module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 298,
"column": 27
} | {
"line": 298,
"column": 31
} | {
"line": 298,
"column": 31
} | [
{
"pp": "b : ℝ≥0\nx✝ : ∞ ≠ ∞ ∨ ↑b ≠ ∞\nx : ℝ≥0\ny : ℝ≥0∞ × ℝ≥0∞\nhy : ↑(b + 1 + x) < y.1 ∧ y.2 ≤ ↑(b + 1)\n⊢ y.2 + ↑x < y.1",
"ppTerm": "?m.161",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"le_refl",
"ENNReal.ofNNReal",
"Preorder.toLT",
... | [
"b : ℝ≥0\nx✝ : ∞ ≠ ∞ ∨ ↑b ≠ ∞\nx : ℝ≥0\ny : ℝ≥0∞ × ℝ≥0∞\nhy : ↑(b + 1 + x) < y.1 ∧ y.2 ≤ ↑(b + 1)\n⊢ ↑(b + 1) + ↑x < y.1"
] | hy.2 | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 332,
"column": 6
} | {
"line": 333,
"column": 13
} | {
"line": 333,
"column": 14
} | [
{
"pp": "case coe.top\na✝ b : ℝ≥0∞\nht : ∀ (b : ℝ≥0∞), b ≠ 0 → Tendsto (fun p ↦ p.1 * p.2) (𝓝 (∞, b)) (𝓝 ∞)\na : ℝ≥0\nhb : ∞ ≠ 0 ∨ ↑a ≠ ∞\nha : ¬↑a = 0\n⊢ Tendsto (fun p ↦ p.1 * p.2) (𝓝 (↑a, ∞)) (𝓝 (↑a * ∞))",
"ppTerm": "?coe.top",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNR... | [
"case coe.top\na✝ b : ℝ≥0∞\nht : ∀ (b : ℝ≥0∞), b ≠ 0 → Tendsto (fun p ↦ p.1 * p.2) (𝓝 (∞, b)) (𝓝 ∞)\na : ℝ≥0\nhb : ∞ ≠ 0 ∨ ↑a ≠ ∞\nha : ¬↑a = 0\n⊢ Tendsto (fun p ↦ p.1 * p.2) (𝓝 (↑a, ∞)) (𝓝 ∞)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 362,
"column": 2
} | {
"line": 362,
"column": 29
} | {
"line": 362,
"column": 30
} | [
{
"pp": "α : Type u_1\nf : Filter α\nm : α → ℝ≥0∞\na b : ℝ≥0∞\nhm : Tendsto m f (𝓝 a)\nha : a ≠ 0 ∨ b ≠ ∞\n⊢ Tendsto (fun x ↦ m x * b) f (𝓝 (a * b))",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"CommSemiring.toNonUnitalCommSemiring",
"con... | [
"α : Type u_1\nf : Filter α\nm : α → ℝ≥0∞\na b : ℝ≥0∞\nhm : Tendsto m f (𝓝 a)\nha : a ≠ 0 ∨ b ≠ ∞\n⊢ Tendsto (fun x ↦ b * m x) f (𝓝 (a * b))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Constructions | {
"line": 214,
"column": 2
} | {
"line": 214,
"column": 48
} | {
"line": 214,
"column": 49
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : CommGroup α\ninst✝¹ : UniformSpace α\ninst✝ : IsUniformGroup α\nγ : β → Type u_4\nf : (b : β) × γ b → α\ng : β → α\na : α\nhf : ∀ (b : β), HasProd (fun c ↦ f ⟨b, c⟩) (g b)\nhg : HasProd g a\nh : CauchySeq fun s ↦ ∏ i ∈ s, f i\nu : Set α\nhu : u ∈ 𝓝 a\ns : Finset ((... | [
"α : Type u_1\nβ : Type u_2\ninst✝² : CommGroup α\ninst✝¹ : UniformSpace α\ninst✝ : IsUniformGroup α\nγ : β → Type u_4\nf : (b : β) × γ b → α\ng : β → α\na : α\nhf : ∀ (b : β), HasProd (fun c ↦ f ⟨b, c⟩) (g b)\nhg : HasProd g a\nh : CauchySeq fun s ↦ ∏ i ∈ s, f i\nu : Set α\nhu : u ∈ 𝓝 a\ns : Finset ((b : β) × γ b... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Constructions | {
"line": 350,
"column": 2
} | {
"line": 350,
"column": 30
} | {
"line": 350,
"column": 31
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nL : SummationFilter β\ninst✝³ : AddCommMonoid α\ninst✝² : TopologicalSpace α\ninst✝¹ : StarAddMonoid α\ninst✝ : ContinuousStar α\nf : β → α\nhf : Summable (fun b ↦ Star.star (f b)) L\n⊢ Summable f L",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
... | [
"α : Type u_1\nβ : Type u_2\nL : SummationFilter β\ninst✝³ : AddCommMonoid α\ninst✝² : TopologicalSpace α\ninst✝¹ : StarAddMonoid α\ninst✝ : ContinuousStar α\nf : β → α\nhf : Summable (fun b ↦ Star.star (f b)) L\n⊢ Summable f L"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.AbstractCompletion | {
"line": 155,
"column": 2
} | {
"line": 156,
"column": 76
} | {
"line": 158,
"column": 0
} | [
{
"pp": "α : Type uα\ninst✝² : UniformSpace α\npkg : AbstractCompletion.{vα, uα} α\nβ : Type uβ\ninst✝¹ : UniformSpace β\nf : α → β\ninst✝ : CompleteSpace β\nh : IsUniformInducing f\n⊢ IsUniformInducing (pkg.extend f)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"c... | [] | rw [extend_def _ h.uniformContinuous]
exact pkg.isDenseInducing.isUniformInducing_extend pkg.isUniformInducing h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.AbstractCompletion | {
"line": 155,
"column": 2
} | {
"line": 156,
"column": 76
} | {
"line": 158,
"column": 0
} | [
{
"pp": "α : Type uα\ninst✝² : UniformSpace α\npkg : AbstractCompletion.{vα, uα} α\nβ : Type uβ\ninst✝¹ : UniformSpace β\nf : α → β\ninst✝ : CompleteSpace β\nh : IsUniformInducing f\n⊢ IsUniformInducing (pkg.extend f)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"c... | [] | rw [extend_def _ h.uniformContinuous]
exact pkg.isDenseInducing.isUniformInducing_extend pkg.isUniformInducing h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UniformSpace.AbstractCompletion | {
"line": 163,
"column": 2
} | {
"line": 163,
"column": 38
} | {
"line": 163,
"column": 39
} | [
{
"pp": "α : Type uα\ninst✝³ : UniformSpace α\npkg : AbstractCompletion.{vα, uα} α\nβ : Type uβ\ninst✝² : UniformSpace β\nf : α → β\ninst✝¹ : CompleteSpace β\ninst✝ : T0Space β\nhf : UniformContinuous f\ng : pkg.space → β\nhg : UniformContinuous g\nh : ∀ (a : α), f a = g (pkg.coe a)\n⊢ ∀ (a : α), pkg.extend f (... | [
"α : Type uα\ninst✝³ : UniformSpace α\npkg : AbstractCompletion.{vα, uα} α\nβ : Type uβ\ninst✝² : UniformSpace β\nf : α → β\ninst✝¹ : CompleteSpace β\ninst✝ : T0Space β\nhf : UniformContinuous f\ng : pkg.space → β\nhg : UniformContinuous g\nh : ∀ (a : α), f a = g (pkg.coe a)\n⊢ ∀ (a : α), f a = g (pkg.coe a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 467,
"column": 21
} | {
"line": 467,
"column": 32
} | {
"line": 467,
"column": 33
} | [
{
"pp": "n : ℕ\n⊢ Continuous fun x ↦ x ^ Int.negSucc n",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"Continuous",
"congrArg",
"zpow_negSucc",
"DivInvMonoid.toZPow",
"id",
"DivInvMonoid.toMonoid",
"inst... | [
"n : ℕ\n⊢ Continuous fun x ↦ (x ^ (n + 1))⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Metrizable.Uniformity | {
"line": 209,
"column": 6
} | {
"line": 209,
"column": 82
} | {
"line": 209,
"column": 83
} | [
{
"pp": "case neg\nX : Type u_2\ninst✝¹ : UniformSpace X\ninst✝ : (𝓤 X).IsCountablyGenerated\nU : ℕ → SetRel X X\nhU_symm : ∀ (n : ℕ), (U n).IsSymm\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : (𝓤 X).HasAntitoneBasis U\nd : X → X → ℝ≥0 := fun x y ↦ if h : ∃ n, (x, y) ∉ U n then (1 / 2) ^ Nat.f... | [
"case neg\nX : Type u_2\ninst✝¹ : UniformSpace X\ninst✝ : (𝓤 X).IsCountablyGenerated\nU : ℕ → SetRel X X\nhU_symm : ∀ (n : ℕ), (U n).IsSymm\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : (𝓤 X).HasAntitoneBasis U\nd : X → X → ℝ≥0 := fun x y ↦ if h : ∃ n, (x, y) ∉ U n then (1 / 2) ^ Nat.find h else 0... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Metrizable.Uniformity | {
"line": 203,
"column": 2
} | {
"line": 209,
"column": 84
} | {
"line": 210,
"column": 2
} | [
{
"pp": "X : Type u_2\ninst✝¹ : UniformSpace X\ninst✝ : (𝓤 X).IsCountablyGenerated\nU : ℕ → SetRel X X\nhU_symm : ∀ (n : ℕ), (U n).IsSymm\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : (𝓤 X).HasAntitoneBasis U\nd : X → X → ℝ≥0 := fun x y ↦ if h : ∃ n, (x, y) ∉ U n then (1 / 2) ^ Nat.find h else... | [
"X : Type u_2\ninst✝¹ : UniformSpace X\ninst✝ : (𝓤 X).IsCountablyGenerated\nU : ℕ → SetRel X X\nhU_symm : ∀ (n : ℕ), (U n).IsSymm\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : (𝓤 X).HasAntitoneBasis U\nd : X → X → ℝ≥0 := fun x y ↦ if h : ∃ n, (x, y) ∉ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ ... | have hd₀ : ∀ {x y}, d x y = 0 ↔ Inseparable x y := by
intro x y
refine Iff.trans ?_ hB.inseparable_iff_uniformity.symm
simp only [d, true_imp_iff]
split_ifs with h
· simp [h, pow_eq_zero_iff']
· simpa only [not_exists, Classical.not_not, eq_self_iff_true, true_iff] using h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.UniformSpace.Completion | {
"line": 269,
"column": 6
} | {
"line": 269,
"column": 27
} | {
"line": 269,
"column": 28
} | [
{
"pp": "α : Type u_1\ninst✝² : UniformSpace α\ninst✝¹ : CompleteSpace α\ninst✝ : T0Space α\nf g : CauchyFilter α\n⊢ (↑f).lim = (↑g).lim ↔ Inseparable f g",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cauchy",
"SProd.sprod",
"congrArg",
