module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Topology.UniformSpace.Closeds | {
"line": 199,
"column": 2
} | {
"line": 202,
"column": 96
} | {
"line": 204,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\n⊢ UniformContinuous fun x ↦ x.1 ×ˢ x.2",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Iff.mpr",
"Set.instSProd",
"entourageProd",
"instUniformSpace... | [] | refine (𝓤 α).basis_sets.uniformity_prod (𝓤 β).basis_sets |>.lift' monotone_hausdorffEntourage
|>.tendsto_right_iff.mpr fun ⟨U, V⟩ ⟨hU, hV⟩ => ?_
filter_upwards [entourageProd_mem_uniformity (Filter.mem_lift' hU) (Filter.mem_lift' hV)]
with ⟨⟨s₁, s₂⟩, ⟨t₁, t₂⟩⟩ ⟨h₁, h₂⟩ using prod_mem_hausdorffEntourage_ento... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Semicontinuity.Hemicontinuity | {
"line": 577,
"column": 2
} | {
"line": 577,
"column": 38
} | {
"line": 579,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\nf : α → Set β\ns : Set α\nι : Type u_3\nF : ι → α → Set β\nl : Filter ι\ninst✝¹ : l.NeBot\ninst✝ : UniformSpace β\nhtendsto : TendstoUniformlyOn F f l s\nhf_compact : ∀ x ∈ s, IsCompact (f x)\nx₀ : α\nhx₀s : x₀ ∈ s\nu : Set β\nhu : IsOpen[inst✝.t... | [
"α : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\nf : α → Set β\ns : Set α\nι : Type u_3\nF : ι → α → Set β\nl : Filter ι\ninst✝¹ : l.NeBot\ninst✝ : UniformSpace β\nhtendsto : TendstoUniformlyOn F f l s\nhf_compact : ∀ x ∈ s, IsCompact (f x)\nx₀ : α\nhx₀s : x₀ ∈ s\nu : Set β\nhu : IsOpen[inst✝.toTopological... | obtain ⟨y₀, hy₀f, hy₀z⟩ := hFNx hzFN | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.UniformSpace.Closeds | {
"line": 260,
"column": 2
} | {
"line": 264,
"column": 73
} | {
"line": 266,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\nhf : UniformContinuous f\n⊢ UniformContinuous fun x ↦ f '' x",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Iff.mpr",
"Filter.tendsto_lift'",
... | [] | refine Filter.tendsto_lift'.mpr fun U hU => ?_
filter_upwards [Filter.mem_lift' (hf hU)] with ⟨s, t⟩ ⟨h₁, h₂⟩
simp_rw [mem_hausdorffEntourage, Set.image_subset_iff]
exact ⟨h₁.trans fun x ⟨y, hy, hxy⟩ => ⟨f y, Set.mem_image_of_mem f hy, hxy⟩,
h₂.trans fun x ⟨y, hy, hxy⟩ => ⟨f y, Set.mem_image_of_mem f hy, hxy⟩... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.Closeds | {
"line": 260,
"column": 2
} | {
"line": 264,
"column": 73
} | {
"line": 266,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\nhf : UniformContinuous f\n⊢ UniformContinuous fun x ↦ f '' x",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Iff.mpr",
"Filter.tendsto_lift'",
... | [] | refine Filter.tendsto_lift'.mpr fun U hU => ?_
filter_upwards [Filter.mem_lift' (hf hU)] with ⟨s, t⟩ ⟨h₁, h₂⟩
simp_rw [mem_hausdorffEntourage, Set.image_subset_iff]
exact ⟨h₁.trans fun x ⟨y, hy, hxy⟩ => ⟨f y, Set.mem_image_of_mem f hy, hxy⟩,
h₂.trans fun x ⟨y, hy, hxy⟩ => ⟨f y, Set.mem_image_of_mem f hy, hxy⟩... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Sets.VietorisTopology | {
"line": 188,
"column": 2
} | {
"line": 198,
"column": 38
} | {
"line": 200,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : T2Space α\nt : TopologicalSpace (Set α)\nh₁ : IsOpen[t] {∅}\nh₂ : ∀ {U : Set α}, IsOpen[inst✝¹] U → IsOpen[t] {s | (s ∩ U).Nonempty}\n⊢ IsClosed[t] (range fun x ↦ {x})",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Pure.pu... | [] | rw [← isOpen_compl_iff, isOpen_iff_mem_nhds]
intro s hs
rcases Set.eq_empty_or_nonempty s with rfl | h
· rwa [(isOpen_singleton_iff_nhds_eq_pure _).mp h₁, Filter.mem_pure]
rcases h.exists_eq_singleton_or_nontrivial with ⟨x, rfl⟩ | ⟨x, hx, y, hy, hxy⟩
· cases hs <| Set.mem_range_self x
obtain ⟨U, V, hU, hV, ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Sets.VietorisTopology | {
"line": 188,
"column": 2
} | {
"line": 198,
"column": 38
} | {
"line": 200,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : T2Space α\nt : TopologicalSpace (Set α)\nh₁ : IsOpen[t] {∅}\nh₂ : ∀ {U : Set α}, IsOpen[inst✝¹] U → IsOpen[t] {s | (s ∩ U).Nonempty}\n⊢ IsClosed[t] (range fun x ↦ {x})",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Pure.pu... | [] | rw [← isOpen_compl_iff, isOpen_iff_mem_nhds]
intro s hs
rcases Set.eq_empty_or_nonempty s with rfl | h
· rwa [(isOpen_singleton_iff_nhds_eq_pure _).mp h₁, Filter.mem_pure]
rcases h.exists_eq_singleton_or_nontrivial with ⟨x, rfl⟩ | ⟨x, hx, y, hy, hxy⟩
· cases hs <| Set.mem_range_self x
obtain ⟨U, V, hU, hV, ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 364,
"column": 6
} | {
"line": 364,
"column": 77
} | {
"line": 364,
"column": 78
} | [
{
"pp": "case neg\n𝕜 : Type u_1\nA : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedRing A\ninst✝ : NormedAlgebra 𝕜 A\na : A\nz : 𝕜\nh : ↑‖z‖₊ < (spectralRadius 𝕜 a)⁻¹\nhz : ¬z = 0\nu : 𝕜ˣ := Units.mk0 z hz\n⊢ spectralRadius 𝕜 a < ↑‖↑u‖₊⁻¹",
"ppTerm": "?neg✝",
"assigned": true,
... | [
"case neg\n𝕜 : Type u_1\nA : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedRing A\ninst✝ : NormedAlgebra 𝕜 A\na : A\nz : 𝕜\nh : ↑‖z‖₊ < (spectralRadius 𝕜 a)⁻¹\nhz : ¬z = 0\nu : 𝕜ˣ := Units.mk0 z hz\n⊢ spectralRadius 𝕜 a < (↑‖↑(Units.mk0 z hz)‖₊)⁻¹"
] | coe_inv (nnnorm_ne_zero_iff.mpr (Units.val_mk0 hz ▸ hz : (u : 𝕜) ≠ 0)), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | {
"line": 107,
"column": 2
} | {
"line": 110,
"column": 16
} | {
"line": 112,
"column": 0
} | [
{
"pp": "case inr\n𝕜 : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedRing A\ninst✝² : StarRing A\ninst✝¹ : NormedAlgebra 𝕜 A\ninst✝ : IsometricContinuousFunctionalCalculus 𝕜 A p\nf : 𝕜 → 𝕜\na : A\nc : ℝ\nhc : 0 ≤ c\nh : ∀ x ∈ σ 𝕜 a, ‖f x‖ ≤ c\nh✝ : Nontrivial A\n⊢ ‖cfc f a‖ ≤ c"... | [] | · refine cfc_cases (‖·‖ ≤ c) a f (by simpa) fun hf ha ↦ ?_
simp only [← cfc_apply f a, isLUB_le_iff (IsGreatest.norm_cfc f a hf ha |>.isLUB)]
rintro - ⟨x, hx, rfl⟩
exact h x hx | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Sets.VietorisTopology | {
"line": 477,
"column": 2
} | {
"line": 478,
"column": 48
} | {
"line": 478,
"column": 49
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\nB : Set (Set α)\nhB : IsTopologicalBasis B\n⊢ IsTopologicalBasis ((fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) '' {u | u.Finite ∧ u ⊆ B})",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"TopologicalSpace.IsTopologicalBasis.... | [
