module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.LinearAlgebra.Complex.Module | {
"line": 628,
"column": 2
} | {
"line": 628,
"column": 47
} | {
"line": 628,
"column": 48
} | [
{
"pp": "A : Type u_1\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : PartialOrder A\ninst✝² : StarOrderedRing A\ninst✝¹ : Module ℂ A\ninst✝ : StarModule ℂ A\na b : A\n⊢ a ≤ b ↔ ℜ a ≤ ℜ b ∧ ℑ a = ℑ b",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instTrivia... | [
"A : Type u_1\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : PartialOrder A\ninst✝² : StarOrderedRing A\ninst✝¹ : Module ℂ A\ninst✝ : StarModule ℂ A\na b : A\n⊢ a ≤ b ↔ ℜ a ≤ ℜ b ∧ ℑ b = ℑ a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 299,
"column": 43
} | {
"line": 299,
"column": 58
} | {
"line": 299,
"column": 59
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nh : I * I = -1\n⊢ im I = 1",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"K : Type u_1\ninst✝ : RCLike K\nh : I * I = -1\n⊢ im I = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.OpenPartialHomeomorph.Basic | {
"line": 130,
"column": 20
} | {
"line": 130,
"column": 79
} | {
"line": 130,
"column": 80
} | [
{
"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... | [
"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 : PartialEquiv X Y... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.OpenPartialHomeomorph.Basic | {
"line": 214,
"column": 4
} | {
"line": 214,
"column": 43
} | {
"line": 214,
"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... | [
"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.source = univ\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.OpenPartialHomeomorph.Basic | {
"line": 216,
"column": 4
} | {
"line": 216,
"column": 44
} | {
"line": 216,
"column": 45
} | [
{
"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... | [
"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.source = univ\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 394,
"column": 2
} | {
"line": 394,
"column": 23
} | {
"line": 394,
"column": 24
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nx : K\ny : ℝ\nhx : IsSelfAdjoint x\n⊢ y = re x ↔ ↑y = x",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"AddMonoid.toAddSemigroup",
"Real.instAddMonoid",
"congrArg",
"AddMonoid.toAddZeroClass",
... | [
"K : Type u_1\ninst✝ : RCLike K\nx : K\ny : ℝ\nhx : IsSelfAdjoint x\n⊢ y = re x ↔ x = ↑y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.OpenPartialHomeomorph.Continuity | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 48
} | {
"line": 71,
"column": 49
} | [
{
"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)",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 680,
"column": 2
} | {
"line": 680,
"column": 38
} | {
"line": 680,
"column": 39
} | [
{
"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‖",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real"... | [
"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‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 684,
"column": 30
} | {
"line": 684,
"column": 66
} | {
"line": 684,
"column": 67
} | [
{
"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‖₊",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHSMul",
"HMu... | [
"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‖₊"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 711,
"column": 2
} | {
"line": 712,
"column": 9
} | {
"line": 712,
"column": 10
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\na : K\nh : ‖a‖ ≤ re a\n⊢ im a = 0",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"K : Type u_1\ninst✝ : RCLike K\na : K\nh : ‖a‖ ≤ re a\n⊢ im a = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 751,
"column": 23
} | {
"line": 751,
"column": 49
} | {
"line": 751,
"column": 50
} | [
{
"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‖",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real",
... | [
"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⊢ |re (↑f j) - re (↑f i)| ≤ ‖↑f j - ↑f i‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 755,
"column": 23
} | {
"line": 755,
"column": 49
} | {
"line": 755,
"column": 50
} | [
{
"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‖",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real",
... | [
"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⊢ |im (↑f j) - im (↑f i)| ≤ ‖↑f j - ↑f i‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 121,
"column": 2
} | {
"line": 128,
"column": 25
} | {
"line": 129,
"column": 2
} | [
{
"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‖",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"le_max_right",