"Filter.Ne... | [
"α : Type u_1\ninst✝² : UniformSpace α\ninst✝¹ : CompleteSpace α\ninst✝ : T0Space α\nf g : CauchyFilter α\n⊢ Inseparable (↑f).lim (↑g).lim ↔ Inseparable f g"
] | ← inseparable_iff_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Group.Continuity | {
"line": 387,
"column": 11
} | {
"line": 387,
"column": 29
} | {
"line": 387,
"column": 30
} | [
{
"pp": "E : Type u_4\nF : Type u_5\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\nj : E →* F\nb : F\nhb : b ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑j.range\nf : ℕ → ℝ\nb_pos : ∀ (n : ℕ), 0 < f n\nv : ℕ → F\nsum_v : Tendsto (fun n ↦ ∏ i ∈ range (n + 1), v i) atTop (𝓝 b)... | [
"E : Type u_4\nF : Type u_5\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\nj : E →* F\nb : F\nhb : b ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑j.range\nf : ℕ → ℝ\nb_pos : ∀ (n : ℕ), 0 < f n\nv : ℕ → F\nsum_v : Tendsto (fun n ↦ ∏ i ∈ range (n + 1), v i) atTop (𝓝 b)\nhv₀ : ‖(v ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.UniformMulAction | {
"line": 111,
"column": 31
} | {
"line": 111,
"column": 65
} | {
"line": 111,
"column": 66
} | [
{
"pp": "R : Type u\nM : Type v\nN : Type w\nX : Type x\nY : Type y\ninst✝⁵ : UniformSpace X\ninst✝⁴ : UniformSpace Y\ninst✝³ : SMul M X\ninst✝² : SMul Mᵐᵒᵖ X\ninst✝¹ : IsCentralScalar M X\ninst✝ : UniformContinuousConstSMul M X\nc : M\n⊢ UniformContinuous fun x ↦ MulOpposite.op c • x",
"ppTerm": "?m.15",
... | [
"R : Type u\nM : Type v\nN : Type w\nX : Type x\nY : Type y\ninst✝⁵ : UniformSpace X\ninst✝⁴ : UniformSpace Y\ninst✝³ : SMul M X\ninst✝² : SMul Mᵐᵒᵖ X\ninst✝¹ : IsCentralScalar M X\ninst✝ : UniformContinuousConstSMul M X\nc : M\n⊢ UniformContinuous fun x ↦ c • x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.UniformMulAction | {
"line": 153,
"column": 2
} | {
"line": 153,
"column": 30
} | {
"line": 153,
"column": 31
} | [
{
"pp": "R : Type u_3\nβ : Type u_4\ninst✝³ : DivisionRing R\ninst✝² : UniformSpace R\ninst✝¹ : UniformContinuousConstSMul Rᵐᵒᵖ R\ninst✝ : UniformSpace β\nf : β → R\nhf : UniformContinuous f\na : R\n⊢ UniformContinuous fun x ↦ f x / a",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"... | [
"R : Type u_3\nβ : Type u_4\ninst✝³ : DivisionRing R\ninst✝² : UniformSpace R\ninst✝¹ : UniformContinuousConstSMul Rᵐᵒᵖ R\ninst✝ : UniformSpace β\nf : β → R\nhf : UniformContinuous f\na : R\n⊢ UniformContinuous fun x ↦ f x * a⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.UniformMulAction | {
"line": 174,
"column": 23
} | {
"line": 174,
"column": 51
} | {
"line": 174,
"column": 52
} | [
{
"pp": "M : Type v\nX : Type x\ninst✝³ : UniformSpace X\ninst✝² : Monoid M\ninst✝¹ : MulAction M X\ninst✝ : UniformContinuousConstSMul M X\nc : M\nhc : IsUnit c\nd : M\nhcd : c * d = 1\ncU : c • 𝓤 X ≤ 𝓤 X\ndU : d • 𝓤 X ≤ 𝓤 X\n⊢ 𝓤 X ≤ c • 𝓤 X",
"ppTerm": "?m.53",
"assigned": false,
"usedConsta... | [
"M : Type v\nX : Type x\ninst✝³ : UniformSpace X\ninst✝² : Monoid M\ninst✝¹ : MulAction M X\ninst✝ : UniformContinuousConstSMul M X\nc : M\nhc : IsUnit c\nd : M\nhcd : c * d = 1\ncU : c • 𝓤 X ≤ 𝓤 X\ndU : d • 𝓤 X ≤ 𝓤 X\n⊢ 𝓤 X ≤ c • 𝓤 X"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Continuity | {
"line": 387,
"column": 40
} | {
"line": 387,
"column": 58
} | {
"line": 387,
"column": 59
} | [
{
"pp": "E : Type u_4\nF : Type u_5\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\nj : E →* F\nb : F\nhb : b ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑j.range\nf : ℕ → ℝ\nb_pos : ∀ (n : ℕ), 0 < f n\nv : ℕ → F\nsum_v : Tendsto (fun n ↦ ∏ i ∈ range (n + 1), v i) atTop (𝓝 b)... | [
"E : Type u_4\nF : Type u_5\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\nj : E →* F\nb : F\nhb : b ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑j.range\nf : ℕ → ℝ\nb_pos : ∀ (n : ℕ), 0 < f n\nv : ℕ → F\nsum_v : Tendsto (fun n ↦ ∏ i ∈ range (n + 1), v i) atTop (𝓝 b)\nhv₀ : ‖(v ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Continuity | {
"line": 388,
"column": 21
} | {
"line": 388,
"column": 37
} | {
"line": 388,
"column": 38
} | [
{
"pp": "E : Type u_4\nF : Type u_5\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\nj : E →* F\nb : F\nhb : b ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑j.range\nf : ℕ → ℝ\nb_pos : ∀ (n : ℕ), 0 < f n\nv : ℕ → F\nsum_v : Tendsto (fun n ↦ ∏ i ∈ range (n + 1), v i) atTop (𝓝 b)... | [
"E : Type u_4\nF : Type u_5\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\nj : E →* F\nb : F\nhb : b ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑j.range\nf : ℕ → ℝ\nb_pos : ∀ (n : ℕ), 0 < f n\nv : ℕ → F\nsum_v : Tendsto (fun n ↦ ∏ i ∈ range (n + 1), v i) atTop (𝓝 b)\nhv₀ : ‖(v ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Continuity | {
"line": 406,
"column": 2
} | {
"line": 406,
"column": 30
} | {
"line": 406,
"column": 31
} | [
{
"pp": "E : Type u_4\ninst✝ : NormedGroup E\na _x : E\nhx : _x ∈ {a}ᶜ\n⊢ _x⁻¹ * a ≠ 1",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InvOneClass.toOne",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
"congrArg",
... | [
"E : Type u_4\ninst✝ : NormedGroup E\na _x : E\nhx : _x ∈ {a}ᶜ\n⊢ ¬_x = a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 850,
"column": 53
} | {
"line": 850,
"column": 86
} | {
"line": 850,
"column": 87
} | [
{
"pp": "ι : Type u_4\nf : Filter ι\nu : ι → ℝ≥0∞\ninst✝ : f.NeBot\nb : ℝ≥0∞\nb_ne_top : b ≠ ∞\nle_b : ∀ᶠ (i : ι) in f, u i ≤ b\nliminf_le : liminf u f ≤ b\naux : ∀ᶠ (i : ι) in f, (u i).toReal = b.truncateToReal (u i)\naux' : (liminf u f).toReal = b.truncateToReal (liminf u f)\n⊢ ∀ᶠ (x : ℝ≥0∞) in map u f, (fun ... | [
"ι : Type u_4\nf : Filter ι\nu : ι → ℝ≥0∞\ninst✝ : f.NeBot\nb : ℝ≥0∞\nb_ne_top : b ≠ ∞\nle_b : ∀ᶠ (i : ι) in f, u i ≤ b\nliminf_le : liminf u f ≤ b\naux : ∀ᶠ (i : ι) in f, (u i).toReal = b.truncateToReal (u i)\naux' : (liminf u f).toReal = b.truncateToReal (liminf u f)\n⊢ ∀ᶠ (a : ι) in f, u a ≤ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 866,
"column": 17
} | {
"line": 866,
"column": 50
} | {
"line": 866,
"column": 51
} | [
{
"pp": "ι : Type u_4\nf : Filter ι\nu : ι → ℝ≥0∞\ninst✝ : f.NeBot\nb : ℝ≥0∞\nb_ne_top : b ≠ ∞\nle_b : ∀ᶠ (i : ι) in f, u i ≤ b\naux : ∀ᶠ (i : ι) in f, (u i).toReal = b.truncateToReal (u i)\naux' : (limsup u f).toReal = b.truncateToReal (limsup u f)\n⊢ ∀ᶠ (x : ℝ≥0∞) in map u f, (fun x1 x2 ↦ x1 ≤ x2) x b",
"... | [
"ι : Type u_4\nf : Filter ι\nu : ι → ℝ≥0∞\ninst✝ : f.NeBot\nb : ℝ≥0∞\nb_ne_top : b ≠ ∞\nle_b : ∀ᶠ (i : ι) in f, u i ≤ b\naux : ∀ᶠ (i : ι) in f, (u i).toReal = b.truncateToReal (u i)\naux' : (limsup u f).toReal = b.truncateToReal (limsup u f)\n⊢ ∀ᶠ (a : ι) in f, u a ≤ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Lipschitz | {
"line": 74,
"column": 23
} | {
"line": 74,
"column": 65
} | {
"line": 74,
"column": 66
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nf : α → β\nh : ∀ (x y : α), dist (f x) (f y) ≤ dist x y\n⊢ ∀ (x y : α), dist (f x) (f y) ≤ ↑1 * dist x y",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"R... | [
"α : Type u\nβ : Type v\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nf : α → β\nh : ∀ (x y : α), dist (f x) (f y) ≤ dist x y\n⊢ ∀ (x y : α), dist (f x) (f y) ≤ dist x y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Lipschitz | {
"line": 86,
"column": 28
} | {
"line": 86,
"column": 64
} | {
"line": 86,
"column": 65
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nf : α → ℝ\nK : ℝ≥0\nh : ∀ (x y : α), f x ≤ f y + ↑K * dist x y\n⊢ LipschitzWith K f",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\ninst✝ : PseudoMetricSpace α\nf : α → ℝ\nK : ℝ≥0\nh : ∀ (x y : α), f x ≤ f y + ↑K * dist x y\n⊢ LipschitzWith K f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Lipschitz | {
"line": 186,
"column": 2
} | {
"line": 186,
"column": 13
} | {
"line": 186,
"column": 14
