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.Analysis.Complex.Basic | {
"line": 682,
"column": 2
} | {
"line": 682,
"column": 13
} | [
{
"pp": "case h\nz : ℂ\nhz : z ∈ Metric.ball 1 1\n⊢ 0 < z.re",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Basic | {
"line": 695,
"column": 34
} | {
"line": 695,
"column": 45
} | [
{
"pp": "r : ℝ\ns : Set ℂ\nhs : s ⊆ sphere 0 r\nhr : -↑r ∈ s\n⊢ -↑r ≤ 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"Real.instLE",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.instZero",
"AddGroupWithO... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Basic | {
"line": 697,
"column": 23
} | {
"line": 697,
"column": 34
} | [
{
"pp": "r : ℝ\ns : Set ℂ\nhs : s ⊆ sphere 0 r\nz : ℂ\nhzs : z ∈ s\nhz : z ≤ 0\n⊢ ‖z‖ = r",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Basic | {
"line": 717,
"column": 36
} | {
"line": 717,
"column": 53
} | [
{
"pp": "A : Type u_1\ninst✝⁴ : SeminormedAddCommGroup A\ninst✝³ : StarAddMonoid A\ninst✝² : NormedSpace ℂ A\ninst✝¹ : StarModule ℂ A\ninst✝ : NormedStarGroup A\nx : A\n⊢ ‖x‖ + ‖star x‖ ≤ 2 * ‖x‖",
"usedConstants": [
"norm_star",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"GroupWithZe... | by simp [two_mul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.Basic | {
"line": 721,
"column": 18
} | {
"line": 722,
"column": 86
} | [
{
"pp": "A : Type u_1\ninst✝⁴ : SeminormedAddCommGroup A\ninst✝³ : StarAddMonoid A\ninst✝² : NormedSpace ℂ A\ninst✝¹ : StarModule ℂ A\ninst✝ : NormedStarGroup A\nx : A\n⊢ ‖realPart (Complex.I • -x)‖ ≤ ‖x‖",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
"Ne... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Complex.Module | {
"line": 417,
"column": 2
} | {
"line": 418,
"column": 85
} | [
{
"pp": "A : Type u_1\ninst✝³ : AddCommGroup A\ninst✝² : Module ℂ A\ninst✝¹ : StarAddMonoid A\ninst✝ : StarModule ℂ A\na : A\n⊢ ↑(ℜ a) + I • ↑(ℑ a) = a",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"instTrivialStarReal",
"neg_smul",
"Real",
"instHSMul",
"NonU... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Complex.Module | {
"line": 506,
"column": 2
} | {
"line": 506,
"column": 34
} | [
{
"pp": "A : Type u_1\ninst✝³ : AddCommGroup A\ninst✝² : Module ℂ A\ninst✝¹ : StarAddMonoid A\ninst✝ : StarModule ℂ A\nx : A\n⊢ ℑ x = 0 ↔ IsSelfAdjoint x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Complex.Module | {
"line": 531,
"column": 2
} | {
"line": 532,
"column": 9
} | [
{
"pp": "z : ↥(selfAdjoint ℂ)\n⊢ ↑(selfAdjointEquiv z) = ↑z",
"usedConstants": [
"instTrivialStarReal",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.toModule",
"instStarRingReal",
"CommRing.toNonUnitalCommRing",
"Complex.commRing",
"selfA... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.OpenPartialHomeomorph.Basic | {
"line": 130,
"column": 20
} | {
"line": 130,
"column": 79
} | [
{
"pp": "X : Type u_1\nX' : Type u_2\nY : Type u_3\nY' : Type u_4\nZ : Type u_5\nZ' : Type u_6\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace X'\ninst✝³ : TopologicalSpace Y\ninst✝² : TopologicalSpace Y'\ninst✝¹ : TopologicalSpace Z\ninst✝ : TopologicalSpace Z'\ne✝ : OpenPartialHomeomorph X Y\ne : Part... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.OpenPartialHomeomorph.Basic | {
"line": 214,
"column": 4
} | {
"line": 214,
"column": 43
} | [
{
"pp": "X : Type u_1\nX' : Type u_2\nY : Type u_3\nY' : Type u_4\nZ : Type u_5\nZ' : Type u_6\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace X'\ninst✝³ : TopologicalSpace Y\ninst✝² : TopologicalSpace Y'\ninst✝¹ : TopologicalSpace Z\ninst✝ : TopologicalSpace Z'\ne : OpenPartialHomeomorph X Y\nh : e.sou... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.OpenPartialHomeomorph.Basic | {
"line": 216,
"column": 4
} | {
"line": 216,
"column": 44
} | [
{
"pp": "X : Type u_1\nX' : Type u_2\nY : Type u_3\nY' : Type u_4\nZ : Type u_5\nZ' : Type u_6\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace X'\ninst✝³ : TopologicalSpace Y\ninst✝² : TopologicalSpace Y'\ninst✝¹ : TopologicalSpace Z\ninst✝ : TopologicalSpace Z'\ne : OpenPartialHomeomorph X Y\nh : e.sou... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.OpenPartialHomeomorph.Continuity | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 48
} | [
{
"pp": "X : Type u_1\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ne : OpenPartialHomeomorph X Y\nx : X\nhx : x ∈ e.source\n⊢ Tendsto (↑e.symm) (𝓝 (↑e x)) (𝓝 x)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 299,
"column": 43
} | {
"line": 299,
"column": 58
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nh : I * I = -1\n⊢ im I = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 394,