"R... | [
"case mpr\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 : ∃ c > 0, ∀ᶠ (x : α) in l, ‖f x‖ ≤ c * ‖g x‖\n⊢ f =O[l] g"
] | 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.RCLike.Basic | {
"line": 844,
"column": 2
} | {
"line": 844,
"column": 38
} | {
"line": 844,
"column": 39
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ 0 ≤ z ↔ 0 ≤ re z ∧ im z = 0",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"AddMonoid.toAddSemigroup",
"Real.instZero",
"Real.instAddMonoid",
"congrArg",
"Add... | [
"K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ 0 ≤ z ↔ 0 ≤ re z ∧ 0 = im z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 847,
"column": 2
} | {
"line": 847,
"column": 38
} | {
"line": 847,
"column": 39
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ 0 < z ↔ 0 < re z ∧ im z = 0",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder.toLT",
"AddMonoid.toAddSemigroup",
"Real.instZero",
"Real.instAddMonoid",
"congrArg",
"A... | [
"K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ 0 < z ↔ 0 < re z ∧ 0 = im z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 850,
"column": 2
} | {
"line": 850,
"column": 29
} | {
"line": 850,
"column": 30
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ z ≤ 0 ↔ re z ≤ 0 ∧ im z = 0",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ z ≤ 0 ↔ re z ≤ 0 ∧ im z = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 853,
"column": 2
} | {
"line": 853,
"column": 29
} | {
"line": 853,
"column": 30
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ z < 0 ↔ re z < 0 ∧ im z = 0",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ z < 0 ↔ re z < 0 ∧ im z = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 853,
"column": 2
} | {
"line": 853,
"column": 60
} | {
"line": 855,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ z < 0 ↔ re z < 0 ∧ im z = 0",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Real",
"Preorder.toLT",
"AddMonoidHom.instAddMonoidHomClass",
"AddMonoid.toAddSemigroup",
"Real.instZero",
"Real.instAddMonoid"... | [] | 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": 853,
"column": 2
} | {
"line": 853,
"column": 60
} | {
"line": 855,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ z < 0 ↔ re z < 0 ∧ im z = 0",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Real",
"Preorder.toLT",
"AddMonoidHom.instAddMonoidHomClass",
"AddMonoid.toAddSemigroup",
"Real.instZero",
"Real.instAddMonoid"... | [] | 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": 853,
"column": 2
} | {
"line": 853,
"column": 60
} | {
"line": 855,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ z < 0 ↔ re z < 0 ∧ im z = 0",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Real",
"Preorder.toLT",
"AddMonoidHom.instAddMonoidHomClass",
"AddMonoid.toAddSemigroup",
"Real.instZero",
"Real.instAddMonoid"... | [] | 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": 903,
"column": 2
} | {
"line": 903,
"column": 41
} | {
"line": 903,
"column": 42
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ re z ≤ -‖z‖ ↔ z = -↑‖z‖",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ re z ≤ -‖z‖ ↔ z = -↑‖z‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 939,
"column": 6
} | {
"line": 939,
"column": 63
} | {
"line": 939,
"column": 64
} | [
{
"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)",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real.instLE",
"Real",
"add_le_add_iff_... | [
"K : Type u_1\ninst✝ : RCLike K\nx y : K\nhxy : re x ≤ re y ∧ im x = im y\nz : K\n⊢ re x ≤ re y ∧ im x = im y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 949,
"column": 20
} | {
"line": 949,
"column": 49
} | {
"line": 951,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\n⊢ 0 ≤ 1",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"RCLike.one_re",
"Real.instLE",
"Real",
"AddMonoidHom.instAddMonoidHomClass",
"AddMonoid.toAddSemigroup",
"Real.instZero",
"Real.instAddMonoid",
"... | [] | simp [@RCLike.le_iff_re_im K] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.RCLike.Basic | {
"line": 949,
"column": 20
} | {
"line": 949,
"column": 49
} | {
"line": 951,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\n⊢ 0 ≤ 1",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"RCLike.one_re",
"Real.instLE",
"Real",
"AddMonoidHom.instAddMonoidHomClass",
"AddMonoid.toAddSemigroup",
"Real.instZero",
"Real.instAddMonoid",
"... | [] | simp [@RCLike.le_iff_re_im K] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.RCLike.Basic | {
"line": 949,
"column": 20
} | {
"line": 949,
"column": 49
} | {