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nf : α → ℝ\nKf : ℝ≥0\nhf : LipschitzWith Kf f\na : ℝ\n⊢ LipschitzWith Kf fun x ↦ max (f x) a",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\ninst✝ : PseudoEMetricSpace α\nf : α → ℝ\nKf : ℝ≥0\nhf : LipschitzWith Kf f\na : ℝ\n⊢ LipschitzWith Kf fun x ↦ max (f x) a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Lipschitz | {
"line": 189,
"column": 2
} | {
"line": 189,
"column": 29
} | {
"line": 189,
"column": 30
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nf : α → ℝ\nKf : ℝ≥0\nhf : LipschitzWith Kf f\na : ℝ\n⊢ LipschitzWith Kf fun x ↦ max a (f x)",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\ninst✝ : PseudoEMetricSpace α\nf : α → ℝ\nKf : ℝ≥0\nhf : LipschitzWith Kf f\na : ℝ\n⊢ LipschitzWith Kf fun x ↦ max a (f x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Lipschitz | {
"line": 192,
"column": 2
} | {
"line": 192,
"column": 13
} | {
"line": 192,
"column": 14
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nf : α → ℝ\nKf : ℝ≥0\nhf : LipschitzWith Kf f\na : ℝ\n⊢ LipschitzWith Kf fun x ↦ min (f x) a",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\ninst✝ : PseudoEMetricSpace α\nf : α → ℝ\nKf : ℝ≥0\nhf : LipschitzWith Kf f\na : ℝ\n⊢ LipschitzWith Kf fun x ↦ min (f x) a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Lipschitz | {
"line": 195,
"column": 2
} | {
"line": 195,
"column": 29
} | {
"line": 195,
"column": 30
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nf : α → ℝ\nKf : ℝ≥0\nhf : LipschitzWith Kf f\na : ℝ\n⊢ LipschitzWith Kf fun x ↦ min a (f x)",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\ninst✝ : PseudoEMetricSpace α\nf : α → ℝ\nKf : ℝ≥0\nhf : LipschitzWith Kf f\na : ℝ\n⊢ LipschitzWith Kf fun x ↦ min a (f x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Lipschitz | {
"line": 225,
"column": 23
} | {
"line": 225,
"column": 65
} | {
"line": 225,
"column": 66
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\ns : Set α\nf : α → β\nh : ∀ x ∈ s, ∀ y ∈ s, dist (f x) (f y) ≤ dist x y\n⊢ ∀ x ∈ s, ∀ y ∈ s, dist (f x) (f y) ≤ ↑1 * dist x y",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"R... | [
"α : Type u\nβ : Type v\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\ns : Set α\nf : α → β\nh : ∀ x ∈ s, ∀ y ∈ s, dist (f x) (f y) ≤ dist x y\n⊢ ∀ x ∈ s, ∀ y ∈ s, dist (f x) (f y) ≤ dist x y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Lipschitz | {
"line": 240,
"column": 2
} | {
"line": 240,
"column": 38
} | {
"line": 240,
"column": 39
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\ns : Set α\nf : α → ℝ\nK : ℝ≥0\nh : ∀ x ∈ s, ∀ y ∈ s, f x ≤ f y + ↑K * dist x y\n⊢ LipschitzOnWith K f s",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\ninst✝ : PseudoMetricSpace α\ns : Set α\nf : α → ℝ\nK : ℝ≥0\nh : ∀ x ∈ s, ∀ y ∈ s, f x ≤ f y + ↑K * dist x y\n⊢ LipschitzOnWith K f s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Lipschitz | {
"line": 289,
"column": 2
} | {
"line": 289,
"column": 24
} | {
"line": 289,
"column": 25
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nf : α → ℝ\nhf : LocallyLipschitz f\na : ℝ\n⊢ LocallyLipschitz fun x ↦ max a (f x)",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\ninst✝ : PseudoEMetricSpace α\nf : α → ℝ\nhf : LocallyLipschitz f\na : ℝ\n⊢ LocallyLipschitz fun x ↦ max a (f x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Lipschitz | {
"line": 295,
"column": 2
} | {
"line": 295,
"column": 24
} | {
"line": 295,
"column": 25
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nf : α → ℝ\nhf : LocallyLipschitz f\na : ℝ\n⊢ LocallyLipschitz fun x ↦ min a (f x)",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\ninst✝ : PseudoEMetricSpace α\nf : α → ℝ\nhf : LocallyLipschitz f\na : ℝ\n⊢ LocallyLipschitz fun x ↦ min a (f x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Lipschitz | {
"line": 327,
"column": 4
} | {
"line": 327,
"column": 68
} | {
"line": 327,
"column": 69
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nf : α → ℝ\ns : Set α\nK : ℝ≥0\nhf : LipschitzOnWith K f s\nhs : s.Nonempty\nthis : Nonempty ↑s\ng : α → ℝ := fun y ↦ ⨅ x, f ↑x + ↑K * dist y ↑x\nB : ∀ (y : α), BddBelow (range fun x ↦ f ↑x + ↑K * dist y ↑x)\nx : α\nhx : x ∈ s\n⊢ g x ≤ f x",
"ppTerm": "?m.271... | [
"α : Type u\ninst✝ : PseudoMetricSpace α\nf : α → ℝ\ns : Set α\nK : ℝ≥0\nhf : LipschitzOnWith K f s\nhs : s.Nonempty\nthis : Nonempty ↑s\ng : α → ℝ := fun y ↦ ⨅ x, f ↑x + ↑K * dist y ↑x\nB : ∀ (y : α), BddBelow (range fun x ↦ f ↑x + ↑K * dist y ↑x)\nx : α\nhx : x ∈ s\n⊢ g x ≤ f x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.Lattice | {
"line": 124,
"column": 17
} | {
"line": 124,
"column": 28
} | {
"line": 124,
"column": 29
} | [
{
"pp": "case singleton\nL : Type u_1\ninst✝² : TopologicalSpace L\nι : Type u_3\nα : Type u_4\ns : Finset ι\nf : ι → α → L\nl : Filter α\ng : ι → L\ninst✝¹ : SemilatticeSup L\ninst✝ : ContinuousSup L\na✝ : ι\nhs : ∀ i ∈ {a✝}, Tendsto (f i) l (𝓝 (g i))\n⊢ Tendsto ({a✝}.sup' ⋯ f) l (𝓝 ({a✝}.sup' ⋯ g))",
"p... | [
"case singleton\nL : Type u_1\ninst✝² : TopologicalSpace L\nι : Type u_3\nα : Type u_4\ns : Finset ι\nf : ι → α → L\nl : Filter α\ng : ι → L\ninst✝¹ : SemilatticeSup L\ninst✝ : ContinuousSup L\na✝ : ι\nhs : ∀ i ∈ {a✝}, Tendsto (f i) l (𝓝 (g i))\n⊢ Tendsto (f a✝) l (𝓝 (g a✝))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.Lattice | {
"line": 126,
"column": 8
} | {
"line": 126,
"column": 23
} | {
"line": 126,
"column": 23
} | [
{
"pp": "case cons\nL : Type u_1\ninst✝² : TopologicalSpace L\nι : Type u_3\nα : Type u_4\ns✝ : Finset ι\nf : ι → α → L\nl : Filter α\ng : ι → L\ninst✝¹ : SemilatticeSup L\ninst✝ : ContinuousSup L\na : ι\ns : Finset ι\nha : a ∉ s\nhne : s.Nonempty\nihs : (∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))) → Tendsto (s.sup' h... | [
"case cons\nL : Type u_1\ninst✝² : TopologicalSpace L\nι : Type u_3\nα : Type u_4\ns✝ : Finset ι\nf : ι → α → L\nl : Filter α\ng : ι → L\ninst✝¹ : SemilatticeSup L\ninst✝ : ContinuousSup L\na : ι\ns : Finset ι\nha : a ∉ s\nhne : s.Nonempty\nihs : (∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))) → Tendsto (s.sup' hne f) l (𝓝 ... | forall_mem_cons | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order.Lattice | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 40
} | {
"line": 133,
"column": 41
} | [
{
"pp": "L : Type u_1\ninst✝² : TopologicalSpace L\nι : Type u_3\nα : Type u_4\ns : Finset ι\nf : ι → α → L\nl : Filter α\ng : ι → L\ninst✝¹ : SemilatticeSup L\ninst✝ : ContinuousSup L\nhne : s.Nonempty\nhs : ∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))\n⊢ Tendsto (fun a ↦ s.sup' hne fun x ↦ f x a) l (𝓝 (s.sup' hne g))... | [
"L : Type u_1\ninst✝² : TopologicalSpace L\nι : Type u_3\nα : Type u_4\ns : Finset ι\nf : ι → α → L\nl : Filter α\ng : ι → L\ninst✝¹ : SemilatticeSup L\ninst✝ : ContinuousSup L\nhne : s.Nonempty\nhs : ∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))\n⊢ Tendsto (fun a ↦ s.sup' hne f a) l (𝓝 (s.sup' hne g))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.Lattice | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 39
} | {
"line": 155,
"column": 40
} | [
{
"pp": "L : Type u_1\ninst✝³ : TopologicalSpace L\nι : Type u_3\nα : Type u_4\ns : Finset ι\nf : ι → α → L\nl : Filter α\ng : ι → L\ninst✝² : SemilatticeSup L\ninst✝¹ : OrderBot L\ninst✝ : ContinuousSup L\nhs : ∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))\n⊢ Tendsto (fun a ↦ s.sup fun x ↦ f x a) l (𝓝 (s.sup g))",
... | [
"L : Type u_1\ninst✝³ : TopologicalSpace L\nι : Type u_3\nα : Type u_4\ns : Finset ι\nf : ι → α → L\nl : Filter α\ng : ι → L\ninst✝² : SemilatticeSup L\ninst✝¹ : OrderBot L\ninst✝ : ContinuousSup L\nhs : ∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))\n⊢ Tendsto (fun a ↦ s.sup f a) l (𝓝 (s.sup g))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.Lattice | {
"line": 259,
"column": 2
} | {
"line": 259,
"column": 40
} | {
"line": 259,
"column": 41
} | [
{