"case refine_1\nα : Type u_1\ninst✝ : TopologicalSpace α\nB : Set (Set α)\nhB : IsTopologicalBasis B\n⊢ ∀ u ∈ (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) '' {u | u.Finite ∧ u ⊆ B}, IsOpen u",
"case refine_2\nα : Type u_1\ninst✝ : TopologicalSpace α\nB : Set (Set α)\nhB : IsTopologicalBasis B\n⊢ ∀\n ... | refine hB.vietoris.isInducing isEmbedding_coe.isInducing
|>.isTopologicalBasis_of_exists_subset ?_ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 711,
"column": 70
} | {
"line": 711,
"column": 72
} | {
"line": 712,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁴ : NormedField 𝕜\ninst✝³ : ProperSpace 𝕜\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra 𝕜 A\ninst✝ : CompleteSpace A\na₀ a : A\n⊢ a ∈ Metric.closedBall a₀ 1 → spectrum 𝕜 a ⊆ Metric.closedBall 0 ((‖a₀‖ + 1) * ‖1‖)",
"ppTerm": "?m.89",
"assigned": true,
... | [
"𝕜 : Type u_1\nA : Type u_2\ninst✝⁴ : NormedField 𝕜\ninst✝³ : ProperSpace 𝕜\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra 𝕜 A\ninst✝ : CompleteSpace A\na₀ a : A\nha : a ∈ Metric.closedBall a₀ 1\n⊢ spectrum 𝕜 a ⊆ Metric.closedBall 0 ((‖a₀‖ + 1) * ‖1‖)"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.Sets.VietorisTopology | {
"line": 768,
"column": 32
} | {
"line": 775,
"column": 72
} | {
"line": 777,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\nK L : Compacts α\nhK : K ≠ ⊥\nhL : IsPreconnected ↑L\n⊢ IsPreconnected (Icc K L)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"False",
"eq_false",
"ne_bot_of_le_ne_bot",
"Topologi... | [] | by
wlog hKL : K ≤ L
· simpa [hKL] using isPreconnected_empty
convert (isPreconnected_nonempty_subsets hL).image (K ⊔ ·) (by fun_prop)
exact subset_antisymm
(fun M hM => ⟨M, ⟨Compacts.coe_nonempty.mpr (ne_bot_of_le_ne_bot hK hM.1), hM.2⟩,
sup_eq_right.mpr hM.1⟩)
(image_subset_iff.mpr fun M ⟨_, hM⟩ ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.TotallyBounded | {
"line": 47,
"column": 2
} | {
"line": 47,
"column": 56
} | {
"line": 48,
"column": 2
} | [
{
"pp": "case right\nE : Type u_1\ns : Set E\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module ℝ E\ninst✝³ : UniformSpace E\ninst✝² : IsUniformAddGroup E\ninst✝¹ : LocallyConvexSpace ℝ E\ninst✝ : ContinuousSMul ℝ E\nhs : ∀ U ∈ nhds 0, ∃ t, t.Finite ∧ s ⊆ ⋃ y ∈ t, y +ᵥ U\nU : Set E\nhU : U ∈ nhds 0\nW : Set E\nhW₁ : W ∈... | [
"case right\nE : Type u_1\ns : Set E\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module ℝ E\ninst✝³ : UniformSpace E\ninst✝² : IsUniformAddGroup E\ninst✝¹ : LocallyConvexSpace ℝ E\ninst✝ : ContinuousSMul ℝ E\nhs : ∀ U ∈ nhds 0, ∃ t, t.Finite ∧ s ⊆ ⋃ y ∈ t, y +ᵥ U\nU : Set E\nhU : U ∈ nhds 0\nW : Set E\nhW₁ : W ∈ nhds 0\nhW₂... | simp only [iUnion_vadd_set, vadd_eq_add] at hts hts' ⊢ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.LocallyConvex.Polar | {
"line": 161,
"column": 2
} | {
"line": 165,
"column": 49
} | {
"line": 166,
"column": 2
} | [
{
"pp": "case mp\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕜 F\nB : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜\nS : Type u_4\ninst✝¹ : SetLike S E\ninst✝ : SMulMemClass S 𝕜 E\nm : S\ny : F\n⊢ y ∈ B.polar... | [
"case mpr\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕜 F\nB : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜\nS : Type u_4\ninst✝¹ : SetLike S E\ninst✝ : SMulMemClass S 𝕜 E\nm : S\ny : F\n⊢ y ∈ {y | ∀ x ∈ m, (B x... | · intro hy x hx
obtain ⟨r, hr⟩ := NormedField.exists_lt_norm 𝕜 ‖B x y‖⁻¹
contrapose! hr
rw [← one_div, le_div_iff₀ (norm_pos_iff.2 hr)]
simpa using hy _ (SMulMemClass.smul_mem r hx) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Sets.VietorisTopology | {
"line": 898,
"column": 4
} | {
"line": 898,
"column": 25
} | {
"line": 899,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝ : TopologicalSpace α\nB : Set (Set α)\nhB : IsTopologicalBasis B\n⊢ ∀ ⦃x : Set (Set α)⦄, x ∈ {u | u.Finite ∧ u.Nonempty ∧ u ⊆ B} → IsOpen {K | ↑K ⊆ ⋃₀ x ∧ ∀ U ∈ x, (↑K ∩ U).Nonempty}",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Topo... | [
"case refine_1\nα : Type u_1\ninst✝ : TopologicalSpace α\nB : Set (Set α)\nhB : IsTopologicalBasis B\nu : Set (Set α)\nhu : u.Finite\nhuB : u ⊆ B\n⊢ IsOpen {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}"
] | rintro u ⟨hu, -, huB⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Analysis.LocallyConvex.AbsConvex | {
"line": 327,
"column": 10
} | {
"line": 327,
"column": 12
} | {
"line": 328,
"column": 2
} | [
{
"pp": "E : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\na : E\n⊢ a ∈ balancedHull ℝ s → a ∈ (convexHull ℝ) (s ∪ -s)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real",
"DistribMulAction.toDistribSMul",
... | [
"E : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\na : E\nha : a ∈ balancedHull ℝ s\n⊢ a ∈ (convexHull ℝ) (s ∪ -s)"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.Sets.VietorisTopology | {
"line": 1128,
"column": 76
} | {
"line": 1139,
"column": 32
} | {
"line": 1141,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\n⊢ LocallyConnectedSpace (NonemptyCompacts α) ↔ LocallyConnectedSpace α",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Iff.mpr",
"TopologicalSpace.NonemptyCompacts.instSetLike",
"Eq.mpr",
... | [] | by
refine ⟨fun h => locallyConnectedSpace_iff_connected_basis.2 fun x ↦ ?_, fun _ => inferInstance⟩
refine (nhds_basis_opens x).to_hasBasis' (fun U ⟨hx, hU⟩ => ?_) (by grind)
obtain ⟨V, ⟨hV₁, hV₂⟩, hxV, hKV⟩ :=
IsTopologicalBasis.isOpen_isPreconnected.exists_subset_of_mem_open
(show {x} ∈ {K : NonemptyC... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Algebra.Basic | {
"line": 61,
"column": 4
} | {
"line": 61,
"column": 14
} | {
"line": 62,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedRing A\ninst✝² : NormedAlgebra 𝕜 A\ninst✝¹ : CompleteSpace A\ninst✝ : ProperSpace 𝕜\n⊢ characterSpace 𝕜 A ⊆ ⇑toStrongDual ⁻¹' Metric.closedBall 0 ‖1‖",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [... | [
"𝕜 : Type u_1\nA : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedRing A\ninst✝² : NormedAlgebra 𝕜 A\ninst✝¹ : CompleteSpace A\ninst✝ : ProperSpace 𝕜\nφ : WeakDual 𝕜 A\nhφ : φ ∈ characterSpace 𝕜 A\n⊢ φ ∈ ⇑toStrongDual ⁻¹' Metric.closedBall 0 ‖1‖"
