module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Topology.EMetricSpace.VariationOnFromTo | {
"line": 106,
"column": 4
} | {
"line": 106,
"column": 51
} | {
"line": 108,
"column": 0
} | [
{
"pp": "case hx\nα✝ : Type u_1\ninst✝³ : LinearOrder α✝\nE✝ : Type u_2\ninst✝² : PseudoEMetricSpace E✝\nf✝ : α✝ → E✝\ns✝ : Set α✝\nα : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns : Set α\nhf : LocallyBoundedVariationOn f s\na b : α\nha : a ∈ s\nhb : b ∈ s\nh : var... | [] | exacts [⟨ha, ⟨le_rfl, h'⟩⟩, ⟨hb, ⟨h', le_rfl⟩⟩] | Batteries.Tactic._aux_Batteries_Tactic_Init___elabRules_Batteries_Tactic_exacts_1 | Batteries.Tactic.exacts |
Mathlib.Topology.EMetricSpace.VariationOnFromTo | {
"line": 200,
"column": 8
} | {
"line": 200,
"column": 48
} | {
"line": 200,
"column": 48
} | [
{
"pp": "α : Type u_1\ninst✝³ : LinearOrder α\ns : Set α\nE : Type u_3\ninst✝² : PseudoMetricSpace E\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nf : α → E\nl : E\na b : α\nha : a ∈ s\nhb : b ∈ s\nhf : LocallyBoundedVariationOn f s\nh'f : Tendsto f (𝓝[s ∩ Iio b] b) (𝓝 l)\nH :\n Tendsto (fun x ↦ var... | [
"α : Type u_1\ninst✝³ : LinearOrder α\ns : Set α\nE : Type u_3\ninst✝² : PseudoMetricSpace E\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nf : α → E\nl : E\na b : α\nha : a ∈ s\nhb : b ∈ s\nhf : LocallyBoundedVariationOn f s\nh'f : Tendsto f (𝓝[s ∩ Iio b] b) (𝓝 l)\nH :\n Tendsto (fun x ↦ variationOnFrom... | variationOnFromTo.sub_left hf ha hb hx.1 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 476,
"column": 2
} | {
"line": 484,
"column": 22
} | {
"line": 485,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\nx : α\nx✝ : x ∈ {x | v.limRatioMeas hρ x = ∞}... | [
"α : Type u_1\ninst✝⁴ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\nx : α\nx✝ : x ∈ {x | v.limRatioMeas hρ x = ∞}\no : Set α\... | have A : ∀ q : ℝ≥0, 1 ≤ q → μ s ≤ (q : ℝ≥0∞)⁻¹ * ρ s := by
intro q hq
rw [mul_comm, ← div_eq_mul_inv, ENNReal.le_div_iff_mul_le _ (Or.inr ρs), mul_comm]
· apply v.mul_measure_le_of_subset_lt_limRatioMeas hρ
intro y hy
have : v.limRatioMeas hρ y = ∞ := hy.1
simp only [this, ENNReal.coe_lt_t... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.BoundedVariation | {
"line": 100,
"column": 2
} | {
"line": 102,
"column": 23
} | {
"line": 104,
"column": 0
} | [
{
"pp": "V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : NormedSpace ℝ V\ninst✝ : FiniteDimensional ℝ V\nf : ℝ → V\nh : LocallyBoundedVariationOn f univ\n⊢ ∀ᵐ (x : ℝ), DifferentiableAt ℝ f x",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Real",
... | [] | filter_upwards [h.ae_differentiableWithinAt_of_mem] with x hx
rw [differentiableWithinAt_univ] at hx
exact hx (mem_univ _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.BoundedVariation | {
"line": 100,
"column": 2
} | {
"line": 102,
"column": 23
} | {
"line": 104,
"column": 0
} | [
{
"pp": "V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : NormedSpace ℝ V\ninst✝ : FiniteDimensional ℝ V\nf : ℝ → V\nh : LocallyBoundedVariationOn f univ\n⊢ ∀ᵐ (x : ℝ), DifferentiableAt ℝ f x",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Real",
... | [] | filter_upwards [h.ae_differentiableWithinAt_of_mem] with x hx
rw [differentiableWithinAt_univ] at hx
exact hx (mem_univ _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.ContinuousMap.StarOrdered | {
"line": 97,
"column": 33
} | {
"line": 97,
"column": 38
} | {
"line": 98,
"column": 8
} | [
{
"pp": "α : Type u_1\ninst✝¹⁰ : TopologicalSpace α\ninst✝⁹ : Zero α\nR : Type u_2\ninst✝⁸ : TopologicalSpace R\ninst✝⁷ : CommSemiring R\ninst✝⁶ : PartialOrder R\ninst✝⁵ : NoZeroDivisors R\ninst✝⁴ : StarRing R\ninst✝³ : StarOrderedRing R\ninst✝² : IsTopologicalSemiring R\ninst✝¹ : ContinuousStar R\ninst✝ : Star... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 279,
"column": 70
} | {
"line": 279,
"column": 94
} | {
"line": 280,
"column": 6
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns : Set α\nx : α\nhx : x ∈ s\nu : ℕ → α\nhu : Monotone u\nus : ∀ (i : ℕ), u i ∈ s\nn : ℕ\nh : u n ≤ x\nv : ℕ → α := fun i ↦ if i ≤ n then u i else x\n⊢ ∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) = ∑ ... | [] | grind [Finset.sum_congr] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 279,
"column": 70
} | {
"line": 279,
"column": 94
} | {
"line": 280,
"column": 6
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns : Set α\nx : α\nhx : x ∈ s\nu : ℕ → α\nhu : Monotone u\nus : ∀ (i : ℕ), u i ∈ s\nn : ℕ\nh : u n ≤ x\nv : ℕ → α := fun i ↦ if i ≤ n then u i else x\n⊢ ∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) = ∑ ... | [] | grind [Finset.sum_congr] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 279,
"column": 70
} | {
"line": 279,
"column": 94
} | {
"line": 280,
"column": 6
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns : Set α\nx : α\nhx : x ∈ s\nu : ℕ → α\nhu : Monotone u\nus : ∀ (i : ℕ), u i ∈ s\nn : ℕ\nh : u n ≤ x\nv : ℕ → α := fun i ↦ if i ≤ n then u i else x\n⊢ ∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) = ∑ ... | [] | grind [Finset.sum_congr] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 318,
"column": 12
} | {
"line": 318,
"column": 36
} | {
"line": 319,
"column": 10
} | [
{
"pp": "case e_a.e_a\nα : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns : Set α\nx : α\nhx : x ∈ s\nu : ℕ → α\nhu : Monotone u\nus : ∀ (i : ℕ), u i ∈ s\nn : ℕ\nh : x < u n\nexists_N : ∃ N ≤ n, x < u N\nN : ℕ := Nat.find exists_N\nhN : N ≤ n ∧ x < u N\nw : ℕ → α := f... | [] | grind [Finset.sum_congr] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 318,
"column": 12
} | {
"line": 318,
"column": 36
} | {
"line": 319,
"column": 10
} | [
{
"pp": "case e_a.e_a\nα : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns : Set α\nx : α\nhx : x ∈ s\nu : ℕ → α\nhu : Monotone u\nus : ∀ (i : ℕ), u i ∈ s\nn : ℕ\nh : x < u n\nexists_N : ∃ N ≤ n, x < u N\nN : ℕ := Nat.find exists_N\nhN : N ≤ n ∧ x < u N\nw : ℕ → α := f... | [] | grind [Finset.sum_congr] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 318,
"column": 12
} | {
"line": 318,
"column": 36
} | {
"line": 319,
"column": 10
} | [
{
"pp": "case e_a.e_a\nα : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns : Set α\nx : α\nhx : x ∈ s\nu : ℕ → α\nhu : Monotone u\nus : ∀ (i : ℕ), u i ∈ s\nn : ℕ\nh : x < u n\nexists_N : ∃ N ≤ n, x < u N\nN : ℕ := Nat.find exists_N\nhN : N ≤ n ∧ x < u N\nw : ℕ → α := f... | [] | grind [Finset.sum_congr] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 322,
"column": 10
} | {
"line": 322,
"column": 34
} | {
"line": 323,
"column": 6
} | [
{
"pp": "case e_a\nα : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns : Set α\nx : α\nhx : x ∈ s\nu : ℕ → α\nhu : Monotone u\nus : ∀ (i : ℕ), u i ∈ s\nn : ℕ\nh : x < u n\nexists_N : ∃ N ≤ n, x < u N\nN : ℕ := Nat.find exists_N\nhN : N ≤ n ∧ x < u N\nw : ℕ → α := fun i... | [] | grind [Finset.sum_congr] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 322,