"pp": "L : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace L\ninst✝² : TopologicalSpace X\nι : Type u_3\ninst✝¹ : SemilatticeSup L\ninst✝ : ContinuousSup L\ns : Finset ι\nf : ι → X → L\nx : X\nhne : s.Nonempty\nhs : ∀ i ∈ s, ContinuousAt (f i) x\n⊢ ContinuousAt (s.sup' hne f) x",
"ppTerm": "?m.19",
... | [
"L : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace L\ninst✝² : TopologicalSpace X\nι : Type u_3\ninst✝¹ : SemilatticeSup L\ninst✝ : ContinuousSup L\ns : Finset ι\nf : ι → X → L\nx : X\nhne : s.Nonempty\nhs : ∀ i ∈ s, ContinuousAt (f i) x\n⊢ ContinuousAt (s.sup' hne f) x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.Lattice | {
"line": 270,
"column": 2
} | {
"line": 270,
"column": 40
} | {
"line": 270,
"column": 41
} | [
{
"pp": "L : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace L\ninst✝² : TopologicalSpace X\nι : Type u_3\ninst✝¹ : SemilatticeSup L\ninst✝ : ContinuousSup L\ns : Finset ι\nf : ι → X → L\nt : Set X\nx : X\nhne : s.Nonempty\nhs : ∀ i ∈ s, ContinuousWithinAt (f i) t x\n⊢ ContinuousWithinAt (s.sup' hne f) t x",
... | [
"L : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace L\ninst✝² : TopologicalSpace X\nι : Type u_3\ninst✝¹ : SemilatticeSup L\ninst✝ : ContinuousSup L\ns : Finset ι\nf : ι → X → L\nt : Set X\nx : X\nhne : s.Nonempty\nhs : ∀ i ∈ s, ContinuousWithinAt (f i) t x\n⊢ ContinuousWithinAt (s.sup' hne f) t x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.Lattice | {
"line": 308,
"column": 2
} | {
"line": 308,
"column": 39
} | {
"line": 308,
"column": 40
} | [
{
"pp": "L : Type u_1\nX : Type u_2\ninst✝⁴ : TopologicalSpace L\ninst✝³ : TopologicalSpace X\nι : Type u_3\ninst✝² : SemilatticeSup L\ninst✝¹ : OrderBot L\ninst✝ : ContinuousSup L\ns : Finset ι\nf : ι → X → L\nx : X\nhs : ∀ i ∈ s, ContinuousAt (f i) x\n⊢ ContinuousAt (s.sup f) x",
"ppTerm": "?m.20",
"a... | [
"L : Type u_1\nX : Type u_2\ninst✝⁴ : TopologicalSpace L\ninst✝³ : TopologicalSpace X\nι : Type u_3\ninst✝² : SemilatticeSup L\ninst✝¹ : OrderBot L\ninst✝ : ContinuousSup L\ns : Finset ι\nf : ι → X → L\nx : X\nhs : ∀ i ∈ s, ContinuousAt (f i) x\n⊢ ContinuousAt (s.sup f) x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.Lattice | {
"line": 319,
"column": 2
} | {
"line": 319,
"column": 39
} | {
"line": 319,
"column": 40
} | [
{
"pp": "L : Type u_1\nX : Type u_2\ninst✝⁴ : TopologicalSpace L\ninst✝³ : TopologicalSpace X\nι : Type u_3\ninst✝² : SemilatticeSup L\ninst✝¹ : OrderBot L\ninst✝ : ContinuousSup L\ns : Finset ι\nf : ι → X → L\nt : Set X\nx : X\nhs : ∀ i ∈ s, ContinuousWithinAt (f i) t x\n⊢ ContinuousWithinAt (s.sup f) t x",
... | [
"L : Type u_1\nX : Type u_2\ninst✝⁴ : TopologicalSpace L\ninst✝³ : TopologicalSpace X\nι : Type u_3\ninst✝² : SemilatticeSup L\ninst✝¹ : OrderBot L\ninst✝ : ContinuousSup L\ns : Finset ι\nf : ι → X → L\nt : Set X\nx : X\nhs : ∀ i ∈ s, ContinuousWithinAt (f i) t x\n⊢ ContinuousWithinAt (s.sup f) t x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.Lattice | {
"line": 356,
"column": 2
} | {
"line": 356,
"column": 40
} | {
"line": 356,
"column": 41
} | [
{
"pp": "L : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace L\ninst✝² : TopologicalSpace X\nι : Type u_3\ninst✝¹ : SemilatticeInf L\ninst✝ : ContinuousInf L\ns : Finset ι\nf : ι → X → L\nx : X\nhne : s.Nonempty\nhs : ∀ i ∈ s, ContinuousAt (f i) x\n⊢ ContinuousAt (s.inf' hne f) x",
"ppTerm": "?m.19",
... | [
"L : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace L\ninst✝² : TopologicalSpace X\nι : Type u_3\ninst✝¹ : SemilatticeInf L\ninst✝ : ContinuousInf L\ns : Finset ι\nf : ι → X → L\nx : X\nhne : s.Nonempty\nhs : ∀ i ∈ s, ContinuousAt (f i) x\n⊢ ContinuousAt (s.inf' hne f) x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.Lattice | {
"line": 367,
"column": 2
} | {
"line": 367,
"column": 40
} | {
"line": 367,
"column": 41
} | [
{
"pp": "L : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace L\ninst✝² : TopologicalSpace X\nι : Type u_3\ninst✝¹ : SemilatticeInf L\ninst✝ : ContinuousInf L\ns : Finset ι\nf : ι → X → L\nt : Set X\nx : X\nhne : s.Nonempty\nhs : ∀ i ∈ s, ContinuousWithinAt (f i) t x\n⊢ ContinuousWithinAt (s.inf' hne f) t x",
... | [
"L : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace L\ninst✝² : TopologicalSpace X\nι : Type u_3\ninst✝¹ : SemilatticeInf L\ninst✝ : ContinuousInf L\ns : Finset ι\nf : ι → X → L\nt : Set X\nx : X\nhne : s.Nonempty\nhs : ∀ i ∈ s, ContinuousWithinAt (f i) t x\n⊢ ContinuousWithinAt (s.inf' hne f) t x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.Lattice | {
"line": 405,
"column": 2
} | {
"line": 405,
"column": 39
} | {
"line": 405,
"column": 40
} | [
{
"pp": "L : Type u_1\nX : Type u_2\ninst✝⁴ : TopologicalSpace L\ninst✝³ : TopologicalSpace X\nι : Type u_3\ninst✝² : SemilatticeInf L\ninst✝¹ : OrderTop L\ninst✝ : ContinuousInf L\ns : Finset ι\nf : ι → X → L\nx : X\nhs : ∀ i ∈ s, ContinuousAt (f i) x\n⊢ ContinuousAt (s.inf f) x",
"ppTerm": "?m.20",
"a... | [
"L : Type u_1\nX : Type u_2\ninst✝⁴ : TopologicalSpace L\ninst✝³ : TopologicalSpace X\nι : Type u_3\ninst✝² : SemilatticeInf L\ninst✝¹ : OrderTop L\ninst✝ : ContinuousInf L\ns : Finset ι\nf : ι → X → L\nx : X\nhs : ∀ i ∈ s, ContinuousAt (f i) x\n⊢ ContinuousAt (s.inf f) x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.Lattice | {
"line": 416,
"column": 2
} | {
"line": 416,
"column": 39
} | {
"line": 416,
"column": 40
} | [
{
"pp": "L : Type u_1\nX : Type u_2\ninst✝⁴ : TopologicalSpace L\ninst✝³ : TopologicalSpace X\nι : Type u_3\ninst✝² : SemilatticeInf L\ninst✝¹ : OrderTop L\ninst✝ : ContinuousInf L\ns : Finset ι\nf : ι → X → L\nt : Set X\nx : X\nhs : ∀ i ∈ s, ContinuousWithinAt (f i) t x\n⊢ ContinuousWithinAt (s.inf f) t x",
... | [
"L : Type u_1\nX : Type u_2\ninst✝⁴ : TopologicalSpace L\ninst✝³ : TopologicalSpace X\nι : Type u_3\ninst✝² : SemilatticeInf L\ninst✝¹ : OrderTop L\ninst✝ : ContinuousInf L\ns : Finset ι\nf : ι → X → L\nt : Set X\nx : X\nhs : ∀ i ∈ s, ContinuousWithinAt (f i) t x\n⊢ ContinuousWithinAt (s.inf f) t x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Order.Lattice | {
"line": 65,
"column": 20
} | {
"line": 65,
"column": 70
} | {
"line": 65,
"column": 71
} | [
{
"pp": "α : Type u_1\ninst✝² : NormedAddCommGroup α\ninst✝¹ : Lattice α\ninst✝ : HasSolidNorm α\nx y : ℤ\nh : |x| ≤ |y|\n⊢ ‖x‖ ≤ ‖y‖",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Int.cast",
"Eq.mpr",
"Real.partialOrder",
"Real.instLE",
"... | [
"α : Type u_1\ninst✝² : NormedAddCommGroup α\ninst✝¹ : Lattice α\ninst✝ : HasSolidNorm α\nx y : ℤ\nh : |x| ≤ |y|\n⊢ |x| ≤ |y|"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Order.Lattice | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 39
} | {
"line": 121,
"column": 40
} | [
{
"pp": "α : Type u_1\ninst✝³ : NormedAddCommGroup α\ninst✝² : Lattice α\ninst✝¹ : HasSolidNorm α\ninst✝ : IsOrderedAddMonoid α\nx y : α\nh : ‖x ⊓ y - 0 ⊓ 0‖ ≤ ‖x - 0‖ + ‖y - 0‖\n⊢ ‖x ⊓ y‖ ≤ ‖x‖ + ‖y‖",
"ppTerm": "?m.65",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": ... | [
"α : Type u_1\ninst✝³ : NormedAddCommGroup α\ninst✝² : Lattice α\ninst✝¹ : HasSolidNorm α\ninst✝ : IsOrderedAddMonoid α\nx y : α\nh : ‖x ⊓ y - 0 ⊓ 0‖ ≤ ‖x - 0‖ + ‖y - 0‖\n⊢ ‖x ⊓ y‖ ≤ ‖x‖ + ‖y‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Order.Lattice | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 39
} | {
"line": 125,
"column": 40
} | [
{
"pp": "α : Type u_1\ninst✝³ : NormedAddCommGroup α\ninst✝² : Lattice α\ninst✝¹ : HasSolidNorm α\ninst✝ : IsOrderedAddMonoid α\nx y : α\nh : ‖x ⊔ y - 0 ⊔ 0‖ ≤ ‖x - 0‖ + ‖y - 0‖\n⊢ ‖x ⊔ y‖ ≤ ‖x‖ + ‖y‖",
"ppTerm": "?m.65",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": ... | [
"α : Type u_1\ninst✝³ : NormedAddCommGroup α\ninst✝² : Lattice α\ninst✝¹ : HasSolidNorm α\ninst✝ : IsOrderedAddMonoid α\nx y : α\nh : ‖x ⊔ y - 0 ⊔ 0‖ ≤ ‖x - 0‖ + ‖y - 0‖\n⊢ ‖x ⊔ y‖ ≤ ‖x‖ + ‖y‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Uniform | {
"line": 73,
