module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral | {
"line": 409,
"column": 4
} | {
"line": 409,
"column": 30
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nF : ℕ → α → β\nf : α → β\nbound : α → ℝ\nF_measurable : ∀ (n : ℕ), AEStronglyMeasurable (F n) μ\nbound_hasFiniteIntegral : HasFiniteIntegral bound μ\nh_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a\nh_... | rw [← ENNReal.ofReal_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Real.Sqrt | {
"line": 166,
"column": 62
} | {
"line": 166,
"column": 78
} | [
{
"pp": "x y : ℝ\nhx : 0 ≤ x\nhy : 0 ≤ y\nh : x = y * y\n⊢ √(y * y) = y",
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"congrArg",
"Real.sqrt_mul_self",
"id",
"Real.instMul",
"Real.sqrt",
"Eq",
"instHMul"
]
}
] | sqrt_mul_self hy | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Real.Sqrt | {
"line": 235,
"column": 54
} | {
"line": 236,
"column": 62
} | [
{
"pp": "x y : ℝ\nhy : 0 < y\n⊢ √x < y ↔ x < y ^ 2",
"usedConstants": [
"Eq.mpr",
"pow_pos",
"Real.partialOrder",
"Real",
"Real.instZero",
"Real.instZeroLEOneClass",
"congrArg",
"Iff.rfl",
"Real.sqrt_lt_sqrt_iff_of_pos",
"Real.instLT",
"Real.... | by
rw [← sqrt_lt_sqrt_iff_of_pos (pow_pos hy _), sqrt_sq hy.le] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Real.Sqrt | {
"line": 353,
"column": 2
} | {
"line": 353,
"column": 94
} | [
{
"pp": "x : ℝ\nhx : 0 ≤ x\ny : ℝ\n⊢ √(x * y) = √x * √y",
"usedConstants": [
"Eq.mpr",
"Real.toNNReal_mul",
"Real",
"HMul.hMul",
"_private.Mathlib.Data.Real.Sqrt.0.Real.sqrt_mul._simp_1_3",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Real.sq... | simp_rw [Real.sqrt, ← NNReal.coe_mul, NNReal.coe_inj, Real.toNNReal_mul hx, NNReal.sqrt_mul] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Data.Real.Sqrt | {
"line": 353,
"column": 2
} | {
"line": 353,
"column": 94
} | [
{
"pp": "x : ℝ\nhx : 0 ≤ x\ny : ℝ\n⊢ √(x * y) = √x * √y",
"usedConstants": [
"Eq.mpr",
"Real.toNNReal_mul",
"Real",
"HMul.hMul",
"_private.Mathlib.Data.Real.Sqrt.0.Real.sqrt_mul._simp_1_3",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Real.sq... | simp_rw [Real.sqrt, ← NNReal.coe_mul, NNReal.coe_inj, Real.toNNReal_mul hx, NNReal.sqrt_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Real.Sqrt | {
"line": 353,
"column": 2
} | {
"line": 353,
"column": 94
} | [
{
"pp": "x : ℝ\nhx : 0 ≤ x\ny : ℝ\n⊢ √(x * y) = √x * √y",
"usedConstants": [
"Eq.mpr",
"Real.toNNReal_mul",
"Real",
"HMul.hMul",
"_private.Mathlib.Data.Real.Sqrt.0.Real.sqrt_mul._simp_1_3",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Real.sq... | simp_rw [Real.sqrt, ← NNReal.coe_mul, NNReal.coe_inj, Real.toNNReal_mul hx, NNReal.sqrt_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Order | {
"line": 78,
"column": 30
} | {
"line": 78,
"column": 60
} | [
{
"pp": "z : ℂ\nh : z.im = 0\n⊢ 0 ≤ z.re ^ 2 - z.im ^ 2 ∧ (z.re = 0 ∨ z.im = 0)",
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"False",
"Real.partialOrder",
"Real.instLE",
"Real",
"IsOrderedRing.toPosMulMono",
"and_true",
"Real.instZero",
... | simpa [h] using sq_nonneg z.re | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Complex.Order | {
"line": 78,
"column": 30
} | {
"line": 78,
"column": 60
} | [
{
"pp": "z : ℂ\nh : z.im = 0\n⊢ 0 ≤ z.re ^ 2 - z.im ^ 2 ∧ (z.re = 0 ∨ z.im = 0)",
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"False",
"Real.partialOrder",
"Real.instLE",
"Real",
"IsOrderedRing.toPosMulMono",
"and_true",
"Real.instZero",
... | simpa [h] using sq_nonneg z.re | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Order | {
"line": 78,
"column": 30
} | {
"line": 78,
"column": 60
} | [
{
"pp": "z : ℂ\nh : z.im = 0\n⊢ 0 ≤ z.re ^ 2 - z.im ^ 2 ∧ (z.re = 0 ∨ z.im = 0)",
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"False",
"Real.partialOrder",
"Real.instLE",
"Real",
"IsOrderedRing.toPosMulMono",
"and_true",
"Real.instZero",
... | simpa [h] using sq_nonneg z.re | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Star.MonoidHom | {
"line": 54,
"column": 47
} | {
"line": 54,
"column": 55
} | [
{
"pp": "case mk.mk\nF : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\nD : Type u_5\ninst✝³ : Monoid A\ninst✝² : Star A\ninst✝¹ : Monoid B\ninst✝ : Star B\ntoMonoidHom✝¹ : A →* B\nmap_star'✝¹ : ∀ (a : A), (↑toMonoidHom✝¹).toFun (star a) = star ((↑toMonoidHom✝¹).toFun a)\ntoMonoidHom✝ : A →* B\nmap_star'✝ ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Star.MonoidHom | {
"line": 192,
"column": 47
} | {
"line": 192,
"column": 55
} | [
{
"pp": "case mk.mk\nF : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\nD : Type u_5\ninst✝⁷ : Mul A\ninst✝⁶ : Mul B\ninst✝⁵ : Mul C\ninst✝⁴ : Mul D\ninst✝³ : Star A\ninst✝² : Star B\ninst✝¹ : Star C\ninst✝ : Star D\ntoMulEquiv✝¹ : A ≃* B\nmap_star'✝¹ : ∀ (a : A), toMulEquiv✝¹.toFun (star a) = star (toMulE... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Normed.Group.Hom | {
"line": 765,
"column": 55
} | {
"line": 768,
"column": 11
} | [
{
"pp": "V : Type u_1\nW : Type u_2\ninst✝¹ : SeminormedAddCommGroup V\ninst✝ : SeminormedAddCommGroup W\nf g : NormedAddGroupHom V W\n⊢ f.comp (ι f g) = g.comp (ι f g)",
"usedConstants": [
"NormedAddGroupHom.ext",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NormedAddGroupHom",
... | by
ext x