] | intro φ hφ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.CStarAlgebra.GelfandDuality | {
"line": 306,
"column": 54
} | {
"line": 306,
"column": 61
} | {
"line": 306,
"column": 61
} | [
{
"pp": "A : Type u_1\ninst✝ : NonUnitalCStarAlgebra A\nι : Type u_2\nf : ι → A\nh0 : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)\nj : ι\ns : Finset ι\nhj : j ∉ s\nih : (∀ i ∈ s, IsSelfAdjoint (f i)) → ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊\nh : ∀ i ∈ insert j s, IsSelfAdjoint (f i)\nthis : f j * ∑ i ∈ s, f i = 0... | [] | cfc_tac | _aux_Mathlib_Tactic_ContinuousFunctionalCalculus___macroRules_cfcTac_1 | cfcTac |
Mathlib.Analysis.CStarAlgebra.GelfandDuality | {
"line": 306,
"column": 54
} | {
"line": 306,
"column": 61
} | {
"line": 306,
"column": 61
} | [
{
"pp": "A : Type u_1\ninst✝ : NonUnitalCStarAlgebra A\nι : Type u_2\nf : ι → A\nh0 : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)\nj : ι\ns : Finset ι\nhj : j ∉ s\nih : (∀ i ∈ s, IsSelfAdjoint (f i)) → ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊\nh : ∀ i ∈ insert j s, IsSelfAdjoint (f i)\nthis : f j * ∑ i ∈ s, f i = 0... | [] | cfc_tac | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.GelfandDuality | {
"line": 306,
"column": 54
} | {
"line": 306,
"column": 61
} | {
"line": 306,
"column": 61
} | [
{
"pp": "A : Type u_1\ninst✝ : NonUnitalCStarAlgebra A\nι : Type u_2\nf : ι → A\nh0 : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)\nj : ι\ns : Finset ι\nhj : j ∉ s\nih : (∀ i ∈ s, IsSelfAdjoint (f i)) → ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊\nh : ∀ i ∈ insert j s, IsSelfAdjoint (f i)\nthis : f j * ∑ i ∈ s, f i = 0... | [] | cfc_tac | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 152,
"column": 49
} | {
"line": 152,
"column": 56
} | {
"line": 152,
"column": 56
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : NonUnitalRing A\ninst✝⁷ : Module ℝ A\ninst✝⁶ : SMulCommClass ℝ A A\ninst✝⁵ : IsScalarTower ℝ A A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : A\nha : Is... | [] | cfc_tac | _aux_Mathlib_Tactic_ContinuousFunctionalCalculus___macroRules_cfcTac_1 | cfcTac |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 152,
"column": 49
} | {
"line": 152,
"column": 56
} | {
"line": 152,
"column": 56
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : NonUnitalRing A\ninst✝⁷ : Module ℝ A\ninst✝⁶ : SMulCommClass ℝ A A\ninst✝⁵ : IsScalarTower ℝ A A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : A\nha : Is... | [] | cfc_tac | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 152,
"column": 49
} | {
"line": 152,
"column": 56
} | {
"line": 152,
"column": 56
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : NonUnitalRing A\ninst✝⁷ : Module ℝ A\ninst✝⁶ : SMulCommClass ℝ A A\ninst✝⁵ : IsScalarTower ℝ A A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : A\nha : Is... | [] | cfc_tac | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 152,
"column": 61
} | {
"line": 152,
"column": 68
} | {
"line": 152,
"column": 68
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : NonUnitalRing A\ninst✝⁷ : Module ℝ A\ninst✝⁶ : SMulCommClass ℝ A A\ninst✝⁵ : IsScalarTower ℝ A A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : A\nha : Is... | [] | cfc_tac | _aux_Mathlib_Tactic_ContinuousFunctionalCalculus___macroRules_cfcTac_1 | cfcTac |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 152,
"column": 61
} | {
"line": 152,
"column": 68
} | {
"line": 152,
"column": 68
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : NonUnitalRing A\ninst✝⁷ : Module ℝ A\ninst✝⁶ : SMulCommClass ℝ A A\ninst✝⁵ : IsScalarTower ℝ A A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : A\nha : Is... | [] | cfc_tac | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 152,
"column": 61
} | {
"line": 152,
"column": 68
} | {
"line": 152,
"column": 68
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : NonUnitalRing A\ninst✝⁷ : Module ℝ A\ninst✝⁶ : SMulCommClass ℝ A A\ninst✝⁵ : IsScalarTower ℝ A A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : A\nha : Is... | [] | cfc_tac | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.UniformConvergence | {
"line": 240,
"column": 13
} | {
"line": 240,
"column": 62
} | {
"line": 241,
"column": 6
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : PseudoEMetricSpace γ\n𝔖 𝔗 : Set (Set α)\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : Finite ↑𝔖\nx✝ : Fintype ↑𝔖 := Fintype.ofFinite ↑𝔖\n⊢ 𝓤 (α →ᵤ[𝔖] β) = ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | edist p.1 p.2 < ε}",
"ppTerm": "?m.20",
"assigned": true,
... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : PseudoEMetricSpace γ\n𝔖 𝔗 : Set (Set α)\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : Finite ↑𝔖\nx✝ : Fintype ↑𝔖 := Fintype.ofFinite ↑𝔖\n⊢ comap\n (fun x ↦\n (fun s ↦ UniformFun.ofFun ((↑s).domRestrict ((toFun 𝔖) x.1)), fun s ↦\n UniformFun.of... | ← isUniformInducing_pi_restrict.comap_uniformity, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic | {
"line": 297,
"column": 2
} | {
"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⊢ ∀ x ∈ spectrum ℝ a, 0 ≤ x",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"IsSelfAdjoint",
"NormedRing.toRing",
"HMul.hMul",
"Ring.toNonAssocRing",
"congrArg",... | [
"A : Type u_1\ninst✝ : CStarAlgebra A\nb : A\na : A := star b * b\na_def : a = star b * b\nha : IsSelfAdjoint a\n⊢ ∀ x ∈ spectrum ℝ a, 0 ≤ x"
] | have ha : IsSelfAdjoint a := by simp [a_def] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic | {
"line": 153,
"column": 2
} | {
"line": 153,
"column": 21
} | {
"line": 154,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝¹¹ : PartialOrder A\ninst✝¹⁰ : NonUnitalRing A\ninst✝⁹ : TopologicalSpace A\ninst✝⁸ : StarRing A\ninst✝⁷ : Module ℝ A\ninst✝⁶ : SMulCommClass ℝ A A\ninst✝⁵ : IsScalarTower ℝ A A\ninst✝⁴ : StarOrderedRing A\ninst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝² : No... | [
"case pos\nA : Type u_1\ninst✝¹¹ : PartialOrder A\ninst✝¹⁰ : NonUnitalRing A\ninst✝⁹ : TopologicalSpace A\ninst✝⁸ : StarRing A\ninst✝⁷ : Module ℝ A\ninst✝⁶ : SMulCommClass ℝ A A\ninst✝⁵ : IsScalarTower ℝ A A\ninst✝⁴ : StarOrderedRing A\ninst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝² : Nonn... | by_cases ha : 0 ≤ a | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic | {
"line": 258,
"column": 68
} | {
"line": 259,
"column": 60
} | {
"line": 261,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝¹¹ : PartialOrder A\ninst✝¹⁰ : NonUnitalRing A\ninst✝⁹ : TopologicalSpace A\ninst✝⁸ : StarRing A\ninst✝⁷ : Module ℝ A\ninst✝⁶ : SMulCommClass ℝ A A\ninst✝⁵ : IsScalarTower ℝ A A\ninst✝⁴ : StarOrderedRing A\ninst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝² : No... | [] | by
rw [sqrt_eq_nnrpow, nnrpow_nnrpow, one_div_mul_eq_div 2 x] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.SpecialFunctions.PosPart | {
"line": 71,
"column": 8
} | {
"line": 71,
"column": 15
} | {
"line": 72,
"column": 4
} | [
{
"pp": "case h.left.«0».left\nA : Type u_2\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : A\n⊢ 0 ≤ (↑(realPart a))⁺",
"ppTerm": "?h.left.«0».left",
"assigned": true,
"usedConstants": [
"NonUnitalCStarAlgebra.toStarModule",
"instTrivialStarReal... | [] | cfc_tac | _aux_Mathlib_Tactic_ContinuousFunctionalCalculus___macroRules_cfcTac_1 | cfcTac |
Mathlib.Analysis.CStarAlgebra.SpecialFunctions.PosPart | {
"line": 71,
"column": 8
} | {
"line": 71,
"column": 15
} | {
"line": 72,
"column": 4
} | [
{
"pp": "case h.left.«1».left\nA : Type u_2\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : A\n⊢ 0 ≤ (↑(imaginaryPart a))⁺",
"ppTerm": "?h.left.«1».left",
"assigned": true,
"usedConstants": [
"NonUnitalCStarAlgebra.toStarModule",
"instTrivialSta... | [] | cfc_tac | _aux_Mathlib_Tactic_ContinuousFunctionalCalculus___macroRules_cfcTac_1 | cfcTac |
Mathlib.Analysis.CStarAlgebra.SpecialFunctions.PosPart | {
"line": 71,
"column": 8
} | {
"line": 71,
"column": 15
} | {
"line": 72,
"column": 4
} | [
{
"pp": "case h.left.«2».left\nA : Type u_2\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : A\n⊢ 0 ≤ (↑(realPart a))⁻",
"ppTerm": "?h.left.«2».left",
"assigned": true,
"usedConstants": [
"NonUnitalCStarAlgebra.toStarModule",
"instTrivialStarReal... | [] | cfc_tac | _aux_Mathlib_Tactic_ContinuousFunctionalCalculus___macroRules_cfcTac_1 | cfcTac |
Mathlib.Analysis.CStarAlgebra.SpecialFunctions.PosPart | {
"line": 71,
"column": 8
} | {
"line": 71,
"column": 15
} | {
"line": 72,
"column": 4
} | [
{
"pp": "case h.left.«3».left\nA : Type u_2\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : A\n⊢ 0 ≤ (↑(imaginaryPart a))⁻",
"ppTerm": "?h.left.«3».left",
"assigned": true,
"usedConstants": [
"NonUnitalCStarAlgebra.toStarModule",
"instTrivialSta... | [] | cfc_tac | _aux_Mathlib_Tactic_ContinuousFunctionalCalculus___macroRules_cfcTac_1 | cfcTac |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic | {
"line": 307,
"column": 2
} | {
"line": 307,
"column": 92
} | {
"line": 309,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝¹¹ : PartialOrder A\ninst✝¹⁰ : NonUnitalRing A\ninst✝⁹ : TopologicalSpace A\ninst✝⁸ : StarRing A\ninst✝⁷ : Module ℝ A\ninst✝⁶ : SMulCommClass ℝ A A\ninst✝⁵ : IsScalarTower ℝ A A\ninst✝⁴ : StarOrderedRing A\ninst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝² : No... | [] | exact cfcₙ_congr fun x hx ↦ Real.mul_self_sqrt <| quasispectrum_nonneg_of_nonneg a ha x hx | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic | {
"line": 444,
"column": 10
} | {
"line": 444,
"column": 12
} | {
"line": 445,
"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\n⊢ a ∈ Set.Ici 0 → (fun a ↦ a ^ 0) a = (fun x ↦ 1) ... | [
"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\nha : a ∈ Set.Ici 0\n⊢ (fun a ↦ a ^ 0) a = (fun x ↦ 1) a"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic | {
"line": 670,
"column": 2
} | {
"line": 670,
"column": 9
} | {
"line": 673,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝¹⁰ : PartialOrder A\ninst✝⁹ : Ring A\ninst✝⁸ : StarRing A\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : StarOrderedRing A\ninst✝⁵ : Algebra ℝ A\ninst✝⁴ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝³ : NonnegSpectrumClass ℝ A\ninst✝² : IsSemitopologicalRing A\ninst✝¹ : T2Space A\ni... | [] | cfc_tac | _aux_Mathlib_Tactic_ContinuousFunctionalCalculus___macroRules_cfcTac_1 | cfcTac |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic | {
"line": 689,
"column": 83
} | {
"line": 698,
"column": 77
} | {
"line": 700,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝⁹ : PartialOrder A\ninst✝⁸ : Ring A\ninst✝⁷ : StarRing A\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : StarOrderedRing A\ninst✝⁴ : Algebra ℝ A\ninst✝³ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝² : NonnegSpectrumClass ℝ A\ninst✝¹ : IsSemitopologicalRing A\ninst✝ : T2Space A\na :... | [] | by
by_cases htriv : 0 ≤ a
case neg => simp [sqrt_eq_cfc, rpow_def, cfc_apply_of_not_predicate a htriv]
case pos =>
cases eq_zero_or_pos x with
| inl hx => simp [hx, rpow_zero _ htriv]
| inr h₁ =>
have h₂ : (x : ℝ) / 2 = NNReal.toReal (x / 2) := by simp
have h₃ : 0 < x / 2 := by positivity
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order | {
"line": 115,
"column": 22
} | {
"line": 115,
"column": 24
} | {
"line": 115,
"column": 25
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nf : A → A\ns : Set A\nhf : ∀ (x : A), IsSelfAdjoint (f x)\nhf₂ : ConvexOn ℝ s (inr ∘ f)\nx : A\nhx : x ∈ s\ny : A\nhy : y ∈ s\na b : ℝ\n⊢ 0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y... | [
"A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nf : A → A\ns : Set A\nhf : ∀ (x : A), IsSelfAdjoint (f x)\nhf₂ : ConvexOn ℝ s (inr ∘ f)\nx : A\nhx : x ∈ s\ny : A\nhy : y ∈ s\na b : ℝ\nha : 0 ≤ a\n⊢ 0 ≤ b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order | {
"line": 123,
"column": 22
} | {
"line": 123,
"column": 24
} | {
"line": 123,
"column": 25
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nf : A → A\ns : Set A\nhf : ∀ (x : A), IsSelfAdjoint (f x)\nhf₂ : ConcaveOn ℝ s (inr ∘ f)\nx : A\nhx : x ∈ s\ny : A\nhy : y ∈ s\na b : ℝ\n⊢ 0 ≤ a → 0 ≤ b → a + b = 1 → a • f x + b • f y ≤ f (a • x + b • y... | [
"A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nf : A → A\ns : Set A\nhf : ∀ (x : A), IsSelfAdjoint (f x)\nhf₂ : ConcaveOn ℝ s (inr ∘ f)\nx : A\nhx : x ∈ s\ny : A\nhy : y ∈ s\na b : ℝ\nha : 0 ≤ a\n⊢ 0 ≤ b → a + b = 1 → a • f x + b • f y ≤ f (a • x + b • y)"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 21
} | {
"line": 134,
"column": 22