"column": 4
} | {
"line": 73,
"column": 61
} | {
"line": 73,
"column": 62
} | [
{
"pp": "𝓕 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : SeminormedGroup E\ninst✝² : SeminormedGroup F\ninst✝¹ : FunLike 𝓕 E F\ninst✝ : MonoidHomClass 𝓕 E F\nf : 𝓕\nC : ℝ\nh : ∀ (x : E), ‖f x‖ ≤ C * ‖x‖\nx y : E\n⊢ dist (f x) (f y) ≤ C * dist x y",
"ppTerm": "?m.29",
"assigned": true,
"usedCo... | [
"𝓕 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : SeminormedGroup E\ninst✝² : SeminormedGroup F\ninst✝¹ : FunLike 𝓕 E F\ninst✝ : MonoidHomClass 𝓕 E F\nf : 𝓕\nC : ℝ\nh : ∀ (x : E), ‖f x‖ ≤ C * ‖x‖\nx y : E\n⊢ ‖(f x)⁻¹ * f y‖ ≤ C * ‖x⁻¹ * y‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Uniform | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 13
} | {
"line": 121,
"column": 14
} | [
{
"pp": "𝓕 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : SeminormedGroup E\ninst✝² : SeminormedGroup F\ninst✝¹ : FunLike 𝓕 E F\ninst✝ : MonoidHomClass 𝓕 E F\nf : 𝓕\nh : ∀ (x y : E), ‖f (x⁻¹ * y)‖ = ‖x⁻¹ * y‖\nx : E\n⊢ ‖f x‖ = ‖x‖",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
... | [
"𝓕 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : SeminormedGroup E\ninst✝² : SeminormedGroup F\ninst✝¹ : FunLike 𝓕 E F\ninst✝ : MonoidHomClass 𝓕 E F\nf : 𝓕\nh : ∀ (x y : E), ‖f (x⁻¹ * y)‖ = ‖x⁻¹ * y‖\nx : E\n⊢ ‖f x‖ = ‖x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Uniform | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 61
} | {
"line": 138,
"column": 62
} | [
{
"pp": "𝓕 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : SeminormedGroup E\ninst✝² : SeminormedGroup F\ninst✝¹ : FunLike 𝓕 E F\ninst✝ : MonoidHomClass 𝓕 E F\nf : 𝓕\nK : ℝ≥0\nh : ∀ (x : E), ‖x‖ ≤ ↑K * ‖f x‖\nx y : E\n⊢ dist x y ≤ ↑K * dist (f x) (f y)",
"ppTerm": "?m.29",
"assigned": true,
"us... | [
"𝓕 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : SeminormedGroup E\ninst✝² : SeminormedGroup F\ninst✝¹ : FunLike 𝓕 E F\ninst✝ : MonoidHomClass 𝓕 E F\nf : 𝓕\nK : ℝ≥0\nh : ∀ (x : E), ‖x‖ ≤ ↑K * ‖f x‖\nx y : E\n⊢ ‖x⁻¹ * y‖ ≤ ↑K * ‖(f x)⁻¹ * f y‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Uniform | {
"line": 142,
"column": 32
} | {
"line": 142,
"column": 69
} | {
"line": 142,
"column": 70
} | [
{
"pp": "E : Type u_2\nF : Type u_3\ninst✝¹ : SeminormedGroup E\ninst✝ : SeminormedGroup F\nf : E → F\nK : ℝ≥0\nh : LipschitzWith K f\nhf : f 1 = 1\nx : E\n⊢ ‖f x‖ ≤ ↑K * ‖x‖",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : Type u_2\nF : Type u_3\ninst✝¹ : SeminormedGroup E\ninst✝ : SeminormedGroup F\nf : E → F\nK : ℝ≥0\nh : LipschitzWith K f\nhf : f 1 = 1\nx : E\n⊢ ‖f x‖ ≤ ↑K * ‖x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Uniform | {
"line": 152,
"column": 2
} | {
"line": 152,
"column": 39
} | {
"line": 152,
"column": 40
} | [
{
"pp": "E : Type u_2\nF : Type u_3\ninst✝¹ : SeminormedGroup E\ninst✝ : SeminormedGroup F\nf : E → F\nK : ℝ≥0\nh : AntilipschitzWith K f\nhf : f 1 = 1\nx : E\n⊢ ‖x‖ ≤ ↑K * ‖f x‖",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : Type u_2\nF : Type u_3\ninst✝¹ : SeminormedGroup E\ninst✝ : SeminormedGroup F\nf : E → F\nK : ℝ≥0\nh : AntilipschitzWith K f\nhf : f 1 = 1\nx : E\n⊢ ‖x‖ ≤ ↑K * ‖f x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Uniform | {
"line": 182,
"column": 2
} | {
"line": 182,
"column": 13
} | {
"line": 182,
"column": 14
} | [
{
"pp": "E : Type u_2\ninst✝ : SeminormedGroup E\n⊢ LipschitzWith 1 norm",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : Type u_2\ninst✝ : SeminormedGroup E\n⊢ LipschitzWith 1 norm"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Uniform | {
"line": 232,
"column": 2
} | {
"line": 232,
"column": 50
} | {
"line": 232,
"column": 51
} | [
{
"pp": "E : Type u_2\ninst✝ : SeminormedCommGroup E\na₁ a₂ b₁ b₂ : E\n⊢ dist (a₁ * a₂) (b₁ * b₂) ≤ dist a₁ b₁ + dist a₂ b₂",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : Type u_2\ninst✝ : SeminormedCommGroup E\na₁ a₂ b₁ b₂ : E\n⊢ dist (a₁ * a₂) (b₁ * b₂) ≤ dist a₁ b₁ + dist a₂ b₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Uniform | {
"line": 241,
"column": 2
} | {
"line": 241,
"column": 49
} | {
"line": 241,
"column": 50
} | [
{
"pp": "E : Type u_2\ninst✝ : SeminormedCommGroup E\na₁ a₂ b₁ b₂ : E\n⊢ dist (a₁ / a₂) (b₁ / b₂) ≤ dist a₁ b₁ + dist a₂ b₂",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
... | [
"E : Type u_2\ninst✝ : SeminormedCommGroup E\na₁ a₂ b₁ b₂ : E\n⊢ dist (a₁ * a₂⁻¹) (b₁ * b₂⁻¹) ≤ dist a₁ b₁ + dist a₂ b₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Uniform | {
"line": 251,
"column": 2
} | {
"line": 251,
"column": 64
} | {
"line": 252,
"column": 4
} | [
{
"pp": "E : Type u_2\ninst✝ : SeminormedCommGroup E\na₁ a₂ b₁ b₂ : E\n⊢ |dist a₁ b₁ - dist a₂ b₂| ≤ dist (a₁ * a₂) (b₁ * b₂)",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : Type u_2\ninst✝ : SeminormedCommGroup E\na₁ a₂ b₁ b₂ : E\n⊢ |dist a₁ b₁ - dist a₂ b₂| ≤ dist (a₁ * a₂) (b₁ * b₂)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Uniform | {
"line": 310,
"column": 2
} | {
"line": 310,
"column": 38
} | {
"line": 310,
"column": 39
} | [
{
"pp": "α : Type u_4\nE : Type u_5\ninst✝¹ : SeminormedCommGroup E\ninst✝ : PseudoEMetricSpace α\nKf Kg : ℝ≥0\nf g : α → E\nhf : LipschitzWith Kf f\nhg : LipschitzWith Kg g\n⊢ LipschitzWith (Kf + Kg) fun x ↦ f x * g x",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [
"α : Type u_4\nE : Type u_5\ninst✝¹ : SeminormedCommGroup E\ninst✝ : PseudoEMetricSpace α\nKf Kg : ℝ≥0\nf g : α → E\nhf : LipschitzWith Kf f\nhg : LipschitzWith Kg g\n⊢ LipschitzOnWith (Kf + Kg) (fun x ↦ f x * g x) Set.univ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Uniform | {
"line": 323,
"column": 2
} | {
"line": 323,
"column": 41
} | {
"line": 323,
"column": 42
} | [
{
"pp": "α : Type u_4\nE : Type u_5\ninst✝¹ : SeminormedCommGroup E\ninst✝ : PseudoEMetricSpace α\nf g : α → E\nhf : LocallyLipschitz f\nhg : LocallyLipschitz g\n⊢ LocallyLipschitz fun x ↦ f x * g x",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"M... | [
"α : Type u_4\nE : Type u_5\ninst✝¹ : SeminormedCommGroup E\ninst✝ : PseudoEMetricSpace α\nf g : α → E\nhf : LocallyLipschitz f\nhg : LocallyLipschitz g\n⊢ LocallyLipschitzOn Set.univ fun x ↦ f x * g x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Uniform | {
"line": 322,
"column": 42
} | {
"line": 323,
"column": 89
} | {
"line": 325,
"column": 0
} | [
{
"pp": "α : Type u_4\nE : Type u_5\ninst✝¹ : SeminormedCommGroup E\ninst✝ : PseudoEMetricSpace α\nf g : α → E\nhf : LocallyLipschitz f\nhg : LocallyLipschitz g\n⊢ LocallyLipschitz fun x ↦ f x * g x",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"M... | [] | by
simpa [← locallyLipschitzOn_univ] using hf.locallyLipschitzOn.mul hg.locallyLipschitzOn | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Group.Uniform | {
"line": 363,
"column": 2
} | {
"line": 363,
"column": 49
} | {
"line": 363,
"column": 50
} | [
{
"pp": "α : Type u_4\nE : Type u_5\ninst✝¹ : SeminormedCommGroup E\ninst✝ : PseudoEMetricSpace α\nKf Kg : ℝ≥0\nf g : α → E\nhf : AntilipschitzWith Kf f\nhg : LipschitzWith Kg (g / f)\nhK : Kg < Kf⁻¹\n⊢ AntilipschitzWith (Kf⁻¹ - Kg)⁻¹ g",
"ppTerm": "?m.29",
"assigned": false,
"usedConstants": [],
... | [
"α : Type u_4\nE : Type u_5\ninst✝¹ : SeminormedCommGroup E\ninst✝ : PseudoEMetricSpace α\nKf Kg : ℝ≥0\nf g : α → E\nhf : AntilipschitzWith Kf f\nhg : LipschitzWith Kg (g / f)\nhK : Kg < Kf⁻¹\n⊢ AntilipschitzWith (Kf⁻¹ - Kg)⁻¹ g"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Uniform | {
"line": 443,
"column": 23
} | {
"line": 443,
"column": 34
} | {
"line": 443,
"column": 35
} | [
{
"pp": "G : Type u_4\ninst✝ : SeminormedGroup G\nu : ℕ → G\nhu : CauchySeq u\nC : ℝ\nhC : ∀ (m n : ℕ), ‖(u m)⁻¹ * u n‖ < C\nthis : ∀ (n : ℕ), ‖u n‖ ≤ C + ‖u 0‖\n⊢ ∀ y ∈ Set.range fun n ↦ ‖u n‖, y ≤ C + ‖u 0‖",
"ppTerm": "?m.85",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr"... | [
