module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.NumberTheory.DirichletCharacter.Basic | {
"line": 281,
"column": 55
} | {
"line": 287,
"column": 25
} | {
"line": 289,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommMonoidWithZero R\nn : ℕ\nχ : DirichletCharacter R n\nd : ℕ\nhd : d ∈ χ.conductorSet\n⊢ χ.conductor ≤ (Classical.choose ⋯).conductor",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"DirichletCharacter.conductor",
"Eq.mpr",
... | [] | by
refine Nat.sInf_le <| (mem_conductorSet_iff χ).mpr <|
⟨dvd_trans (conductor_dvd_level _) hd.1,
(factorsThrough_conductor (Classical.choose hd.2)).2.choose, ?_⟩
rw [changeLevel_trans _ (conductor_dvd_level _) hd.dvd,
← (factorsThrough_conductor (Classical.choose hd.2)).2.choose_spec]
exact hd.eq_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 855,
"column": 39
} | {
"line": 855,
"column": 52
} | {
"line": 855,
"column": 53
} | [
{
"pp": "S : Set ℕ\nK : Type w\ninst✝¹ : Field K\ninst✝ : IsSepClosed K\nh : ∀ a ∈ S, a ≠ 0 → NeZero ↑a\na : ℕ\nha : a ∈ S\nha' : a ≠ 0\nthis : NeZero ↑a\nr : K\nhr : eval r (cyclotomic a K) = 0\n⊢ IsPrimitiveRoot r a",
"ppTerm": "?m.82",
"assigned": true,
"usedConstants": [
"Polynomial.eval",... | [
"S : Set ℕ\nK : Type w\ninst✝¹ : Field K\ninst✝ : IsSepClosed K\nh : ∀ a ∈ S, a ≠ 0 → NeZero ↑a\na : ℕ\nha : a ∈ S\nha' : a ≠ 0\nthis : NeZero ↑a\nr : K\nhr : (cyclotomic a K).IsRoot r\n⊢ IsPrimitiveRoot r a"
] | ← IsRoot.def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 897,
"column": 6
} | {
"line": 897,
"column": 54
} | {
"line": 898,
"column": 6
} | [
{
"pp": "case a\nn : ℕ\ninst✝⁴ : NeZero n\nA : Type u\nB : Type v\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsDomain B\nC : Subalgebra A B\nζ : B\nhζ : IsPrimitiveRoot ζ n\nx : B\nx✝ : x ∈ {b | n ≠ 0 ∧ b ^ n = 1}\nleft✝ : n ≠ 0\nhx : x ^ n = 1\n⊢ x ∈ ↑A[ζ]",
"ppTerm": "?a✝",
... | [
"case a\nn : ℕ\ninst✝⁴ : NeZero n\nA : Type u\nB : Type v\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsDomain B\nC : Subalgebra A B\nζ : B\nhζ : IsPrimitiveRoot ζ n\nleft✝¹ : n ≠ 0\nk : ℕ\nleft✝ : k < n\nx✝ : ζ ^ k ∈ {b | n ≠ 0 ∧ b ^ n = 1}\nhx : (ζ ^ k) ^ n = 1\n⊢ ζ ^ k ∈ ↑A[ζ]"
] | obtain ⟨k, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.NumberTheory.GaussSum | {
"line": 169,
"column": 2
} | {
"line": 169,
"column": 34
} | {
"line": 171,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝² : Field R\ninst✝¹ : Fintype R\nR' : Type v\ninst✝ : CommRing R'\nχ : MulChar R R'\nψ : AddChar R R'\n⊢ χ ↑(-1) * gaussSum χ (ψ.mulShift ↑(-1)) = gaussSum χ ψ",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | [] | exact gaussSum_mulShift χ ψ (-1) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.InnerProductSpace.NormPow | {
"line": 114,
"column": 13
} | {
"line": 114,
"column": 16
} | {
"line": 114,
"column": 17
} | [
{
"pp": "case pos\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\np : ℝ\nhp : 1 < p\nx : E\nhx : x = 0\n⊢ ContinuousAt (fderiv ℝ fun x ↦ ‖x‖ ^ p) x",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"InnerProductSpace.toNor... | [
"case pos\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\np : ℝ\nhp : 1 < p\nx : E\nhx : x = 0\n⊢ ContinuousAt (fderiv ℝ fun x ↦ ‖x‖ ^ p) 0"
] | hx, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Data.Finset.Grade | {
"line": 127,
"column": 4
} | {
"line": 128,
"column": 12
} | {
"line": 129,
"column": 2
} | [
{
"pp": "case mp\nα : Type u_1\ns t : Finset α\ninst✝ : DecidableEq α\n⊢ (∃ a ∉ t, insert a t = s) → t ⊆ s ∧ #(s \\ t) = 1",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"False",
"eq_false",
"congrArg",
"and_self",
"Finset",
"PartialOrder.toPreorder",
... | [] | rintro ⟨a, ha, rfl⟩
simp [*] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finset.Grade | {
"line": 127,
"column": 4
} | {
"line": 128,
"column": 12
} | {
"line": 129,
"column": 2
} | [
{
"pp": "case mp\nα : Type u_1\ns t : Finset α\ninst✝ : DecidableEq α\n⊢ (∃ a ∉ t, insert a t = s) → t ⊆ s ∧ #(s \\ t) = 1",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"False",
"eq_false",
"congrArg",
"and_self",
"Finset",
"PartialOrder.toPreorder",
... | [] | rintro ⟨a, ha, rfl⟩
simp [*] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.GaussSum | {
"line": 224,
"column": 50
} | {
"line": 228,
"column": 5
} | {
"line": 230,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝³ : Field R\ninst✝² : Fintype R\nR' : Type v\ninst✝¹ : CommRing R'\ninst✝ : IsDomain R'\nχ : MulChar R R'\nhχ₁ : χ ≠ 1\nhχ₂ : χ.IsQuadratic\nψ : AddChar R R'\nhψ : ψ.IsPrimitive\n⊢ gaussSum χ ψ ^ 2 = χ (-1) * ↑(Fintype.card R)",
"ppTerm": "?m.44",
"assigned": true,
"usedCon... | [] | by
rw [pow_two, ← gaussSum_mul_gaussSum_eq_card hχ₁ hψ, hχ₂.inv, mul_rotate']
congr
rw [mul_comm, ← gaussSum_mulShift _ _ (-1 : Rˣ), inv_mulShift]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Finset.Interval | {
"line": 59,
"column": 10
} | {
"line": 59,
"column": 100
} | {
"line": 60,
"column": 6
} | [
{
"pp": "case pos.mpr\nα : Type u_1\nβ : Type u_2\ninst✝ : DecidableEq α\ns t u : Finset α\nhst : s ⊆ t\nhsu : s ⊆ u\nhut : u ⊆ t\n⊢ ∃ a ⊆ t, Disjoint a s ∧ a ∪ s = u",
"ppTerm": "?pos.mpr✝",
"assigned": true,
"usedConstants": [
"Finset.instGeneralizedBooleanAlgebra",
"Finset.instUnion",... | [] | exact ⟨u \ s, sdiff_subset.trans hut, disjoint_sdiff_self_left, sdiff_union_of_subset hsu⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Hofer | {
"line": 35,
"column": 52
} | {
"line": 35,
"column": 80
} | {
"line": 35,
"column": 81
} | [
{
"pp": "X : Type u_1\ninst✝¹ : MetricSpace X\ninst✝ : CompleteSpace X\nx : X\nε : ℝ\nε_pos : 0 < ε\nϕ : X → ℝ\ncont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ϕ\nnonneg : ∀ (y : X), 0 ≤ ϕ y\nH : ∀ ε' > 0, ∀ (x' : X), ε' ≤ ε → d x' x ≤ 2 * ε → ε * ϕ x ≤ ε' * ϕ x' → ∃ y, d x' y ≤ ε' ∧ 2... | [
"X : Type u_1\ninst✝¹ : MetricSpace X\ninst✝ : CompleteSpace X\nx : X\nε : ℝ\nε_pos : 0 < ε\nϕ : X → ℝ\ncont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ϕ\nnonneg : ∀ (y : X), 0 ≤ ϕ y\nH : ∀ ε' > 0, ∀ (x' : X), ε' ≤ ε → d x' x ≤ 2 * ε → ε * ϕ x ≤ ε' * ϕ x' → ∃ y, d x' y ≤ ε' ∧ 2 * ϕ x' < ϕ ... | mul_le_mul_iff_right₀ ε_pos, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 264,
"column": 2
} | {
"line": 268,
"column": 83
} | {
"line": 271,
"column": 0
} | [
{
"pp": "ι : Type u_1\nA : ι → Type u_2\ninst✝³ : (i : ι) → MeasurableSpace (A i)\nμ : (i : ι) → Measure (A i)\ninst✝² : DecidableEq ι\ninst✝¹ : Fintype ι\ninst✝ : ∀ (i : ι), SigmaFinite (μ i)\np : ℝ\nhp₀ : 0 ≤ p\nhp : (↑#ι - 1) * p ≤ 1\nf : ((i : ι) → A i) → ℝ≥0∞\nhf : Measurable f\n⊢ ∫⁻ (x : (i : ι) → A i), f... | [] | cases isEmpty_or_nonempty (∀ i, A i)
· simp
inhabit ∀ i, A i
have H : (∅ : Finset ι) ≤ Finset.univ := Finset.empty_subset _
simpa [lmarginal_univ] using GridLines.T_lmarginal_antitone μ hp₀ hp hf H default | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 264,
"column": 2
} | {
"line": 268,
"column": 83
} | {
"line": 271,
"column": 0
} | [
{
"pp": "ι : Type u_1\nA : ι → Type u_2\ninst✝³ : (i : ι) → MeasurableSpace (A i)\nμ : (i : ι) → Measure (A i)\ninst✝² : DecidableEq ι\ninst✝¹ : Fintype ι\ninst✝ : ∀ (i : ι), SigmaFinite (μ i)\np : ℝ\nhp₀ : 0 ≤ p\nhp : (↑#ι - 1) * p ≤ 1\nf : ((i : ι) → A i) → ℝ≥0∞\nhf : Measurable f\n⊢ ∫⁻ (x : (i : ι) → A i), f... | [] | cases isEmpty_or_nonempty (∀ i, A i)
· simp
inhabit ∀ i, A i
have H : (∅ : Finset ι) ≤ Finset.univ := Finset.empty_subset _
simpa [lmarginal_univ] using GridLines.T_lmarginal_antitone μ hp₀ hp hf H default | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Coalgebra | {
"line": 158,
"column": 12
} | {
"line": 158,
"column": 26
} | {
"line": 158,
"column": 27
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\ninst✝ : Coalgebra 𝕜 E\nx y : 𝕜\n⊢ (adjoint counit) (x * y) = x • (adjoint counit) y",
"ppTerm": "?m.79",
"assigned": true,
"usedConstants": [
... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\ninst✝ : Coalgebra 𝕜 E\nx y : 𝕜\n⊢ (adjoint counit) (x • y) = x • (adjoint counit) y"
] | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 389,
"column": 44
} | {
"line": 389,
"column": 56
} | {
"line": 390,
"column": 2
} | [
{
"pp": "F : Type u_3\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nu : E → F\nhu : ContDiff ℝ 1 u\nh2u... | [] | rwa [hιcard] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 554,
"column": 4
} | {
"line": 554,
"column": 18
} | {
"line": 554,
"column": 19