"column": 10
} | {
"line": 322,
"column": 34
} | {
"line": 323,
"column": 6
} | [
{
"pp": "case e_a\nα : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns : Set α\nx : α\nhx : x ∈ s\nu : ℕ → α\nhu : Monotone u\nus : ∀ (i : ℕ), u i ∈ s\nn : ℕ\nh : x < u n\nexists_N : ∃ N ≤ n, x < u N\nN : ℕ := Nat.find exists_N\nhN : N ≤ n ∧ x < u N\nw : ℕ → α := fun i... | [] | grind [Finset.sum_congr] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 322,
"column": 10
} | {
"line": 322,
"column": 34
} | {
"line": 323,
"column": 6
} | [
{
"pp": "case e_a\nα : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns : Set α\nx : α\nhx : x ∈ s\nu : ℕ → α\nhu : Monotone u\nus : ∀ (i : ℕ), u i ∈ s\nn : ℕ\nh : x < u n\nexists_N : ∃ N ≤ n, x < u N\nN : ℕ := Nat.find exists_N\nhN : N ≤ n ∧ x < u N\nw : ℕ → α := fun i... | [] | grind [Finset.sum_congr] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 336,
"column": 8
} | {
"line": 337,
"column": 59
} | {
"line": 338,
"column": 8
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns : Set α\nx : α\nhx : x ∈ s\nu : ℕ → α\nhu : Monotone u\nus : ∀ (i : ℕ), u i ∈ s\nn : ℕ\nh : x < u n\nexists_N : ∃ N ≤ n, x < u N\nN : ℕ := Nat.find exists_N\nhN : N ≤ n ∧ x < u N\nw : ℕ → α := fun i ↦ if i < ... | [
"α : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns : Set α\nx : α\nhx : x ∈ s\nu : ℕ → α\nhu : Monotone u\nus : ∀ (i : ℕ), u i ∈ s\nn : ℕ\nh : x < u n\nexists_N : ∃ N ≤ n, x < u N\nN : ℕ := Nat.find exists_N\nhN : N ≤ n ∧ x < u N\nw : ℕ → α := fun i ↦ if i < N then u i e... | rw [Finset.sum_eq_sum_Ico_succ_bot C, Finset.sum_eq_sum_Ico_succ_bot B, A, Finset.Ico_self,
Finset.sum_empty, add_zero, add_comm (edist _ _)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 367,
"column": 8
} | {
"line": 367,
"column": 32
} | {
"line": 368,
"column": 6
} | [
{
"pp": "case e_a\nα : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns t : Set α\nh : ∀ x ∈ s, ∀ y ∈ t, x ≤ y\nhs : ¬s = ∅\nthis✝ : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }\nht : ¬t = ∅\nthis : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ t }\nn : ℕ\nu : ℕ → ... | [] | grind [Finset.sum_congr] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 367,
"column": 8
} | {
"line": 367,
"column": 32
} | {
"line": 368,
"column": 6
} | [
{
"pp": "case e_a\nα : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns t : Set α\nh : ∀ x ∈ s, ∀ y ∈ t, x ≤ y\nhs : ¬s = ∅\nthis✝ : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }\nht : ¬t = ∅\nthis : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ t }\nn : ℕ\nu : ℕ → ... | [] | grind [Finset.sum_congr] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 367,
"column": 8
} | {
"line": 367,
"column": 32
} | {
"line": 368,
"column": 6
} | [
{
"pp": "case e_a\nα : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns t : Set α\nh : ∀ x ∈ s, ∀ y ∈ t, x ≤ y\nhs : ¬s = ∅\nthis✝ : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }\nht : ¬t = ∅\nthis : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ t }\nn : ℕ\nu : ℕ → ... | [] | grind [Finset.sum_congr] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 506,
"column": 2
} | {
"line": 506,
"column": 34
} | {
"line": 508,
"column": 0
} | [
{
"pp": "case e'_3\nα : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns : Set α\n⊢ s = ⇑ofDual '' ⇑ofDual ⁻¹' s",
"ppTerm": "?e'_3",
"assigned": true,
"usedConstants": [
"Equiv.instEquivLike",
"OrderDual.ofDual",
"congrArg",
"Equiv",... | [] | simp only [Equiv.image_preimage] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict | {
"line": 298,
"column": 4
} | {
"line": 302,
"column": 77
} | {
"line": 303,
"column": 4
} | [
{
"pp": "case hom_map_spectrum\nR : Type u_1\nS : Type u_2\nA : Type u_3\np q : A → Prop\ninst✝²³ : Semifield R\ninst✝²² : StarRing R\ninst✝²¹ : MetricSpace R\ninst✝²⁰ : IsTopologicalSemiring R\ninst✝¹⁹ : ContinuousStar R\ninst✝¹⁸ : Field S\ninst✝¹⁷ : StarRing S\ninst✝¹⁶ : MetricSpace S\ninst✝¹⁵ : IsTopological... | [
"case predicate_hom\nR : Type u_1\nS : Type u_2\nA : Type u_3\np q : A → Prop\ninst✝²³ : Semifield R\ninst✝²² : StarRing R\ninst✝²¹ : MetricSpace R\ninst✝²⁰ : IsTopologicalSemiring R\ninst✝¹⁹ : ContinuousStar R\ninst✝¹⁸ : Field S\ninst✝¹⁷ : StarRing S\ninst✝¹⁶ : MetricSpace S\ninst✝¹⁵ : IsTopologicalRing S\ninst✝¹⁴... | case hom_map_spectrum =>
simp only [nonUnitalStarAlgHom_apply, ← @quasispectrum.preimage_algebraMap (R := R) S,
cfcₙHom_map_quasispectrum, Set.ext_iff, Set.mem_preimage, Set.mem_range, comp_apply, coe_mk,
ContinuousMap.coe_mk, StarAlgHom.ofId_apply, halg.injective.eq_iff]
exact fun _ _ ↦ ((h... | Lean.Elab.Tactic.evalCase | Lean.Parser.Tactic.case |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict | {
"line": 363,
"column": 4
} | {
"line": 363,
"column": 65
} | {
"line": 364,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\np q : A → Prop\ninst✝²⁵ : Semifield R\ninst✝²⁴ : StarRing R\ninst✝²³ : MetricSpace R\ninst✝²² : IsTopologicalSemiring R\ninst✝²¹ : ContinuousStar R\ninst✝²⁰ : Field S\ninst✝¹⁹ : StarRing S\ninst✝¹⁸ : MetricSpace S\ninst✝¹⁷ : IsTopologicalRing S\ninst✝¹⁶ : Conti... | [
"R : Type u_1\nS : Type u_2\nA : Type u_3\np q : A → Prop\ninst✝²⁵ : Semifield R\ninst✝²⁴ : StarRing R\ninst✝²³ : MetricSpace R\ninst✝²² : IsTopologicalSemiring R\ninst✝²¹ : ContinuousStar R\ninst✝²⁰ : Field S\ninst✝¹⁹ : StarRing S\ninst✝¹⁸ : MetricSpace S\ninst✝¹⁷ : IsTopologicalRing S\ninst✝¹⁶ : ContinuousStar S\... | rw [cfcₙHom_eq_restrict f ha ((h a).mp ha).1 ((h a).mp ha).2] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.StarSubalgebra | {
"line": 153,
"column": 2
} | {
"line": 153,
"column": 40
} | {
"line": 154,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝¹³ : CommSemiring R\ninst✝¹² : StarRing R\ninst✝¹¹ : TopologicalSpace A\ninst✝¹⁰ : Semiring A\ninst✝⁹ : Algebra R A\ninst✝⁸ : StarRing A\ninst✝⁷ : StarModule R A\ninst✝⁶ : IsSemitopologicalSemiring A\ninst✝⁵ : ContinuousStar A\ninst✝⁴ : TopologicalSpace B\... | [
"R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝¹³ : CommSemiring R\ninst✝¹² : StarRing R\ninst✝¹¹ : TopologicalSpace A\ninst✝¹⁰ : Semiring A\ninst✝⁹ : Algebra R A\ninst✝⁸ : StarRing A\ninst✝⁷ : StarModule R A\ninst✝⁶ : IsSemitopologicalSemiring A\ninst✝⁵ : ContinuousStar A\ninst✝⁴ : TopologicalSpace B\ninst✝³ : Se... | refine Continuous.ext_on this hφ hψ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 675,
"column": 68
} | {
"line": 675,
"column": 92
} | {
"line": 676,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\nε : ℝ≥0∞\ns : Set α\nh : ε < eVariationOn f s\nn : ℕ\nu : ℕ → α\nhu : ε < ∑ x ∈ Finset.range n, edist (f (u (x + 1))) (f (u x))\nu_mono : ∀ ⦃a b : ℕ⦄, a ≤ b → u a ≤ u b\nu_mem : ∀ (i : ℕ), u i ∈ s\nv : ℕ → α :=... | [] | grind [Finset.sum_congr] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 675,
"column": 68
} | {
"line": 675,
"column": 92
} | {