"G : Type u_4\ninst✝ : SeminormedGroup G\nu : ℕ → G\nhu : CauchySeq u\nC : ℝ\nhC : ∀ (m n : ℕ), ‖(u m)⁻¹ * u n‖ < C\nthis : ∀ (n : ℕ), ‖u n‖ ≤ C + ‖u 0‖\n⊢ ∀ (a : ℕ), ‖u a‖ ≤ C + ‖u 0‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Uniform | {
"line": 448,
"column": 2
} | {
"line": 448,
"column": 30
} | {
"line": 448,
"column": 31
} | [
{
"pp": "E : Type u_2\nF : Type u_3\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\nf : E → F\nC : ℝ≥0\ns : Set E\n⊢ LipschitzOnWith C f s ↔ ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ‖f x / f y‖ ≤ ↑C * ‖x / y‖",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Norm.norm",
... | [
"E : Type u_2\nF : Type u_3\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\nf : E → F\nC : ℝ≥0\ns : Set E\n⊢ LipschitzOnWith C f s ↔ ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ‖(f x)⁻¹ * f y‖ ≤ ↑C * ‖x⁻¹ * y‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Algebra | {
"line": 255,
"column": 32
} | {
"line": 255,
"column": 67
} | {
"line": 255,
"column": 68
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁵ : PseudoMetricSpace α\ninst✝⁴ : PseudoMetricSpace β\ninst✝³ : Zero α\ninst✝² : Zero β\ninst✝¹ : SMul α β\ninst✝ : IsBoundedSMul α β\nx y₁ y₂ : ℝ\n⊢ dist (x • y₁) (x • y₂) ≤ dist x 0 * dist y₁ y₂",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁵ : PseudoMetricSpace α\ninst✝⁴ : PseudoMetricSpace β\ninst✝³ : Zero α\ninst✝² : Zero β\ninst✝¹ : SMul α β\ninst✝ : IsBoundedSMul α β\nx y₁ y₂ : ℝ\n⊢ |x * y₁ - x * y₂| ≤ |x| * |y₁ - y₂|"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Algebra | {
"line": 256,
"column": 32
} | {
"line": 256,
"column": 67
} | {
"line": 256,
"column": 68
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁵ : PseudoMetricSpace α\ninst✝⁴ : PseudoMetricSpace β\ninst✝³ : Zero α\ninst✝² : Zero β\ninst✝¹ : SMul α β\ninst✝ : IsBoundedSMul α β\nx₁ x₂ y : ℝ\n⊢ dist (x₁ • y) (x₂ • y) ≤ dist x₁ x₂ * dist y 0",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁵ : PseudoMetricSpace α\ninst✝⁴ : PseudoMetricSpace β\ninst✝³ : Zero α\ninst✝² : Zero β\ninst✝¹ : SMul α β\ninst✝ : IsBoundedSMul α β\nx₁ x₂ y : ℝ\n⊢ |x₁ * y - x₂ * y| ≤ |x₁ - x₂| * |y|"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Bounded | {
"line": 34,
"column": 2
} | {
"line": 34,
"column": 35
} | {
"line": 34,
"column": 36
} | [
{
"pp": "E : Type u_2\ninst✝ : SeminormedGroup E\n⊢ comap norm atTop = cobounded E",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : Type u_2\ninst✝ : SeminormedGroup E\n⊢ comap norm atTop = cobounded E"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Bounded | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 59
} | {
"line": 73,
"column": 60
} | [
{
"pp": "E : Type u_2\ninst✝ : SeminormedGroup E\ns : Set E\n⊢ Bornology.IsBounded s ↔ ∃ C, ∀ x ∈ s, ‖x‖ ≤ C",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : Type u_2\ninst✝ : SeminormedGroup E\ns : Set E\n⊢ Bornology.IsBounded s ↔ ∃ C, ∀ x ∈ s, ‖x‖ ≤ C"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Bounded | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 13
} | {
"line": 105,
"column": 14
} | [
{
"pp": "α : Type u_1\nE : Type u_2\ninst✝¹ : SeminormedGroup E\ninst✝ : TopologicalSpace α\nf : α → E\nhf : HasCompactMulSupport f\nh'f : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f\n⊢ ∃ C, ∀ (x : α), ‖f x‖ ≤ C",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": []... | [
"α : Type u_1\nE : Type u_2\ninst✝¹ : SeminormedGroup E\ninst✝ : TopologicalSpace α\nf : α → E\nhf : HasCompactMulSupport f\nh'f : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f\n⊢ ∃ C, ∀ (x : α), ‖f x‖ ≤ C"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Bounded | {
"line": 160,
"column": 2
} | {
"line": 160,
"column": 28
} | {
"line": 160,
"column": 29
} | [
{
"pp": "α : Type u_1\nE : Type u_2\ninst✝¹ : NormedAddGroup E\ninst✝ : TopologicalSpace α\nf : α → E\nhf : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f\nh : HasCompactSupport f\n⊢ ∃ C, ∀ (x : α), ‖f x‖ ≤ C",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
... | [
"α : Type u_1\nE : Type u_2\ninst✝¹ : NormedAddGroup E\ninst✝ : TopologicalSpace α\nf : α → E\nhf : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f\nh : HasCompactSupport f\n⊢ ∃ C, ∀ (x : α), ‖f x‖ ≤ C"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Dilation | {
"line": 161,
"column": 4
} | {
"line": 161,
"column": 23
} | {
"line": 162,
"column": 2
} | [
{
"pp": "case pos.inr\nα : Type u_1\nβ : Type u_2\nF : Type u_4\ninst✝³ : PseudoEMetricSpace α\ninst✝² : PseudoEMetricSpace β\ninst✝¹ : FunLike F α β\ninst✝ : DilationClass F α β\nf : F\nx y : α\nkey : ∀ (x y : α), edist x y = 0 ∨ edist x y = ∞\nr : ℝ≥0\nhne : r ≠ 0\nhr : edist (f x) (f y) = ↑r * edist x y\nh :... | [] | · simp [hr, h, hne] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.MetricSpace.Dilation | {
"line": 180,
"column": 2
} | {
"line": 180,
"column": 70
} | {
"line": 180,
"column": 71
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nF : Type u_4\ninst✝³ : PseudoEMetricSpace α\ninst✝² : PseudoEMetricSpace β\ninst✝¹ : FunLike F α β\ninst✝ : DilationClass F α β\nf : F\nx y : α\nr : ℝ≥0\nh₀ : edist x y ≠ 0\nhtop : edist x y ≠ ∞\nhr : edist (f x) (f y) = ↑r * edist x y\n⊢ r = ratio f",
"ppTerm": "?m.33",... | [
"α : Type u_1\nβ : Type u_2\nF : Type u_4\ninst✝³ : PseudoEMetricSpace α\ninst✝² : PseudoEMetricSpace β\ninst✝¹ : FunLike F α β\ninst✝ : DilationClass F α β\nf : F\nx y : α\nr : ℝ≥0\nh₀ : edist x y ≠ 0\nhtop : edist x y ≠ ∞\nhr : edist (f x) (f y) = ↑r * edist x y\n⊢ r = ratio f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Basic | {
"line": 106,
"column": 2
} | {
"line": 106,
"column": 13
} | {
"line": 106,
"column": 14
} | [
{
"pp": "α : Type u_2\ninst✝ : NormedDivisionRing α\na b : α\nha : a ≠ 0\nhb : b ≠ 0\n⊢ ‖a * b * a⁻¹ * b⁻¹ - 1‖ ≤ 2 * ‖a‖⁻¹ * ‖b‖⁻¹ * ‖a - 1‖ * ‖b - 1‖",
"ppTerm": "?m.75",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_2\ninst✝ : NormedDivisionRing α\na b : α\nha : a ≠ 0\nhb : b ≠ 0\n⊢ ‖a * b * a⁻¹ * b⁻¹ - 1‖ ≤ 2 * ‖a‖⁻¹ * ‖b‖⁻¹ * ‖a - 1‖ * ‖b - 1‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Basic | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 13
} | {
"line": 110,
"column": 14
} | [
{
"pp": "α : Type u_2\ninst✝ : NormedDivisionRing α\na b : α\nha : a ≠ 0\nhb : b ≠ 0\n⊢ ‖a * b * a⁻¹ * b⁻¹ - 1‖₊ ≤ 2 * ‖a‖₊⁻¹ * ‖b‖₊⁻¹ * ‖a - 1‖₊ * ‖b - 1‖₊",
"ppTerm": "?m.75",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_2\ninst✝ : NormedDivisionRing α\na b : α\nha : a ≠ 0\nhb : b ≠ 0\n⊢ ‖a * b * a⁻¹ * b⁻¹ - 1‖₊ ≤ 2 * ‖a‖₊⁻¹ * ‖b‖₊⁻¹ * ‖a - 1‖₊ * ‖b - 1‖₊"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.DilationEquiv | {
"line": 129,
"column": 91
} | {
"line": 130,
"column": 86
} | {
"line": 131,
"column": 4
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝² : PseudoEMetricSpace X\ninst✝¹ : PseudoEMetricSpace Y\ninst✝ : PseudoEMetricSpace Z\ne : X ≃ᵈ Y\ne' : Y ≃ᵈ Z\nhX : ∀ (x y : X), edist x y = 0 ∨ edist x y = ∞\nx y : X\n⊢ edist (e x) (e y) = 0 ∨ edist (e x) (e y) = ∞",
"ppTerm": "?m.83",
"assigned... | [] | by
refine (hX x y).imp (fun h ↦ ?_) fun h ↦ ?_ <;> simp [*, Dilation.ratio_ne_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.AddTorsor | {
"line": 131,
"column": 2
} | {
"line": 136,
"column": 46
} | {
"line": 138,
"column": 0
} | [
{
"pp": "G : Type u_1\nP : Type u_2\ninst✝³ : LE G\ninst✝² : Preorder P\ninst✝¹ : SMul G P\ninst✝ : IsOrderedCancelSMul G P\na b : G\nc d : P\nh₁ : a ≤ b\nh₂ : c < d\n⊢ a • c < b • d",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"lt_of_le_of_lt",
"False",
"instHSMul",
... | [] | refine lt_of_le_of_lt (IsOrderedSMul.smul_le_smul_right a b h₁ c) ?_
refine lt_of_le_not_ge (IsOrderedSMul.smul_le_smul_left c d (le_of_lt h₂) b) ?_
by_contra hbdc
have h : d ≤ c := IsOrderedCancelSMul.le_of_smul_le_smul_left b d c hbdc
rw [@lt_iff_le_not_ge] at h₂