"column": 2
} | {
"line": 394,
"column": 23
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nx : K\ny : ℝ\nhx : IsSelfAdjoint x\n⊢ y = re x ↔ ↑y = x",
"usedConstants": [
"Eq.mpr",
"Real",
"AddMonoid.toAddSemigroup",
"Real.instAddMonoid",
"congrArg",
"AddMonoid.toAddZeroClass",
"AddMonoid.toZero",
"NormedField.t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 680,
"column": 2
} | {
"line": 680,
"column": 38
} | [
{
"pp": "K : Type u_1\nE : Type u_2\ninst✝² : RCLike K\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace K E\nn : ℕ\nx : E\n⊢ ‖n • x‖ = n • ‖x‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"instHSMul",
"HMul.hMul",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 684,
"column": 30
} | {
"line": 684,
"column": 66
} | [
{
"pp": "K : Type u_1\nE : Type u_2\ninst✝² : RCLike K\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace K E\nn : ℕ\nx : E\n⊢ ‖n • x‖₊ = n • ‖x‖₊",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHSMul",
"HMul.hMul",
"congrArg",
"SeminormedA... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 711,
"column": 2
} | {
"line": 712,
"column": 9
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\na : K\nh : ‖a‖ ≤ re a\n⊢ im a = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 748,
"column": 23
} | {
"line": 748,
"column": 49
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nf : CauSeq K norm\nx✝ : ℝ\nε0 : x✝ > 0\ni : ℕ\nH : ∀ j ≥ i, ‖↑f j - ↑f i‖ < x✝\nj : ℕ\nij : j ≥ i\n⊢ |(fun n ↦ re (↑f n)) j - (fun n ↦ re (↑f n)) i| ≤ ‖↑f j - ↑f i‖",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Real",
"Norm... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 752,
"column": 23
} | {
"line": 752,
"column": 49
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nf : CauSeq K norm\nx✝ : ℝ\nε0 : x✝ > 0\ni : ℕ\nH : ∀ j ≥ i, ‖↑f j - ↑f i‖ < x✝\nj : ℕ\nij : j ≥ i\n⊢ |(fun n ↦ im (↑f n)) j - (fun n ↦ im (↑f n)) i| ≤ ‖↑f j - ↑f i‖",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Real",
"Norm... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 800,
"column": 2
} | {
"line": 800,
"column": 42
} | [
{
"pp": "K : Type u_1\nE : Type u_2\ninst✝³ : RCLike K\ninst✝² : NormedField E\ninst✝¹ : CharZero E\ninst✝ : NormedSpace K E\nq : ℚ≥0\nx : E\n⊢ ‖q • x‖ = q • ‖x‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 804,
"column": 2
} | {
"line": 804,
"column": 42
} | [
{
"pp": "K : Type u_1\nE : Type u_2\ninst✝³ : RCLike K\ninst✝² : NormedField E\ninst✝¹ : CharZero E\ninst✝ : NormedSpace K E\nq : ℚ≥0\nx : E\n⊢ ‖q • x‖₊ = q • ‖x‖₊",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 826,
"column": 2
} | {
"line": 826,
"column": 38
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ 0 ≤ z ↔ 0 ≤ re z ∧ im z = 0",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"AddMonoid.toAddSemigroup",
"Real.instZero",
"Real.instAddMonoid",
"congrArg",
"AddMonoid.toAddZeroClass",
"AddMonoid.toZe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 829,
"column": 2
} | {
"line": 829,
"column": 38
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ 0 < z ↔ 0 < re z ∧ im z = 0",
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder.toLT",
"AddMonoid.toAddSemigroup",
"Real.instZero",
"Real.instAddMonoid",
"congrArg",
"AddMonoid.toAddZeroClass",
"AddMonoid.to... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 832,
"column": 2
} | {
"line": 832,
"column": 29
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ z ≤ 0 ↔ re z ≤ 0 ∧ im z = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 835,
"column": 2
} | {
"line": 835,
"column": 29
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ z < 0 ↔ re z < 0 ∧ im z = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 835,
"column": 2
} | {
"line": 835,
"column": 60
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ z < 0 ↔ re z < 0 ∧ im z = 0",
"usedConstants": [
"Real",
"Preorder.toLT",
"AddMonoidHom.instAddMonoidHomClass",
"AddMonoid.toAddSemigroup",
"Real.instZero",
"Real.instAddMonoid",
"congrArg",
"AddMonoid.toAddZer... | simpa only [map_zero] using lt_iff_re_im (z := z) (w := 0) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.RCLike.Basic | {
"line": 835,
"column": 2
} | {
"line": 835,
"column": 60
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ z < 0 ↔ re z < 0 ∧ im z = 0",
"usedConstants": [
"Real",
"Preorder.toLT",
"AddMonoidHom.instAddMonoidHomClass",
"AddMonoid.toAddSemigroup",
"Real.instZero",
"Real.instAddMonoid",
"congrArg",
"AddMonoid.toAddZer... | simpa only [map_zero] using lt_iff_re_im (z := z) (w := 0) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.RCLike.Basic | {