"line": 676,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\nε : ℝ≥0∞\ns : Set α\nh : ε < eVariationOn f s\nn : ℕ\nu : ℕ → α\nhu : ε < ∑ x ∈ Finset.range n, edist (f (u (x + 1))) (f (u x))\nu_mono : ∀ ⦃a b : ℕ⦄, a ≤ b → u a ≤ u b\nu_mem : ∀ (i : ℕ), u i ∈ s\nv : ℕ → α :=... | [] | grind [Finset.sum_congr] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 675,
"column": 68
} | {
"line": 675,
"column": 92
} | {
"line": 676,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\nε : ℝ≥0∞\ns : Set α\nh : ε < eVariationOn f s\nn : ℕ\nu : ℕ → α\nhu : ε < ∑ x ∈ Finset.range n, edist (f (u (x + 1))) (f (u x))\nu_mono : ∀ ⦃a b : ℕ⦄, a ≤ b → u a ≤ u b\nu_mem : ∀ (i : ℕ), u i ∈ s\nv : ℕ → α :=... | [] | grind [Finset.sum_congr] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 667,
"column": 2
} | {
"line": 684,
"column": 18
} | {
"line": 686,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\nε : ℝ≥0∞\ns : Set α\nh : ε < eVariationOn f s\n⊢ ∃ a ∈ s, ∃ b ∈ s, a < b ∧ ε < eVariationOn f (s ∩ Icc a b)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOr... | [] | obtain ⟨n, u, ⟨u_mono, u_mem⟩, hu⟩ : ∃ n u, (Monotone u ∧ ∀ (i : ℕ), u i ∈ s) ∧
ε < ∑ x ∈ Finset.range n, edist (f (u (x + 1))) (f (u x)) := by
simpa [eVariationOn, lt_iSup_iff] using h
have A : ε < eVariationOn f (s ∩ Icc (u 0) (u n)) := by
apply hu.trans_le
simp only [Monotone] at u_mono
let v... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 667,
"column": 2
} | {
"line": 684,
"column": 18
} | {
"line": 686,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\nε : ℝ≥0∞\ns : Set α\nh : ε < eVariationOn f s\n⊢ ∃ a ∈ s, ∃ b ∈ s, a < b ∧ ε < eVariationOn f (s ∩ Icc a b)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOr... | [] | obtain ⟨n, u, ⟨u_mono, u_mem⟩, hu⟩ : ∃ n u, (Monotone u ∧ ∀ (i : ℕ), u i ∈ s) ∧
ε < ∑ x ∈ Finset.range n, edist (f (u (x + 1))) (f (u x)) := by
simpa [eVariationOn, lt_iSup_iff] using h
have A : ε < eVariationOn f (s ∩ Icc (u 0) (u n)) := by
apply hu.trans_le
simp only [Monotone] at u_mono
let v... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 765,
"column": 33
} | {
"line": 765,
"column": 38
} | {
"line": 766,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁰ : Semifield R\ninst✝⁹ : StarRing R\ninst✝⁸ : MetricSpace R\ninst✝⁷ : IsTopologicalSemiring R\ninst✝⁶ : ContinuousStar R\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ : Ring A\ninst✝³ : StarRing A\ninst✝² : Algebra R A\ninst✝¹ : ContinuousFunctionalCalculus R A p... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 765,
"column": 33
} | {
"line": 765,
"column": 38
} | {
"line": 766,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁰ : Semifield R\ninst✝⁹ : StarRing R\ninst✝⁸ : MetricSpace R\ninst✝⁷ : IsTopologicalSemiring R\ninst✝⁶ : ContinuousStar R\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ : Ring A\ninst✝³ : StarRing A\ninst✝² : Algebra R A\ninst✝¹ : ContinuousFunctionalCalculus R A p... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 765,
"column": 33
} | {
"line": 765,
"column": 38
} | {
"line": 766,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁰ : Semifield R\ninst✝⁹ : StarRing R\ninst✝⁸ : MetricSpace R\ninst✝⁷ : IsTopologicalSemiring R\ninst✝⁶ : ContinuousStar R\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ : Ring A\ninst✝³ : StarRing A\ninst✝² : Algebra R A\ninst✝¹ : ContinuousFunctionalCalculus R A p... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 768,
"column": 33
} | {
"line": 768,
"column": 38
} | {
"line": 770,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁰ : Semifield R\ninst✝⁹ : StarRing R\ninst✝⁸ : MetricSpace R\ninst✝⁷ : IsTopologicalSemiring R\ninst✝⁶ : ContinuousStar R\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ : Ring A\ninst✝³ : StarRing A\ninst✝² : Algebra R A\ninst✝¹ : ContinuousFunctionalCalculus R A p... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 768,
"column": 33
} | {
"line": 768,
"column": 38
} | {
"line": 770,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁰ : Semifield R\ninst✝⁹ : StarRing R\ninst✝⁸ : MetricSpace R\ninst✝⁷ : IsTopologicalSemiring R\ninst✝⁶ : ContinuousStar R\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ : Ring A\ninst✝³ : StarRing A\ninst✝² : Algebra R A\ninst✝¹ : ContinuousFunctionalCalculus R A p... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 768,
"column": 33
} | {
"line": 768,
"column": 38
} | {
"line": 770,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁰ : Semifield R\ninst✝⁹ : StarRing R\ninst✝⁸ : MetricSpace R\ninst✝⁷ : IsTopologicalSemiring R\ninst✝⁶ : ContinuousStar R\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ : Ring A\ninst✝³ : StarRing A\ninst✝² : Algebra R A\ninst✝¹ : ContinuousFunctionalCalculus R A p... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 885,
"column": 31
} | {
"line": 885,
"column": 60
} | {
"line": 887,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁹ : CommRing R\ninst✝⁸ : StarRing R\ninst✝⁷ : MetricSpace R\ninst✝⁶ : IsTopologicalRing R\ninst✝⁵ : ContinuousStar R\ninst✝⁴ : TopologicalSpace A\ninst✝³ : Ring A\ninst✝² : StarRing A\ninst✝¹ : Algebra R A\ninst✝ : ContinuousFunctionalCalculus R A p\nf : R... | [] | by simpa using hf_neg.fun_neg | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.Bernstein | {
"line": 115,
"column": 4
} | {
"line": 115,
"column": 27
} | {
"line": 116,
"column": 4
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝ : CommRing R\nn ν : ℕ\n⊢ ↑((n + 1).choose (ν + 1)) * (↑(ν + 1) * X ^ ν) * (1 - X) ^ (n - ν) =\n ↑(n + 1) * (↑(n.choose ν) * X ^ ν * (1 - X) ^ (n - ν))",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.instOn... | [
"case refine_1\nR : Type u_1\ninst✝ : CommRing R\nn ν : ℕ\n⊢ ↑((n + 1).choose (ν + 1)) * ↑(ν + 1) * X ^ ν * (1 - X) ^ (n - ν) =\n ↑(n + 1) * ↑(n.choose ν) * X ^ ν * (1 - X) ^ (n - ν)"
] | simp only [← mul_assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 981,
"column": 2
} | {
"line": 981,
"column": 19
} | {
"line": 982,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹³ : CommSemiring R\ninst✝¹² : PartialOrder R\ninst✝¹¹ : StarRing R\ninst✝¹⁰ : MetricSpace R\ninst✝⁹ : IsTopologicalSemiring R\ninst✝⁸ : ContinuousStar R\ninst✝⁷ : ContinuousSqrt R\ninst✝⁶ : StarOrderedRing R\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ : Ring A\n... | [
"R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹³ : CommSemiring R\ninst✝¹² : PartialOrder R\ninst✝¹¹ : StarRing R\ninst✝¹⁰ : MetricSpace R\ninst✝⁹ : IsTopologicalSemiring R\ninst✝⁸ : ContinuousStar R\ninst✝⁷ : ContinuousSqrt R\ninst✝⁶ : StarOrderedRing R\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ : Ring A\ninst✝³ : Sta... | rw [← cfc_id R a] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 986,
"column": 2
} | {
"line": 986,
"column": 19
} | {
"line": 987,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹³ : CommSemiring R\ninst✝¹² : PartialOrder R\ninst✝¹¹ : StarRing R\ninst✝¹⁰ : MetricSpace R\ninst✝⁹ : IsTopologicalSemiring R\ninst✝⁸ : ContinuousStar R\ninst✝⁷ : ContinuousSqrt R\ninst✝⁶ : StarOrderedRing R\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ : Ring A\n... | [