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\nT : E →L[𝕜] E\nhT : T.IsPositive\na : Fin (Module.finrank 𝕜 E) → E := fun i ↦ ↑√(⋯.eigenvalues ⋯ i) • (⋯.eigenvectorBasis ⋯) i\nx✝ : E\n⊢ T x✝ = ∑ x, (⟪(⋯.e... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\nT : E →L[𝕜] E\nhT : T.IsPositive\na : Fin (Module.finrank 𝕜 E) → E := fun i ↦ ↑√(⋯.eigenvalues ⋯ i) • (⋯.eigenvectorBasis ⋯) i\nx✝ : E\n⊢ T x✝ = ∑ x, (⟪(⋯.eigenvectorBa... | ← smul_eq_mul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.InnerProductSpace.LinearPMap | {
"line": 95,
"column": 23
} | {
"line": 95,
"column": 68
} | {
"line": 95,
"column": 68
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nT : E →ₗ.[𝕜] F\nS : F →ₗ.[𝕜] E\na✝ b✝ : F\nhx : a✝ ∈ {y | Continuous ⇑((innerₛₗ 𝕜) y ∘ₗ T.toFun)}\nhy : b✝ ∈ {... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nT : E →ₗ.[𝕜] F\nS : F →ₗ.[𝕜] E\na✝ b✝ : F\nhx : Continuous ⇑((innerₛₗ 𝕜) a✝ ∘ₗ T.toFun)\nhy : Continuous ⇑((innerₛₗ 𝕜) b✝... | rw [Set.mem_setOf_eq, LinearMap.map_add] at * | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.TensorProduct | {
"line": 441,
"column": 22
} | {
"line": 441,
"column": 25
} | {
"line": 441,
"column": 26
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : InnerProductSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : InnerProductSpace 𝕜 G\ninst✝¹ : NormedAddCo... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : InnerProductSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : InnerProductSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ni... | hx, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 573,
"column": 68
} | {
"line": 576,
"column": 27
} | {
"line": 577,
"column": 6
} | [
{
"pp": "E : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : MeasurableSpace E\ninst✝⁴ : BorelSpace E\ninst✝³ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝² : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : InnerProductSpace ℝ F'\nu : E → F'\nhu : ContDiff ... | [] | by
rw [eLpNorm_nnreal_eq_lintegral h0p]
congr
norm_cast at this ⊢ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.HausdorffDimension | {
"line": 205,
"column": 6
} | {
"line": 205,
"column": 24
} | {
"line": 205,
"column": 25
} | [
{
"pp": "X : Type u_2\ninst✝ : EMetricSpace X\nS : Set (Set X)\nhS : S.Countable\n⊢ dimH (⋃₀ S) = ⨆ s ∈ S, dimH s",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"iSup",
"Set.sUnion",
"Membership.mem",
"id",
"ConditionallyComp... | [
"X : Type u_2\ninst✝ : EMetricSpace X\nS : Set (Set X)\nhS : S.Countable\n⊢ dimH (⋃ i ∈ S, i) = ⨆ s ∈ S, dimH s"
] | sUnion_eq_biUnion, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.HausdorffDimension | {
"line": 286,
"column": 2
} | {
"line": 288,
"column": 14
} | {
"line": 289,
"column": 2
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝¹ : EMetricSpace X\ninst✝ : EMetricSpace Y\nC r : ℝ≥0\nf : X → Y\ns : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nthis✝³ : MeasurableSpace X := borel X\nthis✝² : BorelSpace X\nthis✝¹ : MeasurableSpace Y := borel Y\nthis✝ : BorelSpace Y\nd : ℝ≥0\nhd : μH[↑d] (f '' s) = ... | [
"X : Type u_2\nY : Type u_3\ninst✝¹ : EMetricSpace X\ninst✝ : EMetricSpace Y\nC r : ℝ≥0\nf : X → Y\ns : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nthis✝³ : MeasurableSpace X := borel X\nthis✝² : BorelSpace X\nthis✝¹ : MeasurableSpace Y := borel Y\nthis✝ : BorelSpace Y\nd : ℝ≥0\nhd : μH[↑d] (f '' s) = ∞\nthis : ↑(... | have Hrd : μH[(r * d : ℝ≥0)] s = ⊤ := by
contrapose this
finiteness | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 332,
"column": 6
} | {
"line": 332,
"column": 20
} | {
"line": 332,
"column": 21
} | [
{
"pp": "X : Type u_2\ninst✝ : EMetricSpace X\nm₁ m₂ : ℝ≥0∞ → ℝ≥0∞\nc : ℝ≥0∞\nhc : c ≠ ∞\nh0 : c ≠ 0\nhle : m₁ ≤ᶠ[𝓝[≥] 0] c • m₂\nr : ℝ≥0∞\nhr0 : r ∈ Ioi 0\nhr : Ico 0 r ⊆ {x | (fun x ↦ m₁ x ≤ (c • m₂) x) x}\ns : Set X\nr' : ℝ≥0∞\nhr' : r' ∈ Ioo 0 r\n⊢ (boundedBy (extend fun s x ↦ m₁ (ediam s))) s ≤ c * (bound... | [
"X : Type u_2\ninst✝ : EMetricSpace X\nm₁ m₂ : ℝ≥0∞ → ℝ≥0∞\nc : ℝ≥0∞\nhc : c ≠ ∞\nh0 : c ≠ 0\nhle : m₁ ≤ᶠ[𝓝[≥] 0] c • m₂\nr : ℝ≥0∞\nhr0 : r ∈ Ioi 0\nhr : Ico 0 r ⊆ {x | (fun x ↦ m₁ x ≤ (c • m₂) x) x}\ns : Set X\nr' : ℝ≥0∞\nhr' : r' ∈ Ioo 0 r\n⊢ (boundedBy (extend fun s x ↦ m₁ (ediam s))) s ≤ c • (boundedBy (extend... | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 354,
"column": 2
} | {
"line": 363,
"column": 61
} | {
"line": 365,
"column": 0
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝¹ : EMetricSpace X\ninst✝ : EMetricSpace Y\nm : ℝ≥0∞ → ℝ≥0∞\nf : X → Y\nhf : Isometry f\nH : Monotone m ∨ Surjective f\n⊢ (comap f) (mkMetric m) = mkMetric m",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
... | [] | simp only [mkMetric, mkMetric', mkMetric'.pre, comap_iSup]
refine surjective_id.iSup_congr id fun ε => surjective_id.iSup_congr id fun hε => ?_
rw [comap_boundedBy _ (H.imp _ id)]
· congr with s : 1
apply extend_congr <;> simp [hf.ediam_image]
· intro h_mono s t hst
simp only [extend, le_iInf_iff]
i... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 354,
"column": 2
} | {
"line": 363,
"column": 61
} | {
"line": 365,
"column": 0
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝¹ : EMetricSpace X\ninst✝ : EMetricSpace Y\nm : ℝ≥0∞ → ℝ≥0∞\nf : X → Y\nhf : Isometry f\nH : Monotone m ∨ Surjective f\n⊢ (comap f) (mkMetric m) = mkMetric m",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
... | [] | simp only [mkMetric, mkMetric', mkMetric'.pre, comap_iSup]
refine surjective_id.iSup_congr id fun ε => surjective_id.iSup_congr id fun hε => ?_
rw [comap_boundedBy _ (H.imp _ id)]
· congr with s : 1
apply extend_congr <;> simp [hf.ediam_image]
· intro h_mono s t hst
simp only [extend, le_iInf_iff]
i... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Volume.Measure | {
"line": 349,
"column": 45
} | {
"line": 349,
"column": 67
} | {
"line": 349,
"column": 67
} | [
{
"pp": "V : Type u_3\nP : Type u_4\ninst✝⁸ : NormedAddCommGroup V\ninst✝⁷ : InnerProductSpace ℝ V\ninst✝⁶ : MeasurableSpace V\ninst✝⁵ : BorelSpace V\ninst✝⁴ : FiniteDimensional ℝ V\ninst✝³ : MetricSpace P\ninst✝² : MeasurableSpace P\ninst✝¹ : BorelSpace P\ninst✝ : NormedAddTorsor V P\np : P\nv : V\nhv : v ≠ 0\... | [] | by simpa [v'] using hv | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Volume.Measure | {
"line": 351,
"column": 85
} | {
"line": 351,
"column": 88
} | {
"line": 352,
"column": 4
} | [
{
"pp": "V : Type u_3\nP : Type u_4\ninst✝⁸ : NormedAddCommGroup V\ninst✝⁷ : InnerProductSpace ℝ V\ninst✝⁶ : MeasurableSpace V\ninst✝⁵ : BorelSpace V\ninst✝⁴ : FiniteDimensional ℝ V\ninst✝³ : MetricSpace P\ninst✝² : MeasurableSpace P\ninst✝¹ : BorelSpace P\ninst✝ : NormedAddTorsor V P\np : P\nv : V\nhv : v ≠ 0\... | [
"V : Type u_3\nP : Type u_4\ninst✝⁸ : NormedAddCommGroup V\ninst✝⁷ : InnerProductSpace ℝ V\ninst✝⁶ : MeasurableSpace V\ninst✝⁵ : BorelSpace V\ninst✝⁴ : FiniteDimensional ℝ V\ninst✝³ : MetricSpace P\ninst✝² : MeasurableSpace P\ninst✝¹ : BorelSpace P\ninst✝ : NormedAddTorsor V P\np : P\nv : V\nhv : v ≠ 0\nt : Set P\n... | hx, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 127,
"column": 4
} | {
"line": 133,
"column": 55
} | {
"line": 135,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\nh : f.ker ≠ ⊥\n⊢ f.normDet = 0",
"ppTerm": "?m.51",
"ass... | [] | suffices ¬ Nonempty (OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 f.range) by
simp [normDet, this]
contrapose h
obtain ⟨b⟩ := h
have hrank : finrank 𝕜 f.range = finrank 𝕜 U := by
simpa using finrank_eq_card_basis b.toBasis
simpa [hrank] using f.finrank_range_add_finrank_ker | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 127,
"column": 4
} | {
"line": 133,
"column": 55
} | {
"line": 135,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\nh : f.ker ≠ ⊥\n⊢ f.normDet = 0",
"ppTerm": "?m.51",
"ass... | [] | suffices ¬ Nonempty (OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 f.range) by
simp [normDet, this]
contrapose h
obtain ⟨b⟩ := h
have hrank : finrank 𝕜 f.range = finrank 𝕜 U := by
simpa using finrank_eq_card_basis b.toBasis
simpa [hrank] using f.finrank_range_add_finrank_ker | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 586,
"column": 2
} | {
"line": 587,
"column": 100
} | {
"line": 588,
"column": 2
} | [
{
"pp": "X : Type u_2\ninst✝² : EMetricSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nd₁ d₂ : ℝ\nh : d₁ < d₂\ns : Set X\nH : μH[d₂] s ≠ 0 ∧ μH[d₁] s ≠ ∞\nc : ℝ≥0\nhc : c ≠ 0\nthis : 0 < ↑c ^ (d₂ - d₁)⁻¹\nr : ℝ≥0\nhr₀ : 0 ≤ ↑r\nhrc : ↑r < ↑c ^ (d₂ - d₁)⁻¹\n⊢ ↑r ^ d₂ ≤ (↑c • fun r ↦ r ^ d₁) ↑r",
"... | [
"X : Type u_2\ninst✝² : EMetricSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nd₁ d₂ : ℝ\nh : d₁ < d₂\ns : Set X\nH : μH[d₂] s ≠ 0 ∧ μH[d₁] s ≠ ∞\nc : ℝ≥0\nhc : c ≠ 0\nthis : 0 < ↑c ^ (d₂ - d₁)⁻¹\nr : ℝ≥0\nhr₀ : 0 ≤ ↑r\nhrc : ↑r < ↑c ^ (d₂ - d₁)⁻¹\n⊢ ↑r ^ d₂ / ↑r ^ d₁ ≤ ↑c"