"line": 835,
"column": 2
} | {
"line": 835,
"column": 60
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ z < 0 ↔ re z < 0 ∧ im z = 0",
"usedConstants": [
"Real",
"Preorder.toLT",
"AddMonoidHom.instAddMonoidHomClass",
"AddMonoid.toAddSemigroup",
"Real.instZero",
"Real.instAddMonoid",
"congrArg",
"AddMonoid.toAddZer... | simpa only [map_zero] using lt_iff_re_im (z := z) (w := 0) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.RCLike.Basic | {
"line": 885,
"column": 2
} | {
"line": 885,
"column": 41
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ re z ≤ -‖z‖ ↔ z = -↑‖z‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 921,
"column": 6
} | {
"line": 921,
"column": 63
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nx y : K\nhxy : re x ≤ re y ∧ im x = im y\nz : K\n⊢ re (z + x) ≤ re (z + y) ∧ im (z + x) = im (z + y)",
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real.instLE",
"Real",
"add_le_add_iff_left._simp_1",
"AddMonoidHom.instAddMon... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 931,
"column": 20
} | {
"line": 931,
"column": 49
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\n⊢ 0 ≤ 1",
"usedConstants": [
"RCLike.one_re",
"NormedCommRing.toNormedRing",
"Real.instLE",
"Real",
"AddMonoidHom.instAddMonoidHomClass",
"NormedRing.toRing",
"AddMonoid.toAddSemigroup",
"Real.instZero",
"Real.ins... | simp [@RCLike.le_iff_re_im K] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.RCLike.Basic | {
"line": 931,
"column": 20
} | {
"line": 931,
"column": 49
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\n⊢ 0 ≤ 1",
"usedConstants": [
"RCLike.one_re",
"NormedCommRing.toNormedRing",
"Real.instLE",
"Real",
"AddMonoidHom.instAddMonoidHomClass",
"NormedRing.toRing",
"AddMonoid.toAddSemigroup",
"Real.instZero",
"Real.ins... | simp [@RCLike.le_iff_re_im K] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.RCLike.Basic | {
"line": 931,
"column": 20
} | {
"line": 931,
"column": 49
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\n⊢ 0 ≤ 1",
"usedConstants": [
"RCLike.one_re",
"NormedCommRing.toNormedRing",
"Real.instLE",
"Real",
"AddMonoidHom.instAddMonoidHomClass",
"NormedRing.toRing",
"AddMonoid.toAddSemigroup",
"Real.instZero",
"Real.ins... | simp [@RCLike.le_iff_re_im K] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.RCLike.Basic | {
"line": 958,
"column": 56
} | {
"line": 958,
"column": 67
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\ny z : K\nr : ℝ\nhx : 0 ≤ re ↑r ∧ im ↑r = 0\nhyz : r * re y < r * re z ∧ (im y = im z ∨ r = 0)\n⊢ 0 ≤ r",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 966,
"column": 4
} | {
"line": 966,
"column": 76
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nr : ℝ\nhr : 0 < r\na b : K\nhab : a < b\n⊢ r • a < r • b",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"False",
"Real.partialOrder",
"Real",
"instHSMul",
"Preorder.toLT",
"RCLike.toNormedAlgebra"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 980,
"column": 4
} | {
"line": 981,
"column": 23
} | [
{
"pp": "case inr.inr\nK : Type u_1\ninst✝ : RCLike K\nx : ℝ\nz : K\nhx : 0 < x\n⊢ 0 < x * re z ∧ x * im z = 0 ↔ x < 0 ∧ re z < 0 ∧ im z = 0 ∨ 0 < x ∧ 0 < re z ∧ im z = 0",
"usedConstants": [
"NormedCommRing.toNormedRing",
"False",
"Real.partialOrder",
"Real",
"Preorder.toLT",
... | simp only [mul_pos_iff, hx, true_and, not_lt_of_gt hx, false_and, or_false, mul_eq_zero,
hx.ne', false_or] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.RCLike.Basic | {
"line": 980,
"column": 4
} | {
"line": 981,
"column": 23
} | [
{
"pp": "case inr.inr\nK : Type u_1\ninst✝ : RCLike K\nx : ℝ\nz : K\nhx : 0 < x\n⊢ 0 < x * re z ∧ x * im z = 0 ↔ x < 0 ∧ re z < 0 ∧ im z = 0 ∨ 0 < x ∧ 0 < re z ∧ im z = 0",
"usedConstants": [
"NormedCommRing.toNormedRing",
"False",
"Real.partialOrder",
"Real",
"Preorder.toLT",
... | simp only [mul_pos_iff, hx, true_and, not_lt_of_gt hx, false_and, or_false, mul_eq_zero,
hx.ne', false_or] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.RCLike.Basic | {
"line": 980,
"column": 4
} | {
"line": 981,
"column": 23
} | [
{
"pp": "case inr.inr\nK : Type u_1\ninst✝ : RCLike K\nx : ℝ\nz : K\nhx : 0 < x\n⊢ 0 < x * re z ∧ x * im z = 0 ↔ x < 0 ∧ re z < 0 ∧ im z = 0 ∨ 0 < x ∧ 0 < re z ∧ im z = 0",
"usedConstants": [
"NormedCommRing.toNormedRing",
"False",
"Real.partialOrder",
"Real",
"Preorder.toLT",
... | simp only [mul_pos_iff, hx, true_and, not_lt_of_gt hx, false_and, or_false, mul_eq_zero,
hx.ne', false_or] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.RCLike.Basic | {
"line": 985,
"column": 2
} | {
"line": 985,
"column": 54