"line": 951,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\n⊢ 0 ≤ 1",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"RCLike.one_re",
"Real.instLE",
"Real",
"AddMonoidHom.instAddMonoidHomClass",
"AddMonoid.toAddSemigroup",
"Real.instZero",
"Real.instAddMonoid",
"... | [] | simp [@RCLike.le_iff_re_im K] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.RCLike.Basic | {
"line": 976,
"column": 56
} | {
"line": 976,
"column": 67
} | {
"line": 976,
"column": 68
} | [
{
"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",
"ppTerm": "?m.126",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 984,
"column": 4
} | {
"line": 984,
"column": 76
} | {
"line": 984,
"column": 77
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nr : ℝ\nhr : 0 < r\na b : K\nhab : a < b\n⊢ r • a < r • b",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"False",
"Real.partialOrder",
"Real",
"instHSMul",
"P... | [
"K : Type u_1\ninst✝ : RCLike K\nr : ℝ\nhr : 0 < r\na b : K\nhab : a < b\n⊢ re a < re b ∧ im a = im b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 155,
"column": 20
} | {
"line": 155,
"column": 46
} | {
"line": 155,
"column": 47
} | [
{
"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‖",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.in... | [
"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‖ ≤ ‖g x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 998,
"column": 4
} | {
"line": 999,
"column": 23
} | {
"line": 1001,
"column": 0
} | [
{
"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",
"ppTerm": "?inr.inr",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"False",
"Real.part... | [] | 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": 998,
"column": 4
} | {
"line": 999,
"column": 23
} | {
"line": 1001,
"column": 0
} | [
{
"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",
"ppTerm": "?inr.inr",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"False",
"Real.part... | [] | 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": 998,
"column": 4
} | {
"line": 999,
"column": 23
} | {
"line": 1001,
"column": 0
} | [
{
"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",
"ppTerm": "?inr.inr",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"False",
"Real.part... | [] | 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": 1003,
"column": 2
} | {
"line": 1003,
"column": 54
} | {
"line": 1003,
"column": 55
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nx : ℝ\nz : K\n⊢ ↑x * z < 0 ↔ x < 0 ∧ 0 < z ∨ 0 < x ∧ z < 0",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"K : Type u_1\ninst✝ : RCLike K\nx : ℝ\nz : K\n⊢ ↑x * z < 0 ↔ x < 0 ∧ 0 < z ∨ 0 < x ∧ z < 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 212,
"column": 2
} | {
"line": 212,
"column": 13
} | {
"line": 212,
"column": 14
} | [
{
"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‖",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 281,
"column": 4
} | {
"line": 281,
"column": 69
} | {
"line": 281,
"column": 70
} | [
{
"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' ... | [
"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' x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 1277,
"column": 2
} | {
"line": 1277,
"column": 41
} | {
"line": 1277,
"column": 42
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ im z ≤ -‖z‖ ↔ z = -(I * ↑‖z‖)",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ im z ≤ -‖z‖ ↔ z = -(I * ↑‖z‖)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Theta | {
"line": 209,
"column": 14
} | {
"line": 209,
"column": 40
} | {
"line": 209,
"column": 41
} | [
{
"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",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals":... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Theta | {
"line": 213,
"column": 2
} | {
"line": 213,
"column": 35
} | {
"line": 213,
"column": 36
} | [
{
"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",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
... | [
"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)⁻¹) =Θ[l] fun x ↦ g₁ x * (g₂ x)⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Theta | {
"line": 222,
"column": 4
} | {
"line": 222,
"column": 57
} | {
"line": 222,
"column": 58
} | [
{
"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✝",
"ppTerm": "?ofNat",
"assigned": true,
"usedConstants":... | [
"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 ^ a✝) =Θ[l] fun x ↦ g x ^ a✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Theta | {
"line": 223,
"column": 4
} | {
"line": 223,
"column": 35
} | {
"line": 223,
"column": 36
} | [
{
"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✝",