rw [comp_apply, comp_apply, ← sub_eq_zero, ← NormedAddGroupHom.sub_apply]
exact x.2 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.Basic | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 41
} | [
{
"pp": "E : Type u_2\ninst✝² : NonUnitalNormedRing E\ninst✝¹ : StarRing E\ninst✝ : CStarRing E\nx : E\n⊢ x⋆ * x = 0 ↔ x = 0",
"usedConstants": [
"norm_eq_zero",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
"NonUnitalNormedRing.toNorm",
"Real",
"HMul.hMul"... | rw [← norm_eq_zero, norm_star_mul_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.InfiniteSum.Module | {
"line": 219,
"column": 2
} | {
"line": 219,
"column": 14
} | [
{
"pp": "case h.h\nα : Type u_1\nβ : Type u_2\nM : Type u_11\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : T2Space M\nR : Type u_12\ninst✝⁴ : DivisionRing R\ninst✝³ : Module R M\ninst✝² : ContinuousConstSMul R M\ninst✝¹ : Group α\ninst✝ : MulAction α β\nf : β → M\ng : Quotient (orbitRel α β) ... | simp_rw [H₁] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Topology.Algebra.InfiniteSum.Module | {
"line": 237,
"column": 2
} | {
"line": 237,
"column": 14
} | [
{
"pp": "case h.h\nα : Type u_1\nβ : Type u_2\nM : Type u_11\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : T2Space M\nR : Type u_12\ninst✝⁴ : DivisionRing R\ninst✝³ : Module R M\ninst✝² : ContinuousConstSMul R M\ninst✝¹ : AddGroup α\ninst✝ : AddAction α β\nf : β → M\ng : Quotient (orbitRel α ... | simp_rw [H₁] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.Complex.Basic | {
"line": 417,
"column": 53
} | {
"line": 417,
"column": 73
} | [
{
"pp": "z : ℂ\nE : Type u_1\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\n𝕜 : Type u_2\ninst✝ : RCLike 𝕜\nh : RCLike.im RCLike.I = 1\nx✝ : 𝕜\n⊢ ‖{ toFun := (RCLike.complexRingEquiv h).toFun, map_add' := ⋯, map_smul' := ⋯,\n invFun := (RCLike.complexRingEquiv h).invFun, left_inv :=... | ← normSq_eq_norm_sq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Basic | {
"line": 463,
"column": 2
} | {
"line": 464,
"column": 33
} | [
{
"pp": "x : ℂ\nhx : 0 ≤ x\n⊢ ↑‖x‖ = x",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"congrArg",
"Complex.instNorm",
"id",
"Complex.instRCLike",
"RCLike.ofReal",
"Complex.ofReal",
"RCLike.ofReal_eq_complex_ofReal",
"Complex",
"Eq.... | rw [← RCLike.ofReal_eq_complex_ofReal]
exact RCLike.norm_of_nonneg' hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Basic | {
"line": 463,
"column": 2
} | {
"line": 464,
"column": 33
} | [
{
"pp": "x : ℂ\nhx : 0 ≤ x\n⊢ ↑‖x‖ = x",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"congrArg",
"Complex.instNorm",
"id",
"Complex.instRCLike",
"RCLike.ofReal",
"Complex.ofReal",
"RCLike.ofReal_eq_complex_ofReal",
"Complex",
"Eq.... | rw [← RCLike.ofReal_eq_complex_ofReal]
exact RCLike.norm_of_nonneg' hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Complex.Module | {
"line": 530,
"column": 36
} | {
"line": 532,
"column": 72
} | [
{
"pp": "z : ↥(selfAdjoint ℂ)\n⊢ ↑(selfAdjointEquiv z) = ↑z",
"usedConstants": [
"instTrivialStarReal",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.toModule",
"instStarRingReal",
"CommRing.toNonUnitalCommRing",
"Complex.commRing",
"selfA... | by
simpa [selfAdjointEquiv_symm_apply]
using (congr_arg Subtype.val <| Complex.selfAdjointEquiv.left_inv z) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Complex.Module | {
"line": 581,
"column": 30
} | {
"line": 581,
"column": 65
} | [
{
"pp": "A : Type u_1\ninst✝⁵ : NonUnitalNonAssocRing A\ninst✝⁴ : StarRing A\ninst✝³ : Module ℂ A\ninst✝² : IsScalarTower ℂ A A\ninst✝¹ : SMulCommClass ℂ A A\ninst✝ : StarModule ℂ A\nx : A\n⊢ 2 • I • (↑(ℜ x) * ↑(ℑ x) - ↑(ℑ x) * ↑(ℜ x)) = 0 ↔ Commute ↑(ℜ x) ↑(ℑ x)",
"usedConstants": [
"AddGroup.toSubtr... | smul_eq_zero_iff_right two_ne_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 728,
"column": 69
} | {
"line": 728,
"column": 80
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nhI : I ≠ 0\n⊢ ‖I‖ * ‖I‖ = 1",
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"NormedDivisi... | ← norm_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 1182,
"column": 7
} | {
"line": 1184,
"column": 77
} | [] | ‖re x - re y‖ₑ
_ = ‖re (x - y)‖ₑ := by rw [map_sub re x y]
_ ≤ ‖x - y‖ₑ := by rw [enorm_le_iff_norm_le]; exact norm_re_le_norm (x - y) | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Analysis.RCLike.Basic | {
"line": 1252,
"column": 37
} | {
"line": 1252,
"column": 45
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\nh : im I = 1\nx✝ : I = 0\n⊢ False",
"usedConstants": [
"False",
"Real",
"AddMonoidHom.instAddMonoidHomClass",
"AddMonoid.toAddSemigroup",
"Real.instZero",
"Real.instAddMonoid",
"congrArg",
"False.elim",
"Ad... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.RCLike.Basic | {
"line": 1252,
"column": 37
} | {
"line": 1252,
"column": 45
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\nh : im I = 1\nx✝ : I = 0\n⊢ False",
"usedConstants": [
"False",
"Real",
"AddMonoidHom.instAddMonoidHomClass",
"AddMonoid.toAddSemigroup",
"Real.instZero",
"Real.instAddMonoid",
"congrArg",
"False.elim",
"Ad... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.RCLike.Basic | {
"line": 1252,
"column": 37
} | {
"line": 1252,
"column": 45
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\nh : im I = 1\nx✝ : I = 0\n⊢ False",