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nx : ℝ\nz : K\n⊢ ↑x * z < 0 ↔ x < 0 ∧ 0 < z ∨ 0 < x ∧ z < 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 1259,
"column": 2
} | {
"line": 1259,
"column": 41
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ im z ≤ -‖z‖ ↔ z = -(I * ↑‖z‖)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 121,
"column": 2
} | {
"line": 128,
"column": 25
} | [
{
"pp": "case mp\nα : Type u_1\nE : Type u_3\nE''' : Type u_12\ninst✝¹ : Norm E\ninst✝ : SeminormedAddGroup E'''\nf : α → E\nl : Filter α\ng : α → E'''\nh : f =O[l] g\n⊢ ∃ c > 0, ∀ᶠ (x : α) in l, ‖f x‖ ≤ c * ‖g x‖",
"usedConstants": [
"le_max_right",
"Real.instIsOrderedRing",
"Norm.norm",
... | case mp =>
rw [isBigO_iff] at h
obtain ⟨c, hc⟩ := h
refine ⟨max c 1, zero_lt_one.trans_le (le_max_right _ _), ?_⟩
filter_upwards [hc] with x hx
apply hx.trans
gcongr
exact le_max_left _ _ | Lean.Elab.Tactic.evalCase | Lean.Parser.Tactic.case |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 155,
"column": 20
} | {
"line": 155,
"column": 46
} | [
{
"pp": "α : Type u_1\nE : Type u_3\nF : Type u_4\ninst✝¹ : Norm E\ninst✝ : Norm F\nf : α → E\ng : α → F\nl : Filter α\nh : ∀ᶠ (x : α) in l, ‖f x‖ ≤ ‖g x‖\n⊢ ∀ᶠ (x : α) in l, ‖f x‖ ≤ 1 * ‖g x‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"HMul.hMul",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 212,
"column": 2
} | {
"line": 212,
"column": 13
} | [
{
"pp": "α : Type u_1\nE : Type u_3\nF : Type u_4\ninst✝¹ : Norm E\ninst✝ : Norm F\nf : α → E\ng : α → F\nl : Filter α\nh : f =o[l] g\n⊢ ∀ᶠ (x : α) in l, ‖f x‖ ≤ ‖g x‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 281,
"column": 4
} | {
"line": 281,
"column": 69
} | [
{
"pp": "α : Type u_1\nE : Type u_3\nF' : Type u_7\ninst✝¹ : Norm E\ninst✝ : SeminormedAddCommGroup F'\nf : α → E\ng' : α → F'\nl : Filter α\nι : Sort u_18\np : ι → Prop\ns : ι → Set α\nh✝ : f =O[l] g'\nhb : l.HasBasis p s\nc : ℝ\nh : c > 0 ∧ IsBigOWith c l f g'\n⊢ c > 0 ∧ ∃ i, p i ∧ ∀ x ∈ s i, ‖f x‖ ≤ c * ‖g' ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Theta | {
"line": 209,
"column": 14
} | {
"line": 209,
"column": 40
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_14\n𝕜' : Type u_15\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nl : Filter α\nf : α → 𝕜\ng : α → 𝕜'\nh : (fun x ↦ (f x)⁻¹) =Θ[l] fun x ↦ (g x)⁻¹\n⊢ f =Θ[l] g",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Theta | {
"line": 213,
"column": 2
} | {
"line": 213,
"column": 35
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_14\n𝕜' : Type u_15\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nl : Filter α\nf₁ f₂ : α → 𝕜\ng₁ g₂ : α → 𝕜'\nh₁ : f₁ =Θ[l] g₁\nh₂ : f₂ =Θ[l] g₂\n⊢ (fun x ↦ f₁ x / f₂ x) =Θ[l] fun x ↦ g₁ x / g₂ x",
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Theta | {
"line": 222,
"column": 4
} | {
"line": 222,
"column": 57
} | [
{
"pp": "case ofNat\nα : Type u_1\n𝕜 : Type u_14\n𝕜' : Type u_15\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nl : Filter α\nf : α → 𝕜\ng : α → 𝕜'\nh : f =Θ[l] g\na✝ : ℕ\n⊢ (fun x ↦ f x ^ Int.ofNat a✝) =Θ[l] fun x ↦ g x ^ Int.ofNat a✝",
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Theta | {
"line": 223,
"column": 4
} | {
"line": 223,
"column": 35
} | [
{
"pp": "case negSucc\nα : Type u_1\n𝕜 : Type u_14\n𝕜' : Type u_15\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nl : Filter α\nf : α → 𝕜\ng : α → 𝕜'\nh : f =Θ[l] g\na✝ : ℕ\n⊢ (fun x ↦ f x ^ Int.negSucc a✝) =Θ[l] fun x ↦ g x ^ Int.negSucc a✝",
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Theta | {
"line": 232,
"column": 2
} | {
"line": 232,
"column": 63
} | [
{
"pp": "α : Type u_1\nE'' : Type u_9\nF'' : Type u_10\ninst✝² : NormedAddCommGroup E''\ninst✝¹ : NormedAddCommGroup F''\nl : Filter α\ninst✝ : l.NeBot\nc₁ : E''\nc₂ : F''\n⊢ ((fun x ↦ c₁) =Θ[l] fun x ↦ c₂) ↔ (c₁ = 0 ↔ c₂ = 0)",
"usedConstants": [
"_private.Mathlib.Analysis.Asymptotics.Theta.0.Asympto... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Theta | {
"line": 256,
"column": 2
} | {
"line": 256,
"column": 34
} | [
{
"pp": "α : Type u_1\nF : Type u_4\n𝕜 : Type u_14\ninst✝¹ : Norm F\ninst✝ : NormedField 𝕜\ng : α → F\nl : Filter α\nc : 𝕜\nf : α → 𝕜\nhc : c ≠ 0\n⊢ (fun x ↦ c * f x) =Θ[l] g ↔ f =Θ[l] g",
"usedConstants": [
"HMul.hMul",
"NormedField.toField",
"NormedField.toNorm",
"id",
"A... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Theta | {
"line": 262,
"column": 2
} | {
"line": 262,
"column": 34
} | [
{
"pp": "α : Type u_1\nE : Type u_3\n𝕜 : Type u_14\ninst✝¹ : Norm E\ninst✝ : NormedField 𝕜\nf : α → E\nl : Filter α\nc : 𝕜\ng : α → 𝕜\nhc : c ≠ 0\n⊢ (f =Θ[l] fun x ↦ c * g x) ↔ f =Θ[l] g",