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : A\n⊢ CFC.sqrt ↑a = ↑(CFC.sqrt a)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"NonUnitalCStarAlgebra.toStarModule",
"Unitization.instAlgebra",
"NormedCommRi... | [
"case pos\nA : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : A\nha : 0 ≤ a\n⊢ CFC.sqrt ↑a = ↑(CFC.sqrt a)",
"case neg\nA : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : A\nha : ¬0 ≤ a\n⊢ CFC.sqrt ↑a = ↑(CFC... | by_cases ha : 0 ≤ a | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic | {
"line": 768,
"column": 2
} | {
"line": 768,
"column": 9
} | {
"line": 770,
"column": 0
} | [
{
"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\nha : IsStrictlyPositive a\n⊢ IsStrictlyPositive (a... | [] | cfc_tac | _aux_Mathlib_Tactic_ContinuousFunctionalCalculus___macroRules_cfcTac_1 | cfcTac |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order | {
"line": 455,
"column": 12
} | {
"line": 455,
"column": 14
} | {
"line": 455,
"column": 15
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na✝ b✝ : A\nha : 0 ≤ a✝\nhab : a✝ ≤ b✝\na b : Unitization ℂ A\n⊢ 0 ≤ a → a ≤ b → ‖a‖ ≤ ‖b‖",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"NonUnitalCStarAlgebra.toNonUnitalNor... | [
"A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na✝ b✝ : A\nha✝ : 0 ≤ a✝\nhab : a✝ ≤ b✝\na b : Unitization ℂ A\nha : 0 ≤ a\n⊢ a ≤ b → ‖a‖ ≤ ‖b‖"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order | {
"line": 563,
"column": 15
} | {
"line": 563,
"column": 17
} | {
"line": 563,
"column": 18
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na✝ e✝ : A\nhe✝ : IsStarProjection e✝\nha : 0 ≤ a✝\nhae : a✝ ≤ e✝\na e : Unitization ℂ A\nhe : IsStarProjection e\n⊢ 0 ≤ a → a ≤ e → a * e = a",
"ppTerm": "?m.91",
"assigned": true,
"usedConst... | [
"A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na✝ e✝ : A\nhe✝ : IsStarProjection e✝\nha✝ : 0 ≤ a✝\nhae : a✝ ≤ e✝\na e : Unitization ℂ A\nhe : IsStarProjection e\nha : 0 ≤ a\n⊢ a ≤ e → a * e = a"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.CStarAlgebra.Module.Constructions | {
"line": 114,
"column": 75
} | {
"line": 119,
"column": 38
} | {
"line": 121,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝⁹ : NonUnitalCStarAlgebra A\ninst✝⁸ : PartialOrder A\nE : Type u_2\nF : Type u_3\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : Module ℂ E\ninst✝⁵ : SMul A E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : Module ℂ F\ninst✝² : SMul A F\ninst✝¹ : CStarModule A E\ninst✝ : CStarModule A F\nx : C⋆ᵐᵒᵈ... | [] | by
refine abs_le_of_sq_le_sq' ?_ (by positivity) |>.2
calc ‖x‖ ^ 2 ≤ ‖⟪x.1, x.1⟫_A‖ + ‖⟪x.2, x.2⟫_A‖ := prod_norm_sq x ▸ norm_add_le _ _
_ = ‖x.1‖ ^ 2 + 0 + ‖x.2‖ ^ 2 := by simp [norm_sq_eq A]
_ ≤ ‖x.1‖ ^ 2 + 2 * ‖x.1‖ * ‖x.2‖ + ‖x.2‖ ^ 2 := by gcongr; positivity
_ = (‖x.1‖ + ‖x.2‖) ^ 2 := by ring | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.ApproximateUnit | {
"line": 55,
"column": 10
} | {
"line": 55,
"column": 12
} | {
"line": 55,
"column": 13
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : A\n⊢ a ∈ Set.Ici 0 → ∀ ⦃b : A⦄, b ∈ Set.Ici 0 → a ≤ b → cfcₙ (fun x ↦ 1 - (1 + x)⁻¹) a ≤ cfcₙ (fun x ↦ 1 - (1 + x)⁻¹) b",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"No... | [
"A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : A\nha : a ∈ Set.Ici 0\n⊢ ∀ ⦃b : A⦄, b ∈ Set.Ici 0 → a ≤ b → cfcₙ (fun x ↦ 1 - (1 + x)⁻¹) a ≤ cfcₙ (fun x ↦ 1 - (1 + x)⁻¹) b"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.UniformSpace.Matrix | {
"line": 42,
"column": 2
} | {
"line": 42,
"column": 41
} | {
"line": 44,
"column": 0
} | [
{
"pp": "m : Type u_1\nn : Type u_2\n𝕜 : Type u_3\ninst✝¹ : UniformSpace 𝕜\nβ : Type u_4\ninst✝ : UniformSpace β\nf : β → Matrix m n 𝕜\n⊢ (∀ (i : m) (i_1 : n), Filter.Tendsto ((fun a ↦ (a.1 i i_1, a.2 i i_1)) ∘ fun x ↦ (f x.1, f x.2)) (𝓤 β) (𝓤 𝕜)) ↔\n ∀ (i : m) (j : n), Filter.Tendsto (fun x ↦ (f x.1 i... | [] | apply Iff.intro <;> intro a <;> apply a | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Analysis.CStarAlgebra.CompletelyPositiveMap | {
"line": 99,
"column": 12
} | {
"line": 99,
"column": 14
} | {
"line": 100,
"column": 2
} | [
{
"pp": "F : Type u_1\nA₁ : Type u_2\nA₂ : Type u_3\ninst✝⁷ : NonUnitalCStarAlgebra A₁\ninst✝⁶ : NonUnitalCStarAlgebra A₂\ninst✝⁵ : PartialOrder A₁\ninst✝⁴ : PartialOrder A₂\ninst✝³ : StarOrderedRing A₁\ninst✝² : StarOrderedRing A₂\ninst✝¹ : FunLike F A₁ A₂\ninst✝ : LinearMapClass F ℂ A₁ A₂\nh : ∀ (φ : F) (k : ... | [
"F : Type u_1\nA₁ : Type u_2\nA₂ : Type u_3\ninst✝⁷ : NonUnitalCStarAlgebra A₁\ninst✝⁶ : NonUnitalCStarAlgebra A₂\ninst✝⁵ : PartialOrder A₁\ninst✝⁴ : PartialOrder A₂\ninst✝³ : StarOrderedRing A₁\ninst✝² : StarOrderedRing A₂\ninst✝¹ : FunLike F A₁ A₂\ninst✝ : LinearMapClass F ℂ A₁ A₂\nh : ∀ (φ : F) (k : ℕ) (M : CSta... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Commute | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 21
} | {
"line": 116,
"column": 2
} | [
{
"pp": "A : Type u_2\ninst✝⁹ : Ring A\ninst✝⁸ : StarRing A\ninst✝⁷ : Algebra ℝ A\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁴ : IsTopologicalRing A\ninst✝³ : T2Space A\ninst✝² : PartialOrder A\ninst✝¹ : NonnegSpectrumClass ℝ A\ninst✝ : StarOrderedRing A\na b : A... | [
"case pos\nA : Type u_2\ninst✝⁹ : Ring A\ninst✝⁸ : StarRing A\ninst✝⁷ : Algebra ℝ A\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁴ : IsTopologicalRing A\ninst✝³ : T2Space A\ninst✝² : PartialOrder A\ninst✝¹ : NonnegSpectrumClass ℝ A\ninst✝ : StarOrderedRing A\na b : A\n... | by_cases ha : 0 ≤ a | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Commute | {
"line": 202,
"column": 2
} | {
"line": 202,
"column": 21
} | {
"line": 203,
"column": 2
} | [
{
"pp": "A : Type u_2\ninst✝¹¹ : NonUnitalRing A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : Module ℝ A\ninst✝⁸ : IsScalarTower ℝ A A\ninst✝⁷ : SMulCommClass ℝ A A\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁴ : IsTopologicalRing A\ninst✝³ : T2Space A\ninst✝² : Parti... | [
"case pos\nA : Type u_2\ninst✝¹¹ : NonUnitalRing A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : Module ℝ A\ninst✝⁸ : IsScalarTower ℝ A A\ninst✝⁷ : SMulCommClass ℝ A A\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁴ : IsTopologicalRing A\ninst✝³ : T2Space A\ninst✝² : Partial... | by_cases ha : 0 ≤ a | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 1114,
