s ∈ s1, MeasurableSet s\nhs2 : ∀ s ∈ s2, MeasurableSet s\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nhs1_eq : ∀ s ∈ s1, (fun ω ↦ (((condExpKernel μ m') ω) s).toReal) =ᵐ[μ] μ... | [
"case refine_2\nΩ : Type u_1\nm' mΩ : MeasurableSpace Ω\ninst✝¹ : StandardBorelSpace Ω\nhm' : m' ≤ mΩ\ns1 s2 : Set (Set Ω)\nhs1 : ∀ s ∈ s1, MeasurableSet s\nhs2 : ∀ s ∈ s2, MeasurableSet s\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nhs1_eq : ∀ s ∈ s1, (fun ω ↦ (((condExpKernel μ m') ω) s).toReal) =ᵐ[μ] μ[s.indicator... | · have h' := ae_eq_of_ae_eq_trim h
filter_upwards [hs1_eq s hs, hs2_eq t ht, hs12_eq s hs t ht, h'] with ω hs_eq ht_eq hst_eq h'
rw [← hst_eq, Pi.mul_apply, ← hs_eq, ← ht_eq, h', ENNReal.toReal_mul] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Independence.BoundedContinuousFunction | {
"line": 182,
"column": 4
} | {
"line": 182,
"column": 15
} | {
"line": 183,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nG : Type u_6\nH : Type u_7\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : BorelSpace G\ninst✝⁵ : HasOuterApproxClosed G\ninst✝⁴ : TopologicalSpace H\ninst✝³ : MeasurableSpace H\ninst✝² : BorelSpace H\ninst✝¹ : HasOuterApproxClosed... | [] | exact h f g | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Independence.BoundedContinuousFunction | {
"line": 182,
"column": 4
} | {
"line": 182,
"column": 15
} | {
"line": 183,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nG : Type u_6\nH : Type u_7\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : BorelSpace G\ninst✝⁵ : HasOuterApproxClosed G\ninst✝⁴ : TopologicalSpace H\ninst✝³ : MeasurableSpace H\ninst✝² : BorelSpace H\ninst✝¹ : HasOuterApproxClosed... | [] | exact h f g | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Independence.BoundedContinuousFunction | {
"line": 182,
"column": 4
} | {
"line": 182,
"column": 15
} | {
"line": 183,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nG : Type u_6\nH : Type u_7\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : BorelSpace G\ninst✝⁵ : HasOuterApproxClosed G\ninst✝⁴ : TopologicalSpace H\ninst✝³ : MeasurableSpace H\ninst✝² : BorelSpace H\ninst✝¹ : HasOuterApproxClosed... | [] | exact h f g | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Martingale.OptionalSampling | {
"line": 210,
"column": 6
} | {
"line": 210,
"column": 24
} | {
"line": 211,
"column": 4
} | [
{
"pp": "Ω : 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\nin... | [] | exact hx hx_mem.le | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Independence.BoundedContinuousFunction | {
"line": 214,
"column": 20
} | {
"line": 214,
"column": 29
} | {
"line": 214,
"column": 29
} | [
{
"pp": "case refine_2\nΩ : Type u_1\nS : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nE : S → Type u_4\ninst✝⁵ : (s : S) → TopologicalSpace (E s)\ninst✝⁴ : (s : S) → MeasurableSpace (E s)\ninst✝³ : ∀ (s : S), BorelSpace (E s)\ninst✝² : ∀ (s : S), HasOuterApproxClosed (E s)\nX : (s : S) → Ω → E s\ninst✝¹ : ... | [
"case refine_2\nΩ : Type u_1\nS : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nE : S → Type u_4\ninst✝⁵ : (s : S) → TopologicalSpace (E s)\ninst✝⁴ : (s : S) → MeasurableSpace (E s)\ninst✝³ : ∀ (s : S), BorelSpace (E s)\ninst✝² : ∀ (s : S), HasOuterApproxClosed (E s)\nX : (s : S) → Ω → E s\ninst✝¹ : Fintype S\ni... | norm_prod | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Martingale.OptionalSampling | {
"line": 203,
"column": 4
} | {
"line": 221,
"column": 85
} | {
"line": 223,
"column": 0
} | [
{
"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... | [] | suffices μ[stoppedValue f τ | (hσ.min hτ).measurableSpace] =ᵐ[μ.restrict {x | τ x ≤ σ x}]
μ[stoppedValue f τ | hσ.measurableSpace] by
rw [ae_restrict_iff' (hσ.measurableSpace_le _ (hσ.measurableSet_le_stopping_time hτ).compl)]
rw [Filter.EventuallyEq, ae_restrict_iff'] at this
swap; · exact hτ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Martingale.OptionalSampling | {
"line": 203,
"column": 4
} | {
"line": 221,
"column": 85
} | {
"line": 223,
"column": 0
} | [
{
"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... | [] | suffices μ[stoppedValue f τ | (hσ.min hτ).measurableSpace] =ᵐ[μ.restrict {x | τ x ≤ σ x}]
μ[stoppedValue f τ | hσ.measurableSpace] by
rw [ae_restrict_iff' (hσ.measurableSpace_le _ (hσ.measurableSet_le_stopping_time hτ).compl)]
rw [Filter.EventuallyEq, ae_restrict_iff'] at this
swap; · exact hτ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Moments.SubGaussian | {
"line": 191,
"column": 2
} | {
"line": 202,
"column": 14
} | {
"line": 204,
"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\nt : ℝ\np : ℝ≥0\n⊢ MemLp (fun ω ↦ rexp (t * X ω)) (↑p) (⇑κ ∘ₘ ν)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
... | [] | by_cases hp0 : p = 0
· simpa [hp0] using (h.integrable_exp_mul t).1
constructor
· exact (h.integrable_exp_mul t).1
· rw [eLpNorm_lt_top_iff_lintegral_rpow_enorm_lt_top (mod_cast hp0) (by simp)]