] | rw [Pi.smul_apply, smul_eq_mul,
← ENNReal.div_le_iff_le_mul (Or.inr ENNReal.coe_ne_top) (Or.inr <| mt ENNReal.coe_eq_zero.1 hc)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 630,
"column": 6
} | {
"line": 630,
"column": 27
} | {
"line": 631,
"column": 6
} | [
{
"pp": "X : Type u_2\ninst✝² : EMetricSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nx : X\nthis :\n 1 ≤\n ⨅ t, ⨅ (_ : {x} ⊆ ⋃ n, t n), ⨅ (_ : ∀ (n : ℕ), ediam (t n) ≤ 1), ∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), ediam (t n) ^ 0\n⊢ 1 ≤\n ⨆ r,\n ⨆ (_ : 0 < r),\n ⨅ t,\n ⨅ (_ : ... | [
"X : Type u_2\ninst✝² : EMetricSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nx : X\nthis :\n 1 ≤\n ⨅ t, ⨅ (_ : {x} ⊆ ⋃ n, t n), ⨅ (_ : ∀ (n : ℕ), ediam (t n) ≤ 1), ∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), ediam (t n) ^ 0\n⊢ ⨅ t, ⨅ (_ : {x} ⊆ ⋃ n, t n), ⨅ (_ : ∀ (n : ℕ), ediam (t n) ≤ 1), ∑' (n : ℕ), ⨆ (... | apply le_trans this _ | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 1008,
"column": 64
} | {
"line": 1011,
"column": 64
} | {
"line": 1013,
"column": 0
} | [
{
"pp": "⊢ μH[1] = volume",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"MeasurableEquiv.instEquivLike",
"Eq.mpr",
"emetricSpacePi",
"Real",
"MeasureTheory.Measure",
"congrArg",
"PUnit.instUnique",
"MeasureTheory.Measure.hausdorffMeasure",
... | [] | by
rw [← (volume_preserving_funUnique Unit ℝ).map_eq,
← (hausdorffMeasure_measurePreserving_funUnique Unit ℝ 1).map_eq,
← hausdorffMeasure_pi_real, Fintype.card_unit, Nat.cast_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.TwoDim | {
"line": 194,
"column": 6
} | {
"line": 217,
"column": 67
} | {
"line": 217,
"column": 68
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\nx : E\n⊢ ‖o.rightAngleRotationAux₁ x‖ = ‖x‖",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Nontrivial",
"AddGroup.toSubtractionMo... | [] | refine le_antisymm ?_ ?_
· rcases eq_or_lt_of_le (norm_nonneg (o.rightAngleRotationAux₁ x)) with h | h
· rw [← h]
positivity
refine le_of_mul_le_mul_right ?_ h
rw [← real_inner_self_eq_norm_mul_norm, o.inner_rightAngleRotationAux₁_left]
exact o.areaForm_le x (o.rightAngle... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.TwoDim | {
"line": 194,
"column": 6
} | {
"line": 217,
"column": 67
} | {
"line": 217,
"column": 68
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\nx : E\n⊢ ‖o.rightAngleRotationAux₁ x‖ = ‖x‖",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Nontrivial",
"AddGroup.toSubtractionMo... | [] | refine le_antisymm ?_ ?_
· rcases eq_or_lt_of_le (norm_nonneg (o.rightAngleRotationAux₁ x)) with h | h
· rw [← h]
positivity
refine le_of_mul_le_mul_right ?_ h
rw [← real_inner_self_eq_norm_mul_norm, o.inner_rightAngleRotationAux₁_left]
exact o.areaForm_le x (o.rightAngle... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Matrix.LDL | {
"line": 118,
"column": 4
} | {
"line": 118,
"column": 46
} | {
"line": 118,
"column": 46
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : RCLike 𝕜\nn : Type u_2\ninst✝³ : LinearOrder n\ninst✝² : WellFoundedLT n\ninst✝¹ : LocallyFiniteOrderBot n\nS : Matrix n n 𝕜\ninst✝ : Fintype n\nhS : S.PosDef\n⊢ diag hS * (lowerInv hS)ᴴ⁻¹ = lowerInv hS * S",
"ppTerm": "?m.93",
"assigned": true,
"usedConstants": [
... | [
"𝕜 : Type u_1\ninst✝⁴ : RCLike 𝕜\nn : Type u_2\ninst✝³ : LinearOrder n\ninst✝² : WellFoundedLT n\ninst✝¹ : LocallyFiniteOrderBot n\nS : Matrix n n 𝕜\ninst✝ : Fintype n\nhS : S.PosDef\n⊢ diag hS = lowerInv hS * S * (lowerInv hS)ᴴ"
] | Matrix.mul_inv_eq_iff_eq_mul_of_invertible | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.LocallyConvex.WeakSpace | {
"line": 75,
"column": 4
} | {
"line": 78,
"column": 70
} | {
"line": 79,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹⁶ : RCLike 𝕜\ninst✝¹⁵ : AddCommGroup E\ninst✝¹⁴ : Module 𝕜 E\ninst✝¹³ : AddCommGroup F\ninst✝¹² : Module 𝕜 F\ninst✝¹¹ : Module ℝ E\ninst✝¹⁰ : IsScalarTower ℝ 𝕜 E\ninst✝⁹ : Module ℝ F\ninst✝⁸ : IsScalarTower ℝ 𝕜 F\ninst✝⁷ : TopologicalSpace E\ninst✝⁶... | [] | have h_convex : Convex ℝ (e '' s) := hs.linear_image (F := F) e
rw [← Set.image_subset_image_iff (toWeakSpace 𝕜 F).injective, h_convex.toWeakSpace_closure 𝕜]
simpa only [Set.image_image, ← hs.toWeakSpace_closure 𝕜, LinearEquiv.symm_apply_apply]
using he'.continuousOn.image_closure (s := toWeakSpace 𝕜 ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.LocallyConvex.WeakSpace | {
"line": 75,
"column": 4
} | {
"line": 78,
"column": 70
} | {
"line": 79,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹⁶ : RCLike 𝕜\ninst✝¹⁵ : AddCommGroup E\ninst✝¹⁴ : Module 𝕜 E\ninst✝¹³ : AddCommGroup F\ninst✝¹² : Module 𝕜 F\ninst✝¹¹ : Module ℝ E\ninst✝¹⁰ : IsScalarTower ℝ 𝕜 E\ninst✝⁹ : Module ℝ F\ninst✝⁸ : IsScalarTower ℝ 𝕜 F\ninst✝⁷ : TopologicalSpace E\ninst✝⁶... | [] | have h_convex : Convex ℝ (e '' s) := hs.linear_image (F := F) e
rw [← Set.image_subset_image_iff (toWeakSpace 𝕜 F).injective, h_convex.toWeakSpace_closure 𝕜]
simpa only [Set.image_image, ← hs.toWeakSpace_closure 𝕜, LinearEquiv.symm_apply_apply]
using he'.continuousOn.image_closure (s := toWeakSpace 𝕜 ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.MellinInversion | {
"line": 63,
"column": 57
} | {
"line": 63,
"column": 71
} | {
"line": 63,
"column": 72
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℝ → E\ns : ℂ\nu : ℝ\n⊢ (cexp (-(↑s.im * I) * ↑u) * cexp (-↑s.re * ↑u)) • f (rexp (-u)) =\n cexp (-↑s.im * ↑u * I) • rexp (-s.re * u) • f (rexp (-u))",
"ppTerm": "?m.240",
"assigned": true,
"usedConstants": [
... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℝ → E\ns : ℂ\nu : ℝ\n⊢ (cexp (-(↑s.im * I) * ↑u) • cexp (-↑s.re * ↑u)) • f (rexp (-u)) =\n cexp (-↑s.im * ↑u * I) • rexp (-s.re * u) • f (rexp (-u))"
] | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.MellinTransform | {
"line": 212,
"column": 4
} | {
"line": 213,
"column": 23
} | {
"line": 214,
"column": 4
} | [
{
"pp": "case refine_1\nf : ℝ → ℝ\nhfc : AEStronglyMeasurable f (volume.restrict (Ioi 0))\na s : ℝ\nhf : f =O[atTop] fun x ↦ x ^ (-a)\nhs : s < a\nd e : ℝ\nhe : ∀ (b : ℝ), e ≤ b → ‖f b‖ ≤ d * ‖b ^ (-a)‖\nhe' : 0 < max e 1\n⊢ AEStronglyMeasurable (fun t ↦ t ^ (s - 1)) (volume.restrict (Ioi (max e 1)))",
"ppT... | [
"case refine_1\nf : ℝ → ℝ\nhfc : AEStronglyMeasurable f (volume.restrict (Ioi 0))\na s : ℝ\nhf : f =O[atTop] fun x ↦ x ^ (-a)\nhs : s < a\nd e : ℝ\nhe : ∀ (b : ℝ), e ≤ b → ‖f b‖ ≤ d * ‖b ^ (-a)‖\nhe' : 0 < max e 1\nt : ℝ\nht : t ∈ Ioi (max e 1)\n⊢ ContinuousAt (fun t ↦ t ^ (s - 1)) t"
] | refine (continuousOn_of_forall_continuousAt fun t ht => ?_).aestronglyMeasurable
measurableSet_Ioi | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.SpecialFunctions.Complex.LogBounds | {
"line": 198,
"column": 6
} | {
"line": 198,
"column": 20
} | {
"line": 198,
"column": 21
} | [
{
"pp": "n : ℕ\nz : ℂ\nhz1 : ‖z‖ < 1\nhz12 : ‖z‖ < 1 / 2\nthis : (1 - ‖z‖)⁻¹ ≤ 2\n⊢ ‖z‖ ^ (n + 1) * (1 - ‖z‖)⁻¹ / (↑n + 1) ≤ 2 / (↑n + 1) * ‖z ^ (n + 1)‖",
"ppTerm": "?m.201",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"instHDiv",
"HMul.hMul",
... | [
"n : ℕ\nz : ℂ\nhz1 : ‖z‖ < 1\nhz12 : ‖z‖ < 1 / 2\nthis : (1 - ‖z‖)⁻¹ ≤ 2\n⊢ ‖z‖ ^ (n + 1) * ((1 - ‖z‖)⁻¹ / (↑n + 1)) ≤ 2 / (↑n + 1) * ‖z ^ (n + 1)‖"
] | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Complex.LogBounds | {
"line": 316,
"column": 22
} | {
"line": 316,
"column": 59
} | {
"line": 316,
"column": 59
} | [
{
"pp": "g : ℝ → ℂ\nt : ℂ\nhg : Tendsto (fun x ↦ ↑x * g x) atTop (𝓝 t)\nx : ℝ\nhx0 : x ≠ 0\n⊢ ↑x * g x ^ 2 = (↑x * g x) ^ 2 * (↑x)⁻¹",
"ppTerm": "?m.229",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Real",
"GroupWithZero.toDivisionMonoid... | [
"g : ℝ → ℂ\nt : ℂ\nhg : Tendsto (fun x ↦ ↑x * g x) atTop (𝓝 t)\nx : ℝ\nhx0 : x ≠ 0\n⊢ ↑x * g x ^ 2 * ↑x = (↑x * g x) ^ 2"
] | eq_mul_inv_iff_mul_eq₀ (mod_cast hx0) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.MellinTransform | {
"line": 329,
"column": 4
} | {
"line": 332,
"column": 99
} | {
"line": 333,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b : ℝ\nf : ℝ → E\ns : ℂ\nhfc : LocallyIntegrableOn f (Ioi 0) volume\nhf_top : f =O[atTop] fun x ↦ x ^ (-a)\nhs_top : s.re < a\nhf_bot : f =O[𝓝[>] 0] fun x ↦ x ^ (-b)\nhs_bot : b < s.re\nF : ℂ → ℝ → E := fun z t ↦ ↑t ^ (z - 1) • f ... | [] | obtain ⟨w, hw1, hw2⟩ := exists_between (sub_pos.mpr hs_top)
obtain ⟨w', hw1', hw2'⟩ := exists_between (sub_pos.mpr hs_bot)
exact
⟨min w w', lt_min hw1 hw1', (min_le_right _ _).trans_lt hw2', (min_le_left _ _).trans_lt hw2⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.MellinTransform | {
"line": 329,
"column": 4
} | {
"line": 332,
"column": 99
} | {