"column": 2
} | {
"line": 1115,
"column": 96
} | {
"line": 1117,
"column": 0
} | [
{
"pp": "X : Type u_1\nA : Type u_2\ninst✝¹² : NonUnitalNormedRing A\ninst✝¹¹ : StarRing A\ninst✝¹⁰ : NormedSpace ℝ A\ninst✝⁹ : IsScalarTower ℝ A A\ninst✝⁸ : SMulCommClass ℝ A A\ninst✝⁷ : ContinuousStar A\ninst✝⁶ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁵ : PartialOrder A\ninst✝⁴... | [] | rw [← continuousOn_univ] at ha_cont ⊢
exact ha_cont.cfcₙ_nnreal f (fun x _ ↦ hs x) (fun x _ ↦ by simpa using ha x) (fun x _ ↦ ha' x) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 1114,
"column": 2
} | {
"line": 1115,
"column": 96
} | {
"line": 1117,
"column": 0
} | [
{
"pp": "X : Type u_1\nA : Type u_2\ninst✝¹² : NonUnitalNormedRing A\ninst✝¹¹ : StarRing A\ninst✝¹⁰ : NormedSpace ℝ A\ninst✝⁹ : IsScalarTower ℝ A A\ninst✝⁸ : SMulCommClass ℝ A A\ninst✝⁷ : ContinuousStar A\ninst✝⁶ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁵ : PartialOrder A\ninst✝⁴... | [] | rw [← continuousOn_univ] at ha_cont ⊢
exact ha_cont.cfcₙ_nnreal f (fun x _ ↦ hs x) (fun x _ ↦ by simpa using ha x) (fun x _ ↦ ha' x) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart | {
"line": 27,
"column": 30
} | {
"line": 27,
"column": 51
} | {
"line": 27,
"column": 51
} | [
{
"pp": "A : Type u_1\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : Module ℂ A\ninst✝³ : IsScalarTower ℂ A A\ninst✝² : SMulCommClass ℂ A A\ninst✝¹ : StarModule ℂ A\ninst✝ : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal\na : A\nha : IsStarNormal a\nx : ℂ\nhx : x ∈... | [
"A : Type u_1\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : Module ℂ A\ninst✝³ : IsScalarTower ℂ A A\ninst✝² : SMulCommClass ℂ A A\ninst✝¹ : StarModule ℂ A\ninst✝ : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal\na : A\nha : IsStarNormal a\nx : ℂ\nhx : x ∈ quasispectr... | ← smul_one_smul ℂ 2⁻¹ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart | {
"line": 48,
"column": 2
} | {
"line": 48,
"column": 48
} | {
"line": 51,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : Module ℂ A\ninst✝³ : IsScalarTower ℂ A A\ninst✝² : SMulCommClass ℂ A A\ninst✝¹ : StarModule ℂ A\ninst✝ : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal\na : A\nha : IsStarNormal a\n⊢ quasispectrum... | [] | rw [← cfcₙ_im_id a, cfcₙ_map_quasispectrum ..] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart | {
"line": 48,
"column": 2
} | {
"line": 48,
"column": 48
} | {
"line": 51,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : Module ℂ A\ninst✝³ : IsScalarTower ℂ A A\ninst✝² : SMulCommClass ℂ A A\ninst✝¹ : StarModule ℂ A\ninst✝ : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal\na : A\nha : IsStarNormal a\n⊢ quasispectrum... | [] | rw [← cfcₙ_im_id a, cfcₙ_map_quasispectrum ..] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart | {
"line": 48,
"column": 2
} | {
"line": 48,
"column": 48
} | {
"line": 51,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : Module ℂ A\ninst✝³ : IsScalarTower ℂ A A\ninst✝² : SMulCommClass ℂ A A\ninst✝¹ : StarModule ℂ A\ninst✝ : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal\na : A\nha : IsStarNormal a\n⊢ quasispectrum... | [] | rw [← cfcₙ_im_id a, cfcₙ_map_quasispectrum ..] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Integral | {
"line": 245,
"column": 2
} | {
"line": 245,
"column": 34
} | {
"line": 247,
"column": 0
} | [
{
"pp": "X : Type u_1\n𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : MeasurableSpace X\nμ : Measure X\ninst✝⁶ : NonUnitalNormedRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : NormedSpace 𝕜 A\ninst✝³ : IsScalarTower 𝕜 A A\ninst✝² : SMulCommClass 𝕜 A A\ninst✝¹ : NonUnitalContinuousFunctionalCal... | [] | exact integrable_cfcₙ' _ _ hf ha | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Integral | {
"line": 245,
"column": 2
} | {
"line": 245,
"column": 34
} | {
"line": 247,
"column": 0
} | [
{
"pp": "X : Type u_1\n𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : MeasurableSpace X\nμ : Measure X\ninst✝⁶ : NonUnitalNormedRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : NormedSpace 𝕜 A\ninst✝³ : IsScalarTower 𝕜 A A\ninst✝² : SMulCommClass 𝕜 A A\ninst✝¹ : NonUnitalContinuousFunctionalCal... | [] | exact integrable_cfcₙ' _ _ hf ha | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Integral | {
"line": 245,
"column": 2
} | {
"line": 245,
"column": 34
} | {
"line": 247,
"column": 0
} | [
{
"pp": "X : Type u_1\n𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : MeasurableSpace X\nμ : Measure X\ninst✝⁶ : NonUnitalNormedRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : NormedSpace 𝕜 A\ninst✝³ : IsScalarTower 𝕜 A A\ninst✝² : SMulCommClass 𝕜 A A\ninst✝¹ : NonUnitalContinuousFunctionalCal... | [] | exact integrable_cfcₙ' _ _ hf ha | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart | {
"line": 108,
"column": 30
} | {
"line": 108,
"column": 51
} | {
"line": 108,
"column": 51
} | [
{
"pp": "A : Type u_1\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ : Ring A\ninst✝³ : StarRing A\ninst✝² : Algebra ℂ A\ninst✝¹ : StarModule ℂ A\ninst✝ : ContinuousFunctionalCalculus ℂ A IsStarNormal\na : A\nhp : IsStarNormal a\nx : ℂ\nhx : x ∈ spectrum ℂ a\n⊢ (x + (starRingEnd ℂ) x) / 2 = 2⁻¹ • (x + star x)",
"ppTe... | [
"A : Type u_1\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ : Ring A\ninst✝³ : StarRing A\ninst✝² : Algebra ℂ A\ninst✝¹ : StarModule ℂ A\ninst✝ : ContinuousFunctionalCalculus ℂ A IsStarNormal\na : A\nhp : IsStarNormal a\nx : ℂ\nhx : x ∈ spectrum ℂ a\n⊢ (x + (starRingEnd ℂ) x) / 2 = (2⁻¹ • 1) • (x + star x)"
] | ← smul_one_smul ℂ 2⁻¹ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart | {
"line": 105,
"column": 34
} | {
"line": 109,
"column": 23
} | {
"line": 111,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ : Ring A\ninst✝³ : StarRing A\ninst✝² : Algebra ℂ A\ninst✝¹ : StarModule ℂ A\ninst✝ : ContinuousFunctionalCalculus ℂ A IsStarNormal\na : A\nhp : IsStarNormal a\n⊢ cfc (fun x ↦ ↑x.re) a = ↑(ℜ a)",
"ppTerm": "?m.43",
"assigned": true,
"usedCon... | [] | by
conv_rhs => rw [realPart_apply_coe, ← cfc_id' ℂ a, ← cfc_star, ← cfc_add .., ← cfc_smul ..]