simp only [ENNReal.coe_toReal]
have h' := (h.integrable_exp_mul (p * t)).2
rw [hasFiniteIntegral_def] at h... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Moments.SubGaussian | {
"line": 191,
"column": 2
} | {
"line": 202,
"column": 14
} | {
"line": 204,
"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\nt : ℝ\np : ℝ≥0\n⊢ MemLp (fun ω ↦ rexp (t * X ω)) (↑p) (⇑κ ∘ₘ ν)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
... | [] | by_cases hp0 : p = 0
· simpa [hp0] using (h.integrable_exp_mul t).1
constructor
· exact (h.integrable_exp_mul t).1
· rw [eLpNorm_lt_top_iff_lintegral_rpow_enorm_lt_top (mod_cast hp0) (by simp)]
simp only [ENNReal.coe_toReal]
have h' := (h.integrable_exp_mul (p * t)).2
rw [hasFiniteIntegral_def] at h... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Moments.SubGaussian | {
"line": 385,
"column": 7
} | {
"line": 388,
"column": 27
} | {
"line": 390,
"column": 0
} | [] | [] | (κ ω') {ω | X ω ≠ 0}
_ = (κ ω') {ω | X ω < 0 ∨ 0 < X ω} := by simp_rw [ne_iff_lt_or_gt]
_ ≤ (κ ω') {ω | X ω < 0} + (κ ω') {ω | 0 < X ω} := measure_union_le _ _
_ = 0 := by simp [h1, h2] | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Probability.Moments.SubGaussian | {
"line": 828,
"column": 2
} | {
"line": 828,
"column": 39
} | {
"line": 829,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\ninst✝ : IsProbabilityMeasure μ\na b t : ℝ\nhm : AEMeasurable X μ\nhb : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b\nhc : ∫ (x : Ω), X x ∂μ = 0\nht : 0 < t\nhi : ∀ (u : ℝ), Integrable (fun ω ↦ rexp (u * X ω)) μ\nhs : Set.Icc 0 t ⊆ interior (integrable... | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\ninst✝ : IsProbabilityMeasure μ\na b t : ℝ\nhm : AEMeasurable X μ\nhb : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b\nhc : ∫ (x : Ω), X x ∂μ = 0\nht : 0 < t\nhi : ∀ (u : ℝ), Integrable (fun ω ↦ rexp (u * X ω)) μ\nhs : Set.Icc 0 t ⊆ interior (integrableExpSet X μ)\... | rw [← exp_cgf (hi t), exp_le_exp, h2] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Moments.SubGaussian | {
"line": 849,
"column": 4
} | {
"line": 849,
"column": 17
} | {
"line": 850,
"column": 4
} | [
{
"pp": "case inr.inl\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\ninst✝ : IsProbabilityMeasure μ\na b : ℝ\nhm : AEMeasurable X μ\nhb : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b\nhc : ∫ (x : Ω), X x ∂μ = 0\nt : ℝ\nht : 0 = t\n⊢ mgf X μ t ≤ rexp (↑((‖b - a‖₊ / 2) ^ 2) * t ^ 2 / 2)",
"ppTerm": "?in... | [
"case inr.inr\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\ninst✝ : IsProbabilityMeasure μ\na b : ℝ\nhm : AEMeasurable X μ\nhb : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b\nhc : ∫ (x : Ω), X x ∂μ = 0\nt : ℝ\nht : t < 0\n⊢ mgf X μ t ≤ rexp (↑((‖b - a‖₊ / 2) ^ 2) * t ^ 2 / 2)"
] | · simp [← ht] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.StrongLaw | {
"line": 337,
"column": 8
} | {
"line": 338,
"column": 12
} | {
"line": 339,
"column": 6
} | [
{
"pp": "case ha\nΩ : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : Ω → ℝ\nhint : Integrable X ℙ\nhnonneg : 0 ≤ X\nK : ℕ\nY : ℕ → Ω → ℝ := ⋯\nρ : Measure ℝ := ⋯\nY2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0..↑n, x ^ 2 ∂ρ\nk : ℕ\na✝ : k ∈ range K\n⊢ 0 ≤ ∫ (x : ℝ) in ↑k..↑(k + ... | [] | refine intervalIntegral.integral_nonneg_of_forall ?_ fun u => sq_nonneg _
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.StrongLaw | {
"line": 337,
"column": 8
} | {
"line": 338,
"column": 12
} | {
"line": 339,
"column": 6
} | [
{
"pp": "case ha\nΩ : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : Ω → ℝ\nhint : Integrable X ℙ\nhnonneg : 0 ≤ X\nK : ℕ\nY : ℕ → Ω → ℝ := ⋯\nρ : Measure ℝ := ⋯\nY2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0..↑n, x ^ 2 ∂ρ\nk : ℕ\na✝ : k ∈ range K\n⊢ 0 ≤ ∫ (x : ℝ) in ↑k..↑(k + ... | [] | refine intervalIntegral.integral_nonneg_of_forall ?_ fun u => sq_nonneg _
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RepresentationTheory.Intertwining | {
"line": 424,
"column": 43
} | {
"line": 424,
"column": 49
} | {
"line": 424,
"column": 50
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : Semiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G W\nτ... | [
"A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : Semiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G W\nτ : Represent... | φ.1.2, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RepresentationTheory.Irreducible | {
"line": 35,
"column": 6
} | {
"line": 35,
"column": 24
} | {
"line": 35,
"column": 24
} | [
{
"pp": "G : Type u_1\nk : Type u_2\nV : Type u_3\ninst✝³ : Monoid G\ninst✝² : Field k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nρ : Representation k G V\n⊢ ρ.IsIrreducible ↔ IsSimpleModule k[G] ρ.asModule",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule... | [
"G : Type u_1\nk : Type u_2\nV : Type u_3\ninst✝³ : Monoid G\ninst✝² : Field k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nρ : Representation k G V\n⊢ ρ.IsIrreducible ↔ IsSimpleOrder (Submodule k[G] ρ.asModule)"