"usedConstants": [
"False",
"Real",
"AddMonoidHom.instAddMonoidHomClass",
"AddMonoid.toAddSemigroup",
"Real.instZero",
"Real.instAddMonoid",
"congrArg",
"False.elim",
"Ad... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 316,
"column": 33
} | {
"line": 316,
"column": 87
} | [
{
"pp": "α : Type u_1\nE' : Type u_6\nF' : Type u_7\ninst✝² : SeminormedAddCommGroup E'\ninst✝¹ : SeminormedAddCommGroup F'\nf' : α → E'\ng' : α → F'\nl : Filter α\ninst✝ : Subsingleton E'\nc : ℝ\nhc : 0 < c\n⊢ ∀ᶠ (x : α) in l, ‖f' x‖ ≤ c * ‖g' x‖",
"usedConstants": [
"Real.instIsOrderedRing",
"... | by simp [Subsingleton.elim (f' _) 0, mul_nonneg hc.le] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.Asymptotics | {
"line": 58,
"column": 89
} | {
"line": 58,
"column": 97
} | [
{
"pp": "f g : ℂ → ℂ\nx : ℝ\nh : f =O[𝓝[≠] ↑x] g\nx✝¹ : ℝ\nx✝ : x✝¹ ∈ {x}ᶜ\n⊢ ↑x✝¹ ∈ {↑x}ᶜ",
"usedConstants": [
"False",
"Real",
"eq_false",
"congrArg",
"Compl.compl",
"Membership.mem",
"Eq.mp",
"Set.instSingletonSet",
"id",
"Complex.ofReal",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Complex.Asymptotics | {
"line": 58,
"column": 89
} | {
"line": 58,
"column": 97
} | [
{
"pp": "f g : ℂ → ℂ\nx : ℝ\nh : f =O[𝓝[≠] ↑x] g\nx✝¹ : ℝ\nx✝ : x✝¹ ∈ {x}ᶜ\n⊢ ↑x✝¹ ∈ {↑x}ᶜ",
"usedConstants": [
"False",
"Real",
"eq_false",
"congrArg",
"Compl.compl",
"Membership.mem",
"Eq.mp",
"Set.instSingletonSet",
"id",
"Complex.ofReal",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Asymptotics | {
"line": 58,
"column": 89
} | {
"line": 58,
"column": 97
} | [
{
"pp": "f g : ℂ → ℂ\nx : ℝ\nh : f =O[𝓝[≠] ↑x] g\nx✝¹ : ℝ\nx✝ : x✝¹ ∈ {x}ᶜ\n⊢ ↑x✝¹ ∈ {↑x}ᶜ",
"usedConstants": [
"False",
"Real",
"eq_false",
"congrArg",
"Compl.compl",
"Membership.mem",
"Eq.mp",
"Set.instSingletonSet",
"id",
"Complex.ofReal",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.CauSeq.BigOperators | {
"line": 113,
"column": 10
} | {
"line": 113,
"column": 13
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : ℕ → β\nha : IsCauSeq abs fun m ↦ ∑ n ∈ range m, abv (f n)\nhb : IsCauSeq abv fun m ↦ ∑ n ∈ range m, g n\nε : α\nε0 : 0 < ε\nP : α\nhP : ... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Exponential | {
"line": 234,
"column": 51
} | {
"line": 234,
"column": 59
} | [
{
"pp": "x : ℝ\nh : ↑(cexp ↑x).re = ↑0\n⊢ cexp ↑x = 0",
"usedConstants": [
"Complex.exp_ne_zero._simp_1",
"False",
"Real",
"Real.instZero",
"congrArg",
"False.elim",
"Complex.instZero",
"Eq.mp",
"Complex.ofReal",
"Complex.re",
"Complex.ofReal... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Complex.Exponential | {
"line": 252,
"column": 54
} | {
"line": 252,
"column": 60
} | [
{
"pp": "x : ℝ\nhx : 0 ≤ x\nn : ℕ\n⊢ (cauSeqRe (exp' ↑x)).lim = (exp' ↑x).lim.re",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Complex.instIsComplete",
"Real",
"Real.lattice",
"abs",
"congrArg",
"IsAbsoluteValue.abs_isAbsoluteValue",
"Real.instIsCompleteA... | lim_re | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Tactic.NormNum.NatFactorial | {
"line": 77,
"column": 2
} | {
"line": 78,
"column": 28
} | [
{
"pp": "n x l y : ℕ\nh₁ : IsNat n x\nh₂ : IsNat l y\na : ℕ\np : x.ascFactorial y = a\n⊢ IsNat (n.ascFactorial l) a",
"usedConstants": [
"congrArg",
"Mathlib.Meta.NormNum.IsNat.mk",
"Nat.ascFactorial",
"AddMonoidWithOne.toNatCast",
"Nat.instAddMonoidWithOne",
"Nat.cast",
... | constructor
simp [h₁.out, h₂.out, ← p] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Tactic.NormNum.NatFactorial | {
"line": 77,
"column": 2
} | {
"line": 78,
"column": 28
} | [
{
"pp": "n x l y : ℕ\nh₁ : IsNat n x\nh₂ : IsNat l y\na : ℕ\np : x.ascFactorial y = a\n⊢ IsNat (n.ascFactorial l) a",
"usedConstants": [
"congrArg",
"Mathlib.Meta.NormNum.IsNat.mk",
"Nat.ascFactorial",
"AddMonoidWithOne.toNatCast",
"Nat.instAddMonoidWithOne",
"Nat.cast",
... | constructor
simp [h₁.out, h₂.out, ← p] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 134,
"column": 19
} | {
"line": 134,
"column": 28
} | [
{
"pp": "x y : ℂ\n⊢ (cexp x - cexp (-x)) * (2 * cosh y) + 2 * (cosh x * (cexp y - cexp (-y))) =\n 2 * (cexp x * cexp y - cexp (-x) * cexp (-y))",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"Complex.commRing",
"congrA... | two_cosh, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 147,
"column": 41
} | {
"line": 147,
"column": 50
} | [
{
"pp": "x y : ℂ\n⊢ 2 * cosh (x + y) = 2 * (cosh x * cosh y + sinh x * sinh y)",
"usedConstants": [
"Eq.mpr",
"Complex.sinh",
"HMul.hMul",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"Complex.instMul",
"id",
"instOfNatNat",
"Complex.instNatCast",
"C... | two_cosh, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 148,
"column": 15
} | {
"line": 148,
"column": 24
} | [
{
"pp": "x y : ℂ\n⊢ 2 * cosh x * cosh y + 2 * (sinh x * sinh y) = cexp x * cexp y + cexp (-x) * cexp (-y)",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Semigroup.toMul",
"Complex.sinh",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"Complex.commRing",
"... | two_cosh, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 149,
"column": 19
} | {
"line": 149,
"column": 28
} | [
{
"pp": "x y : ℂ\n⊢ (cexp x + cexp (-x)) * (2 * cosh y) + 2 * ((cexp x - cexp (-x)) * sinh y) =\n 2 * (cexp x * cexp y + cexp (-x) * cexp (-y))",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Semigroup.toMul",