"usedConstants": [
"HMul.hMul",
"NormedField.toField",
"NormedField.toNorm",
"id",
"A... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Theta | {
"line": 261,
"column": 45
} | {
"line": 262,
"column": 62
} | [
{
"pp": "α : Type u_1\nE : Type u_3\n𝕜 : Type u_14\ninst✝¹ : Norm E\ninst✝ : NormedField 𝕜\nf : α → E\nl : Filter α\nc : 𝕜\ng : α → 𝕜\nhc : c ≠ 0\n⊢ (f =Θ[l] fun x ↦ c * g x) ↔ f =Θ[l] g",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"HMul.hMul",
"NormedField.toField",
... | by
simpa only [← smul_eq_mul] using isTheta_const_smul_right hc | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 847,
"column": 2
} | {
"line": 847,
"column": 42
} | [
{
"pp": "α : Type u_1\nE' : Type u_6\nF' : Type u_7\ninst✝¹ : SeminormedAddCommGroup E'\ninst✝ : SeminormedAddCommGroup F'\nl : Filter α\nf' : α → E' × F'\n⊢ (fun x ↦ (f' x).1) =O[l] f'",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"HMul.hMul",
"Asymp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 850,
"column": 2
} | {
"line": 850,
"column": 42
} | [
{
"pp": "α : Type u_1\nE' : Type u_6\nF' : Type u_7\ninst✝¹ : SeminormedAddCommGroup E'\ninst✝ : SeminormedAddCommGroup F'\nl : Filter α\nf' : α → E' × F'\n⊢ (fun x ↦ (f' x).2) =O[l] f'",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"HMul.hMul",
"Asymp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Asymptotics | {
"line": 24,
"column": 22
} | {
"line": 24,
"column": 33
} | [
{
"pp": "α : Type u_1\nf : α → ℝ\nl : Filter α\n⊢ (fun x ↦ ‖↑(f x)‖) =Θ[l] f",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"Real.lattice",
"abs",
"congrArg",
"Complex.instNormedField",
"Complex.instNorm",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 971,
"column": 66
} | {
"line": 971,
"column": 82
} | [
{
"pp": "α : Type u_1\nE'' : Type u_9\nF'' : Type u_10\ninst✝¹ : NormedAddCommGroup E''\ninst✝ : NormedAddCommGroup F''\nc : ℝ\nf'' : α → E''\ng'' : α → F''\nl : Filter α\nh : IsBigOWith c l f'' g''\nx : α\nhx : ‖f'' x‖ ≤ c * ‖g'' x‖\nhg : g'' x = 0\n⊢ ‖f'' x‖ ≤ 0",
"usedConstants": [
"AddGroup.toSubt... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 159,
"column": 55
} | {
"line": 159,
"column": 68
} | [
{
"pp": "α : Type u_1\nE'' : Type u_9\nF'' : Type u_10\ninst✝¹ : NormedAddCommGroup E''\ninst✝ : NormedAddCommGroup F''\nf'' : α → E''\nl : Filter α\nc : F''\nh : f'' =O[l] fun _x ↦ c\nhc : c = 0\n⊢ f'' =O[l] fun _x ↦ 0",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Asymptotics.IsBigO",
... | by rwa [← hc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 1040,
"column": 2
} | {
"line": 1040,
"column": 35
} | [
{
"pp": "α : Type u_1\nF : Type u_4\nE' : Type u_6\ninst✝¹ : Norm F\ninst✝ : SeminormedAddCommGroup E'\nc₁ c₂ : ℝ\ng : α → F\nl : Filter α\nf₁ f₂ : α → E'\nh₁ : IsBigOWith c₁ l f₁ g\nh₂ : IsBigOWith c₂ l f₂ g\n⊢ IsBigOWith (c₁ + c₂) l (fun x ↦ f₁ x - f₂ x) g",
"usedConstants": [
"Eq.mpr",
"Real"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 1044,
"column": 2
} | {
"line": 1044,
"column": 35
} | [
{
"pp": "α : Type u_1\nF : Type u_4\nE' : Type u_6\ninst✝¹ : Norm F\ninst✝ : SeminormedAddCommGroup E'\nc₁ c₂ : ℝ\ng : α → F\nl : Filter α\nf₁ f₂ : α → E'\nh₁ : IsBigOWith c₁ l f₁ g\nh₂ : f₂ =o[l] g\nhc : c₁ < c₂\n⊢ IsBigOWith c₂ l (fun x ↦ f₁ x - f₂ x) g",
"usedConstants": [
"Eq.mpr",
"Asymptot... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 1047,
"column": 2
} | {
"line": 1047,
"column": 35
} | [
{
"pp": "α : Type u_1\nF : Type u_4\nE' : Type u_6\ninst✝¹ : Norm F\ninst✝ : SeminormedAddCommGroup E'\ng : α → F\nl : Filter α\nf₁ f₂ : α → E'\nh₁ : f₁ =O[l] g\nh₂ : f₂ =O[l] g\n⊢ (fun x ↦ f₁ x - f₂ x) =O[l] g",
"usedConstants": [
"Eq.mpr",
"congrArg",
"AddMonoid.toAddZeroClass",
"A... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 1050,
"column": 2
} | {
"line": 1050,
"column": 35
} | [
{
"pp": "α : Type u_1\nF : Type u_4\nE' : Type u_6\ninst✝¹ : Norm F\ninst✝ : SeminormedAddCommGroup E'\ng : α → F\nl : Filter α\nf₁ f₂ : α → E'\nh₁ : f₁ =o[l] g\nh₂ : f₂ =o[l] g\n⊢ (fun x ↦ f₁ x - f₂ x) =o[l] g",
"usedConstants": [
"Eq.mpr",
"congrArg",
"AddMonoid.toAddZeroClass",
"s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 1102,
"column": 2
} | {
"line": 1102,
"column": 28
} | [
{
"pp": "α : Type u_1\nF : Type u_4\nE' : Type u_6\ninst✝¹ : Norm F\ninst✝ : SeminormedAddCommGroup E'\ng : α → F\nl : Filter α\nf₁ f₂ : α → E'\nh : (fun x ↦ f₁ x - f₂ x) =o[l] g\n⊢ (fun x ↦ f₂ x - f₁ x) =o[l] g",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 1139,
"column": 26
} | {
"line": 1139,
"column": 37
} | [
{