"ppTerm": "?negSucc",
"assigned": true,
"usedCon... | [
"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 ^ (a✝ + 1))⁻¹) =Θ[l] fun x ↦ (g x ^ (a✝ + 1))⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Theta | {
"line": 232,
"column": 2
} | {
"line": 232,
"column": 63
} | {
"line": 232,
"column": 64
} | [
{
"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)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"_private... | [
"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⊢ (c₂ = 0 ↔ c₁ = 0) ↔ (c₁ = 0 ↔ c₂ = 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Theta | {
"line": 256,
"column": 2
} | {
"line": 256,
"column": 34
} | {
"line": 256,
"column": 35
} | [
{
"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",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"NormedField.toField",
... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Theta | {
"line": 262,
"column": 2
} | {
"line": 262,
"column": 34
} | {
"line": 262,
"column": 35
} | [
{
"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",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"NormedField.toField",
... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Theta | {
"line": 261,
"column": 45
} | {
"line": 262,
"column": 62
} | {
"line": 264,
"column": 0
} | [
{
"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",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
... | [] | by
simpa only [← smul_eq_mul] using isTheta_const_smul_right hc | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.Asymptotics | {
"line": 24,
"column": 22
} | {
"line": 24,
"column": 33
} | {
"line": 24,
"column": 34
} | [
{
"pp": "α : Type u_1\nf : α → ℝ\nl : Filter α\n⊢ (fun x ↦ ‖↑(f x)‖) =Θ[l] f",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"Real.lattice",
"abs",
"congrArg",
"Complex.ins... | [
"α : Type u_1\nf : α → ℝ\nl : Filter α\n⊢ (fun x ↦ |f x|) =Θ[l] f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 971,
"column": 66
} | {
"line": 971,
"column": 82
} | {
"line": 971,
"column": 83
} | [
{
"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",
"ppTerm": "?m.37",
"assigned": true,
... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 159,
"column": 55
} | {
"line": 159,
"column": 68
} | {
"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",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | by rwa [← hc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 1040,
"column": 2
} | {
"line": 1040,
"column": 35
} | {
"line": 1040,
"column": 36
} | [
{
"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",
"ppTerm": "?m.27",
"assigned": true,
"u... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 1044,
"column": 2
} | {
"line": 1044,
"column": 35
} | {
"line": 1044,
"column": 36
} | [
{
"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",
"ppTerm": "?m.25",
"assigned": true,
"used... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 1047,
"column": 2
} | {
"line": 1047,
"column": 35
} | {
"line": 1047,
"column": 36
} | [
{
"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",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 1050,
"column": 2
} | {
"line": 1050,
"column": 35
} | {
"line": 1050,
"column": 36
} | [
{
"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",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 1102,
"column": 2
} | {
"line": 1102,
"column": 28
} | {
"line": 1102,
"column": 29
} | [
{
"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",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"u... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 1139,
"column": 26
} | {
"line": 1139,
"column": 37
} | {
"line": 1139,
"column": 38
} | [
{
"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}",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Norm.norm",
... | [
"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⊢ 0 ≤ c * ‖g' x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 1142,
"column": 47
} | {
"line": 1142,
"column": 58
} | {
"line": 1142,
"column": 59
} | [
{
"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}",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Norm.norm",
... | [
"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⊢ 0 ≤ c * ‖g' x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 1347,
"column": 4
} | {
"line": 1347,
"column": 15
} | {
"line": 1347,
"column": 16
} | [
{
"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)... | [
"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 ‖1‖ l (fun x ↦ 1) fun x ↦ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 1349,
"column": 16
} | {
"line": 1349,
"column": 38