"line": 333,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b : ℝ\nf : ℝ → E\ns : ℂ\nhfc : LocallyIntegrableOn f (Ioi 0) volume\nhf_top : f =O[atTop] fun x ↦ x ^ (-a)\nhs_top : s.re < a\nhf_bot : f =O[𝓝[>] 0] fun x ↦ x ^ (-b)\nhs_bot : b < s.re\nF : ℂ → ℝ → E := fun z t ↦ ↑t ^ (z - 1) • f ... | [] | obtain ⟨w, hw1, hw2⟩ := exists_between (sub_pos.mpr hs_top)
obtain ⟨w', hw1', hw2'⟩ := exists_between (sub_pos.mpr hs_bot)
exact
⟨min w w', lt_min hw1 hw1', (min_le_right _ _).trans_lt hw2', (min_le_left _ _).trans_lt hw2⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.MellinTransform | {
"line": 357,
"column": 26
} | {
"line": 357,
"column": 46
} | {
"line": 357,
"column": 46
} | [
{
"pp": "case hbc.inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b : ℝ\nf : ℝ → E\ns : ℂ\nhfc : LocallyIntegrableOn f (Ioi 0) volume\nhf_top : f =O[atTop] fun x ↦ x ^ (-a)\nhs_top : s.re < a\nhf_bot : f =O[𝓝[>] 0] fun x ↦ x ^ (-b)\nhs_bot : b < s.re\nF : ℂ → ℝ → E := fun z t ↦ ↑t ... | [
"case hbc.inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b : ℝ\nf : ℝ → E\ns : ℂ\nhfc : LocallyIntegrableOn f (Ioi 0) volume\nhf_top : f =O[atTop] fun x ↦ x ^ (-a)\nhs_top : s.re < a\nhf_bot : f =O[𝓝[>] 0] fun x ↦ x ^ (-b)\nhs_bot : b < s.re\nF : ℂ → ℝ → E := fun z t ↦ ↑t ^ (z - 1) • ... | sub_le_sub_iff_right | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Beta | {
"line": 172,
"column": 36
} | {
"line": 172,
"column": 55
} | {
"line": 173,
"column": 6
} | [
{
"pp": "u v : ℂ\nhu : 0 < u.re\nhv : 0 < v.re\nF : ℝ → ℂ := ⋯\nhu' : 0 < (u + 1).re\nhv' : 0 < (v + 1).re\nhc : ContinuousOn F (Icc 0 1)\nx : ℝ\nhx : x ∈ Ioo 0 1\nU : HasDerivAt (fun y ↦ y ^ u) (u * ↑x ^ (u - 1)) ↑x\nA : HasDerivAt (fun x ↦ x ^ v) (v * (1 - ↑x) ^ (v - 1) * 1) (1 - ↑x)\n⊢ HasDerivAt (fun x ↦ x)... | [] | apply hasDerivAt_id | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd | {
"line": 192,
"column": 2
} | {
"line": 238,
"column": 22
} | {
"line": 240,
"column": 0
} | [
{
"pp": "case inr\nz : ℂ\nn : ℕ\nhz : z ≠ 0\n⊢ Complex.sin (↑π * z) =\n ((↑π * z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *\n ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /\n ↑(∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * n))",
"ppTerm": "?inr",
"assigned": true... | [] | induction n with
| zero =>
simp_rw [mul_zero, pow_zero, mul_one, Finset.prod_range_zero, mul_one,
integral_one, sub_zero]
rw [integral_cos_mul_complex (mul_ne_zero two_ne_zero hz), Complex.ofReal_zero,
mul_zero, Complex.sin_zero, zero_div, sub_zero,
(by push_cast; ring : 2 * z * ↑(π / 2) = π... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Analysis.Normed.Affine.AsymptoticCone | {
"line": 70,
"column": 11
} | {
"line": 70,
"column": 25
} | {
"line": 70,
"column": 26
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : FiniteDimensional ℝ V\ns : Set P\nh : asymptoticCone ℝ s ⊆ {0}\n⊢ Bornology.IsBounded s",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": ... | [
"V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : FiniteDimensional ℝ V\ns : Set P\nh : asymptoticCone ℝ s ⊆ {0}\n⊢ sᶜ ∈ cobounded P"
] | isBounded_def, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Algebra.AsymptoticCone | {
"line": 184,
"column": 6
} | {
"line": 184,
"column": 15
} | {
"line": 184,
"column": 15
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝¹² : Field k\ninst✝¹¹ : LinearOrder k\ninst✝¹⁰ : AddCommGroup V\ninst✝⁹ : Module k V\ninst✝⁸ : AddTorsor V P\ninst✝⁷ : TopologicalSpace V\ninst✝⁶ : TopologicalSpace k\ninst✝⁵ : OrderTopology k\ninst✝⁴ : IsStrictOrderedRing k\ninst✝³ : IsTopologicalAddGroup... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝¹² : Field k\ninst✝¹¹ : LinearOrder k\ninst✝¹⁰ : AddCommGroup V\ninst✝⁹ : Module k V\ninst✝⁸ : AddTorsor V P\ninst✝⁷ : TopologicalSpace V\ninst✝⁶ : TopologicalSpace k\ninst✝⁵ : OrderTopology k\ninst✝⁴ : IsStrictOrderedRing k\ninst✝³ : IsTopologicalAddGroup V\ninst✝² :... | vadd_vsub | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Beta | {
"line": 284,
"column": 4
} | {
"line": 286,
"column": 82
} | {
"line": 287,
"column": 4
} | [
{
"pp": "s : ℂ\nhs : 0 < s.re\nf : ℕ → ℝ → ℂ := fun n ↦ (Ioc 0 ↑n).indicator fun x ↦ ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)\nn : ℕ\n⊢ Integrable (f n) (volume.restrict (Ioi 0))",
"ppTerm": "?m.104",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Ioc",
"NormedCommRing.toSeminormedC... | [
"s : ℂ\nhs : 0 < s.re\nf : ℕ → ℝ → ℂ := fun n ↦ (Ioc 0 ↑n).indicator fun x ↦ ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)\nn : ℕ\n⊢ IntervalIntegrable (fun x ↦ ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)) volume 0 ↑n"
] | rw [integrable_indicator_iff (measurableSet_Ioc : MeasurableSet (Ioc (_ : ℝ) _)), IntegrableOn,
Measure.restrict_restrict_of_subset Ioc_subset_Ioi_self, ← IntegrableOn, ←
intervalIntegrable_iff_integrableOn_Ioc_of_le (by positivity : (0 : ℝ) ≤ n)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.AsymptoticCone | {
"line": 381,
"column": 59
} | {
"line": 381,
"column": 68
} | {
"line": 381,
"column": 68
} | [
{
"pp": "k : Type u_1\nV : Type u_2\ninst✝⁹ : Field k\ninst✝⁸ : LinearOrder k\ninst✝⁷ : IsStrictOrderedRing k\ninst✝⁶ : TopologicalSpace k\ninst✝⁵ : OrderTopology k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : TopologicalSpace V\ninst✝¹ : IsTopologicalAddGroup V\ninst✝ : ContinuousSMul k V\ns : Set V... | [
"k : Type u_1\nV : Type u_2\ninst✝⁹ : Field k\ninst✝⁸ : LinearOrder k\ninst✝⁷ : IsStrictOrderedRing k\ninst✝⁶ : TopologicalSpace k\ninst✝⁵ : OrderTopology k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : TopologicalSpace V\ninst✝¹ : IsTopologicalAddGroup V\ninst✝ : ContinuousSMul k V\ns : Set V\nc : k\nv p... | vadd_vsub | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.AsymptoticCone | {
"line": 382,
"column": 2
} | {
"line": 382,
"column": 65
} | {
"line": 384,
"column": 0
} | [
{
"pp": "k : Type u_1\nV : Type u_2\ninst✝⁹ : Field k\ninst✝⁸ : LinearOrder k\ninst✝⁷ : IsStrictOrderedRing k\ninst✝⁶ : TopologicalSpace k\ninst✝⁵ : OrderTopology k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : TopologicalSpace V\ninst✝¹ : IsTopologicalAddGroup V\ninst✝ : ContinuousSMul k V\ns : Set V... | [] | exact (SameRay.sameRay_nonneg_smul_left _ hc).pos_smul_right ht | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Affine.MazurUlam | {
"line": 59,
"column": 48
} | {
"line": 59,
"column": 51
} | {
"line": 59,
"column": 52
} | [
{
"pp": "E : Type u_1\nPE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : MetricSpace PE\ninst✝ : NormedAddTorsor E PE\nx y : PE\nz : PE := midpoint ℝ x y\ns : Set (PE ≃ᵢ PE) := {e | e x = x ∧ e y = y}\nthis : Nonempty ↑s\ne : PE ≃ᵢ PE\nhx : e x = x\nright✝ : e y = y\n⊢ dist (e z) ... | [
"E : Type u_1\nPE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : MetricSpace PE\ninst✝ : NormedAddTorsor E PE\nx y : PE\nz : PE := midpoint ℝ x y\ns : Set (PE ≃ᵢ PE) := {e | e x = x ∧ e y = y}\nthis : Nonempty ↑s\ne : PE ≃ᵢ PE\nhx : e x = x\nright✝ : e y = y\n⊢ dist (e z) x + dist x z... | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Beta | {
"line": 405,
"column": 30
} | {
"line": 405,
"column": 71
} | {
"line": 405,
"column": 72
} | [
{
"pp": "case pos\nz : ℂ\npi_ne : ↑π ≠ 0\nk : ℤ\nhk : -z = ↑k ∨ ↑π = 0\n⊢ Gamma z * Gamma (1 - z) = 0",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Int.cast",
"False",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.pi",
... | [
"case pos\nz : ℂ\npi_ne : ↑π ≠ 0\nk : ℤ\nhk : -z = ↑k ∨ False\n⊢ Gamma z * Gamma (1 - z) = 0"
] | eq_false (ofReal_ne_zero.mpr pi_pos.ne'), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Series | {
"line": 80,
"column": 66
} | {
"line": 80,
"column": 80
} | {
"line": 80,
"column": 81
} | [
{
"pp": "z : ℂ\n⊢ (fun n ↦ (-1) ^ n * (z ^ 2) ^ n * z / ↑(2 * n + 1)!) = fun n ↦ (z ^ 2) ^ n * z * (-1) ^ n * I / ↑(2 * n + 1)! / I",
"ppTerm": "?m.118",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
... | [
"z : ℂ\n⊢ (fun n ↦ (-1) ^ n * (z ^ 2) ^ n * (z / ↑(2 * n + 1)!)) = fun n ↦ (z ^ 2) ^ n * z * (-1) ^ n * (I / ↑(2 * n + 1)! / I)"
] | mul_div_assoc, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Series | {
"line": 81,
"column": 72
} | {
"line": 81,
"column": 86
} | {
"line": 81,
"column": 87
} | [
{
"pp": "z : ℂ\n⊢ (fun n ↦ (-1) ^ n * (z ^ 2) ^ n * (z / ↑(2 * n + 1)!)) = fun n ↦ (-1) ^ n * ((z ^ 2) ^ n * z) / ↑(2 * n + 1)!",
"ppTerm": "?m.123",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"Semigroup.toMul",
... | [
"z : ℂ\n⊢ (fun n ↦ (-1) ^ n * (z ^ 2) ^ n * (z / ↑(2 * n + 1)!)) = fun n ↦ (-1) ^ n * ((z ^ 2) ^ n * (z / ↑(2 * n + 1)!))"