"R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹³ : CommSemiring R\ninst✝¹² : PartialOrder R\ninst✝¹¹ : StarRing R\ninst✝¹⁰ : MetricSpace R\ninst✝⁹ : IsTopologicalSemiring R\ninst✝⁸ : ContinuousStar R\ninst✝⁷ : ContinuousSqrt R\ninst✝⁶ : StarOrderedRing R\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ : Ring A\ninst✝³ : Sta... | rw [← cfc_id R a] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.ContinuousMap.Polynomial | {
"line": 216,
"column": 4
} | {
"line": 217,
"column": 39
} | {
"line": 218,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : TopologicalSpace R\ninst✝ : IsTopologicalSemiring R\ns : Set R\np : R[X]\nr : R\n⊢ (toContinuousMapOnAlgHom s) (Polynomial.C r) ∈ R[(toContinuousMapOnAlgHom s) X]",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
... | [] | rw [Polynomial.C_eq_algebraMap, AlgHomClass.commutes]
exact Subalgebra.algebraMap_mem _ r | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.ContinuousMap.Polynomial | {
"line": 216,
"column": 4
} | {
"line": 217,
"column": 39
} | {
"line": 218,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : TopologicalSpace R\ninst✝ : IsTopologicalSemiring R\ns : Set R\np : R[X]\nr : R\n⊢ (toContinuousMapOnAlgHom s) (Polynomial.C r) ∈ R[(toContinuousMapOnAlgHom s) X]",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
... | [] | rw [Polynomial.C_eq_algebraMap, AlgHomClass.commutes]
exact Subalgebra.algebraMap_mem _ r | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Bernstein | {
"line": 377,
"column": 4
} | {
"line": 377,
"column": 27
} | {
"line": 378,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\np :\n ∑ ν ∈ Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν +\n (1 - (2 * n) • X) * ∑ ν ∈ Finset.range (n + 1), ν • bernsteinPolynomial R n ν +\n n ^ 2 • X ^ 2 * ∑ ν ∈ Finset.range (n + 1), bernsteinPolynomial R n ν =\n ∑ ν ∈ Fins... | [
"R : Type u_1\ninst✝ : CommRing R\nn : ℕ\np :\n ∑ ν ∈ Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν +\n (1 - (2 * n) • X) * ∑ ν ∈ Finset.range (n + 1), ν • bernsteinPolynomial R n ν +\n n ^ 2 • X ^ 2 * ∑ ν ∈ Finset.range (n + 1), bernsteinPolynomial R n ν =\n ∑ ν ∈ Finset.range (n ... | simp only [← mul_assoc] | Lean.Elab.Tactic.Conv.evalSimp | Lean.Parser.Tactic.Conv.simp |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 583,
"column": 31
} | {
"line": 583,
"column": 60
} | {
"line": 584,
"column": 4
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹² : CommRing R\ninst✝¹¹ : Nontrivial R\ninst✝¹⁰ : StarRing R\ninst✝⁹ : MetricSpace R\ninst✝⁸ : IsTopologicalRing R\ninst✝⁷ : ContinuousStar R\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : Module R A\ninst✝² : IsScal... | [] | by simpa using hf_neg.fun_neg | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique | {
"line": 317,
"column": 6
} | {
"line": 318,
"column": 23
} | {
"line": 319,
"column": 6
} | [
{
"pp": "case neg\nX : Type u_1\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : Zero X\nA : Type u_2\ninst✝⁴ : NonUnitalRing A\ninst✝³ : StarRing A\ninst✝² : Module ℝ A\ninst✝¹ : TopologicalSpace A\ninst✝ : IsSemitopologicalRing A\nφ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A\nr✝ : ℝ\nf : C(X, ℝ)₀\nr : ℝ≥0\nhr : 0 < ↑r\n⊢ φ (-↑r • f).toN... | [
"case neg\nX : Type u_1\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : Zero X\nA : Type u_2\ninst✝⁴ : NonUnitalRing A\ninst✝³ : StarRing A\ninst✝² : Module ℝ A\ninst✝¹ : TopologicalSpace A\ninst✝ : IsSemitopologicalRing A\nφ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A\nr✝ : ℝ\nf : C(X, ℝ)₀\nr : ℝ≥0\nhr : 0 < ↑r\n⊢ r • φ (-f).toNNReal - r • φ... | simp only [neg_smul, ← smul_def, toNNReal_neg_smul, map_smul, toNNReal_smul, smul_sub,
sub_neg_eq_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.ContinuousMap.StoneWeierstrass | {
"line": 674,
"column": 11
} | {
"line": 677,
"column": 64
} | {
"line": 679,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : RCLike 𝕜\ns : Set 𝕜\ninst✝¹ : Fact (0 ∈ s)\ninst✝ : CompactSpace ↑s\np : C(↑s, 𝕜)₀ → Prop\nzero : p 0\nid : p (ContinuousMapZero.id s)\nstar_id : p (star (ContinuousMapZero.id s))\nadd : ∀ (f g : C(↑s, 𝕜)₀), p f → p g → p (f + g)\nmul : ∀ (f g : C(↑s, 𝕜)₀), p f → p g → p (f... | [] | by
refine f.induction_on zero id star_id add mul smul fun h f ↦ frequently f ?_
have := (ContinuousMapZero.adjoin_id_dense s).closure_eq ▸ Set.mem_univ (x := f)
exact mem_closure_iff_frequently.mp this |>.mp <| .of_forall h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Semicontinuity.Hemicontinuity | {
"line": 114,
"column": 2
} | {
"line": 114,
"column": 7
} | {
"line": 116,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\n⊢ (∀ {u : Set α}, IsOpen[inst✝¹] u → IsOpen[inst✝] (kernImage f u)) ↔\n ∀ (u : Set α), IsOpen[inst✝¹] u → IsOpen[inst✝] ((fun x ↦ f ⁻¹' {x}) ⁻¹' Iic u)",
"ppTerm": "?m.28",
"assigned": true,
"... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Semicontinuity.Hemicontinuity | {
"line": 143,
"column": 2
} | {
"line": 143,
"column": 7
} | {
"line": 145,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\n⊢ (∀ {u : Set α}, IsClosed[inst✝¹] u → IsClosed[inst✝] (kernImage f u)) ↔\n ∀ (u : Set α), IsClosed[inst✝¹] u → IsClosed[inst✝] ((fun x ↦ f ⁻¹' {x}) ⁻¹' Iic u)",
"ppTerm": "?m.28",
"assigned": tru... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Semicontinuity.Hemicontinuity | {
"line": 245,
"column": 2
} | {
"line": 245,
"column": 7
} | {
"line": 247,
"column": 0
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf g : α → Set β\ns : Set α\nx : α\nhf : ∀ t ∈ 𝓝ˢ (f x), ∀ᶠ (x' : α) in 𝓝[s] x, t ∈ 𝓝ˢ (f x')\nhg : ∀ t ∈ 𝓝ˢ (g x), ∀ᶠ (x' : α) in 𝓝[s] x, t ∈ 𝓝ˢ (g x')\n⊢ ∀ t ∈ 𝓝ˢ (f x ∪ g x), ∀ᶠ (x' : α) in 𝓝[s] x, t ∈ 𝓝ˢ (f... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Semicontinuity.Hemicontinuity | {
"line": 288,
"column": 2
} | {
"line": 288,
"column": 19
} | {
"line": 290,
"column": 0
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → Set β\nx : α\nhf : UpperHemicontinuousWithinAt f univ x\nu : Set β\nhu : IsClosed[inst✝] u\n⊢ UpperHemicontinuousWithinAt (fun x ↦ f x ∩ u) univ x",
"ppTerm": "?m.34",
"assigned": true,
"usedConstan... | [] | exact hf.inter hu | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Semicontinuity.Hemicontinuity | {
"line": 411,
"column": 2
} | {
"line": 411,
"column": 30
} | {
"line": 412,
"column": 2
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\ninst✝ : TopologicalSpace α\nf : α → Set β\n⊢ HasOpenLowerSections f ↔ ∀ (b : β), IsClosed[inst✝] (f ⁻¹' Iic {b}ᶜ)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Compl.compl",
"PartialOrder.toPreorder",
... | [
"α : Type u_3\nβ : Type u_4\ninst✝ : TopologicalSpace α\nf : α → Set β\n⊢ HasOpenLowerSections f ↔ ∀ (b : β), IsOpen[inst✝] (f ⁻¹' Iic {b}ᶜ)ᶜ"