"Complex.sinh",
"HMul.hMul",
"CommRing.toNonUnitalCom... | two_cosh, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 220,
"column": 50
} | {
"line": 220,
"column": 59
} | [
{
"pp": "x : ℂ\n⊢ 2 * cosh x + 2 * sinh x = 2 * cexp x",
"usedConstants": [
"Eq.mpr",
"Complex.sinh",
"HMul.hMul",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"Complex.instMul",
"id",
"instOfNatNat",
"Complex.instNatCast",
"Complex.two_cosh",
... | two_cosh, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 235,
"column": 50
} | {
"line": 235,
"column": 59
} | [
{
"pp": "x : ℂ\n⊢ 2 * cosh x - 2 * sinh x = 2 * cexp (-x)",
"usedConstants": [
"Eq.mpr",
"Complex.sinh",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"Complex.commRing",
"congrArg",
"NonUnitalNonAssocRing.toAddC... | two_cosh, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 292,
"column": 41
} | {
"line": 292,
"column": 50
} | [
{
"pp": "x : ℂ\n⊢ 2 * cosh (x * I) = 2 * cos x",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Complex.cos",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"Complex.instMul",
"id",
"instOfNatNat",
"Complex.instNatCast",
"Complex.two_cosh",
"instHAdd... | two_cosh, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 790,
"column": 29
} | {
"line": 790,
"column": 37
} | [
{
"pp": "x : ℝ\n⊢ sinh x / cosh x = (rexp x - rexp (-x)) / (rexp x + rexp (-x))",
"usedConstants": [
"Real.sinh_eq",
"Eq.mpr",
"Real",
"instHDiv",
"congrArg",
"Real.instDivInvMonoid",
"Real.instSub",
"Nat.instAtLeastTwoHAddOfNat",
"HSub.hSub",
"Rea... | sinh_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 876,
"column": 93
} | {
"line": 877,
"column": 94
} | [
{
"pp": "x : ℝ\nhx : |x| ≤ 1\n⊢ |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96)",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"instHDiv",
"HMul.hMul",
"Real.lattice",
"abs",
"congrArg",
"Complex.sin_bound",
"Real.instDivIn... | by
simpa [← ofReal_sin, ← norm_eq_abs, ← norm_real] using Complex.sin_bound (x := x) (by simpa) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 901,
"column": 25
} | {
"line": 901,
"column": 62
} | [
{
"pp": "x : ℝ\nhx0 : 0 < x\nhx : x ≤ 1\n⊢ |x| ≤ 1",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"MulOne.toOne",
"Real",
"Real.lattice",
"abs",
"congrArg",
"PartialOrder.toPreorder",
"le_of_lt",
"Preorder.toLE",
"SemilatticeI... | by rwa [abs_of_nonneg (le_of_lt hx0)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 908,
"column": 50
} | {
"line": 908,
"column": 87
} | [
{
"pp": "x : ℝ\nhx0 : 0 < x\nhx : x ≤ 1\n⊢ |x| ≤ 1",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Real.instLE",
"Real",
"Real.lattice",
"abs",
"congrArg",
"PartialOrder.toPreorder",
"le_of_lt",
"SemilatticeInf.toPartialOrder",
... | by rwa [abs_of_nonneg (le_of_lt hx0)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 932,
"column": 6
} | {
"line": 932,
"column": 28
} | [
{
"pp": "case h.hbc.ha\n⊢ 0 ≤ cos 1",
"usedConstants": [
"Real",
"Real.instZero",
"Real.cos",
"LT.lt.le",
"Real.cos_one_pos",
"Real.instOne",
"One.toOfNat1",
"Zero.toOfNat0",
"OfNat.ofNat",
"Real.instPreorder"
]
},
{
"pp": "case h.hbc.h... | · exact cos_one_pos.le | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Exp | {
"line": 264,
"column": 37
} | {
"line": 264,
"column": 85
} | [
{
"pp": "n : ℕ\nC : ℝ\nhC₁ : 1 ≤ C\nhC₀ : 0 < C\nthis : 0 < (rexp 1 * C)⁻¹\nN : ℕ\nhN : ∀ k ≥ N, ↑k ^ n < rexp ↑k / (rexp 1 * C)\nx : ℝ\nhx : ↑N < x\nhx₀ : 0 < x\n⊢ rexp ↑⌈x⌉₊ / (rexp 1 * C) ≤ rexp (x + 1) / (rexp 1 * C)",
"usedConstants": [
"Real.instIsOrderedRing",
"NonAssocSemiring.toAddCommM... | by gcongr; exact (Nat.ceil_lt_add_one hx₀.le).le | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Group.Pointwise | {
"line": 32,
"column": 93
} | {
"line": 37,
"column": 48
} | [
{
"pp": "E : Type u_1\ninst✝ : SeminormedGroup E\ns t : Set E\nhs : IsBounded s\nht : IsBounded t\n⊢ IsBounded (s * t)",
"usedConstants": [
"Iff.mpr",
"Norm.norm",
"isBounded_iff_forall_norm_le'",
"Real.instLE",
"Real",
"HMul.hMul",
"PseudoMetricSpace.toBornology",
... | by
obtain ⟨Rs, hRs⟩ : ∃ R, ∀ x ∈ s, ‖x‖ ≤ R := hs.exists_norm_le'
obtain ⟨Rt, hRt⟩ : ∃ R, ∀ x ∈ t, ‖x‖ ≤ R := ht.exists_norm_le'
refine isBounded_iff_forall_norm_le'.2 ⟨Rs + Rt, ?_⟩
rintro z ⟨x, hx, y, hy, rfl⟩
exact norm_mul_le_of_le' (hRs x hx) (hRt y hy) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Group.Pointwise | {
"line": 183,
"column": 2
} | {
"line": 183,
"column": 37
} | [
{
"pp": "case h.e'_2.h.e'_5\nE : Type u_1\ninst✝ : SeminormedCommGroup E\nδ : ℝ\ns : Set E\n⊢ s = ⋃ i ∈ s, {i}",
"usedConstants": [
"Membership.mem",
"Set.biUnion_of_singleton",
"Set.instSingletonSet",
"Singleton.singleton",
"Eq.symm",
"Set.instMembership",
"Set.iUn... | · exact s.biUnion_of_singleton.symm | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Module.Ball.Pointwise | {
"line": 62,
"column": 4
} | {
"line": 63,
"column": 91
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : NormedDivisionRing 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : NormSMulClass 𝕜 E\nc : 𝕜\nhc : c ≠ 0\ns : Set E\nx : E\nthis : Function.Surjective fun x ↦ c • x\n⊢ ⨅ y ∈ s, ‖c‖₊ • edist x y = ‖c‖₊ • ⨅ y ∈ s, edist x y",