] | isSimpleModule_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RepresentationTheory.Irreducible | {
"line": 41,
"column": 6
} | {
"line": 41,
"column": 24
} | {
"line": 41,
"column": 24
} | [
{
"pp": "G : Type u_1\nk : Type u_2\ninst✝³ : Monoid G\ninst✝² : Field k\nM : Type u_5\ninst✝¹ : AddCommGroup M\ninst✝ : Module k[G] M\n⊢ IsSimpleModule k[G] M ↔ (ofModule M).IsIrreducible",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"MonoidAlgebra.semiring",
"Eq.mpr",
... | [
"G : Type u_1\nk : Type u_2\ninst✝³ : Monoid G\ninst✝² : Field k\nM : Type u_5\ninst✝¹ : AddCommGroup M\ninst✝ : Module k[G] M\n⊢ IsSimpleOrder (Submodule k[G] M) ↔ (ofModule M).IsIrreducible"
] | isSimpleModule_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RepresentationTheory.Coinvariants | {
"line": 110,
"column": 4
} | {
"line": 111,
"column": 60
} | {
"line": 113,
"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 : ... | [] | simpa only [← hy, SetLike.mem_coe, LinearMap.mem_ker, map_sub, sub_eq_zero, LinearMap.coe_comp,
Function.comp_apply] using LinearMap.ext_iff.1 (h g) y | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RepresentationTheory.Coinvariants | {
"line": 110,
"column": 4
} | {
"line": 111,
"column": 60
} | {
"line": 113,
"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 : ... | [] | simpa only [← hy, SetLike.mem_coe, LinearMap.mem_ker, map_sub, sub_eq_zero, LinearMap.coe_comp,
Function.comp_apply] using LinearMap.ext_iff.1 (h g) y | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RepresentationTheory.Coinvariants | {
"line": 110,
"column": 4
} | {
"line": 111,
"column": 60
} | {
"line": 113,
"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 : ... | [] | simpa only [← hy, SetLike.mem_coe, LinearMap.mem_ker, map_sub, sub_eq_zero, LinearMap.coe_comp,
Function.comp_apply] using LinearMap.ext_iff.1 (h g) y | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RepresentationTheory.Homological.ContCohomology.Functoriality | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 11
} | {
"line": 123,
"column": 2
} | [
{
"pp": "k : Type u\nG H K : Type v\ninst✝¹⁰ : Ring k\ninst✝⁹ : TopologicalSpace k\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : IsTopologicalGroup G\ninst✝⁵ : Group H\ninst✝⁴ : TopologicalSpace H\ninst✝³ : IsTopologicalGroup H\ninst✝² : Group K\ninst✝¹ : TopologicalSpace K\ninst✝ : IsTopologicalGrou... | [
"k : Type u\nG H K : Type v\ninst✝¹⁰ : Ring k\ninst✝⁹ : TopologicalSpace k\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : IsTopologicalGroup G\ninst✝⁵ : Group H\ninst✝⁴ : TopologicalSpace H\ninst✝³ : IsTopologicalGroup H\ninst✝² : Group K\ninst✝¹ : TopologicalSpace K\ninst✝ : IsTopologicalGroup K\nX : Top... | ext i v x | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic | {
"line": 129,
"column": 25
} | {
"line": 129,
"column": 42
} | {
"line": 129,
"column": 43
} | [
{
"pp": "case a.a\nk G : Type u\ninst✝¹ : CommRing k\nn✝ : ℕ\ninst✝ : Group G\nA : Rep k G\nn : ℕ\n| ((barComplex k G).linearYonedaObj k A).d n (n + 1) ≫\n 𝟙 (((barComplex k G).linearYonedaObj k A).X (n + 1)) ≫ ((barComplex k G).linearYonedaObj k A).d (n + 1) (n + 1 + 1)",
"ppTerm": "?a.a",
"assigne... | [
"case a.a\nk G : Type u\ninst✝¹ : CommRing k\nn✝ : ℕ\ninst✝ : Group G\nA : Rep k G\nn : ℕ\n| ((barComplex k G).linearYonedaObj k A).d n (n + 1) ≫ ((barComplex k G).linearYonedaObj k A).d (n + 1) (n + 1 + 1)",
"case a.a\nk G : Type u\ninst✝¹ : CommRing k\nn✝ : ℕ\ninst✝ : Group G\nA : Rep k G\nn : ℕ\n| (freeLiftLEq... | Category.id_comp, | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree | {
"line": 110,
"column": 6
} | {
"line": 110,
"column": 20
} | {
"line": 110,
"column": 21
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\nA : Rep k G\ninst✝ : A.IsTrivial\nx✝¹ : ↑(ModuleCat.of k ↑A)\nx✝ : G\n⊢ (ModuleCat.Hom.hom (d₀₁ A)) x✝¹ x✝ = (ModuleCat.Hom.hom 0) x✝¹ x✝",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.Function.module"... | [
"k G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\nA : Rep k G\ninst✝ : A.IsTrivial\nx✝¹ : ↑(ModuleCat.of k ↑A)\nx✝ : G\n⊢ (A.ρ x✝) x✝¹ - x✝¹ = (ModuleCat.Hom.hom 0) x✝¹ x✝"
] | d₀₁_hom_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree | {
"line": 140,
"column": 8
} | {
"line": 142,
"column": 39
} | {
"line": 143,
"column": 4
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nx y : G × G → ↑A\ng : G × G × G\n⊢ (A.ρ g.1) ((x + y) (g.2.1, g.2.2)) - (x + y) (g.1 * g.2.1, g.2.2) + (x + y) (g.1, g.2.1 * g.2.2) -\n (x + y) (g.1, g.2.1) =\n ((fun g ↦ (A.ρ g.1) (x (g.2.1, g.2.2)) - x (g.1 * g.2.1, g.2.2) + x (... | [] | dsimp
rw [map_add, add_sub_add_comm (A.ρ _ _), add_sub_assoc, add_sub_add_comm, add_add_add_comm,
add_sub_assoc, add_sub_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree | {
"line": 140,
"column": 8
} | {
"line": 142,
"column": 39
} | {
"line": 143,
"column": 4