refine cfc_congr fun x hx ↦ ?_
rw [Complex.re_eq_add_conj, ← smul_one_smul ℂ 2⁻¹]
simp [div_eq_inv_mul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart | {
"line": 146,
"column": 2
} | {
"line": 147,
"column": 36
} | {
"line": 149,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : Ring A\ninst✝⁴ : StarRing A\ninst✝³ : Algebra ℂ A\ninst✝² : StarModule ℂ A\ninst✝¹ : ContinuousFunctionalCalculus ℂ A IsStarNormal\ninst✝ : ContinuousMap.UniqueHom ℂ A\nf : ℂ → ℂ\na : A\nhf : ContinuousOn f (spectrum ℂ ↑(ℑ a))\nha : IsStarNormal a\n⊢ ... | [] | rw [spectrum_imaginaryPart a] at hf
rw [← cfc_im_id a, ← cfc_comp' ..] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart | {
"line": 146,
"column": 2
} | {
"line": 147,
"column": 36
} | {
"line": 149,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : Ring A\ninst✝⁴ : StarRing A\ninst✝³ : Algebra ℂ A\ninst✝² : StarModule ℂ A\ninst✝¹ : ContinuousFunctionalCalculus ℂ A IsStarNormal\ninst✝ : ContinuousMap.UniqueHom ℂ A\nf : ℂ → ℂ\na : A\nhf : ContinuousOn f (spectrum ℂ ↑(ℑ a))\nha : IsStarNormal a\n⊢ ... | [] | rw [spectrum_imaginaryPart a] at hf
rw [← cfc_im_id a, ← cfc_comp' ..] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Extreme | {
"line": 206,
"column": 4
} | {
"line": 206,
"column": 41
} | {
"line": 207,
"column": 4
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : Semiring 𝕜\ninst✝⁴ : PartialOrder 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns : Set E\nt : Set F\nx : E\ny : F\nhx : (x, y) ∈ s ×ˢ t\nh : ∀ ⦃x₁ : E × F⦄, x₁ ∈ s ×ˢ t → ∀ ⦃x₂ : E × ... | [
"case refine_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : Semiring 𝕜\ninst✝⁴ : PartialOrder 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns : Set E\nt : Set F\nx : E\ny : F\nhx : (x, y) ∈ s ×ˢ t\nh : ∀ ⦃x₁ : E × F⦄, x₁ ∈ s ×ˢ t → ∀ ⦃x₂ : E × F⦄, x₂ ∈ s ×... | rw [← Prod.image_mk_openSegment_left] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.CStarAlgebra.Extreme | {
"line": 63,
"column": 8
} | {
"line": 65,
"column": 31
} | {
"line": 66,
"column": 6
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\ne : A\nhe : IsStarProjection e\na : A\nha : 0 ≤ a\nha1 : ‖a‖ ≤ 1\nb : A\nhb : 0 ≤ b\nhb1 : ‖b‖ ≤ 1\nx✝ : e ∈ openSegment ℝ a b\nt s : ℝ\nh0t : 0 < t\nh0s : 0 < s\nhts : t + s = 1\nhlin : t • a + s • b = ... | [] | simp only [mul_one_sub_mul, he.inr.isIdempotentElem.eq, smul_sub, mul_add,
Algebra.mul_smul_comm, add_mul, Algebra.smul_mul_assoc]
match_scalars <;> grind | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.Extreme | {
"line": 63,
"column": 8
} | {
"line": 65,
"column": 31
} | {
"line": 66,
"column": 6
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\ne : A\nhe : IsStarProjection e\na : A\nha : 0 ≤ a\nha1 : ‖a‖ ≤ 1\nb : A\nhb : 0 ≤ b\nhb1 : ‖b‖ ≤ 1\nx✝ : e ∈ openSegment ℝ a b\nt s : ℝ\nh0t : 0 < t\nh0s : 0 < s\nhts : t + s = 1\nhlin : t • a + s • b = ... | [] | simp only [mul_one_sub_mul, he.inr.isIdempotentElem.eq, smul_sub, mul_add,
Algebra.mul_smul_comm, add_mul, Algebra.smul_mul_assoc]
match_scalars <;> grind | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.Matrix | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 81
} | {
"line": 77,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nn : Type u_3\ninst✝² : RCLike 𝕜\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nU : Matrix n n 𝕜\nhU : U ∈ Matrix.unitaryGroup n 𝕜\ni j : n\nnorm_sum : ‖U i j‖ ^ 2 ≤ ∑ x, ‖U i x‖ ^ 2\ndiag_eq_norm_sum : (U * Uᴴ) i i = ↑(∑ x, ‖U i x‖ ^ 2)\nre_diag_eq_norm_sum : RCLike.re ((U * Uᴴ) i i) = ∑... | [
"𝕜 : Type u_1\nn : Type u_3\ninst✝² : RCLike 𝕜\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nU : Matrix n n 𝕜\nhU : U ∈ Matrix.unitaryGroup n 𝕜\ni j : n\nnorm_sum : ‖U i j‖ ^ 2 ≤ ∑ x, ‖U i x‖ ^ 2\ndiag_eq_norm_sum : (U * Uᴴ) i i = ↑(∑ x, ‖U i x‖ ^ 2)\nre_diag_eq_norm_sum : RCLike.re ((U * Uᴴ) i i) = ∑ x, ‖U i x‖ ... | rw [← sq_le_one_iff₀ (norm_nonneg (U i j)), ← diag_eq_one, re_diag_eq_norm_sum] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Field.UnitBall | {
"line": 214,
"column": 4
} | {
"line": 214,
"column": 93
} | {
"line": 216,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : NormedDivisionRing 𝕜\nx : ↑(sphere 0 1)\nn : ℤ\n⊢ ↑x ^ n ∈ sphere 0 1",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"Real",
"Subtype.co... | [] | rw [mem_sphere_zero_iff_norm, norm_zpow, mem_sphere_zero_iff_norm.1 x.coe_prop, one_zpow] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Field.UnitBall | {
"line": 214,
"column": 4
} | {
"line": 214,
"column": 93
} | {
"line": 216,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : NormedDivisionRing 𝕜\nx : ↑(sphere 0 1)\nn : ℤ\n⊢ ↑x ^ n ∈ sphere 0 1",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"Real",
"Subtype.co... | [] | rw [mem_sphere_zero_iff_norm, norm_zpow, mem_sphere_zero_iff_norm.1 x.coe_prop, one_zpow] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Field.UnitBall | {
"line": 214,
"column": 4
} | {
"line": 214,
"column": 93
} | {
"line": 216,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : NormedDivisionRing 𝕜\nx : ↑(sphere 0 1)\nn : ℤ\n⊢ ↑x ^ n ∈ sphere 0 1",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"Real",
"Subtype.co... | [] | rw [mem_sphere_zero_iff_norm, norm_zpow, mem_sphere_zero_iff_norm.1 x.coe_prop, one_zpow] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.OpenPartialHomeomorph.Constructions | {
"line": 282,
"column": 6
} | {
"line": 282,
"column": 47
} | {
"line": 282,
"column": 48
} | [
{
"pp": "X : Type u_1\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ne : OpenPartialHomeomorph X Y\ns : Opens X\nhs : Nonempty ↥s\nx : ↥s\nhxe : ↑x ∈ e.source\n⊢ ↑e ↑x ∈ ↑e.symm ⁻¹' (s.openPartialHomeomorphSubtypeCoe hs).target",
"ppTerm": "?m.31",
"assigned": true,
"usedCon... | [
"X : Type u_1\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ne : OpenPartialHomeomorph X Y\ns : Opens X\nhs : Nonempty ↥s\nx : ↥s\nhxe : ↑x ∈ e.source\n⊢ ↑e ↑x ∈ ↑e.symm ⁻¹' ↑s"
] | s.openPartialHomeomorphSubtypeCoe_target, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 530,
"column": 4
} | {
"line": 530,
"column": 29
} | {
"line": 531,
"column": 4
} | [
{
"pp": "case neg\nB : Type u_1\nF : Type u_2\nZ : Type u_4\ninst✝² : TopologicalSpace B\ninst✝¹ : TopologicalSpace F\nproj : Z → B\ninst✝ : TopologicalSpace Z\ne : Trivialization F proj\nα : Type u_5\nl : Filter α\nf : α → Z\nz : Z\nhz : z ∈ e.source\nhl : ¬∀ᶠ (x : α) in l, f x ∈ e.source\n⊢ Tendsto (proj ∘ f)... | [