} | {
"line": 1349,
"column": 39
} | [
{
"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... | [
"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 (c ^ n * c * c) l (fun x ↦ f x ^ n * f x * f x) fun x ↦ g x ^ n * g x * g x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 1354,
"column": 12
} | {
"line": 1354,
"column": 23
} | {
"line": 1354,
"column": 24
} | [
{
"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",
"ppTe... | [
"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 1 l (fun x ↦ 1) fun x ↦ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 377,
"column": 8
} | {
"line": 377,
"column": 41
} | {
"line": 377,
"column": 42
} | [
{
"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",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
... | [
"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)⁻¹) =o[l] fun x ↦ g x * (g x)⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 383,
"column": 2
} | {
"line": 383,
"column": 98
} | {
"line": 384,
"column": 4
} | [
{
"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)",
"ppTerm": "?m.24",
"assigned": f... | [
"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)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.BigOperators | {
"line": 76,
"column": 4
} | {
"line": 76,
"column": 15
} | {
"line": 76,
"column": 16
} | [
{
"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 : ... | [
"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 : ∀ (i : ℕ), |... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 454,
"column": 2
} | {
"line": 457,
"column": 27
} | {
"line": 459,
"column": 0
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_17\ninst✝³ : NormedField 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\ninst✝ : OrderTopology 𝕜\nl : Filter α\nf g : α → 𝕜\nh : Tendsto (fun x ↦ g x / f x) l atBot\n⊢ f =o[l] g",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [] | refine IsLittleO.of_neg_left (IsLittleO.of_tendsto_div_atTop ?_)
rw [← tendsto_neg_atBot_iff]
convert h
simp [div_neg_eq_neg_div] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 454,
"column": 2
} | {
"line": 457,
"column": 27
} | {
"line": 459,
"column": 0
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_17\ninst✝³ : NormedField 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\ninst✝ : OrderTopology 𝕜\nl : Filter α\nf g : α → 𝕜\nh : Tendsto (fun x ↦ g x / f x) l atBot\n⊢ f =o[l] g",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [] | refine IsLittleO.of_neg_left (IsLittleO.of_tendsto_div_atTop ?_)
rw [← tendsto_neg_atBot_iff]
convert h
simp [div_neg_eq_neg_div] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 473,
"column": 2
} | {
"line": 477,
"column": 8
} | {
"line": 479,
"column": 0
} | [
{
"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",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.norm",
"Eq.mpr",
... | [] | 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": 473,
"column": 2
} | {
"line": 477,
"column": 8
} | {
"line": 479,
"column": 0
} | [
{
"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",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.norm",
"Eq.mpr",
... | [] | 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": 485,
"column": 4
} | {
"line": 485,
"column": 15
} | {
"line": 485,
"column": 16
} | [
{
"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",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
... | [
"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⊢ ‖u y‖ / ‖v y‖ ≤ c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 607,
"column": 2
} | {
"line": 607,
"column": 50
} | {
"line": 607,
"column": 51
} | [
{
"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",
"ppTerm": "?m.28",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 642,
"column": 63
} | {
"line": 642,
"column": 74
} | {
"line": 642,
"column": 75
} | [
{
"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'... | [
"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'' x‖ < ‖f x‖... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 1512,
"column": 73
} | {
"line": 1512,
"column": 89
} | {
"line": 1512,
"column": 90
} | [
{
"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",
"ppTerm": "?m.38",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Tactic.NormNum.BigOperators | {
"line": 258,
"column": 9
} | {
"line": 258,
"column": 20
} | {
"line": 258,
"column": 21
} | [
{
"pp": "α : Type u_1\ninst✝ : Fintype α\nelems : Finset α\ncomplete : ∀ (x : α), x ∈ elems\nx : α\n⊢ x ∈ Finset.univ ↔ x ∈ elems",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.univ",
"congrArg",
"Finset",
"true_iff",