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nx y : G × G → ↑A\ng : G × G × G\n⊢ (A.ρ g.1) ((x + y) (g.2.1, g.2.2)) - (x + y) (g.1 * g.2.1, g.2.2) + (x + y) (g.1, g.2.1 * g.2.2) -\n (x + y) (g.1, g.2.1) =\n ((fun g ↦ (A.ρ g.1) (x (g.2.1, g.2.2)) - x (g.1 * g.2.1, g.2.2) + x (... | [] | dsimp
rw [map_add, add_sub_add_comm (A.ρ _ _), add_sub_assoc, add_sub_add_comm, add_add_add_comm,
add_sub_assoc, add_sub_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RepresentationTheory.Homological.GroupCohomology.Hilbert90 | {
"line": 199,
"column": 6
} | {
"line": 199,
"column": 55
} | {
"line": 200,
"column": 4
} | [
{
"pp": "case refine_2.e_a\nK L : Type\ninst✝¹⁶ : Field K\ninst✝¹⁵ : Field L\ninst✝¹⁴ : Algebra K L\ninst✝¹³ : FiniteDimensional K L\ninst✝¹² : IsGalois K L\ninst✝¹¹ : IsCyclic Gal(L/K)\ng : Gal(L/K)\nA : Type u_1\nB : Type u_2\ninst✝¹⁰ : CommRing A\ninst✝⁹ : CommRing B\ninst✝⁸ : Algebra A B\ninst✝⁷ : Algebra A... | [] | exact algebraMap_galRestrictHom_apply A K L B g a | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RepresentationTheory.Homological.GroupHomology.Basic | {
"line": 164,
"column": 25
} | {
"line": 164,
"column": 42
} | {
"line": 164,
"column": 43
} | [
{
"pp": "case a.a\nk✝ G✝ : Type u\ninst✝³ : CommRing k✝\ninst✝² : Group G✝\nk G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nn✝ n : ℕ\n| (HomologicalComplex.coinvariantsTensorObj A (barComplex k G)).d (n + 1 + 1) (n + 1) ≫\n 𝟙 ((HomologicalComplex.coinvariantsTensorObj A (barComplex k G)).X ... | [
"case a.a\nk✝ G✝ : Type u\ninst✝³ : CommRing k✝\ninst✝² : Group G✝\nk G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nn✝ n : ℕ\n| (HomologicalComplex.coinvariantsTensorObj A (barComplex k G)).d (n + 1 + 1) (n + 1) ≫\n (HomologicalComplex.coinvariantsTensorObj A (barComplex k G)).d (n + 1) n",
"c... | Category.id_comp, | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.RingTheory.MvPowerSeries.Equiv | {
"line": 146,
"column": 9
} | {
"line": 146,
"column": 79
} | {
"line": 147,
"column": 2
} | [
{
"pp": "σ✝ : Type u_1\nR✝ : Type u_2\nn : ℕ\ninst✝³ : CommRing R✝\ninst✝² : Finite σ✝\nσ : Type u_3\nR : Type u_4\ninst✝¹ : Finite σ\ninst✝ : CommRing R\nx✝ : MvPowerSeries σ R\nn✝ : σ →₀ ℕ\n⊢ (coeff n✝) (toAdicCompletionInv σ R ((↑↑(toAdicCompletion σ R).toRingHom).toFun x✝)) = (coeff n✝) x✝",
"ppTerm": "... | [] | simp [coeff_toAdicCompletionInv, coeff_toAdicCompletion_val_apply_out] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 356,
"column": 2
} | {
"line": 356,
"column": 80
} | {
"line": 358,
"column": 0
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\nA : Rep k G\ninst✝ : A.IsTrivial\n⊢ cycles₁ A = ⊤",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"ModuleCat.hom_zero",
"Eq.mpr",
"Submodule",
"Rep.V",
"Finsupp.module",
"CategoryTheory.Catego... | [] | rw [cycles₁, d₁₀_eq_zero_of_isTrivial, ModuleCat.hom_zero, LinearMap.ker_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 356,
"column": 2
} | {
"line": 356,
"column": 80
} | {
"line": 358,
"column": 0
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\nA : Rep k G\ninst✝ : A.IsTrivial\n⊢ cycles₁ A = ⊤",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"ModuleCat.hom_zero",
"Eq.mpr",
"Submodule",
"Rep.V",
"Finsupp.module",
"CategoryTheory.Catego... | [] | rw [cycles₁, d₁₀_eq_zero_of_isTrivial, ModuleCat.hom_zero, LinearMap.ker_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 356,
"column": 2
} | {
"line": 356,
"column": 80
} | {
"line": 358,
"column": 0
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\nA : Rep k G\ninst✝ : A.IsTrivial\n⊢ cycles₁ A = ⊤",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"ModuleCat.hom_zero",
"Eq.mpr",
"Submodule",
"Rep.V",
"Finsupp.module",
"CategoryTheory.Catego... | [] | rw [cycles₁, d₁₀_eq_zero_of_isTrivial, ModuleCat.hom_zero, LinearMap.ker_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 434,
"column": 2
} | {
"line": 434,
"column": 21
} | {
"line": 435,
"column": 2
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\na : ↑A\n⊢ single 1 a ∈ boundaries₁ A",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Rep.V",
"InvOneClass.toOne",
"Finsupp.module",
"DivInvOneMonoid.toInvOneClass",
"CommSemiring.toSemi... | [
"case h\nk G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\na : ↑A\n⊢ (ModuleCat.Hom.hom (d₂₁ A)) (single (1, 1) a) = single 1 a"
] | use single (1, 1) a | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.RingTheory.AdicCompletion.Topology | {
"line": 58,
"column": 4
} | {
"line": 58,
"column": 9
} | {
"line": 59,
"column": 4
} | [
{
"pp": "case mp\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 : ∀ (f : ℕ → R), (∀ {m n : ℕ}, m ≤ n → f m - f n ∈ I ^ m) → ∃ L, ∀ (n : ℕ), f n - L ∈ I ^ n\nu : ℕ → R... | [
"case h\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 : ∀ (f : ℕ → R), (∀ {m n : ℕ}, m ≤ n → f m - f n ∈ I ^ m) → ∃ L, ∀ (n : ℕ), f n - L ∈ I ^ n\nu : ℕ → R\nhu : Cauchy... | use L | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology | {
"line": 218,
"column": 6
} | {
"line": 218,
"column": 20
} | {
"line": 219,
"column": 6
} | [
{
"pp": "case mpr.right\nA : Type u_2\ninst✝² : CommRing A\ninst✝¹ : TopologicalSpace A\ninst✝ : IsTopologicalRing A\na✝ : DiscreteTopology A\n⊢ ∀ s ∈ 𝓝 0, ∃ n, ↑(⊥ ^ n) ⊆ s",
"ppTerm": "?mpr.right",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"CommSemiring.toSemiring"... | [
"case mpr.right\nA : Type u_2\ninst✝² : CommRing A\ninst✝¹ : TopologicalSpace A\ninst✝ : IsTopologicalRing A\na✝ : DiscreteTopology A\nU : Set A\nU_nhds : U ∈ 𝓝 0\n⊢ ∃ n, ↑(⊥ ^ n) ⊆ U"
] | intro U U_nhds | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.AdicCompletion.Completeness | {
"line": 64,
"column": 6
} | {
"line": 64,
"column": 35
} | {
"line": 64,
"column": 36
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\na b c : ℕ\nh : c = b + a\nx : AdicCompletion I ↥(I ^ a • ⊤)\n⊢ ↑((ofPowSMul I M a) x) c = (powSMulQuotInclusion I M h ⊤) (↑x b)",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": ... | [
"R : Type u_1\ninst✝² : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\na b c : ℕ\nh : c = b + a\nx : AdicCompletion I ↥(I ^ a • ⊤)\n⊢ ↑((ofPowSMul I M a) x) c = (powSMulQuotInclusion I M h ⊤) ((transitionMap I ↥(I ^ a • ⊤) ⋯) (↑x c))"
] | ← x.prop (show b ≤ c by lia), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.AdicCompletion.Completeness | {
"line": 192,
"column": 4
} | {
"line": 192,
"column": 9
} | {
"line": 192,
"column": 9
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nh : I.FG\nx : ℕ → AdicCompletion I M\nhx : ∀ {m n : ℕ}, m ≤ n → x m ≡ x n [SMOD I ^ m • ⊤]\nL : AdicCompletion I M := ⟨fun i ↦ ↑(x i) i, ⋯⟩\n⊢ ∃ L, ∀ (n : ℕ), x n ≡ L [SMOD I ^ n • ⊤]",
"ppTer... | [
"case h\nR : Type u_1\ninst✝² : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nh : I.FG\nx : ℕ → AdicCompletion I M\nhx : ∀ {m n : ℕ}, m ≤ n → x m ≡ x n [SMOD I ^ m • ⊤]\nL : AdicCompletion I M := ⋯\n⊢ ∀ (n : ℕ), x n ≡ L [SMOD I ^ n • ⊤]"
] | use L | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.RingTheory.Congruence.Star | {
"line": 30,
"column": 4
} | {
"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 y✝ * Star.star w✝) (Star.star z✝ * Star.star x✝)",
"ppTerm": "?m.379",
"... | [] | exact (h2.star hr).mul (h1.star hr) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Polynomial.ContentIdeal | {
"line": 61,
"column": 2
} | {
"line": 65,
"column": 31
} | {
"line": 67,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\np : R[X]\n⊢ p.contentIdeal = ⊥ ↔ p = 0",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_forall_eq",
"Iff.mpr",
"Polynomial.contentIdeal",
"Eq.mpr",
"False",
"Finset.coe_empty",
"Sem... | [] | simp only [contentIdeal_def, span_eq_bot]
refine ⟨?_, fun h ↦ by simp [h]⟩
contrapose!
exact fun h ↦ ⟨p.leadingCoeff, coeff_mem_coeffs (leadingCoeff_ne_zero.mpr h),
leadingCoeff_ne_zero.mpr h⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.ContentIdeal | {
"line": 61,
"column": 2
} | {
"line": 65,
"column": 31
} | {
"line": 67,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\np : R[X]\n⊢ p.contentIdeal = ⊥ ↔ p = 0",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_forall_eq",
"Iff.mpr",
"Polynomial.contentIdeal",
"Eq.mpr",
"False",
"Finset.coe_empty",
"Sem... | [] | simp only [contentIdeal_def, span_eq_bot]
refine ⟨?_, fun h ↦ by simp [h]⟩
contrapose!
exact fun h ↦ ⟨p.leadingCoeff, coeff_mem_coeffs (leadingCoeff_ne_zero.mpr h),
leadingCoeff_ne_zero.mpr h⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.GaussLemma | {
"line": 39,
"column": 2
} | {
"line": 39,
"column": 73
} | {
"line": 40,
"column": 2
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nb : NNReal\nhb : 1 < b\np : R[X]\nhp0 : ¬p = 0\n⊢ gaussNorm (v.intAdicAbv hb) 1 p < 1 ↔ p.contentIdeal ≤ v.asIdeal",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"CommSemiring.... | [
"case neg\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nb : NNReal\nhb : 1 < b\np : R[X]\nhp0 : ¬p = 0\nhsupp_nonempty : p.support.Nonempty\n⊢ gaussNorm (v.intAdicAbv hb) 1 p < 1 ↔ p.contentIdeal ≤ v.asIdeal"
] | have hsupp_nonempty : p.support.Nonempty := by grind [support_nonempty] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.DedekindDomain.GaussLemma | {
"line": 43,
"column": 2
} | {
"line": 47,
"column": 88
} | {
"line": 48,
"column": 2
} | [
{
"pp": "case neg.mp\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nb : NNReal\nhb : 1 < b\np : R[X]\nhp0 : ¬p = 0\nhsupp_nonempty : p.support.Nonempty\n⊢ (p.support.sup' ⋯ fun x ↦ (v.intAdicAbv hb) (p.coeff x)) < 1 → ∀ x ∈ p.coeffs, (v.intAdicAbv hb) x < 1",
"ppTer... | [
"case neg.mpr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nb : NNReal\nhb : 1 < b\np : R[X]\nhp0 : ¬p = 0\nhsupp_nonempty : p.support.Nonempty\n⊢ (∀ x ∈ p.coeffs, (v.intAdicAbv hb) x < 1) → (p.support.sup' ⋯ fun x ↦ (v.intAdicAbv hb) (p.coeff x)) < 1"
] | · contrapose!