] | mul_div_assoc, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Normed.Algebra.QuaternionExponential | {
"line": 78,
"column": 23
} | {
"line": 78,
"column": 37
} | {
"line": 78,
"column": 38
} | [
{
"pp": "case inr.calc_2.e_a\nq : ℍ\nhq : q.re = 0\nn : ℕ\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(normSq q)\nhqn : ‖q‖ ≠ 0\nk : ℝ := ↑(2 * n + 1)!\n⊢ k⁻¹ * ((-1) ^ n * ‖q‖ ^ (2 * n)) = (-1) ^ n * (‖q‖ ^ (2 * n) * ‖q‖) / k / ‖q‖",
"ppTerm": "?inr.calc_2.e_a",
"assigned": true,
"usedConstants": [
"Norm.n... | [
"case inr.calc_2.e_a\nq : ℍ\nhq : q.re = 0\nn : ℕ\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(normSq q)\nhqn : ‖q‖ ≠ 0\nk : ℝ := ↑(2 * n + 1)!\n⊢ k⁻¹ * ((-1) ^ n * ‖q‖ ^ (2 * n)) = (-1) ^ n * (‖q‖ ^ (2 * n) * (‖q‖ / k / ‖q‖))"
] | mul_div_assoc, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded | {
"line": 99,
"column": 6
} | {
"line": 99,
"column": 26
} | {
"line": 100,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx : R\nhx : 0 < f x\n⊢ 0 ≤ c",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"Real",
"Real.instZero",
"AddGroupWithOne.toAddMonoidWithOne",
... | [
"R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nx : R\nf_mul : f (x * 1) ≤ c * f x * f 1\nhx : 0 < f x\n⊢ 0 ≤ c"
] | specialize f_mul x 1 | Lean.Elab.Tactic.evalSpecialize | Lean.Parser.Tactic.specialize |
Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded | {
"line": 103,
"column": 38
} | {
"line": 103,
"column": 60
} | {
"line": 103,
"column": 60
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nx : R\nf_mul : 1 ≤ c * f 1\nhx : 0 < f x\nf_nonneg : 0 ≤ f 1\nh1 : f 1 = 0\n⊢ 1 ≤ 0",
"ppTerm": "?m.161",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Real",
"HMul.hMul",
"Real.instZero",
"congrArg",
... | [] | simpa [h1] using f_mul | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded | {
"line": 103,
"column": 38
} | {
"line": 103,
"column": 60
} | {
"line": 103,
"column": 60
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nx : R\nf_mul : 1 ≤ c * f 1\nhx : 0 < f x\nf_nonneg : 0 ≤ f 1\nh1 : f 1 = 0\n⊢ 1 ≤ 0",
"ppTerm": "?m.161",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Real",
"HMul.hMul",
"Real.instZero",
"congrArg",
... | [] | simpa [h1] using f_mul | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded | {
"line": 103,
"column": 38
} | {
"line": 103,
"column": 60
} | {
"line": 103,
"column": 60
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nx : R\nf_mul : 1 ≤ c * f 1\nhx : 0 < f x\nf_nonneg : 0 ≤ f 1\nh1 : f 1 = 0\n⊢ 1 ≤ 0",
"ppTerm": "?m.161",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Real",
"HMul.hMul",
"Real.instZero",
"congrArg",
... | [] | simpa [h1] using f_mul | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded | {
"line": 136,
"column": 4
} | {
"line": 136,
"column": 24
} | {
"line": 137,
"column": 4
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx : R\nh1 : f 1 = 0\n⊢ f x ≤ f 1 * seminormFromBounded' f x",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Real",
"Real.instZero",
"A... | [
"case pos\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nx : R\nf_mul : f (x * 1) ≤ c * f x * f 1\nh1 : f 1 = 0\n⊢ f x ≤ f 1 * seminormFromBounded' f x"
] | specialize f_mul x 1 | Lean.Elab.Tactic.evalSpecialize | Lean.Parser.Tactic.specialize |
Mathlib.Analysis.Normed.Unbundled.RingSeminorm | {
"line": 417,
"column": 84
} | {
"line": 417,
"column": 87
} | {
"line": 418,
"column": 16
} | [
{
"pp": "R : Type u_1\nK : Type u_2\ninst✝ : Field K\nf : RingSeminorm K\nhnt : f ≠ 0\nx : K\nhx : f.toFun x = 0\nc : K\nhc : f c ≠ 0\nhn0 : ¬x = 0\n⊢ f.toFun x * f.toFun (c * x⁻¹) ≤ 0",
"ppTerm": "?m.97",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"Real",
... | [
"R : Type u_1\nK : Type u_2\ninst✝ : Field K\nf : RingSeminorm K\nhnt : f ≠ 0\nx : K\nhx : f.toFun x = 0\nc : K\nhc : f c ≠ 0\nhn0 : ¬x = 0\n⊢ 0 * f.toFun (c * x⁻¹) ≤ 0"
] | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded | {
"line": 213,
"column": 10
} | {
"line": 213,
"column": 13
} | {
"line": 213,
"column": 14
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_ne_zero : f ≠ 0\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx : R\nhx : f x = 0\n⊢ f x / f x ≤ 1",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
... | [
"case pos\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_ne_zero : f ≠ 0\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx : R\nhx : f x = 0\n⊢ 0 / 0 ≤ 1"
] | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded | {
"line": 284,
"column": 33
} | {
"line": 284,
"column": 36
} | {
"line": 284,
"column": 37
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx : R\nhx : ∀ (y : R), f (x * y) = f x * f y\n⊢ ⨆ y, f (x * y) / f y = f x",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"ins... | [
"R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx : R\nhx : ∀ (y : R), f (x * y) = f x * f y\n⊢ ⨆ y, f x * f y / f y = f x"
] | hx, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded | {
"line": 288,
"column": 10
} | {
"line": 288,
"column": 13
} | {
"line": 288,
"column": 14
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx✝ : R\nhx✝ : ∀ (y : R), f (x✝ * y) = f x✝ * f y\nx : R\nhx : f x = 0\n⊢ f x✝ * (f x / f x) ≤ f x✝",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
... | [
"case pos\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx✝ : R\nhx✝ : ∀ (y : R), f (x✝ * y) = f x✝ * f y\nx : R\nhx : f x = 0\n⊢ f x✝ * (0 / 0) ≤ f x✝"
] | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Unbundled.InvariantExtension | {
"line": 87,
"column": 2
} | {
"line": 88,
"column": 44
} | {
"line": 90,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝² : NormedField K\nL : Type u_2\ninst✝¹ : Field L\ninst✝ : Algebra K L\nh_fin : FiniteDimensional K L\nhu : IsUltrametricDist K\nσ : L ≃ₐ[K] L\nx y : L\n⊢ (Classical.choose ⋯) (σ x + σ y) ≤ max ((Classical.choose ⋯) (σ x)) ((Classical.choose ⋯) (σ y))",
"ppTerm": "?m.29",
"as... | [] | exact (Classical.choose_spec (exists_nonarchimedean_pow_mul_seminorm_of_finiteDimensional
h_fin hu.isNonarchimedean_norm)).2.2 _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded | {
"line": 319,
"column": 58
} | {
"line": 319,
"column": 72
} | {
"line": 319,
"column": 73
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nf_nonneg : 0 ≤ f\nx : R\nhx : ∀ (y : R), f (x * y) ≤ f x * f y\nh_one : f 1 ≤ 1\ny : R\nhy0 : ¬f y = 0\n⊢ f (x * y) ≤ f x * f y / f y * f y",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.t... | [
"case neg\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nf_nonneg : 0 ≤ f\nx : R\nhx : ∀ (y : R), f (x * y) ≤ f x * f y\nh_one : f 1 ≤ 1\ny : R\nhy0 : ¬f y = 0\n⊢ f (x * y) ≤ f x * (f y / f y) * f y"
] | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Unbundled.IsPowMulFaithful | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 58
} | {
"line": 94,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : NormedCommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nf₁ : AlgebraNorm R S\nhf₁_pm : IsPowMul ⇑f₁\nf₂ : AlgebraNorm R S\nhf₂_pm : IsPowMul ⇑f₂\nh_eq : ∀ (y : S), ∃ C₁ C₂, ∃ (_ : 0 < C₁) (_ : 0 < C₂), ∀ (x : ↥R[y]), f₁ ↑x ≤ C₁ * f₂ ↑x ∧ f₂ ↑x ≤ C₂ * f₁ ↑x\nx : ... | [
"R : Type u_1\nS : Type u_2\ninst✝² : NormedCommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nf₁ : AlgebraNorm R S\nhf₁_pm : IsPowMul ⇑f₁\nf₂ : AlgebraNorm R S\nhf₂_pm : IsPowMul ⇑f₂\nh_eq : ∀ (y : S), ∃ C₁ C₂, ∃ (_ : 0 < C₁) (_ : 0 < C₂), ∀ (x : ↥R[y]), f₁ ↑x ≤ C₁ * f₂ ↑x ∧ f₂ ↑x ≤ C₂ * f₁ ↑x\nx : S\ng₁ : Alge... | let y : R[(x : S)] := ⟨x, self_mem_adjoin_singleton R x⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.Normed.Unbundled.SeminormFromConst | {
"line": 107,
"column": 18
} | {
"line": 107,
"column": 32
} | {
"line": 107,
"column": 33