"Membership.mem",... | [
"α : Type u_1\ninst✝ : Fintype α\nelems : Finset α\ncomplete : ∀ (x : α), x ∈ elems\nx : α\n⊢ x ∈ elems"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Tactic.NormNum.NatFactorial | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 39
} | {
"line": 94,
"column": 40
} | [
{
"pp": "n 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",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AddMonoid.toAddSemigroup",
"congrArg",
"Nat.ascFactorial",
"id",
... | [
"n 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⊢ (z + y).descFactorial y = (z + 1).ascFactorial y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.LinearIsometry | {
"line": 398,
"column": 6
} | {
"line": 398,
"column": 17
} | {
"line": 398,
"column": 18
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nR₃ : Type u_3\nR₄ : Type u_4\nE : Type u_5\nE₂ : Type u_6\nE₃ : Type u_7\nE₄ : Type u_8\nF : Type u_9\n𝓕 : Type u_10\ninst✝³³ : Semiring R\ninst✝³² : Semiring R₂\ninst✝³¹ : Semiring R₃\ninst✝³⁰ : Semiring R₄\nσ₁₂ : R →+* R₂\nσ₂₁ : R₂ →+* R\nσ₁₃ : R →+* R₃\nσ₃₁ : R₃ →+* R\n... | [
"R : Type u_1\nR₂ : Type u_2\nR₃ : Type u_3\nR₄ : Type u_4\nE : Type u_5\nE₂ : Type u_6\nE₃ : Type u_7\nE₄ : Type u_8\nF : Type u_9\n𝓕 : Type u_10\ninst✝³³ : Semiring R\ninst✝³² : Semiring R₂\ninst✝³¹ : Semiring R₃\ninst✝³⁰ : Semiring R₄\nσ₁₂ : R →+* R₂\nσ₂₁ : R₂ →+* R\nσ₁₃ : R →+* R₃\nσ₃₁ : R₃ →+* R\nσ₁₄ : R →+* ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.LinearIsometry | {
"line": 531,
"column": 40
} | {
"line": 531,
"column": 77
} | {
"line": 531,
"column": 78
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nR₃ : Type u_3\nR₄ : Type u_4\nE : Type u_5\nE₂ : Type u_6\nE₃ : Type u_7\nE₄ : Type u_8\nF : Type u_9\n𝓕 : Type u_10\ninst✝³³ : Semiring R\ninst✝³² : Semiring R₂\ninst✝³¹ : Semiring R₃\ninst✝³⁰ : Semiring R₄\nσ₁₂ : R →+* R₂\nσ₂₁ : R₂ →+* R\nσ₁₃ : R →+* R₃\nσ₃₁ : R₃ →+* R\n... | [
"R : Type u_1\nR₂ : Type u_2\nR₃ : Type u_3\nR₄ : Type u_4\nE : Type u_5\nE₂ : Type u_6\nE₃ : Type u_7\nE₄ : Type u_8\nF : Type u_9\n𝓕 : Type u_10\ninst✝³³ : Semiring R\ninst✝³² : Semiring R₂\ninst✝³¹ : Semiring R₃\ninst✝³⁰ : Semiring R₄\nσ₁₂ : R →+* R₂\nσ₂₁ : R₂ →+* R\nσ₁₃ : R →+* R₃\nσ₃₁ : R₃ →+* R\nσ₁₄ : R →+* ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 720,
"column": 2
} | {
"line": 721,
"column": 80
} | {
"line": 723,
"column": 0
} | [
{
"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‖",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstan... | [] | · 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.Lemmas | {
"line": 720,
"column": 2
} | {
"line": 721,
"column": 80
} | {
"line": 723,
"column": 0
} | [
{
"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‖",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstan... | [] | · 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.Lemmas | {
"line": 802,
"column": 2
} | {
"line": 802,
"column": 13
} | {
"line": 802,
"column": 14
} | [
{
"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‖",
"ppTerm": "?m.34",
"assigned": true,
... | [
"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] f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.BigOperators | {
"line": 123,
"column": 4
} | {
"line": 123,
"column": 29
} | {
"line": 123,
"column": 30
} | [
{
"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 : ... | [
"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 : ∀ (i : ℕ), |... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Field.Power | {
"line": 43,
"column": 68
} | {
"line": 43,
"column": 79
} | {
"line": 43,
"column": 80
} | [
{
"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",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"id",
"Ne",
"Int",
"instOfNat",
"OfNat.ofNat"
],
"usedFVars": [
"... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Field.Power | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 13
} | {
"line": 103,
"column": 14
} | [
{
"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",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.InfiniteSum | {
"line": 37,
"column": 7
} | {
"line": 37,
"column": 44
} | {
"line": 37,
"column": 45
} | [
{
"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",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnita... | [
"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 ↦ f x * ∑' (i : ι'), g i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Exponential | {
"line": 530,
"column": 2
} | {
"line": 530,
"column": 29
} | {
"line": 530,
"column": 30
} | [
{
"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 / ... | [