] | simp_rw [← isOpen_compl_iff] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.Complex.Convex | {
"line": 108,
"column": 21
} | {
"line": 108,
"column": 26
} | {
"line": 108,
"column": 26
} | [
{
"pp": "r : ℝ\nx : ℂ\n⊢ x ∈ {c | 0 < c.im} ∪ {c | 0 < c.re} ∪ ({c | c.im < 0} ∪ {c | c.re < 0}) → x ∈ {0}ᶜ",
"ppTerm": "?m.215",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Real",
"Real.instZero",
"congrArg",
"Compl.compl",
"Complex.im",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.Complex.Convex | {
"line": 108,
"column": 21
} | {
"line": 108,
"column": 26
} | {
"line": 108,
"column": 26
} | [
{
"pp": "r : ℝ\nx : ℂ\n⊢ x ∈ {c | 0 < c.im} ∪ {c | 0 < c.re} ∪ ({c | c.im < 0} ∪ {c | c.re < 0}) → x ∈ {0}ᶜ",
"ppTerm": "?m.215",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Real",
"Real.instZero",
"congrArg",
"Compl.compl",
"Complex.im",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Convex | {
"line": 108,
"column": 21
} | {
"line": 108,
"column": 26
} | {
"line": 108,
"column": 26
} | [
{
"pp": "r : ℝ\nx : ℂ\n⊢ x ∈ {c | 0 < c.im} ∪ {c | 0 < c.re} ∪ ({c | c.im < 0} ∪ {c | c.re < 0}) → x ∈ {0}ᶜ",
"ppTerm": "?m.215",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Real",
"Real.instZero",
"congrArg",
"Compl.compl",
"Complex.im",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 230,
"column": 53
} | {
"line": 232,
"column": 81
} | {
"line": 234,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁴ : NormedField 𝕜\ninst✝³ : NormedRing A\ninst✝² : NormedAlgebra 𝕜 A\ninst✝¹ : CompleteSpace A\ninst✝ : ProperSpace 𝕜\na : A\nha : (σ a).Nonempty\n⊢ ∃ k ∈ σ a, ↑‖k‖₊ = spectralRadius 𝕜 a",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"N... | [] | by
obtain ⟨k, hk, h⟩ := (spectrum.isCompact a).exists_isMaxOn ha continuous_nnnorm.continuousOn
exact ⟨k, hk, le_antisymm (le_iSup₂ (α := ℝ≥0∞) k hk) (iSup₂_le <| mod_cast h)⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances | {
"line": 122,
"column": 4
} | {
"line": 122,
"column": 9
} | {
"line": 124,
"column": 0
} | [
{
"pp": "case e'_3\n𝕜 : Type u_1\nA : Type u_2\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : NonUnitalNormedRing A\ninst✝⁶ : StarRing A\ninst✝⁵ : NormedSpace 𝕜 A\ninst✝⁴ : IsScalarTower 𝕜 A A\ninst✝³ : SMulCommClass 𝕜 A A\ninst✝² : StarModule 𝕜 A\np : A → Prop\np₁ : Unitization 𝕜 A → Prop\nhp₁ : ∀ {x : A}, p₁ ↑x ↔ p x\na... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.ContinuousMap.ZeroAtInfty | {
"line": 312,
"column": 4
} | {
"line": 312,
"column": 75
} | {
"line": 313,
"column": 4
} | [
{
"pp": "F : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : TopologicalSpace β\nx : α\nR : Type u_2\ninst✝⁵ : Semiring R\ninst✝⁴ : NonUnitalNonAssocSemiring β\ninst✝³ : IsTopologicalSemiring β\ninst✝² : Module R β\ninst✝¹ : ContinuousConstSMul R β\ninst✝ : IsScalarTower R β ... | [
"F : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : TopologicalSpace β\nx : α\nR : Type u_2\ninst✝⁵ : Semiring R\ninst✝⁴ : NonUnitalNonAssocSemiring β\ninst✝³ : IsTopologicalSemiring β\ninst✝² : Module R β\ninst✝¹ : ContinuousConstSMul R β\ninst✝ : IsScalarTower R β β\nr : R\nf ... | simp only [smul_eq_mul, coe_mul, coe_smul, Pi.mul_apply, Pi.smul_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances | {
"line": 147,
"column": 6
} | {
"line": 150,
"column": 64
} | {
"line": 151,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁹ : RCLike 𝕜\ninst✝⁸ : NonUnitalNormedRing A\ninst✝⁷ : StarRing A\ninst✝⁶ : NormedSpace 𝕜 A\ninst✝⁵ : IsScalarTower 𝕜 A A\ninst✝⁴ : SMulCommClass 𝕜 A A\ninst✝³ : StarModule 𝕜 A\np : A → Prop\np₁ : Unitization 𝕜 A → Prop\nhp₁ : ∀ {x : A}, p₁ ↑x ↔ p x\ninst✝² : Clo... | [] | have h₁ : Function.Injective ⇑(codRestrict (cfcₙAux hp₁ a ha) _
(cfcₙAux_mem_range_inr hp₁ a ha)) :=
(Set.injective_codRestrict _).mpr (cfcₙAux_injective hp₁ a ha)
simpa [ψ] using (inrRangeEquiv 𝕜 A).symm.injective.comp h₁ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances | {
"line": 147,
"column": 6
} | {
"line": 150,
"column": 64
} | {
"line": 151,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁹ : RCLike 𝕜\ninst✝⁸ : NonUnitalNormedRing A\ninst✝⁷ : StarRing A\ninst✝⁶ : NormedSpace 𝕜 A\ninst✝⁵ : IsScalarTower 𝕜 A A\ninst✝⁴ : SMulCommClass 𝕜 A A\ninst✝³ : StarModule 𝕜 A\np : A → Prop\np₁ : Unitization 𝕜 A → Prop\nhp₁ : ∀ {x : A}, p₁ ↑x ↔ p x\ninst✝² : Clo... | [] | have h₁ : Function.Injective ⇑(codRestrict (cfcₙAux hp₁ a ha) _
(cfcₙAux_mem_range_inr hp₁ a ha)) :=
(Set.injective_codRestrict _).mpr (cfcₙAux_injective hp₁ a ha)
simpa [ψ] using (inrRangeEquiv 𝕜 A).symm.injective.comp h₁ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.LocallyConvex.Polar | {
"line": 67,
"column": 71
} | {
"line": 67,
"column": 76
} | {
"line": 70,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NormedCommRing 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nB : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜\ns : Set E\n⊢ B.polar s = ⋂ x ∈ s, ⇑(B x) ⁻¹' Metric.closedBall 0 1",
"ppTerm": "?m.52",
"assigned": tru... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.LocallyConvex.Polar | {
"line": 67,
"column": 71
} | {
"line": 67,
"column": 76
} | {
"line": 70,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NormedCommRing 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nB : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜\ns : Set E\n⊢ B.polar s = ⋂ x ∈ s, ⇑(B x) ⁻¹' Metric.closedBall 0 1",
"ppTerm": "?m.52",
"assigned": tru... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.LocallyConvex.Polar | {
"line": 67,
"column": 71
} | {
"line": 67,
"column": 76
} | {
"line": 70,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NormedCommRing 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nB : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜\ns : Set E\n⊢ B.polar s = ⋂ x ∈ s, ⇑(B x) ⁻¹' Metric.closedBall 0 1",
"ppTerm": "?m.52",
"assigned": tru... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.LocallyConvex.AbsConvexOpen | {
"line": 77,
"column": 2
} | {
"line": 80,
"column": 82
} | {
"line": 82,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : TopologicalSpace E\ninst✝³ : AddCommMonoid E\ninst✝² : SeminormedRing 𝕜\ninst✝¹ : SMul 𝕜 E\ninst✝ : PartialOrder 𝕜\n⊢ Nonempty (AbsConvexOpenSets 𝕜 E)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"trivial",
"c... | [] | rw [← exists_true_iff_nonempty]
dsimp only [AbsConvexOpenSets]
rw [Subtype.exists]
exact ⟨Set.univ, ⟨mem_univ 0, isOpen_univ, balanced_univ, convex_univ⟩, trivial⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.LocallyConvex.AbsConvexOpen | {
"line": 77,
"column": 2
} | {
"line": 80,
"column": 82
} | {
"line": 82,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : TopologicalSpace E\ninst✝³ : AddCommMonoid E\ninst✝² : SeminormedRing 𝕜\ninst✝¹ : SMul 𝕜 E\ninst✝ : PartialOrder 𝕜\n⊢ Nonempty (AbsConvexOpenSets 𝕜 E)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"trivial",