"usedConstants"... | have : (‖c‖₊ : ENNReal) ≠ 0 := by simp [hc]
simp_rw [ENNReal.smul_def, smul_eq_mul, ENNReal.mul_iInf_of_ne this ENNReal.coe_ne_top] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Module.Ball.Pointwise | {
"line": 62,
"column": 4
} | {
"line": 63,
"column": 91
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : NormedDivisionRing 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : NormSMulClass 𝕜 E\nc : 𝕜\nhc : c ≠ 0\ns : Set E\nx : E\nthis : Function.Surjective fun x ↦ c • x\n⊢ ⨅ y ∈ s, ‖c‖₊ • edist x y = ‖c‖₊ • ⨅ y ∈ s, edist x y",
"usedConstants"... | have : (‖c‖₊ : ENNReal) ≠ 0 := by simp [hc]
simp_rw [ENNReal.smul_def, smul_eq_mul, ENNReal.mul_iInf_of_ne this ENNReal.coe_ne_top] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Group.Quotient | {
"line": 167,
"column": 4
} | {
"line": 167,
"column": 33
} | [
{
"pp": "M : Type u_1\ninst✝ : SeminormedCommGroup M\nS : Subgroup M\nx : M\n⊢ infDist ((IsometryEquiv.divLeft x) 1) (⇑(IsometryEquiv.divLeft x) '' {m | ↑m = ↑x}) = infDist x ↑S",
"usedConstants": [
"NormedGroup.to_isIsometricSMul_right",
"Eq.mpr",
"Real",
"InvOneClass.toOne",
... | ← IsometryEquiv.preimage_symm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Connected.PathConnected | {
"line": 396,
"column": 2
} | {
"line": 397,
"column": 21
} | [
{
"pp": "case right\nX : Type u_1\ninst✝ : TopologicalSpace X\nF : Set X\nx : X\nx_in : x ∈ F\nh : pathComponentIn F x = F\n⊢ ∀ ⦃y : X⦄, y ∈ F → JoinedIn F x y",
"usedConstants": [
"congrArg",
"Membership.mem",
"Eq.mp",
"pathComponentIn",
"Eq.symm",
"Set.instMembership",
... | · intro y y_in
rwa [← h] at y_in | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 989,
"column": 2
} | {
"line": 989,
"column": 60
} | [
{
"pp": "⊢ Tendsto sin (𝓝[>] (-(π / 2))) (𝓝 (-1))",
"usedConstants": [
"Eq.mpr",
"Real",
"Set.Ioi",
"instHDiv",
"Real.pi",
"HEq.refl",
"Real.instDivInvMonoid",
"nhdsWithin",
"Nat.instAtLeastTwoHAddOfNat",
"PseudoMetricSpace.toUniformSpace",
... | convert! continuous_sin.continuousWithinAt.tendsto using 2 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 760,
"column": 2
} | {
"line": 760,
"column": 20
} | [
{
"pp": "f : ℕ → ℝ\nhfa : Monotone f\nhf0 : Tendsto f atTop (𝓝 0)\n⊢ CauchySeq fun n ↦ ∑ i ∈ Finset.range n, (-1) ^ i * f i",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMu... | simp_rw [mul_comm] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 773,
"column": 2
} | {
"line": 773,
"column": 20
} | [
{
"pp": "f : ℕ → ℝ\nhfa : Antitone f\nhf0 : Tendsto f atTop (𝓝 0)\n⊢ CauchySeq fun n ↦ ∑ i ∈ Finset.range n, (-1) ^ i * f i",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMu... | simp_rw [mul_comm] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse | {
"line": 53,
"column": 71
} | {
"line": 53,
"column": 81
} | [
{
"pp": "x : ℝ\n⊢ ↑(sinOrderIso.symm (projIcc (-1) 1 ⋯ x)) = ↑(IccExtend arcsin._proof_2 (⇑sinOrderIso.symm) x)",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"Real.instLE",
"Real",
"instHDiv",
"Real.pi",
"Real.instZeroLEO... | IccExtend, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 123,
"column": 6
} | {
"line": 123,
"column": 14
} | [
{
"pp": "x : ℝ\nhx : 0 < x\n⊢ sinh (log x) = (x - x⁻¹) / 2",
"usedConstants": [
"Real.sinh_eq",
"Eq.mpr",
"Real",
"instHDiv",
"congrArg",
"Real.instInv",
"Real.instDivInvMonoid",
"Real.instSub",
"Nat.instAtLeastTwoHAddOfNat",
"HSub.hSub",
"Re... | sinh_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 254,
"column": 37
} | {
"line": 254,
"column": 51
} | [
{
"pp": "x : ℝ\nhx : x < 0\ny : ℝ\nhy : y < 0\nhxy : x < y\n⊢ -y < -x",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"neg_lt_neg_iff",
"Real.partialOrder",
"Real",
"Ad... | neg_lt_neg_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 322,
"column": 4
} | {
"line": 322,
"column": 30
} | [
{
"pp": "case inr.inr.inr.inl\nx : ℂ\nhr : 0 < x.re\nhi : x.im = 0\n⊢ (if 0 ≤ x.re then -Real.arcsin (x.im / ‖x‖)\n else if 0 ≤ -x.im then Real.arcsin (x.im / ‖x‖) + π else Real.arcsin (x.im / ‖x‖) - π) =\n if x.re < 0 ∧ x.im = 0 then π\n else\n -if 0 ≤ x.re then Real.arcsin (x.im / ‖x‖)\n ... | simp [hr.le, hr.le.not_gt] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 322,
"column": 4
} | {
"line": 322,
"column": 30
} | [
{
"pp": "case inr.inr.inr.inl\nx : ℂ\nhr : 0 < x.re\nhi : x.im = 0\n⊢ (if 0 ≤ x.re then -Real.arcsin (x.im / ‖x‖)\n else if 0 ≤ -x.im then Real.arcsin (x.im / ‖x‖) + π else Real.arcsin (x.im / ‖x‖) - π) =\n if x.re < 0 ∧ x.im = 0 then π\n else\n -if 0 ≤ x.re then Real.arcsin (x.im / ‖x‖)\n ... | simp [hr.le, hr.le.not_gt] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 322,
"column": 4
} | {
"line": 322,
"column": 30
} | [
{
"pp": "case inr.inr.inr.inl\nx : ℂ\nhr : 0 < x.re\nhi : x.im = 0\n⊢ (if 0 ≤ x.re then -Real.arcsin (x.im / ‖x‖)\n else if 0 ≤ -x.im then Real.arcsin (x.im / ‖x‖) + π else Real.arcsin (x.im / ‖x‖) - π) =\n if x.re < 0 ∧ x.im = 0 then π\n else\n -if 0 ≤ x.re then Real.arcsin (x.im / ‖x‖)\n ... | simp [hr.le, hr.le.not_gt] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 323,
"column": 4
} | {
"line": 323,
"column": 30
} | [
{