"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 / ↑m.factorial... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Exponential | {
"line": 585,
"column": 37
} | {
"line": 585,
"column": 48
} | {
"line": 585,
"column": 49
} | [
{
"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)",
"ppTerm": "?m.95",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"instHDiv",
"H... | [
"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 - a| ≤ b - |x| / ↑m * ((↑m + 1) / ↑m)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Exponential | {
"line": 597,
"column": 26
} | {
"line": 597,
"column": 37
} | {
"line": 597,
"column": 38
} | [
{
"pp": "x a b : ℝ\nh : |rexp x - expNear 0 x a| ≤ |x| ^ 0 / ↑(Nat.factorial 0) * b\n⊢ |rexp x - a| ≤ b",
"ppTerm": "?m.45",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x a b : ℝ\nh : |rexp x - expNear 0 x a| ≤ |x| ^ 0 / ↑(Nat.factorial 0) * b\n⊢ |rexp x - a| ≤ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.AtTopBot.ModEq | {
"line": 35,
"column": 2
} | {
"line": 35,
"column": 46
} | {
"line": 35,
"column": 47
} | [
{
"pp": "d n : ℕ\nh : d < n\n⊢ ∃ᶠ (m : ℕ) in atTop, m % n = d",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"d n : ℕ\nh : d < n\n⊢ ∃ᶠ (m : ℕ) in atTop, m % n = d"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.AtTopBot.ModEq | {
"line": 38,
"column": 2
} | {
"line": 38,
"column": 29
} | {
"line": 38,
"column": 30
} | [
{
"pp": "⊢ ∃ᶠ (m : ℕ) in atTop, Even m",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"Nat.instMod",
"instHMod",
"instOfNatNat",
"Filter.Frequently",
"Filter.atTop",
"funext",
"HMod.hMod",
"Nat.in... | [
"⊢ ∃ᶠ (m : ℕ) in atTop, m % 2 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.AtTopBot.ModEq | {
"line": 41,
"column": 2
} | {
"line": 41,
"column": 28
} | {
"line": 41,
"column": 29
} | [
{
"pp": "⊢ ∃ᶠ (m : ℕ) in atTop, Odd m",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Odd",
"id",
"Nat.instMod",
"instHMod",
"instOfNatNat",
"Filter.Frequently",
"Filter.atTop",
"funext",
"HMod.hMod",
... | [
"⊢ ∃ᶠ (m : ℕ) in atTop, m % 2 = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 551,
"column": 6
} | {
"line": 551,
"column": 31
} | {
"line": 552,
"column": 4
} | [
{
"pp": "x : ℂ\nhx : ‖x‖ ≤ 1\n⊢ ‖(cexp (x * I) + cexp (-(x * I))) / 2 - (1 - x ^ 2 / 2)‖ =\n ‖(cexp (-(x * I)) - (1 + -(x * I) + (x * I) ^ 2 / 2 + (-(x * I)) ^ 3 / 6)) / 2 +\n (cexp (x * I) - (1 + x * I + (x * I) ^ 2 / 2 + (x * I) ^ 3 / 6)) / 2‖",
"ppTerm": "?m.383",
"assigned": true,
"use... | [] | grind [I_sq, two_ne_zero] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.Analysis.Complex.Exponential | {
"line": 628,
"column": 2
} | {
"line": 629,
"column": 9
} | {
"line": 629,
"column": 10
} | [
{
"pp": "case inr.inr\nx : ℝ\nhx✝ : x ≠ 0\nhx : x < 0\nh' : -x < 1\nhx' : 0 < x + 1\n⊢ x + 1 < rexp x",
"ppTerm": "?inr.inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instLT",
"id",
"Real.exp",
"Real.instAdd",
"Real.instOne",
"instH... | [
"case inr.inr\nx : ℝ\nhx✝ : x ≠ 0\nhx : x < 0\nh' : -x < 1\nhx' : 0 < x + 1\n⊢ x + 1 < rexp x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Exponential | {
"line": 651,
"column": 28
} | {
"line": 651,
"column": 47
} | {
"line": 651,
"column": 48
} | [
{
"pp": "n : ℕ\nt : ℝ\nht' : t ≤ ↑n\nhn : n ≠ 0\n⊢ rexp (-(t / ↑n)) ^ n = rexp (-t)",
"ppTerm": "?m.86",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"HMul.hMul",
"congrArg",
"Real.instDivInvMonoid",
"id",
"HDiv.hDiv",
"Real... | [
"n : ℕ\nt : ℝ\nht' : t ≤ ↑n\nhn : n ≠ 0\n⊢ rexp (↑n * -(t / ↑n)) = rexp (-t)"
] | ← Real.exp_nat_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 567,
"column": 6
} | {
"line": 567,
"column": 31
} | {
"line": 568,
"column": 4
} | [
{
"pp": "x : ℂ\nhx : ‖x‖ ≤ 1\n⊢ ‖(cexp (-(x * I)) - cexp (x * I)) * I / 2 - (x - x ^ 3 / 6)‖ =\n ‖(cexp (-(x * I)) - (1 + -(x * I) + (x * I) ^ 2 / 2 + (-(x * I)) ^ 3 / 6 + (-(x * I)) ^ 4 / 24)) * I / 2 -\n (cexp (x * I) - (1 + x * I + (x * I) ^ 2 / 2 + (x * I) ^ 3 / 6 + (x * I) ^ 4 / 24)) * I / 2‖",
... | [] | grind [I_sq, two_ne_zero] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 727,
"column": 21
} | {
"line": 727,
"column": 32
} | {
"line": 727,
"column": 33
} | [
{
"pp": "x : ℝ\nhx : cos x ≠ 0\nthis : Complex.cos ↑x ≠ 0\n⊢ ↑(1 + tan x ^ 2)⁻¹ = ↑(cos x ^ 2)",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Complex.cos",
"Real.cos",
"congrArg",
"Real.instInv",
"Complex.ofReal_add",
"id"... | [
"x : ℝ\nhx : cos x ≠ 0\nthis : Complex.cos ↑x ≠ 0\n⊢ (1 + Complex.tan ↑x ^ 2)⁻¹ = Complex.cos ↑x ^ 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 874,
"column": 2
} | {
"line": 874,
"column": 56
} | {
"line": 874,
"column": 57
} | [