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝ : CommRing R\nc : R\nf : RingSeminorm R\nhf1 : f 1 ≤ 1\nhc : f c ≠ 0\nhpm : IsPowMul ⇑f\nx : R\nm n : ℕ\nhmn : m ≤ n\nhc_pos : 0 < f c\nhlt : m < n\nh1 : 1 ≤ n - m\n⊢ f (x * c ^ m) * f c ^ (n - m) / f c ^ n ≤ f (x * c ^ m) / f c ^ m",
"ppTerm": "?inr",
"assigned": ... | [
"case inr\nR : Type u_1\ninst✝ : CommRing R\nc : R\nf : RingSeminorm R\nhf1 : f 1 ≤ 1\nhc : f c ≠ 0\nhpm : IsPowMul ⇑f\nx : R\nm n : ℕ\nhmn : m ≤ n\nhc_pos : 0 < f c\nhlt : m < n\nh1 : 1 ≤ n - m\n⊢ f (x * c ^ m) * (f c ^ (n - m) / f c ^ n) ≤ f (x * c ^ m) / f c ^ m"
] | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Unbundled.SeminormFromConst | {
"line": 240,
"column": 71
} | {
"line": 240,
"column": 85
} | {
"line": 241,
"column": 8
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nc : R\nf : RingSeminorm R\nhf1 : f 1 ≤ 1\nhc : f c ≠ 0\nhpm : IsPowMul ⇑f\nn : ℕ\n⊢ f c * f c ^ n / f c ^ n = f c",
"ppTerm": "?m.87",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
... | [
"R : Type u_1\ninst✝ : CommRing R\nc : R\nf : RingSeminorm R\nhf1 : f 1 ≤ 1\nhc : f c ≠ 0\nhpm : IsPowMul ⇑f\nn : ℕ\n⊢ f c * (f c ^ n / f c ^ n) = f c"
] | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 205,
"column": 8
} | {
"line": 205,
"column": 22
} | {
"line": 205,
"column": 23
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\nx : R\nhx : μ x ≠ 0\nL : ℝ := ⨅ n, μ (x ^ ↑n) ^ (1 / ↑↑n)\nhL0 : 0 ≤ L\nε : ℝ\nhε : ε > 0\nm1 : ℕ+\nhm1 : μ (x ^ ↑m1) < (L + ε / 2) ^ ↑↑m1\nm2 : ℕ\nhm2 : ∀ n ≥ m2, (L + ε / 2) ^ (-(↑(n % ↑m1) / ↑n)) * (μ x ^ (n % ↑m1)) ^ (1 / ↑n) - 1 ... | [
"R : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\nx : R\nhx : μ x ≠ 0\nL : ℝ := ⨅ n, μ (x ^ ↑n) ^ (1 / ↑↑n)\nhL0 : 0 ≤ L\nε : ℝ\nhε : ε > 0\nm1 : ℕ+\nhm1 : μ (x ^ ↑m1) < (L + ε / 2) ^ ↑↑m1\nm2 : ℕ\nhm2 : ∀ n ≥ m2, (L + ε / 2) ^ (-(↑(n % ↑m1) / ↑n)) * (μ x ^ (n % ↑m1)) ^ (1 / ↑n) - 1 ≤ ε / (2 * (... | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 254,
"column": 8
} | {
"line": 254,
"column": 68
} | {
"line": 255,
"column": 6
} | [
{
"pp": "K : Type u_2\ninst✝² : NormedField K\nL : Type u_3\ninst✝¹ : Field L\ninst✝ : Algebra K L\nf : AlgebraNorm K L\nhf_pm : IsPowMul ⇑f\nhf_na : IsNonarchimedean ⇑f\np : K[X]\nhp : p.Monic\nx : L\nhx : (aeval x) p = 0\nhx0 : ¬f x = 0\nh_ge : ∀ x_1 ∈ Set.range (spectralValueTerms p), x_1 < f x\nn : ℕ\nhn : ... | [] | exact h_ge (‖p.coeff n‖₊ ^ (1 / (p.natDegree - n : ℝ))) h_rg | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Group.HomCompletion | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 84
} | {
"line": 99,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝ : SeminormedAddCommGroup G\nx : Completion G\n⊢ (id G).completion x = (id (Completion G)) x",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"NormedAddGroupHom.completion_def",
"UniformSpace.Completion.map",
"Eq.mpr",
"NormedAddGroupHom",... | [
"G : Type u_1\ninst✝ : SeminormedAddCommGroup G\nx : Completion G\n⊢ _root_.id x = (id (Completion G)) x"
] | rw [NormedAddGroupHom.completion_def, NormedAddGroupHom.coe_id, Completion.map_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Group.SemiNormedGrp | {
"line": 166,
"column": 52
} | {
"line": 166,
"column": 67
} | {
"line": 166,
"column": 68
} | [
{
"pp": "V W : SemiNormedGrp\ni : V ≅ W\nh1 : (Hom.hom i.hom).NormNoninc\nh2 : (Hom.hom i.inv).NormNoninc\nv : V.carrier\n⊢ ‖v‖ = ‖(ConcreteCategory.hom (i.hom ≫ i.inv)) v‖",
"ppTerm": "?m.84",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"NormedAddGrou... | [
"V W : SemiNormedGrp\ni : V ≅ W\nh1 : (Hom.hom i.hom).NormNoninc\nh2 : (Hom.hom i.inv).NormNoninc\nv : V.carrier\n⊢ ‖v‖ = ‖(ConcreteCategory.hom (𝟙 V)) v‖"
] | Iso.hom_inv_id, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Group.SemiNormedGrp | {
"line": 384,
"column": 52
} | {
"line": 384,
"column": 67
} | {
"line": 384,
"column": 68
} | [
{
"pp": "V W : SemiNormedGrp₁\ni : V ≅ W\nv : V.carrier\n⊢ ‖v‖ = ‖↑(Hom.hom (i.hom ≫ i.inv)) v‖",
"ppTerm": "?m.117",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"NormedAddGroupHom",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
... | [
"V W : SemiNormedGrp₁\ni : V ≅ W\nv : V.carrier\n⊢ ‖v‖ = ‖↑(Hom.hom (𝟙 V)) v‖"
] | Iso.hom_inv_id, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 605,
"column": 4
} | {
"line": 605,
"column": 83
} | {
"line": 606,
"column": 2
} | [
{
"pp": "K : Type u_2\ninst✝³ : NormedField K\nL : Type u_3\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : IsUltrametricDist K\nk : K\ny : L\nhy : IsAlgebraic K y\nE : IntermediateField K L := K⟮y⟯\nh_finiteDimensional_E : FiniteDimensional K ↥E\ng : ↥K⟮y⟯ := AdjoinSimple.gen K y\nhgy : k • y = (algebraMap (↥... | [] | rw [Algebra.algebraMap_eq_smul_one, Algebra.algebraMap_eq_smul_one, smul_assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Lp.Finsupp | {
"line": 52,
"column": 4
} | {
"line": 60,
"column": 27
} | {
"line": 62,
"column": 0
} | [
{
"pp": "ι : Type u_1\nX : Type u_2\ninst✝² : Zero X\np : ℝ≥0\ninst✝¹ : Fact (1 ≤ p)\ninst✝ : PseudoEMetricSpace X\nf g h : WithLp (↑p) (ι →₀ X)\n⊢ ((zipWith edist ⋯ f.ofLp h.ofLp).sum fun i r ↦ r ^ ↑p) ^ (↑p)⁻¹ ≤\n ((zipWith edist ⋯ f.ofLp g.ofLp).sum fun i r ↦ r ^ ↑p) ^ (↑p)⁻¹ +\n ((zipWith edist ⋯ g.... | [] | have : 0 < p := zero_lt_one.trans_le Fact.out
let s := f.ofLp.support ∪ g.ofLp.support ∪ h.ofLp.support
rw [sum_of_support_subset (s := s) _ (by grind [support_zipWith]) _ (by simp [*]),
sum_of_support_subset (s := s) _ (by grind [support_zipWith]) _ (by simp [*]),
sum_of_support_subset (s := s) _ (... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Lp.Finsupp | {
"line": 52,
"column": 4
} | {
"line": 60,
"column": 27
} | {
"line": 62,
"column": 0
} | [
{
"pp": "ι : Type u_1\nX : Type u_2\ninst✝² : Zero X\np : ℝ≥0\ninst✝¹ : Fact (1 ≤ p)\ninst✝ : PseudoEMetricSpace X\nf g h : WithLp (↑p) (ι →₀ X)\n⊢ ((zipWith edist ⋯ f.ofLp h.ofLp).sum fun i r ↦ r ^ ↑p) ^ (↑p)⁻¹ ≤\n ((zipWith edist ⋯ f.ofLp g.ofLp).sum fun i r ↦ r ^ ↑p) ^ (↑p)⁻¹ +\n ((zipWith edist ⋯ g.... | [] | have : 0 < p := zero_lt_one.trans_le Fact.out
let s := f.ofLp.support ∪ g.ofLp.support ∪ h.ofLp.support
rw [sum_of_support_subset (s := s) _ (by grind [support_zipWith]) _ (by simp [*]),
sum_of_support_subset (s := s) _ (by grind [support_zipWith]) _ (by simp [*]),
sum_of_support_subset (s := s) _ (... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Module.Bases | {
"line": 227,
"column": 2
} | {
"line": 227,
"column": 14
} | {
"line": 229,
"column": 2
} | [
{
"pp": "case h\n𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\nX : Type u_2\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\nβ : Type u_3\nb : UnconditionalSchauderBasis β 𝕜 X\ninst✝ : CompleteSpace X\nx : X\nA₀ : Finset β\nhA₀ : ∀ (t : Finset β), Disjoint t A₀ → ‖∑ i ∈ t, (b.coord i) x • ↑b i‖... | [
"case h\n𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\nX : Type u_2\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\nβ : Type u_3\nb : UnconditionalSchauderBasis β 𝕜 X\ninst✝ : CompleteSpace X\nx : X\nA₀ : Finset β\nhA₀ : ∀ (t : Finset β), Disjoint t A₀ → ‖∑ i ∈ t, (b.coord i) x • ↑b i‖ < 1\nA : Fi... | rw [hdecomp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 104,
"column": 52
} | {
"line": 125,
"column": 65
} | {
"line": 126,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝³ : NormedAddCommGroup X\nM : Type u_2\ninst✝² : Ring M\ninst✝¹ : Module M X\ninst✝ : FaithfulSMul M X\nP Q : M\nh₁ : IsLprojection X P\nh₂ : IsLprojection X Q\nR : M\nh₃ : IsLprojection X R\nx : X\n⊢ (P * R) • x = (R * P * R) • x",
"ppTerm": "?m.44",
"assigned": true,
"u... | [] | by
rw [← norm_sub_eq_zero_iff]
have e1 : ‖R • x‖ ≥ ‖R • x‖ + 2 • ‖(P * R) • x - (R * P * R) • x‖ :=
calc
‖R • x‖ = ‖R • P • R • x‖ + ‖(1 - R) • P • R • x‖ +
(‖(R * R) • x - R • P • R • x‖ + ‖(1 - R) • (1 - P) • R • x‖) := by
rw [h₁.Lnorm, h₃.Lnorm, h₃.Lnorm ((1 - ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 142,