simp only [mem_coeffs_iff, mem_support_iff, ↓existsAndEq, and_true, forall_exists_index,
and_imp]
intro _ h1 h2
exact Finset.le_sup'_of_le (fun n ↦ (v.intAdicAbv hb) (p.coeff n)) (by simp [h1]) h2 | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.DividedPowers.Basic | {
"line": 98,
"column": 24
} | {
"line": 101,
"column": 40
} | {
"line": 102,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nn : ℕ\na : A\nha : a ∉ ⊥\n⊢ (if a = 0 ∧ n = 0 then 1 else 0) = 0",
"ppTerm": "?m.67",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semiring.toModule",
"congrArg",
"... | [] | by
simp only [mem_bot] at ha
rw [if_neg]
exact not_and_of_not_left (n = 0) ha | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.DividedPowers.Basic | {
"line": 122,
"column": 10
} | {
"line": 122,
"column": 20
} | {
"line": 122,
"column": 21
} | [
{
"pp": "case pos\nA : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nn : ℕ\nx✝¹ x✝ : A\nhx : x✝ ∈ ⊥\nhn : n = 0\n⊢ (if n = 0 then 1 else 0) = if n = 0 then x✝¹ ^ n else 0",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [
"case pos\nA : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nn : ℕ\nx✝¹ x✝ : A\nhx : x✝ ∈ ⊥\nhn : n = 0\n⊢ 1 = if n = 0 then x✝¹ ^ n else 0"
] | if_pos hn, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DividedPowers.Basic | {
"line": 235,
"column": 41
} | {
"line": 235,
"column": 47
} | {
"line": 235,
"column": 48
} | [
{
"pp": "A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\na : A\nn : ℕ\nhn : n ≠ 0\nhnI : ∀ {y : A}, y ∈ I → n • y = 0\nhI : DividedPowers I\nha : a ∈ I\nh_fac : ↑n ! * hI.dpow n a = n • ↑(n - 1)! * hI.dpow n a\n⊢ ↑n ! * hI.dpow n a = 0",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
... | [
"A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\na : A\nn : ℕ\nhn : n ≠ 0\nhnI : ∀ {y : A}, y ∈ I → n • y = 0\nhI : DividedPowers I\nha : a ∈ I\nh_fac : ↑n ! * hI.dpow n a = n • ↑(n - 1)! * hI.dpow n a\n⊢ n • ↑(n - 1)! * hI.dpow n a = 0"
] | h_fac, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DividedPowers.Basic | {
"line": 297,
"column": 13
} | {
"line": 297,
"column": 27
} | {
"line": 298,
"column": 6
} | [
{
"pp": "case insert\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}... | [
"case insert\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 → y ... | sum_insert ha, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Depth.Rees | {
"line": 141,
"column": 10
} | {
"line": 142,
"column": 71
} | {
"line": 144,
"column": 8
} | [
{
"pp": "R : 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 ... | [] | exact (Ext.postcomp_smul_id_mono_iff (a ^ k) (i + 1)).mpr <|
((Ext.postcomp_smul_id_mono_iff a (i + 1)).mp mono_g).pow k | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.DividedPowers.Basic | {
"line": 371,
"column": 8
} | {
"line": 371,
"column": 28
} | {
"line": 372,
"column": 8
} | [
{
"pp": "A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nι : Type u_3\nr : ι → A\nn : ℕ\na : ι\ns : Finset ι\nhas : a ∉ s\nhrec : s.Nonempty → (∀ i ∈ s, r i ∈ I) → hI.dpow n (∏ i ∈ s, r i) = n ! ^ (#s - 1) • ∏ i ∈ s, hI.dpow n (r i)\nhs : (insert a s).Nonempty\nhs' : ∀ i ∈ insert a s, r... | [
"A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nι : Type u_3\nr : ι → A\nn : ℕ\na : ι\ns : Finset ι\nhas : a ∉ s\nhrec : s.Nonempty → (∀ i ∈ s, r i ∈ I) → hI.dpow n (∏ i ∈ s, r i) = n ! ^ (#s - 1) • ∏ i ∈ s, hI.dpow n (r i)\nhs : (insert a s).Nonempty\nhs' : ∀ i ∈ insert a s, r i ∈ I\nh : ... | nth_rewrite 2 [this] | Mathlib.Tactic._aux_Mathlib_Tactic_NthRewrite___macroRules_Mathlib_Tactic_tacticNth_rewrite______1 | Mathlib.Tactic.tacticNth_rewrite_____ |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 164,
"column": 8
} | {
"line": 164,
"column": 18
} | {
"line": 164,
"column": 19
} | [
{
"pp": "case inl\nR : Type u_2\nM : Type u_3\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nn : ℕ\nhn : n = 0\n⊢ dp R n 0 = if n = 0 then 1 else 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AddMonoidAlgebra.instAddMonoid",
"Nat.ins... | [
"case inl\nR : Type u_2\nM : Type u_3\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nn : ℕ\nhn : n = 0\n⊢ dp R n 0 = 1"
] | if_pos hn, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 69,
"column": 76
} | {
"line": 69,
"column": 94
} | {
"line": 71,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : DecidablePred fun x ↦ x ∈ I\nm : ℕ\nx : A\nhx : x ∉ I\n⊢ dpow I m x = 0",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semiring.toModule",
"HMul.hMul",
... | [] | by simp [dpow, hx] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 71,
"column": 67
} | {
"line": 71,
"column": 85
} | {
"line": 73,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : DecidablePred fun x ↦ x ∈ I\nm : ℕ\nx : A\nhx : x ∉ I\n⊢ dpow I m x = 0",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semiring.toModule",
"HMul.hMul",
... | [] | by simp [dpow, hx] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 73,
"column": 59
} | {
"line": 73,
"column": 77
} | {
"line": 75,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : DecidablePred fun x ↦ x ∈ I\nx : A\nhx : x ∈ I\n⊢ dpow I 0 x = 1",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"Semiring.toModule",