{
"pp": "x : ℝ\nhx : |x| ≤ 1\n⊢ |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96)",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"instHDiv",
"HMul.hMul",
"Real.lattice",
"Real.cos",
"abs",
... | [
"x : ℝ\nhx : |x| ≤ 1\n⊢ ‖↑(cos x) - (1 - ↑x ^ 2 / 2)‖ ≤ ‖↑x‖ ^ 4 * (5 / 96)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 877,
"column": 2
} | {
"line": 877,
"column": 56
} | {
"line": 877,
"column": 57
} | [
{
"pp": "x : ℝ\nhx : |x| ≤ 1\n⊢ |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 5 / 100",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"instHDiv",
"Real.lattice",
"abs",
"congrArg",
"Real.instDivInvMonoi... | [
"x : ℝ\nhx : |x| ≤ 1\n⊢ ‖↑(sin x) - (↑x - ↑x ^ 3 / 6)‖ ≤ ‖↑x‖ ^ 5 / 100"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.BigOperators | {
"line": 136,
"column": 19
} | {
"line": 136,
"column": 40
} | {
"line": 137,
"column": 4
} | [
{
"pp": "case 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 < ε\nP :... | [] | 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
} | {
"line": 137,
"column": 4
} | [
{
"pp": "case 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 < ε\nP :... | [] | exact le_of_lt (hQ _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.Exp | {
"line": 72,
"column": 8
} | {
"line": 72,
"column": 54
} | {
"line": 72,
"column": 55
} | [
{
"pp": "x y : ℂ\n⊢ dist y x < 1 → dist (cexp y) (cexp x) ≤ 2 * ‖cexp x‖ * dist y x",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.instLE",
"Real",
"dist_eq_norm",
"HMul.hMul",
... | [
"x y : ℂ\n⊢ ‖y - x‖ < 1 → ‖cexp y - cexp x‖ ≤ 2 * ‖cexp x‖ * ‖y - x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Exp | {
"line": 80,
"column": 4
} | {
"line": 80,
"column": 15
} | {
"line": 80,
"column": 16
} | [
{
"pp": "case inl\n⊢ (fun x ↦ cexp x - ∑ i ∈ Finset.range 0, x ^ i / ↑i !) =O[𝓝 0] fun x ↦ x ^ 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"MulOne.toOne",
"Nat.instMulZeroClass",
"Real.i... | [
"case inl\n⊢ IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (𝓝 0) fun x ↦ ‖cexp x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.BigOperators | {
"line": 163,
"column": 52
} | {
"line": 163,
"column": 69
} | {
"line": 163,
"column": 70
} | [
{
"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... | [
"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, f n > a - ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 42,
"column": 2
} | {
"line": 42,
"column": 44
} | {
"line": 42,
"column": 45
} | [
{
"pp": "α : Type u_1\ninst✝² : NormedRing α\ninst✝¹ : NormSMulClass ℤ α\ninst✝ : Nontrivial α\n⊢ Tendsto (fun x ↦ ‖↑x‖) atTop atTop",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"norm_natCast_eq_mul_norm_one",
... | [
"α : Type u_1\ninst✝² : NormedRing α\ninst✝¹ : NormSMulClass ℤ α\ninst✝ : Nontrivial α\n⊢ Tendsto (fun x ↦ ↑x * ‖1‖) atTop atTop"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 49,
"column": 2
} | {
"line": 49,
"column": 48
} | {
"line": 49,
"column": 49
} | [
{
"pp": "α : Type u_1\ninst✝² : NormedRing α\ninst✝¹ : NormSMulClass ℤ α\ninst✝ : Nontrivial α\n⊢ Tendsto (fun x ↦ ‖↑x‖) (atBot ⊔ atTop) atTop",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Int.cast",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"Semi... | [
"α : Type u_1\ninst✝² : NormedRing α\ninst✝¹ : NormSMulClass ℤ α\ninst✝ : Nontrivial α\n⊢ Tendsto (fun x ↦ |↑x| * ‖1‖) atBot atTop ∧ Tendsto (fun x ↦ |↑x| * ‖1‖) atTop atTop"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Exp | {
"line": 136,
"column": 35
} | {
"line": 136,
"column": 76
} | {
"line": 136,
"column": 77
} | [
{
"pp": "a ε : ℝ\nhε : ε > 0\nthis : ∀ (a : ℂ) (ε : ℝ), 0 < ε → ∀ᶠ (x : ℂ) in 𝓝 a, dist (cexp x) (cexp a) < ε\nha : 0 < ε / (2 * Real.exp a)\nδ : ℝ\nhδ : δ > 0 ∧ ∀ ⦃y : ℂ⦄, dist y 0 < δ → dist (cexp y) (cexp 0) < ε / (2 * Real.exp a)\nx : ℂ\na✝ : x ∈ {x | x.re ≤ a}\ny : ℂ\nhy : y ∈ {x | x.re ≤ a}\nhxy : dist x... | [
"a ε : ℝ\nhε : ε > 0\nthis : ∀ (a : ℂ) (ε : ℝ), 0 < ε → ∀ᶠ (x : ℂ) in 𝓝 a, dist (cexp x) (cexp a) < ε\nha : 0 < ε / (2 * Real.exp a)\nδ : ℝ\nhδ : δ > 0 ∧ ∀ ⦃y : ℂ⦄, dist y 0 < δ → dist (cexp y) (cexp 0) < ε / (2 * Real.exp a)\nx : ℂ\na✝ : x ∈ {x | x.re ≤ a}\ny : ℂ\nhy : y ∈ {x | x.re ≤ a}\nhxy : dist x y < δ\n⊢ ‖x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.BigOperators | {
"line": 181,
"column": 52
} | {
"line": 181,
"column": 63
} | {
"line": 181,
"column": 64
} | [
{
"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",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"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 : ℕ), m ≤ n → |f n| ≤ a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.