simp_all only [not_true_eq_false, and_false] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.AddTorsor | {
"line": 131,
"column": 2
} | {
"line": 136,
"column": 46
} | {
"line": 138,
"column": 0
} | [
{
"pp": "G : Type u_1\nP : Type u_2\ninst✝³ : LE G\ninst✝² : Preorder P\ninst✝¹ : SMul G P\ninst✝ : IsOrderedCancelSMul G P\na b : G\nc d : P\nh₁ : a ≤ b\nh₂ : c < d\n⊢ a • c < b • d",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"lt_of_le_of_lt",
"False",
"instHSMul",
... | [] | refine lt_of_le_of_lt (IsOrderedSMul.smul_le_smul_right a b h₁ c) ?_
refine lt_of_le_not_ge (IsOrderedSMul.smul_le_smul_left c d (le_of_lt h₂) b) ?_
by_contra hbdc
have h : d ≤ c := IsOrderedCancelSMul.le_of_smul_le_smul_left b d c hbdc
rw [@lt_iff_le_not_ge] at h₂
simp_all only [not_true_eq_false, and_false] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.ENNReal.Action | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 60
} | {
"line": 99,
"column": 2
} | [
{
"pp": "r : ℝ≥0\ns : ℝ≥0∞\n⊢ (r • s).toReal = r • s.toReal",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"ENNReal.ofNNReal",
"instHSMul",
"instSMulOfMul",
"HMul.hMul",
"ENNReal.smul_def",
"NNReal.instSMulOfReal",
"c... | [
"r : ℝ≥0\ns : ℝ≥0∞\n⊢ ↑r * s.toReal = r • s.toReal"
] | rw [ENNReal.smul_def, smul_eq_mul, toReal_mul, coe_toReal] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.MulAction | {
"line": 35,
"column": 2
} | {
"line": 35,
"column": 25
} | {
"line": 35,
"column": 26
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : SeminormedAddGroup α\ninst✝² : SeminormedAddGroup β\ninst✝¹ : SMulZeroClass α β\ninst✝ : IsBoundedSMul α β\nr : α\nx : β\n⊢ ‖r • x‖ ≤ ‖r‖ * ‖x‖",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\ninst✝³ : SeminormedAddGroup α\ninst✝² : SeminormedAddGroup β\ninst✝¹ : SMulZeroClass α β\ninst✝ : IsBoundedSMul α β\nr : α\nx : β\n⊢ ‖r • x‖ ≤ ‖r‖ * ‖x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.Basic | {
"line": 207,
"column": 6
} | {
"line": 207,
"column": 31
} | {
"line": 207,
"column": 32
} | [
{
"pp": "G : Type u_1\nα✝ : Type u_2\nβ : Type u_3\nι✝ : Type u_4\nι : Type u_5\nα : ι → Type u_6\ninst✝⁴ : Nonempty ι\ninst✝³ : Fintype ι\ninst✝² : (i : ι) → SeminormedAddCommGroup (α i)\ninst✝¹ : (i : ι) → One (α i)\ninst✝ : ∀ (i : ι), NormOneClass (α i)\n⊢ ‖1‖ = 1",
"ppTerm": "?m.12",
"assigned": tru... | [
"G : Type u_1\nα✝ : Type u_2\nβ : Type u_3\nι✝ : Type u_4\nι : Type u_5\nα : ι → Type u_6\ninst✝⁴ : Nonempty ι\ninst✝³ : Fintype ι\ninst✝² : (i : ι) → SeminormedAddCommGroup (α i)\ninst✝¹ : (i : ι) → One (α i)\ninst✝ : ∀ (i : ι), NormOneClass (α i)\n⊢ (Finset.univ.sup fun b ↦ 1) = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.MulAction | {
"line": 43,
"column": 2
} | {
"line": 43,
"column": 40
} | {
"line": 43,
"column": 41
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : SeminormedAddGroup α\ninst✝² : SeminormedAddGroup β\ninst✝¹ : SMulZeroClass α β\ninst✝ : IsBoundedSMul α β\nr : α\nx : β\n⊢ ‖r • x‖ₑ ≤ ‖r‖ₑ * ‖x‖ₑ",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.m... | [
"α : Type u_1\nβ : Type u_2\ninst✝³ : SeminormedAddGroup α\ninst✝² : SeminormedAddGroup β\ninst✝¹ : SMulZeroClass α β\ninst✝ : IsBoundedSMul α β\nr : α\nx : β\n⊢ ‖r • x‖₊ ≤ ‖r‖₊ * ‖x‖₊"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.Basic | {
"line": 236,
"column": 8
} | {
"line": 236,
"column": 34
} | {
"line": 236,
"column": 35
} | [
{
"pp": "β : Type u_5\ninst✝¹ : NormedRing β\ninst✝ : Nontrivial β\n⊢ ‖1‖ ≤ ‖1‖ * ‖1‖",
"ppTerm": "?m.34",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"β : Type u_5\ninst✝¹ : NormedRing β\ninst✝ : Nontrivial β\n⊢ ‖1‖ ≤ ‖1‖ * ‖1‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.MulAction | {
"line": 46,
"column": 2
} | {
"line": 46,
"column": 61
} | {
"line": 46,
"column": 62
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : SeminormedAddGroup α\ninst✝² : SeminormedAddGroup β\ninst✝¹ : SMulZeroClass α β\ninst✝ : IsBoundedSMul α β\ns : α\nx y : β\n⊢ dist (s • x) (s • y) ≤ ‖s‖ * dist x y",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMon... | [
"α : Type u_1\nβ : Type u_2\ninst✝³ : SeminormedAddGroup α\ninst✝² : SeminormedAddGroup β\ninst✝¹ : SMulZeroClass α β\ninst✝ : IsBoundedSMul α β\ns : α\nx y : β\n⊢ ‖-(s • x) + s • y‖ ≤ ‖s‖ * ‖-x + y‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.MulAction | {
"line": 62,
"column": 32
} | {
"line": 62,
"column": 67
} | {
"line": 62,
"column": 68
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : NonUnitalSeminormedRing α\nx y₁ y₂ : α\n⊢ dist (x • y₁) (x • y₂) ≤ dist x 0 * dist y₁ y₂",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"instHSMul",
"instSMulOfMu... | [
"α : Type u_1\nβ : Type u_2\ninst✝ : NonUnitalSeminormedRing α\nx y₁ y₂ : α\n⊢ ‖x * y₁ - x * y₂‖ ≤ ‖x‖ * ‖y₁ - y₂‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.MulAction | {
"line": 63,
"column": 32
} | {
"line": 63,
"column": 67
} | {
"line": 63,
"column": 68
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : NonUnitalSeminormedRing α\nx₁ x₂ y : α\n⊢ dist (x₁ • y) (x₂ • y) ≤ dist x₁ x₂ * dist y 0",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"instHSMul",
"instSMulOfMu... | [
"α : Type u_1\nβ : Type u_2\ninst✝ : NonUnitalSeminormedRing α\nx₁ x₂ y : α\n⊢ ‖x₁ * y - x₂ * y‖ ≤ ‖x₁ - x₂‖ * ‖y‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.MulAction | {
"line": 79,
"column": 41
} | {
"line": 79,
"column": 77
} | {
"line": 79,
"column": 78
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : SeminormedRing α\ninst✝¹ : SeminormedAddCommGroup β\ninst✝ : Module α β\nh : ∀ (r : α) (x : β), ‖r • x‖ ≤ ‖r‖ * ‖x‖\na : α\nb₁ b₂ : β\n⊢ dist (a • b₁) (a • b₂) ≤ dist a 0 * dist b₁ b₂",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Norm.... | [
"α : Type u_1\nβ : Type u_2\ninst✝² : SeminormedRing α\ninst✝¹ : SeminormedAddCommGroup β\ninst✝ : Module α β\nh : ∀ (r : α) (x : β), ‖r • x‖ ≤ ‖r‖ * ‖x‖\na : α\nb₁ b₂ : β\n⊢ ‖a • b₁ - a • b₂‖ ≤ ‖a‖ * ‖b₁ - b₂‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.MulAction | {
"line": 80,
"column": 41
} | {
"line": 80,
"column": 77
} | {
"line": 80,
"column": 78
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : SeminormedRing α\ninst✝¹ : SeminormedAddCommGroup β\ninst✝ : Module α β\nh : ∀ (r : α) (x : β), ‖r • x‖ ≤ ‖r‖ * ‖x‖\na₁ a₂ : α\nb : β\n⊢ dist (a₁ • b) (a₂ • b) ≤ dist a₁ a₂ * dist b 0",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Norm.... | [
"α : Type u_1\nβ : Type u_2\ninst✝² : SeminormedRing α\ninst✝¹ : SeminormedAddCommGroup β\ninst✝ : Module α β\nh : ∀ (r : α) (x : β), ‖r • x‖ ≤ ‖r‖ * ‖x‖\na₁ a₂ : α\nb : β\n⊢ ‖a₁ • b - a₂ • b‖ ≤ ‖a₁ - a₂‖ * ‖b‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.MulAction | {
"line": 196,
"column": 4
} | {
"line": 196,
"column": 28
} | {
"line": 196,
"column": 29
} | [
{
"pp": "case mp\nα : Type u_1\nβ : Type u_2\ninst✝³ : NormedDivisionRing α\ninst✝² : SeminormedAddCommGroup β\ninst✝¹ : Module α β\ninst✝ : NormSMulClass α β\ns : α\nhs : s ≠ 0\nx : β\nε : ℝ\ny : β\nh1 : dist y x < ε\n⊢ dist (s • y) (s • x) < ‖s‖ * ε",
"ppTerm": "?mp",
"assigned": true,
"usedConsta... | [
"case mp\nα : Type u_1\nβ : Type u_2\ninst✝³ : NormedDivisionRing α\ninst✝² : SeminormedAddCommGroup β\ninst✝¹ : Module α β\ninst✝ : NormSMulClass α β\ns : α\nhs : s ≠ 0\nx : β\nε : ℝ\ny : β\nh1 : dist y x < ε\n⊢ ‖s‖ * dist y x < ‖s‖ * ε"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.MulAction | {
"line": 200,
"column": 4
} | {
"line": 200,
"column": 31
} | {
"line": 200,
"column": 32
} | [
{
"pp": "case mpr\nα : Type u_1\nβ : Type u_2\ninst✝³ : NormedDivisionRing α\ninst✝² : SeminormedAddCommGroup β\ninst✝¹ : Module α β\ninst✝ : NormSMulClass α β\ns : α\nhs : s ≠ 0\nx : β\nε : ℝ\np : β\nh : dist p (s • x) < ‖s‖ * ε\n⊢ dist (s • s⁻¹ • p) (s • x) < ‖s‖ * ε",
"ppTerm": "?mpr",
"assigned": tr... | [