"pp": "α : Type u_1\nE' : Type u_6\nF' : Type u_7\ninst✝¹ : SeminormedAddCommGroup E'\ninst✝ : SeminormedAddCommGroup F'\ng' : α → F'\nl : Filter α\nc : ℝ\nhc : 0 < c\nx : α\n⊢ x ∈ {x | (fun x ↦ ‖0‖ ≤ c * ‖g' x‖) x}",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 1142,
"column": 47
} | {
"line": 1142,
"column": 58
} | [
{
"pp": "α : Type u_1\nE' : Type u_6\nF' : Type u_7\ninst✝¹ : SeminormedAddCommGroup E'\ninst✝ : SeminormedAddCommGroup F'\nc : ℝ\ng' : α → F'\nl : Filter α\nhc : 0 ≤ c\nx : α\n⊢ x ∈ {x | (fun x ↦ ‖0‖ ≤ c * ‖g' x‖) x}",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 367,
"column": 8
} | {
"line": 367,
"column": 41
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_15\ninst✝ : NormedDivisionRing 𝕜\nl : Filter α\nf g : α → 𝕜\nh : f =o[l] g\n⊢ (fun x ↦ f x / g x) =o[l] fun x ↦ g x / g x",
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 1347,
"column": 4
} | {
"line": 1347,
"column": 15
} | [
{
"pp": "α : Type u_1\nR : Type u_13\ninst✝³ : SeminormedRing R\nS : Type u_17\ninst✝² : NormedRing S\ninst✝¹ : NormMulClass S\nc : ℝ\nl : Filter α\ninst✝ : NormOneClass S\nf : α → R\ng : α → S\nh : IsBigOWith c l f g\nthis : Nontrivial S\n⊢ IsBigOWith (Nat.casesOn 0 ‖1‖ fun n ↦ c ^ (n + 1)) l (fun x ↦ f x ^ 0)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 373,
"column": 2
} | {
"line": 373,
"column": 98
} | [
{
"pp": "α : Type u_1\nE' : Type u_6\n𝕜 : Type u_15\ninst✝³ : SeminormedAddCommGroup E'\ninst✝² : NormedDivisionRing 𝕜\ninst✝¹ : Module 𝕜 E'\ninst✝ : NormSMulClass 𝕜 E'\nf : α → E'\ng : α → 𝕜\nl : Filter α\nh : f =o[l] g\n⊢ Tendsto (fun x ↦ (g x)⁻¹ • f x) l (𝓝 0)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 1349,
"column": 16
} | {
"line": 1349,
"column": 38
} | [
{
"pp": "α : Type u_1\nR : Type u_13\ninst✝³ : SeminormedRing R\nS : Type u_17\ninst✝² : NormedRing S\ninst✝¹ : NormMulClass S\nc : ℝ\nl : Filter α\ninst✝ : NormOneClass S\nf : α → R\ng : α → S\nh : IsBigOWith c l f g\nn : ℕ\n⊢ IsBigOWith (Nat.casesOn (n + 2) ‖1‖ fun n ↦ c ^ (n + 1)) l (fun x ↦ f x ^ (n + 2)) f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 1354,
"column": 12
} | {
"line": 1354,
"column": 23
} | [
{
"pp": "α : Type u_1\nR : Type u_13\ninst✝⁴ : SeminormedRing R\nS : Type u_17\ninst✝³ : NormedRing S\ninst✝² : NormMulClass S\nc : ℝ\nl : Filter α\ninst✝¹ : NormOneClass R\ninst✝ : NormOneClass S\nf : α → R\ng : α → S\nh : IsBigOWith c l f g\n⊢ IsBigOWith (c ^ 0) l (fun x ↦ f x ^ 0) fun x ↦ g x ^ 0",
"used... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 463,
"column": 2
} | {
"line": 467,
"column": 8
} | [
{
"pp": "α : Type u_1\nR : Type u_13\ninst✝ : SeminormedRing R\nc : ℝ\nl : Filter α\nu v φ : α → R\nhφ : ∀ᶠ (x : α) in l, ‖φ x‖ ≤ c\nh : u =ᶠ[l] φ * v\n⊢ IsBigOWith c l u v",
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.norm",
"Eq.mpr",
"SeminormedRing.toNorm",
"Real.part... | simp only [IsBigOWith_def]
refine h.symm.rw (fun x a => ‖a‖ ≤ c * ‖v x‖) (hφ.mono fun x hx => ?_)
simp only [Pi.mul_apply]
refine (norm_mul_le _ _).trans ?_
gcongr | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 463,
"column": 2
} | {
"line": 467,
"column": 8
} | [
{
"pp": "α : Type u_1\nR : Type u_13\ninst✝ : SeminormedRing R\nc : ℝ\nl : Filter α\nu v φ : α → R\nhφ : ∀ᶠ (x : α) in l, ‖φ x‖ ≤ c\nh : u =ᶠ[l] φ * v\n⊢ IsBigOWith c l u v",
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.norm",
"Eq.mpr",
"SeminormedRing.toNorm",
"Real.part... | simp only [IsBigOWith_def]
refine h.symm.rw (fun x a => ‖a‖ ≤ c * ‖v x‖) (hφ.mono fun x hx => ?_)
simp only [Pi.mul_apply]
refine (norm_mul_le _ _).trans ?_
gcongr | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 475,
"column": 4
} | {
"line": 475,
"column": 15
} | [
{
"pp": "case h\nα : Type u_1\n𝕜 : Type u_15\ninst✝ : NormedDivisionRing 𝕜\nc : ℝ\nl : Filter α\nu v : α → 𝕜\nhc : 0 ≤ c\nh : IsBigOWith c l u v\ny : α\nhy : ‖u y‖ ≤ c * ‖v y‖\n⊢ ‖(fun x ↦ u x / v x) y‖ ≤ c",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 1393,
"column": 4
} | {
"line": 1393,
"column": 74
} | [
{
"pp": "case inr\nα : Type u_1\n𝕜 : Type u_15\n𝕜' : Type u_16\ninst✝¹ : NormedDivisionRing 𝕜\ninst✝ : NormedDivisionRing 𝕜'\nc : ℝ\nl : Filter α\nf : α → 𝕜\ng : α → 𝕜'\nh : IsBigOWith c l f g\nh₀✝ : ∀ᶠ (x : α) in l, f x = 0 → g x = 0\nx : α\nh₀ : f x = 0 → g x = 0\nhx : f x ≠ 0\nhc : 0 < c\nhle : (c * ‖g... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 597,
"column": 2
} | {
"line": 597,
"column": 50
} | [
{
"pp": "E' : Type u_6\ninst✝ : SeminormedAddCommGroup E'\nx₀ : E'\nm : ℕ\nh : 1 < m\n⊢ (fun x ↦ ‖x - x₀‖ ^ m) =o[𝓝 x₀] fun x ↦ x - x₀",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 632,
"column": 63
} | {
"line": 632,