"HMu... | [] | by simp [dpow, hx] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.DividedPowers.Basic | {
"line": 403,
"column": 20
} | {
"line": 407,
"column": 32
} | {
"line": 408,
"column": 2
} | [
{
"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\nx✝ y✝ : B\nhx : x✝ ∈ J\nhy : y✝ ∈ J\n⊢ e (hI.dpow n✝ (e.symm (x✝ + y✝))) = ∑ k ∈ antidiagonal n✝, e (hI.dpow k.1 (e.symm x✝)) * e (hI.... | [] | by
simp only [map_add]
rw [hI.dpow_add (symm_apply_mem_of_equiv_iff.mpr (h ▸ hx))
(symm_apply_mem_of_equiv_iff.mpr (h ▸ hy))]
simp only [map_sum, map_mul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 104,
"column": 6
} | {
"line": 104,
"column": 54
} | {
"line": 105,
"column": 6
} | [
{
"pp": "A : 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\nh_sub : I ^ m ≤ I ^ n\n⊢ (x + y) ^ m = 0",
"ppTerm": "?m.121",
"assigned": true,
"usedConstan... | [
"A : 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\nh_sub : I ^ m ≤ I ^ n\n⊢ (x + y) ^ m ∈ I ^ n"
] | rw [← Ideal.mem_bot, ← Ideal.zero_eq_bot, ← hnI] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Regular.RegularSequence | {
"line": 661,
"column": 2
} | {
"line": 661,
"column": 23
} | {
"line": 662,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsLocalRing R\ninst✝ : IsNoetherian R M\nrs rs' : List R\nh1 : IsRegular M rs\nh2 : rs ~ rs'\n⊢ IsRegular M rs'",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Submodule"... | [
"R : Type u_1\nM : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsLocalRing R\ninst✝ : IsNoetherian R M\nrs rs' : List R\nh2 : rs ~ rs'\nh3 : IsWeaklyRegular M rs\nh4 : ⊤ ≠ Ideal.ofList rs • ⊤\n⊢ IsRegular M rs'"
] | obtain ⟨h3, h4⟩ := h1 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 111,
"column": 4
} | {
"line": 111,
"column": 52
} | {
"line": 112,
"column": 4
} | [
{
"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\nh_sub : I ^ m ≤ I ^ n\nhxy : (x + y) ^ m = 0\nk : ℕ × ℕ\nhk : k ∈ Finset.antidiagonal m\n⊢ y ^ ... | [
"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\nh_sub : I ^ m ≤ I ^ n\nhxy : (x + y) ^ m = 0\nk : ℕ × ℕ\nhk : k ∈ Finset.antidiagonal m\n⊢ y ^ k.2 * x ^ k.... | rw [← Ideal.mem_bot, ← Ideal.zero_eq_bot, ← hnI] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 118,
"column": 26
} | {
"line": 118,
"column": 45
} | {
"line": 118,
"column": 46
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : DecidablePred fun x ↦ x ∈ I\nm : ℕ\na x : A\nhx : x ∈ I\n⊢ (↑m !)⁻¹ʳ * a ^ m * x ^ m = a ^ m * ((↑m !)⁻¹ʳ * x ^ m)",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWit... | [
"A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : DecidablePred fun x ↦ x ∈ I\nm : ℕ\na x : A\nhx : x ∈ I\n⊢ a ^ m * (↑m !)⁻¹ʳ * x ^ m = a ^ m * ((↑m !)⁻¹ʳ * x ^ m)"
] | mul_comm _ (a ^ m), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DividedPowers.Padic | {
"line": 158,
"column": 2
} | {
"line": 158,
"column": 40
} | {
"line": 159,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\nx : ℤ_[p]\n⊢ (dividedPowers p).dpow n x = if hx : x ∈ Ideal.span {↑p} then ⟨dpow' p n ↑x, ⋯⟩ else 0",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real.instLE",
"Real",
"Semiring.toModule",
"PadicI... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\nx : ℤ_[p]\n⊢ (if hx : x ∈ Ideal.span {↑p} then ⋯.choose else 0) = if hx : x ∈ Ideal.span {↑p} then ⟨dpow' p n ↑x, ⋯⟩ else 0"
] | simp only [dividedPowers, ofInjective] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 159,
"column": 6
} | {
"line": 159,
"column": 31
} | {
"line": 159,
"column": 31
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : DecidablePred fun x ↦ x ∈ I\nn : ℕ\nhn_fac : IsUnit ↑(n - 1)!\nm k : ℕ\nhk : k ≠ 0\nhkm : m * k < n\nx : A\nhx : x ∈ I\nhmn : m < n\nhm0 : ¬m = 0\nhkn : k < n\n⊢ ↑(m * k)! = ↑k ! ^ m * (↑m ! * ↑(m.uniformBell k))",
"ppTerm": "?m.189",
... | [
"A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : DecidablePred fun x ↦ x ∈ I\nn : ℕ\nhn_fac : IsUnit ↑(n - 1)!\nm k : ℕ\nhk : k ≠ 0\nhkm : m * k < n\nx : A\nhx : x ∈ I\nhmn : m < n\nhm0 : ¬m = 0\nhkn : k < n\n⊢ ↑(m.uniformBell k * k ! ^ m * m !) = ↑k ! ^ m * (↑m ! * ↑(m.uniformBell k))"
] | ← uniformBell_mul_eq _ hk | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 149,
"column": 2
} | {
"line": 161,
"column": 11
} | {
"line": 163,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : DecidablePred fun x ↦ x ∈ I\nn : ℕ\nhn_fac : IsUnit ↑(n - 1)!\nm k : ℕ\nhk : k ≠ 0\nhkm : m * k < n\nx : A\nhx : x ∈ I\n⊢ dpow I m (dpow I k x) = ↑(m.uniformBell k) * dpow I (m * k) x",
"ppTerm": "?m.47",
"assigned": true,
"usedCon... | [] | have hmn : m < n := lt_of_le_of_lt (Nat.le_mul_of_pos_right _ (Nat.pos_of_ne_zero hk)) hkm
rw [dpow_eq_of_mem (m := m * k) hx, dpow_eq_of_mem (dpow_mem hk hx)]
by_cases hm0 : m = 0
· simp only [hm0, zero_mul, _root_.pow_zero, mul_one, uniformBell_zero_left, cast_one, one_mul]
· have hkn : k < n := lt_of_le_of_l... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 149,
"column": 2
} | {
"line": 161,
"column": 11
} | {
"line": 163,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : DecidablePred fun x ↦ x ∈ I\nn : ℕ\nhn_fac : IsUnit ↑(n - 1)!\nm k : ℕ\nhk : k ≠ 0\nhkm : m * k < n\nx : A\nhx : x ∈ I\n⊢ dpow I m (dpow I k x) = ↑(m.uniformBell k) * dpow I (m * k) x",
"ppTerm": "?m.47",
"assigned": true,