"pp": "case inr.inr.inr.inr\nx : ℂ\nhr : 0 < x.re\nhi : 0 < x.im\n⊢ (if 0 ≤ x.re then -Real.arcsin (x.im / ‖x‖)\n else if 0 ≤ -x.im then Real.arcsin (x.im / ‖x‖) + π else Real.arcsin (x.im / ‖x‖) - π) =\n if x.re < 0 ∧ x.im = 0 then π\n else\n -if 0 ≤ x.re then Real.arcsin (x.im / ‖x‖)\n ... | simp [hr.le, hr.le.not_gt] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 323,
"column": 4
} | {
"line": 323,
"column": 30
} | [
{
"pp": "case inr.inr.inr.inr\nx : ℂ\nhr : 0 < x.re\nhi : 0 < x.im\n⊢ (if 0 ≤ x.re then -Real.arcsin (x.im / ‖x‖)\n else if 0 ≤ -x.im then Real.arcsin (x.im / ‖x‖) + π else Real.arcsin (x.im / ‖x‖) - π) =\n if x.re < 0 ∧ x.im = 0 then π\n else\n -if 0 ≤ x.re then Real.arcsin (x.im / ‖x‖)\n ... | simp [hr.le, hr.le.not_gt] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 323,
"column": 4
} | {
"line": 323,
"column": 30
} | [
{
"pp": "case inr.inr.inr.inr\nx : ℂ\nhr : 0 < x.re\nhi : 0 < x.im\n⊢ (if 0 ≤ x.re then -Real.arcsin (x.im / ‖x‖)\n else if 0 ≤ -x.im then Real.arcsin (x.im / ‖x‖) + π else Real.arcsin (x.im / ‖x‖) - π) =\n if x.re < 0 ∧ x.im = 0 then π\n else\n -if 0 ≤ x.re then Real.arcsin (x.im / ‖x‖)\n ... | simp [hr.le, hr.le.not_gt] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 331,
"column": 62
} | {
"line": 331,
"column": 98
} | [
{
"pp": "x : ℂ\n⊢ |x⁻¹.arg| = |x.arg|",
"usedConstants": [
"Eq.mpr",
"abs_neg",
"Real",
"Real.pi",
"Real.lattice",
"abs",
"congrArg",
"Complex.arg",
"id",
"Real.instAddGroup",
"if_pos",
"dite",
"Inv.inv",
"Complex.arg_inv",
... | rw [arg_inv]; split_ifs <;> simp [*] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 331,
"column": 62
} | {
"line": 331,
"column": 98
} | [
{
"pp": "x : ℂ\n⊢ |x⁻¹.arg| = |x.arg|",
"usedConstants": [
"Eq.mpr",
"abs_neg",
"Real",
"Real.pi",
"Real.lattice",
"abs",
"congrArg",
"Complex.arg",
"id",
"Real.instAddGroup",
"if_pos",
"dite",
"Inv.inv",
"Complex.arg_inv",
... | rw [arg_inv]; split_ifs <;> simp [*] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 392,
"column": 39
} | {
"line": 392,
"column": 47
} | [
{
"pp": "s : Multiset ℝ\nh : ∀ x ∈ s, x ≠ 0\n⊢ ∀ x ∈ s.toList, x ≠ 0",
"usedConstants": [
"False",
"Real",
"eq_false",
"Real.instZero",
"congrArg",
"Multiset.mem_toList._simp_1",
"Membership.mem",
"Multiset",
"id",
"Multiset.toList",
"Multise... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 392,
"column": 39
} | {
"line": 392,
"column": 47
} | [
{
"pp": "s : Multiset ℝ\nh : ∀ x ∈ s, x ≠ 0\n⊢ ∀ x ∈ s.toList, x ≠ 0",
"usedConstants": [
"False",
"Real",
"eq_false",
"Real.instZero",
"congrArg",
"Multiset.mem_toList._simp_1",
"Membership.mem",
"Multiset",
"id",
"Multiset.toList",
"Multise... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 392,
"column": 39
} | {
"line": 392,
"column": 47
} | [
{
"pp": "s : Multiset ℝ\nh : ∀ x ∈ s, x ≠ 0\n⊢ ∀ x ∈ s.toList, x ≠ 0",
"usedConstants": [
"False",
"Real",
"eq_false",
"Real.instZero",
"congrArg",
"Multiset.mem_toList._simp_1",
"Membership.mem",
"Multiset",
"id",
"Multiset.toList",
"Multise... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 398,
"column": 43
} | {
"line": 398,
"column": 51
} | [
{
"pp": "α : Type u_1\ns : Finset α\nf : α → ℝ\nhf : ∀ x ∈ s, f x ≠ 0\n⊢ ∀ x ∈ List.map (fun i ↦ f i) s.toList, x ≠ 0",
"usedConstants": [
"False",
"Real",
"eq_false",
"Real.instZero",
"congrArg",
"Finset",
"List.map",
"Membership.mem",
"Exists",
"... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 398,
"column": 43
} | {
"line": 398,
"column": 51
} | [
{
"pp": "α : Type u_1\ns : Finset α\nf : α → ℝ\nhf : ∀ x ∈ s, f x ≠ 0\n⊢ ∀ x ∈ List.map (fun i ↦ f i) s.toList, x ≠ 0",
"usedConstants": [
"False",
"Real",
"eq_false",
"Real.instZero",
"congrArg",
"Finset",
"List.map",
"Membership.mem",
"Exists",
"... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 398,
"column": 43
} | {
"line": 398,
"column": 51
} | [
{
"pp": "α : Type u_1\ns : Finset α\nf : α → ℝ\nhf : ∀ x ∈ s, f x ≠ 0\n⊢ ∀ x ∈ List.map (fun i ↦ f i) s.toList, x ≠ 0",
"usedConstants": [
"False",
"Real",
"eq_false",
"Real.instZero",
"congrArg",
"Finset",
"List.map",
"Membership.mem",
"Exists",
"... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 459,
"column": 44
} | {
"line": 459,
"column": 63
} | [
{
"pp": "case inr.inr\nx : ℂ\nhx : x ≠ 0\nhi : 0 < x.im\n⊢ ↑(x.arg - π) = ↑x.arg + ↑π",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.pi",
"Real.Angle",
"Real.Angle.coe",
"congrArg",
"AddCommGroup.toAddCommMonoid",
"Real.instSub",
"HSub.hSub",
"AddCom... | Real.Angle.coe_sub, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 505,
"column": 8
} | {
"line": 505,
"column": 22
} | [
{
"pp": "case mpr\nθ : Angle\nh : θ ≠ ↑π\n⊢ -π < -θ.toReal ∧ -θ.toReal ≤ π",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"neg_lt_neg_iff",
"Real.partialOrder",
"Real.instLE... | neg_lt_neg_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 763,
"column": 12
} | {
"line": 763,
"column": 25
} | [
{
"pp": "θ : Angle\n⊢ θ.toReal < 0 ↔ SignType.sign θ.sin = -1",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"SignType.instOne",
"congrArg",
"PartialOrder.toPreorder",
"Real.decidableLT",
"SignType.instLinearOrder",
"Real.instLT",
"Semilat... | ← sin_toReal, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 775,
"column": 14
} | {
"line": 775,
"column": 27
} | [
{
"pp": "case inr.inr\nθ : Angle\nh : θ ≠ ↑π\nht : 0 < θ.toReal\n⊢ SignType.sign θ.toReal = SignType.sign θ.sin",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"congrArg",