"c... | [] | rw [← exists_true_iff_nonempty]
dsimp only [AbsConvexOpenSets]
rw [Subtype.exists]
exact ⟨Set.univ, ⟨mem_univ 0, isOpen_univ, balanced_univ, convex_univ⟩, trivial⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | {
"line": 388,
"column": 4
} | {
"line": 393,
"column": 49
} | {
"line": 395,
"column": 0
} | [
{
"pp": "case refine_2\nR : Type u_1\nS : Type u_2\nA : Type u_3\np q : A → Prop\ninst✝²⁵ : Semifield R\ninst✝²⁴ : StarRing R\ninst✝²³ : MetricSpace R\ninst✝²² : IsTopologicalSemiring R\ninst✝²¹ : ContinuousStar R\ninst✝²⁰ : Field S\ninst✝¹⁹ : StarRing S\ninst✝¹⁸ : MetricSpace S\ninst✝¹⁷ : IsTopologicalRing S\n... | [] | · let x' : σₙ S a := Subtype.map (algebraMap R S) (fun _ ↦ quasispectrum.algebraMap_mem S) x
apply le_of_eq_of_le ?_ <| ContinuousMap.dist_apply_le_dist x'
simp only [ContinuousMapZero.comp_apply, ContinuousMapZero.coe_mk,
ContinuousMap.coe_mk, StarAlgHom.ofId_apply, halg.dist_eq, x']
congr! 2... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.UrysohnsLemma | {
"line": 180,
"column": 71
} | {
"line": 185,
"column": 81
} | {
"line": 187,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nP : Set X → Set X → Prop\nc : CU P\nn : ℕ\nx : X\n⊢ 0 ≤ approx n c x",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real.instIsOrderedRing",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"mul_nonneg... | [] | by
induction n generalizing c with
| zero => exact indicator_nonneg (fun _ _ => zero_le_one) _
| succ n ihn =>
simp only [approx, midpoint_eq_smul_add]
refine mul_nonneg (inv_nonneg.2 zero_le_two) (add_nonneg ?_ ?_) <;> apply ihn | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Affine.AddTorsor | {
"line": 263,
"column": 50
} | {
"line": 263,
"column": 92
} | {
"line": 265,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : NormedDivisionRing 𝕜\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : NormSMulClass 𝕜 E\nP : Type u_3\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor E P\nc : P\nk : 𝕜\nhk : k ≠ 0\nx y : E\nh : dist x y ≠ 0\n⊢ dist ((smulTorsor c hk) x) ... | [] | simp [dist_eq_norm, ← smul_sub, norm_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Normed.Affine.AddTorsor | {
"line": 263,
"column": 50
} | {
"line": 263,
"column": 92
} | {
"line": 265,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : NormedDivisionRing 𝕜\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : NormSMulClass 𝕜 E\nP : Type u_3\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor E P\nc : P\nk : 𝕜\nhk : k ≠ 0\nx y : E\nh : dist x y ≠ 0\n⊢ dist ((smulTorsor c hk) x) ... | [] | simp [dist_eq_norm, ← smul_sub, norm_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Affine.AddTorsor | {
"line": 263,
"column": 50
} | {
"line": 263,
"column": 92
} | {
"line": 265,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : NormedDivisionRing 𝕜\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : NormSMulClass 𝕜 E\nP : Type u_3\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor E P\nc : P\nk : 𝕜\nhk : k ≠ 0\nx y : E\nh : dist x y ≠ 0\n⊢ dist ((smulTorsor c hk) x) ... | [] | simp [dist_eq_norm, ← smul_sub, norm_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.Ordered | {
"line": 94,
"column": 13
} | {
"line": 94,
"column": 38
} | {
"line": 94,
"column": 38
} | [
{
"pp": "k : Type u_1\nE : Type u_2\ninst✝⁸ : Ring k\ninst✝⁷ : PartialOrder k\ninst✝⁶ : IsOrderedRing k\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : PartialOrder E\ninst✝³ : IsOrderedAddMonoid E\ninst✝² : Module k E\ninst✝¹ : IsStrictOrderedModule k E\na b : E\nr : k\ninst✝ : PosSMulReflectLT k E\nh : r < 1\n⊢ (lineMap a... | [] | by rw [lineMap_apply_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.AffineSpace.Ordered | {
"line": 175,
"column": 13
} | {
"line": 175,
"column": 38
} | {
"line": 175,
"column": 38
} | [
{
"pp": "k : Type u_1\nE : Type u_2\ninst✝⁸ : Field k\ninst✝⁷ : LinearOrder k\ninst✝⁶ : IsStrictOrderedRing k\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : PartialOrder E\ninst✝³ : IsOrderedAddMonoid E\ninst✝² : Module k E\ninst✝¹ : IsStrictOrderedModule k E\ninst✝ : PosSMulReflectLE k E\na b : E\nr : k\nh : r < 1\n⊢ (lin... | [] | by rw [lineMap_apply_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Module.WeakDual | {
"line": 222,
"column": 4
} | {
"line": 222,
"column": 84
} | {
"line": 223,
"column": 4
} | [
{
"pp": "case mpr\n𝕜 : Type u_1\nE : Type u_3\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : CompleteSpace E\ns : Set (WeakDual 𝕜 E)\nh_vN : IsVonNBounded 𝕜 s\nh_ptwise : ∀ (i : E), ∃ r > 0, ∀ x ∈ s, (seminormFamily 𝕜 E i) x < r\nC : ℝ\nhC : ∀ (i ... | [
"case mpr\n𝕜 : Type u_1\nE : Type u_3\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : CompleteSpace E\ns : Set (WeakDual 𝕜 E)\nh_vN : IsVonNBounded 𝕜 s\nh_ptwise : ∀ (i : E), ∃ r > 0, ∀ x ∈ s, (seminormFamily 𝕜 E i) x < r\nC : ℝ\nhC : ∀ (i : ↑s), ‖toSt... | rw [← isBounded_toWeakDual_preimage_iff_isBounded, isBounded_iff_forall_norm_le] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.ContinuousMap.Ideals | {
"line": 354,
"column": 2
} | {
"line": 354,
"column": 60
} | {
"line": 356,
"column": 0
} | [
{
"pp": "X : Type u_1\n𝕜 : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nI : Ideal C(X, 𝕜)\nhI : I.IsMaximal\nh : (idealOfSet 𝕜 (setOfIdeal I)).IsMaximal\nx : X\nhx : opensOfIdeal I = {x}.compl\n⊢ ∃ x, setOfIdeal I = {x}ᶜ",
"ppTerm": "?m.82",
"... | [] | exact ⟨x, congr_arg (fun (s : Opens X) => (s : Set X)) hx⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.CStarAlgebra.GelfandDuality | {
"line": 182,
"column": 2
} | {
"line": 183,
"column": 21
} | {
"line": 186,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝ : CommCStarAlgebra A\nrng : StarSubalgebra ℂ C(↑(characterSpace ℂ A), ℂ) :=\n let __Subalgebra := (gelfandTransform ℂ A).range;\n { toSubalgebra := __Subalgebra, star_mem' := ⋯ }\nh : rng.topologicalClosure = rng\n⊢ rng = ⊤",
"ppTerm": "?m.175",
"assigned": true,
"used... | [
"A : Type u_1\ninst✝ : CommCStarAlgebra A\nrng : StarSubalgebra ℂ C(↑(characterSpace ℂ A), ℂ) :=\n let __Subalgebra := (gelfandTransform ℂ A).range;\n { toSubalgebra := __Subalgebra, star_mem' := ⋯ }\nh : rng.topologicalClosure = rng\nx✝¹ x✝ : ↑(characterSpace ℂ A)\n⊢ x✝¹ ≠ x✝ → ∃ f ∈ (fun f ↦ ⇑f) '' ↑rng.toSubal... | refine h ▸ ContinuousMap.starSubalgebra_topologicalClosure_eq_top_of_separatesPoints
_ (fun _ _ => ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.CStarAlgebra.GelfandDuality | {
"line": 258,
"column": 41
} | {
"line": 258,
"column": 46
} | {
"line": 260,
"column": 0
} | [
{
"pp": "case refine_1\nA : Type u_1\ninst✝ : NonUnitalCStarAlgebra A\na b : A\nha : IsStarNormal a\nhb : IsStarNormal b\nhcomm : Commute a b\nhab : a * b = 0\nS : NonUnitalStarSubalgebra ℂ A := ⋯\nhS : IsClosed ↑S\nhcomm₁ : Commute (star a) b\nhcomm₂ : Commute a (star b)\nthis : IsMulCommutative ↥(adjoin ℂ {a,... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.CStarAlgebra.GelfandDuality | {