"usedCon... | [] | have hmn : m < n := lt_of_le_of_lt (Nat.le_mul_of_pos_right _ (Nat.pos_of_ne_zero hk)) hkm
rw [dpow_eq_of_mem (m := m * k) hx, dpow_eq_of_mem (dpow_mem hk hx)]
by_cases hm0 : m = 0
· simp only [hm0, zero_mul, _root_.pow_zero, mul_one, uniformBell_zero_left, cast_one, one_mul]
· have hkn : k < n := lt_of_le_of_l... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 303,
"column": 41
} | {
"line": 303,
"column": 55
} | {
"line": 303,
"column": 56
} | [
{
"pp": "A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nx✝ : ℕ\nhn : x✝ ≠ 0\nx : A\nhx : x ∈ ⊥\n⊢ hI.dpow x✝ x ∈ ⊥",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Ideal.mem_bot",
"Eq.mpr",
"Semiring.toModule",
"congrArg",
"CommSemiri... | [
"A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nx✝ : ℕ\nhn : x✝ ≠ 0\nx : A\nhx : x ∈ ⊥\n⊢ hI.dpow x✝ 0 ∈ ⊥"
] | mem_bot.mp hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Frobenius | {
"line": 86,
"column": 4
} | {
"line": 86,
"column": 53
} | {
"line": 87,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ ψ : S →ₐ[R] S\nQ : Ideal S\nH : φ.IsArithFrobAt Q\nx : R ⧸ Ideal.under R Q\n⊢ (↑↑(Ideal.quotientMap Q ↑φ ⋯)).toFun ((algebraMap (R ⧸ Ideal.under R Q) (S ⧸ Q)) x) =\n (algebraMap (R ⧸ Ideal.under R Q) (S ⧸ Q)... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ ψ : S →ₐ[R] S\nQ : Ideal S\nH : φ.IsArithFrobAt Q\nx : R\n⊢ (↑↑(Ideal.quotientMap Q ↑φ ⋯)).toFun\n ((algebraMap (R ⧸ Ideal.under R Q) (S ⧸ Q)) ((Ideal.Quotient.mk (Ideal.under R Q)) x)) =\n (algebraMap (R ⧸ Ideal.u... | obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Frobenius | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 51
} | {
"line": 92,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : S →ₐ[R] S\nQ : Ideal S\nH : φ.IsArithFrobAt Q\nx : S ⧸ Q\n⊢ H.restrict x = x ^ Nat.card (R ⧸ Ideal.under R Q)",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Ideal.Quotient.commSe... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : S →ₐ[R] S\nQ : Ideal S\nH : φ.IsArithFrobAt Q\nx : S\n⊢ H.restrict ((Ideal.Quotient.mk Q) x) = (Ideal.Quotient.mk Q) x ^ Nat.card (R ⧸ Ideal.under R Q)"
] | obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.HahnSeries.HEval | {
"line": 73,
"column": 18
} | {
"line": 73,
"column": 55
} | {
"line": 73,
"column": 55
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\nV : Type u_4\ninst✝⁵ : AddCommMonoid Γ\ninst✝⁴ : LinearOrder Γ\ninst✝³ : IsOrderedCancelAddMonoid Γ\ninst✝² : CommRing R\ninst✝¹ : CommRing V\ninst✝ : Algebra R V\nf : PowerSeries R\ng : Γ\nhg : ¬g = 0\nn : ℕ\n⊢ ((powerSeriesFamily 0 f) n).coeff g = 0",
"ppTerm": "?m.85"... | [] | by_cases hn : n = 0 <;> simp [hg, hn] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.RingTheory.HahnSeries.HEval | {
"line": 73,
"column": 18
} | {
"line": 73,
"column": 55
} | {
"line": 73,
"column": 55
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\nV : Type u_4\ninst✝⁵ : AddCommMonoid Γ\ninst✝⁴ : LinearOrder Γ\ninst✝³ : IsOrderedCancelAddMonoid Γ\ninst✝² : CommRing R\ninst✝¹ : CommRing V\ninst✝ : Algebra R V\nf : PowerSeries R\ng : Γ\nhg : ¬g = 0\nn : ℕ\n⊢ ((powerSeriesFamily 0 f) n).coeff g = 0",
"ppTerm": "?m.85"... | [] | by_cases hn : n = 0 <;> simp [hg, hn] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.HahnSeries.HEval | {
"line": 73,
"column": 18
} | {
"line": 73,
"column": 55
} | {
"line": 73,
"column": 55
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\nV : Type u_4\ninst✝⁵ : AddCommMonoid Γ\ninst✝⁴ : LinearOrder Γ\ninst✝³ : IsOrderedCancelAddMonoid Γ\ninst✝² : CommRing R\ninst✝¹ : CommRing V\ninst✝ : Algebra R V\nf : PowerSeries R\ng : Γ\nhg : ¬g = 0\nn : ℕ\n⊢ ((powerSeriesFamily 0 f) n).coeff g = 0",
"ppTerm": "?m.85"... | [] | by_cases hn : n = 0 <;> simp [hg, hn] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.HahnSeries.PowerSeries | {
"line": 119,
"column": 4
} | {
"line": 119,
"column": 12
} | {
"line": 120,
"column": 4
} | [
{
"pp": "case pos\nΓ : Type u_1\nR : Type u_2\ninst✝³ : Semiring R\ninst✝² : Semiring Γ\ninst✝¹ : PartialOrder Γ\ninst✝ : IsStrictOrderedRing Γ\nr : R\nn : Γ\nhn : n = 0\n⊢ (embDomain Nat.castOrderEmbedding (toPowerSeries.symm (PowerSeries.C r))).coeff n = r",
"ppTerm": "?pos✝",
"assigned": true,
"u... | [
"case pos\nΓ : Type u_1\nR : Type u_2\ninst✝³ : Semiring R\ninst✝² : Semiring Γ\ninst✝¹ : PartialOrder Γ\ninst✝ : IsStrictOrderedRing Γ\nr : R\n⊢ (embDomain Nat.castOrderEmbedding (toPowerSeries.symm (PowerSeries.C r))).coeff 0 = r"
] | subst hn | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.RingTheory.FormalGroup.Basic | {
"line": 251,
"column": 32
} | {
"line": 251,
"column": 57
} | {
"line": 251,
"column": 58
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nF : FormalGroup R\nthis : Invertible ((PowerSeries.coeff 1) F.Xzero)\naux₀ : PowerSeries.HasSubst F.Xzero\n⊢ F.Xzero = PowerSeries.subst F.Xzero PowerSeries.X",
"ppTerm": "?m.89",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
... | [
"R : Type u_1\ninst✝ : CommRing R\nF : FormalGroup R\nthis : Invertible ((PowerSeries.coeff 1) F.Xzero)\naux₀ : PowerSeries.HasSubst F.Xzero\n⊢ F.Xzero = F.Xzero"
] | PowerSeries.subst_X aux₀, | Lean.Elab.Tactic.evalRewriteSeq | null |
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