"PartialOrder.toPreorder",
"Real.decidableLT",
"SignType.instLinearOrder",
"Semi... | ← sin_toReal, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 909,
"column": 2
} | {
"line": 914,
"column": 95
} | [
{
"pp": "θ ψ : Angle\nhθ : θ ≠ ↑π\nhψ : ψ ≠ ↑π\nhs : θ.sign ≠ ψ.sign ∨ θ.sign = (θ + ψ).sign\n⊢ (θ + ψ).toReal = θ.toReal + ψ.toReal",
"usedConstants": [
"_private.Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle.0.Real.Angle.toReal_add_eq_toReal_add_toReal._proof_1_1",
"Iff.mpr",
"Re... | obtain (hs | hs) := hs
· obtain (h | h | h) := ψ.sign.trichotomy <;> obtain (h | h | h) := θ.sign.trichotomy
all_goals grind [add_comm, toReal_add_of_sign_pos_sign_neg, sign_eq_zero_iff]
· rw [← neg_neg θ.sign, ← sign_neg] at hs
have := toReal_add_of_sign_eq_neg_sign (.inr <| by simpa [neg_eq_iff_eq_neg]) h... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 909,
"column": 2
} | {
"line": 914,
"column": 95
} | [
{
"pp": "θ ψ : Angle\nhθ : θ ≠ ↑π\nhψ : ψ ≠ ↑π\nhs : θ.sign ≠ ψ.sign ∨ θ.sign = (θ + ψ).sign\n⊢ (θ + ψ).toReal = θ.toReal + ψ.toReal",
"usedConstants": [
"_private.Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle.0.Real.Angle.toReal_add_eq_toReal_add_toReal._proof_1_1",
"Iff.mpr",
"Re... | obtain (hs | hs) := hs
· obtain (h | h | h) := ψ.sign.trichotomy <;> obtain (h | h | h) := θ.sign.trichotomy
all_goals grind [add_comm, toReal_add_of_sign_pos_sign_neg, sign_eq_zero_iff]
· rw [← neg_neg θ.sign, ← sign_neg] at hs
have := toReal_add_of_sign_eq_neg_sign (.inr <| by simpa [neg_eq_iff_eq_neg]) h... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 52,
"column": 40
} | {
"line": 52,
"column": 60
} | [
{
"pp": "x : ℝ\nhx : 0 < x\ny : ℝ\n⊢ (if x = 0 then if y = 0 then 1 else 0 else rexp (log x * y)) = rexp (log x * y)",
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"Real.instZero",
"congrArg",
"ne_of_gt",
"id",
"Real.exp",
"Real.log",
"Real.in... | if_neg (ne_of_gt hx) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 196,
"column": 59
} | {
"line": 197,
"column": 43
} | [
{
"pp": "α : Type u_1\nl : Filter α\nf g : α → ℂ\nhl : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun x ↦ |(g x).im|\n⊢ (have this := fun x ↦ ‖f x‖ ^ (g x).re / 1;\n this) =ᶠ[l]\n have this := fun x ↦ ‖f x‖ ^ (g x).re;\n this",
"usedConstants": [
"Norm.norm",
"Real.instPow",
"Real",
... | by
simp only [div_one, EventuallyEq.rfl] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 208,
"column": 43
} | {
"line": 209,
"column": 43
} | [
{
"pp": "α : Type u_1\nl : Filter α\nf g : α → ℂ\nhl_im : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun x ↦ |(g x).im|\nhl : ∀ᶠ (x : α) in l, f x = 0 → (g x).re = 0 → g x = 0\n⊢ (fun x ↦ ‖f x‖ ^ (g x).re / 1) =ᶠ[l] fun x ↦ ‖f x‖ ^ (g x).re",
"usedConstants": [
"Norm.norm",
"Real.instPow",
"Re... | by
simp only [div_one, EventuallyEq.rfl] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 233,
"column": 32
} | {
"line": 233,
"column": 87
} | [
{
"pp": "α : Type u_1\nr c : ℝ\nl : Filter α\nf g : α → ℝ\nh : IsBigOWith c l f g\nhc : 0 ≤ c\nhr : 0 ≤ r\nhg : 0 ≤ᶠ[l] g\nx : α\nhgx : 0 x ≤ g x\nhx : ‖f x‖ ≤ c * ‖g x‖\n⊢ (c * |g x|) ^ r = c ^ r * |g x ^ r|",
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real",
"HMul.hMul",
"... | rw [mul_rpow hc (abs_nonneg _), abs_rpow_of_nonneg hgx] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 233,
"column": 32
} | {
"line": 233,
"column": 87
} | [
{
"pp": "α : Type u_1\nr c : ℝ\nl : Filter α\nf g : α → ℝ\nh : IsBigOWith c l f g\nhc : 0 ≤ c\nhr : 0 ≤ r\nhg : 0 ≤ᶠ[l] g\nx : α\nhgx : 0 x ≤ g x\nhx : ‖f x‖ ≤ c * ‖g x‖\n⊢ (c * |g x|) ^ r = c ^ r * |g x ^ r|",
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real",
"HMul.hMul",
"... | rw [mul_rpow hc (abs_nonneg _), abs_rpow_of_nonneg hgx] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 233,
"column": 32
} | {
"line": 233,
"column": 87
} | [
{
"pp": "α : Type u_1\nr c : ℝ\nl : Filter α\nf g : α → ℝ\nh : IsBigOWith c l f g\nhc : 0 ≤ c\nhr : 0 ≤ r\nhg : 0 ≤ᶠ[l] g\nx : α\nhgx : 0 x ≤ g x\nhx : ‖f x‖ ≤ c * ‖g x‖\n⊢ (c * |g x|) ^ r = c ^ r * |g x ^ r|",
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real",
"HMul.hMul",
"... | rw [mul_rpow hc (abs_nonneg _), abs_rpow_of_nonneg hgx] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 365,
"column": 4
} | {
"line": 365,
"column": 62
} | [
{
"pp": "case h₁\nx : ℝ\nhx : 0 ≤ x\ny : ℝ\nz : ℂ\n⊢ -π < (log ↑x * ↑y).im",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"Complex.log",
"Real.partialOrder",
"Real",
"Real.pi",
"HMul.hMul",
"Real.instZero",
"co... | rw [← ofReal_log hx, ← ofReal_mul, ofReal_im, neg_lt_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 477,
"column": 67
} | {
"line": 477,
"column": 75
} | [
{
"pp": "case pos\nx y z : ℝ\nhx : 0 ≤ x\nhy : 0 ≤ y\nh_ifs : x * y = 0\nh✝ : z = 0\n⊢ 1 = x ^ z * y ^ z",
"usedConstants": [
"Real.instPow",
"Real",
"HMul.hMul",
"Real.instZero",
"congrArg",
"Real.rpow_zero",
"Real.semiring",
"MulZeroOneClass.toMulOneClass",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.SpecialFunctions.Pow.Continuity | {
"line": 39,
"column": 24
} | {
"line": 39,
"column": 37
} | [
{
"pp": "b : ℂ\nhb : b ≠ 0\nthis : ∀ᶠ (x : ℂ) in 𝓝 b, x ≠ 0\nx : ℂ\nhx : x ≠ 0\n⊢ 0 = 0 x",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Complex.instZero",