"column": 74
} | [
{
"pp": "α : Type u_1\nE : Type u_3\nF'' : Type u_10\ninst✝¹ : Norm E\ninst✝ : NormedAddCommGroup F''\nf : α → E\ng'' : α → F''\nh : f =O[cofinite] g''\nC : ℝ\nC₀ : C > 0\nhC : {x | ¬‖f x‖ ≤ C * ‖g'' x‖}.Finite\nC' : ℝ\nhC' : ∀ i ∈ Finset.image (fun x ↦ ‖f x‖ / ‖g'' x‖) hC.toFinset, i ≤ C'\n⊢ ∀ (x : α), C * ‖g'... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 710,
"column": 2
} | {
"line": 711,
"column": 80
} | [
{
"pp": "case refine_1\nα : Type u_1\n𝕜 : Type u_15\ninst✝ : NormedDivisionRing 𝕜\nl : Filter α\nf g h✝ : α → 𝕜\nc : ℝ\nhf : ∀ᶠ (x : α) in l, f x ≠ 0\nh : ∀ᶠ (x : α) in l, ‖f x * g x‖ ≤ c * ‖h✝ x‖\n⊢ ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h✝ x / f x‖",
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtrac... | · refine h.congr <| Eventually.mp hf <| Eventually.of_forall fun x hx ↦ ?_
rw [norm_mul, norm_div, ← mul_div_assoc, le_div_iff₀' (norm_pos_iff.mpr hx)] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Order.CauSeq.BigOperators | {
"line": 76,
"column": 4
} | {
"line": 76,
"column": 15
} | [
{
"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 : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 710,
"column": 2
} | {
"line": 711,
"column": 80
} | [
{
"pp": "case refine_2\nα : Type u_1\n𝕜 : Type u_15\ninst✝ : NormedDivisionRing 𝕜\nl : Filter α\nf g h✝ : α → 𝕜\nc : ℝ\nhf : ∀ᶠ (x : α) in l, f x ≠ 0\nh : ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h✝ x / f x‖\n⊢ ∀ᶠ (x : α) in l, ‖f x * g x‖ ≤ c * ‖h✝ x‖",
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtrac... | · refine h.congr <| Eventually.mp hf <| Eventually.of_forall fun x hx ↦ ?_
rw [norm_mul, norm_div, ← mul_div_assoc, le_div_iff₀' (norm_pos_iff.mpr hx)] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 1512,
"column": 73
} | {
"line": 1512,
"column": 89
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_15\ninst✝ : NormedDivisionRing 𝕜\nc : ℝ\nl : Filter α\nu v : α → 𝕜\nh : IsBigOWith c l u v\ny : α\nhy : ‖u y‖ ≤ c * ‖v y‖\nhv : v y = 0\n⊢ u y = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.BigOperators | {
"line": 123,
"column": 4
} | {
"line": 123,
"column": 29
} | [
{
"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 : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 792,
"column": 2
} | {
"line": 792,
"column": 13
} | [
{
"pp": "F : Type u_1\nι : Type u_2\ninst✝³ : NormedRing F\ninst✝² : NormMulClass F\ninst✝¹ : NormOneClass F\ninst✝ : CompleteSpace F\nf g : ι → F\nhf : Summable fun n ↦ ‖f n‖\nc : F\nhg : Tendsto g cofinite (𝓝 c)\n⊢ (fun n ↦ f n * g n) =O[cofinite] fun n ↦ ‖f n‖",
"usedConstants": [
"Norm.norm",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 905,
"column": 2
} | {
"line": 905,
"column": 13
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\ninst✝³ : NormedDivisionRing 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : IsBoundedSMul 𝕜 E\nl : Filter ι\nε : ι → 𝕜\nf : ι → E\nhε : ε =o[l] fun _x ↦ 1\nhf : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l (norm ∘ f)\n⊢ (ε • f) =o[l] fun _x ↦ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Tactic.NormNum.NatFactorial | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 39
} | [
{
"pp": "case out\nn x l y z : ℕ\nh₁ : IsNat n x\nh₂ : IsNat l y\nh₃ : x = z + y\na : ℕ\np : (z + 1).ascFactorial y = a\n⊢ n.descFactorial l = ↑a",
"usedConstants": [
"Eq.mpr",
"AddMonoid.toAddSemigroup",
"congrArg",
"Nat.ascFactorial",
"id",
"AddMonoidWithOne.toNatCast",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Exponential | {
"line": 508,
"column": 8
} | {
"line": 508,
"column": 19
} | [
{
"pp": "case e_a.refine_2\nx : ℂ\nn j : ℕ\nhj : j ≥ n\na : ℕ\nha : a ∈ range (j - n)\nb : ℕ\nhb : b ∈ range (j - n)\nhab : (fun m hm ↦ m + n) a ha = (fun m hm ↦ m + n) b hb\n⊢ a = b",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Exponential | {
"line": 543,
"column": 2
} | {
"line": 543,
"column": 29
} | [
{
"pp": "x : ℝ\nh1 : 0 ≤ x\nh2 : x ≤ 1\nn : ℕ\nhn : 0 < n\nh3 : |x| = x\nh4 : |x| ≤ 1\nh' : |rexp x - ∑ m ∈ range n, x ^ m / ↑m.factorial| ≤ x ^ n * (↑n.succ / (↑n.factorial * ↑n))\nh'' : rexp x - ∑ m ∈ range n, x ^ m / ↑m.factorial ≤ x ^ n * (↑n.succ / (↑n.factorial * ↑n))\nt : rexp x ≤ ∑ m ∈ range n, x ^ m / ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Exponential | {
"line": 598,
"column": 37
} | {
"line": 598,
"column": 48
} | [
{
"pp": "n : ℕ\nx a b : ℝ\nm : ℕ\ne₁ : n + 1 = m\nh : |x| ≤ 1\ne : |1 - a| ≤ b - |x| / ↑m * ((↑m + 1) / ↑m)\n⊢ |1 + x / ↑m * 0 - a| ≤ b - |x| / ↑m * ((↑m + 1) / ↑m)",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"instHDiv",
"HMul.hMul",
"Real.lattice",