"case mpr\nα : Type u_1\nβ : Type u_2\ninst✝³ : NormedDivisionRing α\ninst✝² : SeminormedAddCommGroup β\ninst✝¹ : Module α β\ninst✝ : NormSMulClass α β\ns : α\nhs : s ≠ 0\nx : β\nε : ℝ\np : β\nh : dist p (s • x) < ‖s‖ * ε\n⊢ dist p (s • x) < ‖s‖ * ε"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.MulAction | {
"line": 208,
"column": 4
} | {
"line": 208,
"column": 28
} | {
"line": 208,
"column": 29
} | [
{
"pp": "case mp\nα : Type u_1\nβ : Type u_2\ninst✝³ : NormedDivisionRing α\ninst✝² : SeminormedAddCommGroup β\ninst✝¹ : Module α β\ninst✝ : NormSMulClass α β\ns : α\nhs : s ≠ 0\nx : β\nε : ℝ\ny : β\nh1 : dist y x ≤ ε\n⊢ dist (s • y) (s • x) ≤ ‖s‖ * ε",
"ppTerm": "?mp",
"assigned": true,
"usedConsta... | [
"case mp\nα : Type u_1\nβ : Type u_2\ninst✝³ : NormedDivisionRing α\ninst✝² : SeminormedAddCommGroup β\ninst✝¹ : Module α β\ninst✝ : NormSMulClass α β\ns : α\nhs : s ≠ 0\nx : β\nε : ℝ\ny : β\nh1 : dist y x ≤ ε\n⊢ ‖s‖ * dist y x ≤ ‖s‖ * ε"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.Basic | {
"line": 415,
"column": 4
} | {
"line": 415,
"column": 42
} | {
"line": 416,
"column": 6
} | [
{
"pp": "α : Type u_2\ninst✝ : SeminormedRing α\na : α\nn : ℕ\nx✝ : 0 < n + 2\n⊢ ‖a ^ (n + 2)‖₊ ≤ ‖a‖₊ ^ (n + 2)",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Monoid",
"Eq.mpr",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"SeminormedAddGroup.toNN... | [
"α : Type u_2\ninst✝ : SeminormedRing α\na : α\nn : ℕ\nx✝ : 0 < n + 2\n⊢ ‖a * a ^ (n + 1)‖₊ ≤ ‖a‖₊ * ‖a‖₊ ^ (n + 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.MulAction | {
"line": 212,
"column": 4
} | {
"line": 212,
"column": 31
} | {
"line": 212,
"column": 32
} | [
{
"pp": "case mpr\nα : Type u_1\nβ : Type u_2\ninst✝³ : NormedDivisionRing α\ninst✝² : SeminormedAddCommGroup β\ninst✝¹ : Module α β\ninst✝ : NormSMulClass α β\ns : α\nhs : s ≠ 0\nx : β\nε : ℝ\np : β\nh : dist p (s • x) ≤ ‖s‖ * ε\n⊢ dist (s • s⁻¹ • p) (s • x) ≤ ‖s‖ * ε",
"ppTerm": "?mpr",
"assigned": tr... | [
"case mpr\nα : Type u_1\nβ : Type u_2\ninst✝³ : NormedDivisionRing α\ninst✝² : SeminormedAddCommGroup β\ninst✝¹ : Module α β\ninst✝ : NormSMulClass α β\ns : α\nhs : s ≠ 0\nx : β\nε : ℝ\np : β\nh : dist p (s • x) ≤ ‖s‖ * ε\n⊢ dist p (s • x) ≤ ‖s‖ * ε"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.Basic | {
"line": 426,
"column": 2
} | {
"line": 426,
"column": 47
} | {
"line": 426,
"column": 48
} | [
{
"pp": "α : Type u_2\ninst✝ : SeminormedRing α\na : α\nn : ℕ\nh : 0 < n\n⊢ ‖a ^ n‖ ≤ ‖a‖ ^ n",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_2\ninst✝ : SeminormedRing α\na : α\nn : ℕ\nh : 0 < n\n⊢ ‖a ^ n‖ ≤ ‖a‖ ^ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.Basic | {
"line": 454,
"column": 8
} | {
"line": 454,
"column": 33
} | {
"line": 454,
"column": 34
} | [
{
"pp": "α : Type u_2\ninst✝ : SeminormedRing α\na b c : α\nha : ‖a‖ ≤ 1\n⊢ ‖c - a * b‖ ≤ ‖c - a‖ + ‖a * (1 - b)‖",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"SeminormedRing.toNorm",
"Real.instLE",
"Real",
"HMul.hMul",
"R... | [
"α : Type u_2\ninst✝ : SeminormedRing α\na b c : α\nha : ‖a‖ ≤ 1\n⊢ ‖c - a * b‖ ≤ ‖c - a‖ + ‖a - a * b‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.Basic | {
"line": 642,
"column": 2
} | {
"line": 642,
"column": 13
} | {
"line": 642,
"column": 14
} | [
{
"pp": "R : Type u_5\nS : Type u_6\ninst✝² : CommRing R\ninst✝¹ : Ring S\ninst✝ : Algebra R S\nA : Subalgebra R S\nf : S → ℝ\nhf_pm : IsPowMul f\nx : ↥A\nn : ℕ\nhn : 1 ≤ n\n⊢ (fun x ↦ f ↑x) (x ^ n) = (fun x ↦ f ↑x) x ^ n",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Subalgebra.in... | [
"R : Type u_5\nS : Type u_6\ninst✝² : CommRing R\ninst✝¹ : Ring S\ninst✝ : Algebra R S\nA : Subalgebra R S\nf : S → ℝ\nhf_pm : IsPowMul f\nx : ↥A\nn : ℕ\nhn : 1 ≤ n\n⊢ f (↑x ^ n) = f ↑x ^ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.Basic | {
"line": 786,
"column": 4
} | {
"line": 786,
"column": 56
} | {
"line": 786,
"column": 57
} | [
{
"pp": "α : Type u_2\ninst✝³ : NormedAddCommGroup α\ninst✝² : MulOneClass α\ninst✝¹ : NormMulClass α\ninst✝ : Nontrivial α\nu : α\nhu : u ≠ 0\n⊢ ‖1‖ = 1",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_2\ninst✝³ : NormedAddCommGroup α\ninst✝² : MulOneClass α\ninst✝¹ : NormMulClass α\ninst✝ : Nontrivial α\nu : α\nhu : u ≠ 0\n⊢ ‖1‖ = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.Basic | {
"line": 801,
"column": 4
} | {
"line": 801,
"column": 69
} | {
"line": 801,
"column": 70
} | [
{
"pp": "G : Type u_1\nα : Type u_2\nβ : Type u_3\nι : Type u_4\ninst✝¹ : NormedRing α\ninst✝ : NormMulClass α\na✝ b✝ : α\nh : a✝ * b✝ = 0\n⊢ a✝ = 0 ∨ b✝ = 0",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"norm_eq_zero",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
... | [
"G : Type u_1\nα : Type u_2\nβ : Type u_3\nι : Type u_4\ninst✝¹ : NormedRing α\ninst✝ : NormMulClass α\na✝ b✝ : α\nh : a✝ * b✝ = 0\n⊢ ‖a✝‖ = 0 ∨ ‖b✝‖ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.Basic | {
"line": 934,
"column": 28
} | {
"line": 934,
"column": 78
} | {
"line": 934,
"column": 79
} | [
{
"pp": "G : Type u_1\nα : Type u_2\nβ : Type u_3\nι : Type u_4\nR : Type u_5\ninst✝ : Ring R\nv : AbsoluteValue R ℝ\nx y z : R\n⊢ v (-x + z) ≤ v (-x + y) + v (-y + z)",
"ppTerm": "?m.86",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Real.partialOrder",
... | [
"G : Type u_1\nα : Type u_2\nβ : Type u_3\nι : Type u_4\nR : Type u_5\ninst✝ : Ring R\nv : AbsoluteValue R ℝ\nx y z : R\n⊢ v (z - x) ≤ v (z - y) + v (y - x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Prod.TProd | {
"line": 150,
"column": 8
} | {
"line": 150,
"column": 18
} | {
"line": 150,
"column": 19
} | [
{
"pp": "ι : Type u\nα : ι → Type v\ni : ι\nl : List ι\nt : (i : ι) → Set (α i)\nf : (i : ι) → α i\nh : TProd.mk l f ∈ Set.tprod l t ↔ ∀ (i : ι), i ∈ l → f i ∈ t i\n⊢ f ∈ TProd.mk (i :: l) ⁻¹' Set.tprod (i :: l) t ↔ f ∈ {i_1 | i_1 ∈ i :: l}.pi t",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants"... | [
"ι : Type u\nα : ι → Type v\ni : ι\nl : List ι\nt : (i : ι) → Set (α i)\nf : (i : ι) → α i\nh : TProd.mk l f ∈ Set.tprod l t ↔ ∀ (i : ι), i ∈ l → f i ∈ t i\n⊢ f ∈ TProd.mk (i :: l) ⁻¹' t i ×ˢ Set.tprod l t ↔ f ∈ {i_1 | i_1 ∈ i :: l}.pi t"
] | Set.tprod, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.MeasurableSpace.Basic | {
"line": 317,
"column": 17
} | {
"line": 317,
"column": 49
} | {
"line": 317,
"column": 50
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns : Set α\nf g : α → β\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nx✝ : DecidablePred fun x ↦ x ∈ s\nhs : MeasurableSet s\nhf : Measurable f\nhg : Measurable g\nt : Set β\nht : MeasurableSet t\n⊢ MeasurableSet (s.piecewise f g ⁻¹' t)",
"ppTerm": "?m.26",
"assigne... | [
"α : Type u_1\nβ : Type u_2\ns : Set α\nf g : α → β\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nx✝ : DecidablePred fun x ↦ x ∈ s\nhs : MeasurableSet s\nhf : Measurable f\nhg : Measurable g\nt : Set β\nht : MeasurableSet t\n⊢ MeasurableSet (s.ite (f ⁻¹' t) (g ⁻¹' t))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Lemmas | {
"line": 217,
"column": 2
} | {
"line": 217,
"column": 33
} | {
"line": 218,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝ : NormedDivisionRing α\nm : ℕ\nhm : -↑m < 0\n⊢ Tendsto (fun x ↦ x ^ (-↑m)) (𝓝[≠] 0) (cobounded α)",
"ppTerm": "?m.75",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"GroupWithZero.... | [
"α : Type u_1\ninst✝ : NormedDivisionRing α\nm : ℕ\nhm : -↑m < 0\n⊢ Tendsto (fun x ↦ (x ^ m)⁻¹) (𝓝[≠] 0) (cobounded α)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Lemmas | {
"line": 279,
"column": 2
} | {
"line": 279,
"column": 13
} | {
"line": 279,
"column": 14
} | [
{
"pp": "𝕜 : Type u_4\ninst✝ : NontriviallyNormedField 𝕜\nx : 𝕜\n⊢ ContinuousAt Inv.inv x ↔ x ≠ 0",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"Conti... | [
"𝕜 : Type u_4\ninst✝ : NontriviallyNormedField 𝕜\nx : 𝕜\n⊢ ContinuousAt Inv.inv x ↔ ¬x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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