"Pi.zero_apply",
"id",
"Pi.instZero",
"Zero.toOfNat0",
"Complex",
"OfNat.ofNat",
"Eq"
]
... | Pi.zero_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 477,
"column": 67
} | {
"line": 477,
"column": 75
} | [
{
"pp": "case neg\nx y z : ℝ\nhx : 0 ≤ x\nhy : 0 ≤ y\nh_ifs : x * y = 0\nh✝ : ¬z = 0\n⊢ 0 = x ^ z * y ^ z",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Real.instPow",
"False",
"Real",
"HMul.hMul",
"Real.rpow_eq_zero._simp_1",
"eq_false",
"MulZeroClass.t... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 477,
"column": 67
} | {
"line": 477,
"column": 75
} | [
{
"pp": "case neg\nx y z : ℝ\nhx : 0 ≤ x\nhy : 0 ≤ y\nh_ifs : ¬x * y = 0\n⊢ rexp (log (x * y) * z) = x ^ z * y ^ z",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Real",
"HMul.hMul",
"Real.instZero",
"congrArg",
"NormedDivisionRing.toNormMulClass",
"Real.semiri... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 477,
"column": 67
} | {
"line": 477,
"column": 75
} | [
{
"pp": "case pos\nx y z : ℝ\nhx : 0 ≤ x\nhy : 0 ≤ y\nh_ifs✝ : ¬x = 0 ∧ ¬y = 0\nh_ifs : x = 0\nh✝ : z = 0\n⊢ rexp (log (x * y) * z) = 1 * y ^ z",
"usedConstants": [
"Real.instPow",
"False",
"Real",
"HMul.hMul",
"Real.instZero",
"congrArg",
"False.elim",
"false... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.SpecialFunctions.Pow.Continuity | {
"line": 44,
"column": 2
} | {
"line": 48,
"column": 42
} | [
{
"pp": "a b : ℂ\nha : a ≠ 0\n⊢ (fun x ↦ x ^ b) =ᶠ[𝓝 a] fun x ↦ cexp (Complex.log x * b)",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Complex.log",
"False",
"HMul.hMul",
"congrArg",
"IsOpen.eventually_mem",
"Filter.Eventually",
... | suffices ∀ᶠ x : ℂ in 𝓝 a, x ≠ 0 from
this.mono fun x hx ↦ by
dsimp only
rw [cpow_def_of_ne_zero hx]
exact IsOpen.eventually_mem isOpen_ne ha | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.Continuity | {
"line": 44,
"column": 2
} | {
"line": 48,
"column": 42
} | [
{
"pp": "a b : ℂ\nha : a ≠ 0\n⊢ (fun x ↦ x ^ b) =ᶠ[𝓝 a] fun x ↦ cexp (Complex.log x * b)",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Complex.log",
"False",
"HMul.hMul",
"congrArg",
"IsOpen.eventually_mem",
"Filter.Eventually",
... | suffices ∀ᶠ x : ℂ in 𝓝 a, x ≠ 0 from
this.mono fun x hx ↦ by
dsimp only
rw [cpow_def_of_ne_zero hx]
exact IsOpen.eventually_mem isOpen_ne ha | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 477,
"column": 67
} | {
"line": 477,
"column": 75
} | [
{
"pp": "case neg\nx y z : ℝ\nhx : 0 ≤ x\nhy : 0 ≤ y\nh_ifs✝ : ¬x = 0 ∧ ¬y = 0\nh_ifs : x = 0\nh✝ : ¬z = 0\n⊢ rexp (log (x * y) * z) = 0 * y ^ z",
"usedConstants": [
"Real.instPow",
"False",
"Real",
"HMul.hMul",
"Real.instZero",
"congrArg",
"False.elim",
"fals... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 477,
"column": 67
} | {
"line": 477,
"column": 75
} | [
{
"pp": "case neg\nx y z : ℝ\nhx : 0 ≤ x\nhy : 0 ≤ y\nh_ifs✝ : ¬x = 0 ∧ ¬y = 0\nh_ifs : ¬x = 0\n⊢ rexp (log (x * y) * z) = rexp (log x * z) * y ^ z",
"usedConstants": [
"False",
"Real",
"eq_false",
"Real.instZero",
"congrArg",
"Real.semiring",
"Eq.mp",
"And",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.SpecialFunctions.Pow.Continuity | {
"line": 436,
"column": 66
} | {
"line": 436,
"column": 74
} | [
{
"pp": "r : ℝ\ns : Set ℝ≥0\nh : 0 ∉ s ∨ 0 ≤ r\nx✝ : 0 ∉ s\n⊢ s ⊆ {0}ᶜ",
"usedConstants": [
"False",
"eq_false",
"congrArg",
"Compl.compl",
"Membership.mem",
"Set.instSingletonSet",
"NNReal",
"HasSubset.Subset",
"Set.instCompl",
"NNReal.instZero",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.SpecialFunctions.Pow.Continuity | {
"line": 436,
"column": 66
} | {
"line": 436,
"column": 74
} | [
{
"pp": "r : ℝ\ns : Set ℝ≥0\nh : 0 ∉ s ∨ 0 ≤ r\nx✝ : 0 ∉ s\n⊢ s ⊆ {0}ᶜ",
"usedConstants": [
"False",
"eq_false",
"congrArg",
"Compl.compl",
"Membership.mem",
"Set.instSingletonSet",
"NNReal",
"HasSubset.Subset",
"Set.instCompl",
"NNReal.instZero",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.Continuity | {
"line": 436,
"column": 66
} | {
"line": 436,
"column": 74
} | [
{
"pp": "r : ℝ\ns : Set ℝ≥0\nh : 0 ∉ s ∨ 0 ≤ r\nx✝ : 0 ∉ s\n⊢ s ⊆ {0}ᶜ",
"usedConstants": [
"False",
"eq_false",
"congrArg",
"Compl.compl",
"Membership.mem",
"Set.instSingletonSet",
"NNReal",
"HasSubset.Subset",
"Set.instCompl",
"NNReal.instZero",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 985,
"column": 22
} | {
"line": 985,
"column": 34
} | [
{
"pp": "case inl\nx : ℝ\nh : 0 ≤ x\n⊢ x = (x ^ (1 / 2)) ^ ↑2",
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real",
"instHDiv",
"HMul.hMul",
"congrArg",
"Real.instDivInvMonoid",
"Nat.instAtLeastTwoHAddOfNat",
"id",
"HDiv.hDiv",
"instOfNatNat... | ← rpow_mul h | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 646,
"column": 4
} | {
"line": 649,
"column": 71
} | [
{
"pp": "case inl\ny z : ℝ\nhyz : 0 < y + z\nhx : 0 ≠ ∞\n⊢ 0 ^ (y + z) = 0 ^ y * 0 ^ z",
"usedConstants": [
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Tactic.Ring.Com... | by_cases hy' : 0 < y
· simp [ENNReal.zero_rpow_of_pos hyz, ENNReal.zero_rpow_of_pos hy']
· have hz' : 0 < z := by linarith
simp [ENNReal.zero_rpow_of_pos hyz, ENNReal.zero_rpow_of_pos hz'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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