"line": 258,
"column": 41
} | {
"line": 258,
"column": 46
} | {
"line": 260,
"column": 0
} | [
{
"pp": "case refine_2\nA : Type u_1\ninst✝ : NonUnitalCStarAlgebra A\na b : A\nha : IsStarNormal a\nhb : IsStarNormal b\nhcomm : Commute a b\nhab : a * b = 0\nS : NonUnitalStarSubalgebra ℂ A := ⋯\nhS : IsClosed ↑S\nhcomm₁ : Commute (star a) b\nhcomm₂ : Commute a (star b)\nthis : IsMulCommutative ↥(adjoin ℂ {a,... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 242,
"column": 29
} | {
"line": 242,
"column": 34
} | {
"line": 242,
"column": 34
} | [
{
"pp": "A : Type u_1\ninst✝¹¹ : NonUnitalRing A\ninst✝¹⁰ : Module ℝ A\ninst✝⁹ : SMulCommClass ℝ A A\ninst✝⁸ : IsScalarTower ℝ A A\ninst✝⁷ : StarRing A\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁴ : PartialOrder A\ninst✝³ : StarOrderedRing A\ninst✝² : No... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 242,
"column": 29
} | {
"line": 242,
"column": 34
} | {
"line": 242,
"column": 34
} | [
{
"pp": "A : Type u_1\ninst✝¹¹ : NonUnitalRing A\ninst✝¹⁰ : Module ℝ A\ninst✝⁹ : SMulCommClass ℝ A A\ninst✝⁸ : IsScalarTower ℝ A A\ninst✝⁷ : StarRing A\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁴ : PartialOrder A\ninst✝³ : StarOrderedRing A\ninst✝² : No... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 242,
"column": 29
} | {
"line": 242,
"column": 34
} | {
"line": 242,
"column": 34
} | [
{
"pp": "A : Type u_1\ninst✝¹¹ : NonUnitalRing A\ninst✝¹⁰ : Module ℝ A\ninst✝⁹ : SMulCommClass ℝ A A\ninst✝⁸ : IsScalarTower ℝ A A\ninst✝⁷ : StarRing A\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁴ : PartialOrder A\ninst✝³ : StarOrderedRing A\ninst✝² : No... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.UniformConvergence | {
"line": 99,
"column": 6
} | {
"line": 99,
"column": 31
} | {
"line": 99,
"column": 31
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : PseudoEMetricSpace γ\ninst✝ : PseudoEMetricSpace β\nf : α → γ → β\nK : ℝ≥0\nh : ∀ (c : α), LipschitzWith K (f c)\n⊢ UniformContinuous (⇑ofFun ∘ Function.swap f)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"LipschitzWith"... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : PseudoEMetricSpace γ\ninst✝ : PseudoEMetricSpace β\nf : α → γ → β\nK : ℝ≥0\nh : LipschitzWith K fun x ↦ ofFun fun c ↦ f c x\n⊢ UniformContinuous (⇑ofFun ∘ Function.swap f)"
] | ← lipschitzWith_ofFun_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic | {
"line": 297,
"column": 34
} | {
"line": 297,
"column": 46
} | {
"line": 299,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝ : CStarAlgebra A\nb : A\na : A := star b * b\na_def : a = star b * b\n⊢ IsSelfAdjoint a",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"IsSelfAdjoint",
"NormedRing.toRing",
"HMul.hMul",
"Ring.toNonAssocRing",
"congrArg",
"CS... | [] | simp [a_def] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic | {
"line": 474,
"column": 4
} | {
"line": 474,
"column": 51
} | {
"line": 475,
"column": 2
} | [
{
"pp": "case pos\nA : Type u_1\ninst✝¹³ : NonUnitalCStarAlgebra A\nR : Type u_2\ninst✝¹² : Semifield R\ninst✝¹¹ : StarRing R\ninst✝¹⁰ : MetricSpace R\ninst✝⁹ : IsTopologicalSemiring R\ninst✝⁸ : ContinuousStar R\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : Algebra R... | [] | exact (inrNonUnitalStarAlgHom ℂ A).map_cfcₙ f a | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 221,
"column": 55
} | {
"line": 221,
"column": 60
} | {
"line": 221,
"column": 60
} | [
{
"pp": "X : Type u_1\n𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedRing A\ninst✝⁴ : StarRing A\ninst✝³ : NormedAlgebra 𝕜 A\ninst✝² : IsometricContinuousFunctionalCalculus 𝕜 A p\ninst✝¹ : ContinuousStar A\ninst✝ : TopologicalSpace X\ns : Set 𝕜\nhs : IsCompact s\na : X → A\nha... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 221,
"column": 55
} | {
"line": 221,
"column": 60
} | {
"line": 221,
"column": 60
} | [
{
"pp": "X : Type u_1\n𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedRing A\ninst✝⁴ : StarRing A\ninst✝³ : NormedAlgebra 𝕜 A\ninst✝² : IsometricContinuousFunctionalCalculus 𝕜 A p\ninst✝¹ : ContinuousStar A\ninst✝ : TopologicalSpace X\ns : Set 𝕜\nhs : IsCompact s\na : X → A\nha... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 221,
"column": 55
} | {
"line": 221,
"column": 60
} | {
"line": 221,
"column": 60
} | [
{
"pp": "X : Type u_1\n𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedRing A\ninst✝⁴ : StarRing A\ninst✝³ : NormedAlgebra 𝕜 A\ninst✝² : IsometricContinuousFunctionalCalculus 𝕜 A p\ninst✝¹ : ContinuousStar A\ninst✝ : TopologicalSpace X\ns : Set 𝕜\nhs : IsCompact s\na : X → A\nha... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic | {
"line": 466,
"column": 21
} | {
"line": 466,
"column": 26
} | {
"line": 467,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝⁷ : PartialOrder A\ninst✝⁶ : Ring A\ninst✝⁵ : StarRing A\ninst✝⁴ : TopologicalSpace A\ninst✝³ : StarOrderedRing A\ninst✝² : Algebra ℝ A\ninst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : NonnegSpectrumClass ℝ A\na : A\nx y : ℝ\nha : IsUnit a\nha' : 0 ∉ spectrum ℝ≥0 a\n... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 230,
"column": 4
} | {
"line": 230,
"column": 9
} | {
"line": 232,
"column": 0
} | [
{
"pp": "case frequently\nX : Type u_1\n𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedRing A\ninst✝⁴ : StarRing A\ninst✝³ : NormedAlgebra 𝕜 A\ninst✝² : IsometricContinuousFunctionalCalculus 𝕜 A p\ninst✝¹ : ContinuousStar A\ninst✝ : TopologicalSpace X\ns : Set 𝕜\nhs : IsCompact... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.CStarAlgebra.ApproximateUnit | {
"line": 207,
"column": 4
} | {
"line": 207,
"column": 67
} | {
"line": 208,
"column": 4
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nx y : A\nhx : 0 ≤ x ∧ ‖x‖ < 1\nhy : 0 ≤ y ∧ ‖y‖ < 1\n⊢ ∃ k, (0 ≤ k ∧ ‖k‖ < 1) ∧ {x | k ≤ x} ⊆ {x_1 | x ≤ x_1} ∩ {x | y ≤ x}",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"No... | [
"case h\nA : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nx y : A\nhx : 0 ≤ x ∧ ‖x‖ < 1\nhy : 0 ≤ y ∧ ‖y‖ < 1\nz : A\nhz : z ∈ {x | 0 ≤ x} ∩ ball 0 1 ∧ (fun x1 x2 ↦ x1 ≤ x2) x z ∧ (fun x1 x2 ↦ x1 ≤ x2) y z\n⊢ (0 ≤ z ∧ ‖z‖ < 1) ∧ {x | z ≤ x} ⊆ {x_1 | x ≤ x_1} ∩ {x |... | peel directedOn_nonneg_ball x (by simpa) y (by simpa) with z hz | Mathlib.Tactic.Peel._aux_Mathlib_Tactic_Peel___elabRules_Mathlib_Tactic_Peel_peel_1 | Mathlib.Tactic.Peel.peel |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order | {
"line": 467,
"column": 41
} | {
"line": 467,
"column": 46
} | {
"line": 467,
"column": 46
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na b : A\nhb : IsSelfAdjoint b\nthis : ∀ (a b : Unitization ℂ A), IsSelfAdjoint b → star a * b * a ≤ ‖b‖ • (star a * a)\n⊢ IsSelfAdjoint (star a * b * a)",
"ppTerm": "?m.66",
"assigned": true,
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order | {
"line": 467,
"column": 41
} | {
"line": 467,
"column": 46
} | {
"line": 467,