"Real.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Exponential | {
"line": 610,
"column": 26
} | {
"line": 610,
"column": 37
} | [
{
"pp": "x a b : ℝ\nh : |rexp x - expNear 0 x a| ≤ |x| ^ 0 / ↑(Nat.factorial 0) * b\n⊢ |rexp x - a| ≤ b",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Exponential | {
"line": 641,
"column": 2
} | {
"line": 642,
"column": 9
} | [
{
"pp": "case inr.inr\nx : ℝ\nhx✝ : x ≠ 0\nhx : x < 0\nh' : -x < 1\nhx' : 0 < x + 1\n⊢ x + 1 < rexp x",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instLT",
"id",
"Real.exp",
"Real.instAdd",
"Real.instOne",
"instHAdd",
"HAdd.hAdd",
"LT.lt",
"O... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Exponential | {
"line": 664,
"column": 28
} | {
"line": 664,
"column": 47
} | [
{
"pp": "n : ℕ\nt : ℝ\nht' : t ≤ ↑n\nhn : n ≠ 0\n⊢ rexp (-(t / ↑n)) ^ n = rexp (-t)",
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"HMul.hMul",
"congrArg",
"Real.instDivInvMonoid",
"id",
"HDiv.hDiv",
"Real.exp",
"Nat.cast",
"Monoid.toPow"... | ← Real.exp_nat_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.CauSeq.BigOperators | {
"line": 136,
"column": 19
} | {
"line": 136,
"column": 40
} | [
{
"pp": "case h.hbc.h₁\nα : 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 < ... | exact le_of_lt (hQ _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Order.CauSeq.BigOperators | {
"line": 136,
"column": 19
} | {
"line": 136,
"column": 40
} | [
{
"pp": "case h.hbc.h₂\nα : 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 < ... | exact le_of_lt (hQ _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Order.Field.Power | {
"line": 43,
"column": 68
} | {
"line": 43,
"column": 79
} | [
{
"pp": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : α\nk : ℤ\nh : k + k ≠ 0\n⊢ k ≠ 0",
"usedConstants": [
"id",
"Ne",
"Int",
"instOfNat",
"OfNat.ofNat"
]
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Field.Power | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 13
} | [
{
"pp": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : α\nn : ℤ\n⊢ a ^ n = -1 ↔ a = -1 ∧ Odd n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.InfiniteSum | {
"line": 37,
"column": 7
} | {
"line": 37,
"column": 44
} | [
{
"pp": "ι : Type u_2\nι' : Type u_3\nf : ι → ℝ\ng : ι' → ℝ\nhf : Summable f\nhg : Summable g\nhf' : 0 ≤ f\nhg' : 0 ≤ g\n⊢ Summable fun x ↦ ∑' (y : ι'), f (x, y).1 * g (x, y).2",
"usedConstants": [
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.InfiniteSum | {
"line": 179,
"column": 6
} | {
"line": 179,
"column": 22
} | [
{
"pp": "case refine_2\nf : ℕ → ℝ\nr : ℝ\nhr : Tendsto (fun n ↦ ∑ i ∈ range n, |f i|) atTop (𝓝 r)\n⊢ Tendsto (fun n ↦ ∑ i ∈ range n, ‖f i‖) atTop (𝓝 r)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.AtTopBot.ModEq | {
"line": 35,
"column": 2
} | {
"line": 35,
"column": 46
} | [
{
"pp": "d n : ℕ\nh : d < n\n⊢ ∃ᶠ (m : ℕ) in atTop, m % n = d",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.AtTopBot.ModEq | {
"line": 38,
"column": 2
} | {
"line": 38,
"column": 29
} | [
{
"pp": "⊢ ∃ᶠ (m : ℕ) in atTop, Even m",
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"Nat.instMod",
"instHMod",
"instOfNatNat",
"Filter.Frequently",
"Filter.atTop",
"funext",
"HMod.hMod",
"Nat.instPreorder",
"Nat",
"Even",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.AtTopBot.ModEq | {
"line": 41,
"column": 2
} | {
"line": 41,
"column": 28
} | [
{
"pp": "⊢ ∃ᶠ (m : ℕ) in atTop, Odd m",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Odd",
"id",
"Nat.instMod",
"instHMod",
"instOfNatNat",
"Filter.Frequently",
"Filter.atTop",
"funext",
"HMod.hMod",
"_private.Mathlib.Order.Filter.AtTopBot... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.BigOperators | {
"line": 163,
"column": 52
} | {
"line": 163,
"column": 69
} | [
{
"pp": "α : Type u_1\ninst✝³ : Field α\ninst✝² : LinearOrder α\ninst✝¹ : IsStrictOrderedRing α\ninst✝ : Archimedean α\nf : ℕ → α\na : α\nm : ℕ\nham : ∀ n ≥ m, |f n| ≤ a\nhnm : ∀ n ≥ m, f n.succ ≤ f n\nε : α\nε0 : ε > 0\nk : ℕ\nhk : a ≤ k • ε\nh : ∃ l, ∀ n ≥ m, a - l • ε < f n\nl : ℕ := Nat.find h\nhl : ∀ n ≥ m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.BigOperators | {
"line": 181,
"column": 52
} | {
"line": 181,
"column": 63
} | [
{
"pp": "α : Type u_1\ninst✝³ : Field α\ninst✝² : LinearOrder α\ninst✝¹ : IsStrictOrderedRing α\ninst✝ : Archimedean α\nf : ℕ → α\na : α\nm : ℕ\nham : ∀ n ≥ m, |f n| ≤ a\nhnm : ∀ n ≥ m, f n ≤ f n.succ\n⊢ ∀ n ≥ m, |(-f) n| ≤ a",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"abs_neg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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