"column": 47
} | {
"line": 142,
"column": 67
} | {
"line": 143,
"column": 6
} | [
{
"pp": "X : Type u_1\ninst✝³ : NormedAddCommGroup X\nM : Type u_2\ninst✝² : Ring M\ninst✝¹ : Module M X\ninst✝ : FaithfulSMul M X\nP Q : M\nh₁ : IsLprojection X P\nh₂ : IsLprojection X Q\nx : X\n⊢ ‖(P * Q) • x + (x - (P * Q) • x)‖ ≤ ‖(P * Q) • x‖ + ‖x - (P * Q) • x‖",
"ppTerm": "?m.122",
"assigned": tr... | [] | by apply norm_add_le | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 242,
"column": 50
} | {
"line": 242,
"column": 78
} | {
"line": 243,
"column": 4
} | [
{
"pp": "X : Type u_1\ninst✝³ : NormedAddCommGroup X\nM : Type u_2\ninst✝² : Ring M\ninst✝¹ : Module M X\ninst✝ : FaithfulSMul M X\nP Q R : { P // IsLprojection X P }\n⊢ ↑P * ↑P + ↑P * (↑(Q ⊓ R) * ↑Pᶜ) + (↑Pᶜ * ↑R * ↑P + ↑Pᶜ * (↑R * ↑(Q ⊓ R) * ↑Pᶜ)) = ↑P + ↑(Q ⊓ R) * ↑Pᶜ",
"ppTerm": "?m.144",
"assigned"... | [
"X : Type u_1\ninst✝³ : NormedAddCommGroup X\nM : Type u_2\ninst✝² : Ring M\ninst✝¹ : Module M X\ninst✝ : FaithfulSMul M X\nP Q R : { P // IsLprojection X P }\n⊢ ↑P * ↑P + ↑P * (↑(Q ⊓ R) * ↑Pᶜ) + (↑R * ↑Pᶜ * ↑P + ↑Pᶜ * (↑R * ↑(Q ⊓ R) * ↑Pᶜ)) = ↑P + ↑(Q ⊓ R) * ↑Pᶜ"
] | (Pᶜ.prop.commute R.prop).eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 283,
"column": 19
} | {
"line": 283,
"column": 47
} | {
"line": 283,
"column": 48
} | [
{
"pp": "X : Type u_1\ninst✝³ : NormedAddCommGroup X\nM : Type u_2\ninst✝² : Ring M\ninst✝¹ : Module M X\ninst✝ : FaithfulSMul M X\nP Q R : { P // IsLprojection X P }\n⊢ ↑P + ↑Q * (↑Pᶜ * ↑R) = ↑P + ↑Q * ↑R * ↑Pᶜ",
"ppTerm": "?m.241",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsLpro... | [
"X : Type u_1\ninst✝³ : NormedAddCommGroup X\nM : Type u_2\ninst✝² : Ring M\ninst✝¹ : Module M X\ninst✝ : FaithfulSMul M X\nP Q R : { P // IsLprojection X P }\n⊢ ↑P + ↑Q * (↑R * ↑Pᶜ) = ↑P + ↑Q * ↑R * ↑Pᶜ"
] | (Pᶜ.prop.commute R.prop).eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Log.Summable | {
"line": 151,
"column": 28
} | {
"line": 151,
"column": 48
} | {
"line": 151,
"column": 48
} | [
{
"pp": "case insert\nι : Type u_1\nR : Type u_2\ninst✝¹ : NormedCommRing R\ninst✝ : NormOneClass R\nf : ι → R\nx : ι\nt : Finset ι\nhx : x ∉ t\nA : ℝ\nh : Real.exp (∑ i ∈ t, ‖f i‖) = A\nIH : ‖∏ i ∈ t, (1 + f i) - 1‖ ≤ A - 1\n⊢ A + ‖f x‖ * ‖∏ x ∈ t, (1 + f x)‖ - 1 ≤ Real.exp ‖f x‖ * A - 1",
"ppTerm": "?inse... | [
"case insert\nι : Type u_1\nR : Type u_2\ninst✝¹ : NormedCommRing R\ninst✝ : NormOneClass R\nf : ι → R\nx : ι\nt : Finset ι\nhx : x ∉ t\nA : ℝ\nh : Real.exp (∑ i ∈ t, ‖f i‖) = A\nIH : ‖∏ i ∈ t, (1 + f i) - 1‖ ≤ A - 1\n⊢ A + ‖f x‖ * ‖∏ x ∈ t, (1 + f x)‖ ≤ Real.exp ‖f x‖ * A"
] | sub_le_sub_iff_right | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Lp.lpHolder | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 26
} | {
"line": 122,
"column": 2
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\nE : ι → Type u_3\nF : ι → Type u_4\nG : ι → Type u_5\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝³ : (i : ι) → NormedAddCommGroup (F i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (F i)\ninst✝¹ : (i : ι) → NormedAdd... | [
"ι : Type u_1\n𝕜 : Type u_2\nE : ι → Type u_3\nF : ι → Type u_4\nG : ι → Type u_5\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝³ : (i : ι) → NormedAddCommGroup (F i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (F i)\ninst✝¹ : (i : ι) → NormedAddCommGroup (G... | grw [← hDf s, s.mul_sum] | Mathlib.Tactic.GRewrite._aux_Mathlib_Tactic_GRewrite_Elab___macroRules_Mathlib_Tactic_GRewrite_grwSeq_1 | Mathlib.Tactic.GRewrite.grwSeq |
Mathlib.Analysis.Normed.Module.PiTensorProduct.InjectiveSeminorm | {
"line": 87,
"column": 61
} | {
"line": 91,
"column": 74
} | {
"line": 93,
"column": 0
} | [
{
"pp": "ι : Type uι\ninst✝³ : Fintype ι\n𝕜 : Type u𝕜\ninst✝² : NontriviallyNormedField 𝕜\nE : ι → Type uE\ninst✝¹ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E i)\n⊢ BddAbove\n {p |\n ∃ G x x_1, p = (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (to... | [] | by
use projectiveSeminorm
simp only [mem_upperBounds, Set.mem_setOf_eq, forall_exists_index]
intro p G _ _ hp x
simpa [hp] using! toDualContinuousMultilinearMap_le_projectiveSeminorm _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Module.PiTensorProduct.InjectiveSeminorm | {
"line": 123,
"column": 2
} | {
"line": 123,
"column": 66
} | {
"line": 124,
"column": 2
} | [
{
"pp": "ι : Type uι\ninst✝⁵ : Fintype ι\n𝕜 : Type u𝕜\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : ι → Type uE\ninst✝³ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (E i)\nF : Type u_1\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : ContinuousMultilinearMap 𝕜 E F\... | [
"ι : Type uι\ninst✝⁵ : Fintype ι\n𝕜 : Type u𝕜\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : ι → Type uE\ninst✝³ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (E i)\nF : Type u_1\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : ContinuousMultilinearMap 𝕜 E F\nx : ⨂[𝕜] (... | set G := (⨂[𝕜] i, E i) ⧸ LinearMap.ker (lift f.toMultilinearMap) | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.Analysis.Normed.Order.UpperLower | {
"line": 102,
"column": 2
} | {
"line": 116,
"column": 50
} | {
"line": 118,
"column": 0
} | [
{
"pp": "ι : Type u_2\ninst✝ : Finite ι\ns : Set (ι → ℝ)\nx y : ι → ℝ\nhs : IsLowerSet s\nhx : x ∈ closure s\nh : ∀ (i : ι), y i < x i\n⊢ y ∈ interior s",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"Iff.mpr",
"Norm... | [] | cases nonempty_fintype ι
obtain ⟨ε, hε, hxy⟩ := Pi.exists_forall_pos_add_lt h
obtain ⟨z, hz, hxz⟩ := Metric.mem_closure_iff.1 hx _ hε
rw [dist_pi_lt_iff hε] at hxz
have hyz : ∀ i, y i < z i := by
refine fun i =>
(lt_sub_iff_add_lt.2 <| hxy _).trans_le (sub_le_comm.1 <| (le_abs_self _).trans ?_)
rw... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Order.UpperLower | {
"line": 102,
"column": 2
} | {
"line": 116,
"column": 50
} | {
"line": 118,
"column": 0
} | [
{
"pp": "ι : Type u_2\ninst✝ : Finite ι\ns : Set (ι → ℝ)\nx y : ι → ℝ\nhs : IsLowerSet s\nhx : x ∈ closure s\nh : ∀ (i : ι), y i < x i\n⊢ y ∈ interior s",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"Iff.mpr",
"Norm... | [] | cases nonempty_fintype ι
obtain ⟨ε, hε, hxy⟩ := Pi.exists_forall_pos_add_lt h
obtain ⟨z, hz, hxz⟩ := Metric.mem_closure_iff.1 hx _ hε
rw [dist_pi_lt_iff hε] at hxz
have hyz : ∀ i, y i < z i := by
refine fun i =>
(lt_sub_iff_add_lt.2 <| hxy _).trans_le (sub_le_comm.1 <| (le_abs_self _).trans ?_)
rw... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Contracting | {
"line": 302,
"column": 2
} | {
"line": 302,
"column": 60
} | {
"line": 303,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝² : MetricSpace α\nK : ℝ≥0\nf : α → α\nhf : ContractingWith K f\ninst✝¹ : Nonempty α\ninst✝ : CompleteSpace α\nx : α\n⊢ Tendsto (fun n ↦ f^[n] x) atTop (𝓝 (fixedPoint f hf))",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HEq.refl",
... | [
"α : Type u_1\ninst✝² : MetricSpace α\nK : ℝ≥0\nf : α → α\nhf : ContractingWith K f\ninst✝¹ : Nonempty α\ninst✝ : CompleteSpace α\nx : α\n⊢ fixedPoint f hf = efixedPoint f hf x ⋯"
] | convert! tendsto_iterate_efixedPoint hf (edist_ne_top x _) | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Topology.MetricSpace.Contracting | {
"line": 321,
"column": 28
} | {
"line": 321,
"column": 48
} | {
"line": 321,
"column": 49
} | [
{
"pp": "α : Type u_1\ninst✝² : MetricSpace α\nK : ℝ≥0\nf : α → α\ninst✝¹ : Nonempty α\ninst✝ : CompleteSpace α\nn : ℕ\nhf : ContractingWith K f^[n]\nx : α := fixedPoint f^[n] hf\nhx : f^[n] x = x\nthis : dist (f^[n] x) (f^[n.succ] x) ≤ ↑K * dist x (f x)\n⊢ IsFixedPt f x",
"ppTerm": "?m.58",
"assigned":... | [
"α : Type u_1\ninst✝² : MetricSpace α\nK : ℝ≥0\nf : α → α\ninst✝¹ : Nonempty α\ninst✝ : CompleteSpace α\nn : ℕ\nhf : ContractingWith K f^[n]\nx : α := fixedPoint f^[n] hf\nhx : f^[n] x = x\nthis : dist (f^[n] x) (f (f^[n] x)) ≤ ↑K * dist x (f x)\n⊢ IsFixedPt f x"