"column": 46
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na b : A\nhb : IsSelfAdjoint b\nthis : ∀ (a b : Unitization ℂ A), IsSelfAdjoint b → star a * b * a ≤ ‖b‖ • (star a * a)\n⊢ IsSelfAdjoint (star a * b * a)",
"ppTerm": "?m.66",
"assigned": true,
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order | {
"line": 467,
"column": 41
} | {
"line": 467,
"column": 46
} | {
"line": 467,
"column": 46
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na b : A\nhb : IsSelfAdjoint b\nthis : ∀ (a b : Unitization ℂ A), IsSelfAdjoint b → star a * b * a ≤ ‖b‖ • (star a * a)\n⊢ IsSelfAdjoint (star a * b * a)",
"ppTerm": "?m.66",
"assigned": true,
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.PositiveLinearMap | {
"line": 77,
"column": 4
} | {
"line": 77,
"column": 19
} | {
"line": 79,
"column": 2
} | [
{
"pp": "case hbc\nA₁ : Type u_1\nA₂ : Type u_2\ninst✝⁵ : NonUnitalCStarAlgebra A₁\ninst✝⁴ : NonUnitalCStarAlgebra A₂\ninst✝³ : PartialOrder A₁\ninst✝² : StarOrderedRing A₁\ninst✝¹ : PartialOrder A₂\ninst✝ : StarOrderedRing A₂\nf : A₁ →ₚ[ℂ] A₂\nC : ℝ≥0\nhmain : ∀ (a : A₁), 0 ≤ a → ‖f a‖ ≤ ↑C * ‖a‖\nx : A₁\ny : ... | [] | exact hy_norm i | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 755,
"column": 55
} | {
"line": 755,
"column": 60
} | {
"line": 755,
"column": 60
} | [
{
"pp": "X : Type u_1\n𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : NonUnitalNormedRing A\ninst✝⁶ : StarRing A\ninst✝⁵ : NormedSpace 𝕜 A\ninst✝⁴ : IsScalarTower 𝕜 A A\ninst✝³ : SMulCommClass 𝕜 A A\ninst✝² : ContinuousStar A\ninst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus �... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 755,
"column": 55
} | {
"line": 755,
"column": 60
} | {
"line": 755,
"column": 60
} | [
{
"pp": "X : Type u_1\n𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : NonUnitalNormedRing A\ninst✝⁶ : StarRing A\ninst✝⁵ : NormedSpace 𝕜 A\ninst✝⁴ : IsScalarTower 𝕜 A A\ninst✝³ : SMulCommClass 𝕜 A A\ninst✝² : ContinuousStar A\ninst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus �... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 755,
"column": 55
} | {
"line": 755,
"column": 60
} | {
"line": 755,
"column": 60
} | [
{
"pp": "X : Type u_1\n𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : NonUnitalNormedRing A\ninst✝⁶ : StarRing A\ninst✝⁵ : NormedSpace 𝕜 A\ninst✝⁴ : IsScalarTower 𝕜 A A\ninst✝³ : SMulCommClass 𝕜 A A\ninst✝² : ContinuousStar A\ninst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus �... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Commute | {
"line": 58,
"column": 15
} | {
"line": 58,
"column": 40
} | {
"line": 59,
"column": 2
} | [
{
"pp": "case star_id\n𝕜 : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : Ring A\ninst✝⁵ : StarRing A\ninst✝⁴ : Algebra 𝕜 A\ninst✝³ : TopologicalSpace A\ninst✝² : ContinuousFunctionalCalculus 𝕜 A p\ninst✝¹ : IsSemitopologicalRing A\ninst✝ : T2Space A\na b : A\nha : p a\nhb₁ : Commute a b\... | [] | rwa [map_star, cfcHom_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Commute | {
"line": 58,
"column": 15
} | {
"line": 58,
"column": 40
} | {
"line": 59,
"column": 2
} | [
{
"pp": "case star_id\n𝕜 : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : Ring A\ninst✝⁵ : StarRing A\ninst✝⁴ : Algebra 𝕜 A\ninst✝³ : TopologicalSpace A\ninst✝² : ContinuousFunctionalCalculus 𝕜 A p\ninst✝¹ : IsSemitopologicalRing A\ninst✝ : T2Space A\na b : A\nha : p a\nhb₁ : Commute a b\... | [] | rwa [map_star, cfcHom_id] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Commute | {
"line": 58,
"column": 15
} | {
"line": 58,
"column": 40
} | {
"line": 59,
"column": 2
} | [
{
"pp": "case star_id\n𝕜 : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : Ring A\ninst✝⁵ : StarRing A\ninst✝⁴ : Algebra 𝕜 A\ninst✝³ : TopologicalSpace A\ninst✝² : ContinuousFunctionalCalculus 𝕜 A p\ninst✝¹ : IsSemitopologicalRing A\ninst✝ : T2Space A\na b : A\nha : p a\nhb₁ : Commute a b\... | [] | rwa [map_star, cfcHom_id] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Matrix.Normed | {
"line": 449,
"column": 14
} | {
"line": 449,
"column": 25
} | {
"line": 449,
"column": 26
} | [
{
"pp": "case inr\nm : Type u_3\nn : Type u_4\nα : Type u_5\ninst✝³ : Fintype m\ninst✝² : Fintype n\ninst✝¹ : NontriviallyNormedField α\ninst✝ : NormedAlgebra ℝ α\nA : Matrix m n α\nN : ℝ≥0\ni : m\nx✝ : i ∈ Finset.univ\nh✝ : Nonempty n\nx : n → α := fun j ↦ unitOf (A i j)\nhxn : ‖x‖₊ = 1\nhN : ‖∑ x, A i x * uni... | [
"case inr\nm : Type u_3\nn : Type u_4\nα : Type u_5\ninst✝³ : Fintype m\ninst✝² : Fintype n\ninst✝¹ : NontriviallyNormedField α\ninst✝ : NormedAlgebra ℝ α\nA : Matrix m n α\nN : ℝ≥0\ni : m\nx✝ : i ∈ Finset.univ\nh✝ : Nonempty n\nx : n → α := fun j ↦ unitOf (A i j)\nhxn : ‖x‖₊ = 1\nhN : ‖∑ x, (algebraMap ℝ α) ↑‖A i ... | mul_unitOf, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 764,
"column": 4
} | {
"line": 764,
"column": 9
} | {
"line": 766,
"column": 0
} | [
{
"pp": "case frequently\nX : Type u_1\n𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : NonUnitalNormedRing A\ninst✝⁶ : StarRing A\ninst✝⁵ : NormedSpace 𝕜 A\ninst✝⁴ : IsScalarTower 𝕜 A A\ninst✝³ : SMulCommClass 𝕜 A A\ninst✝² : ContinuousStar A\ninst✝¹ : NonUnitalIsometricContinuousFun... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.CStarAlgebra.CStarMatrix | {
"line": 494,
"column": 4
} | {
"line": 497,
"column": 51
} | {
"line": 498,
"column": 4
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_3\nS : Type u_4\nA : Type u_5\nB : Type u_6\ninst✝¹ : Fintype m\ninst✝ : NonUnitalCStarAlgebra A\nc : ℂ\nM : CStarMatrix m n A\nx : C⋆ᵐᵒᵈ(A, m → A)\ni : n\n⊢ { toFun := ⇑(WithCStarModule.equivL ℂ).symm ∘ (fun v ↦ v ᵥ* (c • M)) ∘ ⇑(WithCStarModule.equivL ℂ), map_ad... | [
"m : Type u_1\nn : Type u_2\nR : Type u_3\nS : Type u_4\nA : Type u_5\nB : Type u_6\ninst✝¹ : Fintype m\ninst✝ : NonUnitalCStarAlgebra A\nc : ℂ\nM : CStarMatrix m n A\nx : C⋆ᵐᵒᵈ(A, m → A)\ni : n\n⊢ ((WithCStarModule.equiv A (m → A)) x ᵥ* (c • M)) i = c • ((WithCStarModule.equiv A (m → A)) x ᵥ* M) i"
] | simp only [ContinuousLinearMap.coe_mk', LinearMap.coe_mk, AddHom.coe_mk, Function.comp_apply,
WithCStarModule.equivL_apply, WithCStarModule.equivL_symm_apply,
WithCStarModule.equiv_symm_pi_apply, _root_.smul_apply,
WithCStarModule.smul_apply, RingHom.id_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.CStarAlgebra.CStarMatrix | {
"line": 779,
"column": 20
} | {
"line": 782,
"column": 23
} | {
"line": 783,
"column": 6
} | [
{
"pp": "A : Type u_1\ninst✝⁴ : NonUnitalCStarAlgebra A\ninst✝³ : PartialOrder A\ninst✝² : StarOrderedRing A\nm : Type u_2\nn : Type u_3\ninst✝¹ : Fintype m\ninst✝ : Fintype n\nM : CStarMatrix n n A\nv : C⋆ᵐᵒᵈ(A, n → A)\n⊢ ‖toCLM Mᴴ ∘SL toCLM M‖ * ‖v‖ ^ 2 = ‖M * star M‖ * ‖v‖ ^ 2",
"ppTerm": "?m.322",
"... | [] | congr
apply MulOpposite.op_injective
simp only [← toCLMNonUnitalAlgHom_eq_toCLM, map_mul]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.