] | iterate_succ_apply', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.ODE.PicardLindelof | {
"line": 325,
"column": 32
} | {
"line": 325,
"column": 46
} | {
"line": 325,
"column": 47
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E → E\ntmin tmax : ℝ\nt₀ : ↑(Icc tmin tmax)\nx₀ x : E\na r L K : ℝ≥0\nhf : IsPicardLindelof f t₀ x₀ a r L K\nhx : x ∈ closedBall x₀ ↑r\nn : ℕ\nt : ↑(Icc tmin tmax)\nα β : FunSpace t₀ x₀ r L\nh : dist (((next hf hx)^[n] α).toF... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E → E\ntmin tmax : ℝ\nt₀ : ↑(Icc tmin tmax)\nx₀ x : E\na r L K : ℝ≥0\nhf : IsPicardLindelof f t₀ x₀ a r L K\nhx : x ∈ closedBall x₀ ↑r\nn : ℕ\nt : ↑(Icc tmin tmax)\nα β : FunSpace t₀ x₀ r L\nh : dist (((next hf hx)^[n] α).toFun t) (((nex... | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.ODE.PicardLindelof | {
"line": 338,
"column": 8
} | {
"line": 338,
"column": 28
} | {
"line": 338,
"column": 29
} | [
{
"pp": "case succ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E → E\ntmin tmax : ℝ\nt₀ : ↑(Icc tmin tmax)\nx₀ x : E\na r L K : ℝ≥0\nhf : IsPicardLindelof f t₀ x₀ a r L K\nhx : x ∈ closedBall x₀ ↑r\nα β : FunSpace t₀ x₀ r L\nn : ℕ\nhn :\n ∀ (t : ↑(Icc tmin tmax)),\n dist (... | [
"case succ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E → E\ntmin tmax : ℝ\nt₀ : ↑(Icc tmin tmax)\nx₀ x : E\na r L K : ℝ≥0\nhf : IsPicardLindelof f t₀ x₀ a r L K\nhx : x ∈ closedBall x₀ ↑r\nα β : FunSpace t₀ x₀ r L\nn : ℕ\nhn :\n ∀ (t : ↑(Icc tmin tmax)),\n dist (((next hf hx... | iterate_succ_apply', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.ODE.PicardLindelof | {
"line": 338,
"column": 29
} | {
"line": 338,
"column": 49
} | {
"line": 338,
"column": 50
} | [
{
"pp": "case succ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E → E\ntmin tmax : ℝ\nt₀ : ↑(Icc tmin tmax)\nx₀ x : E\na r L K : ℝ≥0\nhf : IsPicardLindelof f t₀ x₀ a r L K\nhx : x ∈ closedBall x₀ ↑r\nα β : FunSpace t₀ x₀ r L\nn : ℕ\nhn :\n ∀ (t : ↑(Icc tmin tmax)),\n dist (... | [
"case succ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E → E\ntmin tmax : ℝ\nt₀ : ↑(Icc tmin tmax)\nx₀ x : E\na r L K : ℝ≥0\nhf : IsPicardLindelof f t₀ x₀ a r L K\nhx : x ∈ closedBall x₀ ↑r\nα β : FunSpace t₀ x₀ r L\nn : ℕ\nhn :\n ∀ (t : ↑(Icc tmin tmax)),\n dist (((next hf hx... | iterate_succ_apply', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.ODE.PicardLindelof | {
"line": 358,
"column": 61
} | {
"line": 358,
"column": 87
} | {
"line": 358,
"column": 88
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E → E\ntmin tmax : ℝ\nt₀ : ↑(Icc tmin tmax)\nx₀ x : E\na r L K : ℝ≥0\nhf : IsPicardLindelof f t₀ x₀ a r L K\nhx : x ∈ closedBall x₀ ↑r\nα β : FunSpace t₀ x₀ r L\nn : ℕ\nhn :\n ∀ (t : ↑(Icc tmin tmax)),\n dist (((next hf h... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E → E\ntmin tmax : ℝ\nt₀ : ↑(Icc tmin tmax)\nx₀ x : E\na r L K : ℝ≥0\nhf : IsPicardLindelof f t₀ x₀ a r L K\nhx : x ∈ closedBall x₀ ↑r\nα β : FunSpace t₀ x₀ r L\nn : ℕ\nhn :\n ∀ (t : ↑(Icc tmin tmax)),\n dist (((next hf hx)^[n] α).to... | integral_pow_abs_sub_uIoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.ODE.PicardLindelof | {
"line": 438,
"column": 2
} | {
"line": 438,
"column": 57
} | {
"line": 439,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E → E\ntmin tmax : ℝ\nt₀ : ↑(Icc tmin tmax)\nx₀ x : E\na r L K : ℝ≥0\nhf : IsPicardLindelof f t₀ x₀ a r L K\nhx : x ∈ closedBall x₀ ↑r\nα : FunSpace t₀ x₀ r L\nm : ℕ\nC : ℝ≥0\nhm : LipschitzWith C (next hf hx)^[m]\nn i : ℕ\nh... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E → E\ntmin tmax : ℝ\nt₀ : ↑(Icc tmin tmax)\nx₀ x : E\na r L K : ℝ≥0\nhf : IsPicardLindelof f t₀ x₀ a r L K\nhx : x ∈ closedBall x₀ ↑r\nα : FunSpace t₀ x₀ r L\nm : ℕ\nC : ℝ≥0\nhm : LipschitzWith C (next hf hx)^[m]\nn i : ℕ\nhi : i < n\n⊢... | apply le_trans <| hm.dist_iterate_succ_le_geometric α i | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.ODE.PicardLindelof | {
"line": 526,
"column": 4
} | {
"line": 536,
"column": 100
} | {
"line": 537,
"column": 2
} | [
{
"pp": "case pos\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : ℝ → E → E\nα : ℝ → E\nu : Set E\nt₀ tmin tmax : ℝ\nht₀ : t₀ ∈ Icc tmin tmax\nn : ℕ\nhf : ContDiffOn ℝ (↑n) (uncurry f) (Icc tmin tmax ×ˢ u)\nhα : ContinuousOn α (Icc tmin tmax)\nhmem : ∀ t ∈ Ic... | [] | induction n with
| zero =>
simp only [Nat.cast_zero, contDiffOn_zero] at *
exact HasDerivWithinAt.continuousOn this
| succ n hn =>
simp only [Nat.cast_add, Nat.cast_one] at *
rw [contDiffOn_succ_iff_derivWithin <| uniqueDiffOn_Icc hlt]
refine ⟨fun t ht ↦ HasDerivWithinAt.differenti... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Analysis.ODE.PicardLindelof | {
"line": 658,
"column": 25
} | {
"line": 658,
"column": 95
} | {
"line": 659,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℝ → E → E\nt₀ ε : ℝ\nhε : 0 < ε\nx₀ : E\na r L K : ℝ≥0\nhf : IsPicardLindelof f ⟨t₀, ⋯⟩ x₀ a r L K\na' r' : ℝ≥0\nha : a' ≤ a\nhr : r' < a'\nha'r' : 0 < ↑a' - ↑r'\nε' : ℝ := min ε ((↑a' - ↑r') / (↑L + 1))\nhε'pos : 0 < ε'\nhε'_le : ε' ≤ ε\n⊢ ↑L / (↑L + 1) ... | [] | by gcongr; rw [div_le_one (by positivity : (0 : ℝ) < L + 1)]; linarith | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.MvPowerSeries.GaussNorm | {
"line": 114,
"column": 6
} | {
"line": 120,
"column": 71
} | {
"line": 121,
"column": 2
} | [
{
"pp": "case inr\nR : Type u_1\nσ : Type u_2\nv : R → ℝ\nc : σ → ℝ\ninst✝ : Semiring R\nf g : MvPowerSeries σ R\nhc : 0 ≤ c\nvNonneg : ∀ (a : R), v a ≥ 0\nhbfd : HasGaussNorm v c f\nhbgd : HasGaussNorm v c g\nH : ∀ (t : σ →₀ ℕ), 0 ≤ ∏ i ∈ t.support, c i ^ t i\nt : σ →₀ ℕ\nhv : v ((coeff t) f + (coeff t) g) ≤ m... | [] | have : max (v ((coeff t) f) * ∏ i ∈ t.support, c ↑i ^ t ↑i)
(v ((coeff t) g) * ∏ i ∈ t.support, c ↑i ^ t ↑i) =
(v ((coeff t) g) * ∏ i ∈ t.support, c ↑i ^ t ↑i) := by
simp only [sup_eq_right]
exact mul_le_mul_of_nonneg (by aesop) (by aesop) (by aesop) (H t)
simp_rw [this]
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.GaussNorm | {
"line": 114,
"column": 6
} | {
"line": 120,
"column": 71
} | {
"line": 121,
"column": 2
} | [
{
"pp": "case inr\nR : Type u_1\nσ : Type u_2\nv : R → ℝ\nc : σ → ℝ\ninst✝ : Semiring R\nf g : MvPowerSeries σ R\nhc : 0 ≤ c\nvNonneg : ∀ (a : R), v a ≥ 0\nhbfd : HasGaussNorm v c f\nhbgd : HasGaussNorm v c g\nH : ∀ (t : σ →₀ ℕ), 0 ≤ ∏ i ∈ t.support, c i ^ t i\nt : σ →₀ ℕ\nhv : v ((coeff t) f + (coeff t) g) ≤ m... | [] | have : max (v ((coeff t) f) * ∏ i ∈ t.support, c ↑i ^ t ↑i)
(v ((coeff t) g) * ∏ i ∈ t.support, c ↑i ^ t ↑i) =
(v ((coeff t) g) * ∏ i ∈ t.support, c ↑i ^ t ↑i) := by
simp only [sup_eq_right]
exact mul_le_mul_of_nonneg (by aesop) (by aesop) (by aesop) (H t)
simp_rw [this]
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.RCLike.ContinuousMap | {
"line": 94,
"column": 75
} | {
"line": 95,
"column": 60
} | {
"line": 97,
"column": 0
} | [
{
"pp": "X : Type u_1\n𝕜 : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : RCLike 𝕜\n⊢ Monotone rclikeToReal",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real.partialOrder",
"Real.instLE",
"Real",
"AddMonoid.toAddSemi... | [] | by
intro a b; simp_all [le_def, RCLike.le_iff_re_im (K := 𝕜)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Polynomial.MahlerMeasure | {
"line": 398,
"column": 7
} | {
"line": 411,
"column": 60
} | {
"line": 413,
"column": 0
} | [] | [] | ∑ x ∈ S.toFinset, count x S * ‖x.prod‖
_ ≤ ∑ x ∈ S.toFinset, count x S * ((p.roots).map (fun a ↦ max 1 ‖a‖)).prod := by
gcongr with x hx
rw [Finset.prod_multiset_map_count, Finset.prod_multiset_count, norm_prod]
simp_rw [norm_pow]
exact this x hx
_ = p.natDegree.choose n * (p.roots.map (... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
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