"case neg\nB : Type u_1\nF : Type u_2\nZ : Type u_4\ninst✝² : TopologicalSpace B\ninst✝¹ : TopologicalSpace F\nproj : Z → B\ninst✝ : TopologicalSpace Z\ne : Trivialization F proj\nα : Type u_5\nl : Filter α\nf : α → Z\nz : Z\nhz : z ∈ proj ⁻¹' e.baseSet\nhl : ¬∀ᶠ (x : α) in l, f x ∈ proj ⁻¹' e.baseSet\n⊢ Tendsto (p... | rw [e.source_eq] at hl hz | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 789,
"column": 4
} | {
"line": 789,
"column": 37
} | {
"line": 790,
"column": 2
} | [
{
"pp": "case refine_2\nB : Type u_1\nF : Type u_2\nZ : Type u_4\ninst✝² : TopologicalSpace B\ninst✝¹ : TopologicalSpace F\nproj : Z → B\ninst✝ : TopologicalSpace Z\ne₁ e₂ : Trivialization F proj\nb : B\nh₁ : b ∈ e₁.baseSet\nh₂ : b ∈ e₂.baseSet\n⊢ ∀ (x : F), (b, x) ∈ e₁.target",
"ppTerm": "?refine_2",
"... | [] | exact fun x => e₁.mem_target.2 h₁ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 789,
"column": 4
} | {
"line": 789,
"column": 37
} | {
"line": 790,
"column": 2
} | [
{
"pp": "case refine_2\nB : Type u_1\nF : Type u_2\nZ : Type u_4\ninst✝² : TopologicalSpace B\ninst✝¹ : TopologicalSpace F\nproj : Z → B\ninst✝ : TopologicalSpace Z\ne₁ e₂ : Trivialization F proj\nb : B\nh₁ : b ∈ e₁.baseSet\nh₂ : b ∈ e₂.baseSet\n⊢ ∀ (x : F), (b, x) ∈ e₁.target",
"ppTerm": "?refine_2",
"... | [] | exact fun x => e₁.mem_target.2 h₁ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 789,
"column": 4
} | {
"line": 789,
"column": 37
} | {
"line": 790,
"column": 2
} | [
{
"pp": "case refine_2\nB : Type u_1\nF : Type u_2\nZ : Type u_4\ninst✝² : TopologicalSpace B\ninst✝¹ : TopologicalSpace F\nproj : Z → B\ninst✝ : TopologicalSpace Z\ne₁ e₂ : Trivialization F proj\nb : B\nh₁ : b ∈ e₁.baseSet\nh₂ : b ∈ e₂.baseSet\n⊢ ∀ (x : F), (b, x) ∈ e₁.target",
"ppTerm": "?refine_2",
"... | [] | exact fun x => e₁.mem_target.2 h₁ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.IsLocalHomeomorph | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 65
} | {
"line": 139,
"column": 2
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\ng : Y → Z\ns : Set X\nx : X\nhx : x ∈ s\nf : OpenPartialHomeomorph X Y\nhxf : x ∈ f.source\nhgf✝ : IsLocalHomeomorphOn (g ∘ ↑f) s\nhf : IsLocalHomeomorphOn (↑f) s\ngf : OpenPa... | [
"case refine_1\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\ng : Y → Z\ns : Set X\nx : X\nhx : x ∈ s\nf : OpenPartialHomeomorph X Y\nhxf : x ∈ f.source\nhgf✝ : IsLocalHomeomorphOn (g ∘ ↑f) s\nhf : IsLocalHomeomorphOn (↑f) s\ngf : Ope... | refine ⟨f.symm.trans gf, ⟨f.map_source hxf, ?_⟩, fun y hy ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Covering.Quotient | {
"line": 108,
"column": 54
} | {
"line": 109,
"column": 86
} | {
"line": 111,
"column": 0
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace X\nf : E → X\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : MulAction G E\nhf : IsQuotientCoveringMap f G\nx : X\ne e' : ↑(f ⁻¹' {x})\ng : G\n⊢ (hf.fiberEquivGroup e) e' = g ↔ ↑e' = g • ↑e",
"ppTerm": "?m.36",
"assigne... | [] | by
rw [fiberEquivGroup, Equiv.symm_apply_eq, Equiv.ofBijective_apply, Subtype.mk.injEq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Covering.Basic | {
"line": 167,
"column": 70
} | {
"line": 167,
"column": 93
} | {
"line": 168,
"column": 2
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalSpace X\nf : E → X\ns : Set X\nI : Type u_3\ninst✝ : TopologicalSpace I\nhs : IsOpen[inst✝¹] s\nhfs : IsOpen[inst✝²] (f ⁻¹' s)\nx : X\nhx : x ∈ s\nh : IsEvenlyCovered (fun e ↦ f ↑e) x I\ninst : DiscreteTopology I\nU : Set X\nh... | [] | ext; simpa using @hUs _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Covering.Basic | {
"line": 167,
"column": 70
} | {
"line": 167,
"column": 93
} | {
"line": 168,
"column": 2
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalSpace X\nf : E → X\ns : Set X\nI : Type u_3\ninst✝ : TopologicalSpace I\nhs : IsOpen[inst✝¹] s\nhfs : IsOpen[inst✝²] (f ⁻¹' s)\nx : X\nhx : x ∈ s\nh : IsEvenlyCovered (fun e ↦ f ↑e) x I\ninst : DiscreteTopology I\nU : Set X\nh... | [] | ext; simpa using @hUs _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Covering.Basic | {
"line": 180,
"column": 13
} | {
"line": 180,
"column": 46
} | {
"line": 180,
"column": 46
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace X\nf : E → X\nI : Type u_3\ninst✝¹ : TopologicalSpace I\nx : X\nE' : Type u_4\ninst✝ : TopologicalSpace E'\ng : E' ≃ₜ E\nh : IsEvenlyCovered (f ∘ ⇑g) x I\n⊢ IsEvenlyCovered f x I",
"ppTerm": "?m.21",
"assigned": ... | [
"E : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace X\nf : E → X\nI : Type u_3\ninst✝¹ : TopologicalSpace I\nx : X\nE' : Type u_4\ninst✝ : TopologicalSpace E'\ng : E' ≃ₜ E\nh : IsEvenlyCovered (f ∘ ⇑g) x I\n⊢ f = (f ∘ ⇑g) ∘ ⇑g.symm"
] | convert! h.comp_homeomorph g.symm | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Topology.Covering.Basic | {
"line": 273,
"column": 13
} | {
"line": 273,
"column": 46
} | {
"line": 273,
"column": 46
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalSpace X\nf : E → X\ns : Set X\nE' : Type u_3\ninst✝ : TopologicalSpace E'\ng : E' ≃ₜ E\nh : IsCoveringMapOn (f ∘ ⇑g) s\n⊢ IsCoveringMapOn f s",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr"... | [
"E : Type u_1\nX : Type u_2\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalSpace X\nf : E → X\ns : Set X\nE' : Type u_3\ninst✝ : TopologicalSpace E'\ng : E' ≃ₜ E\nh : IsCoveringMapOn (f ∘ ⇑g) s\n⊢ f = (f ∘ ⇑g) ∘ ⇑g.symm"
] | convert! h.comp_homeomorph g.symm | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Topology.Covering.Basic | {
"line": 402,
"column": 13
} | {
"line": 402,
"column": 46
} | {
"line": 402,
"column": 46
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalSpace X\nf : E → X\nE' : Type u_3\ninst✝ : TopologicalSpace E'\ng : E' ≃ₜ E\nh : IsCoveringMap (f ∘ ⇑g)\n⊢ IsCoveringMap f",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsCovering... | [
"E : Type u_1\nX : Type u_2\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalSpace X\nf : E → X\nE' : Type u_3\ninst✝ : TopologicalSpace E'\ng : E' ≃ₜ E\nh : IsCoveringMap (f ∘ ⇑g)\n⊢ f = (f ∘ ⇑g) ∘ ⇑g.symm"
] | convert! h.comp_homeomorph g.symm | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Topology.Covering.Quotient | {
"line": 327,
"column": 6
} | {
"line": 333,
"column": 38
} | {
"line": 333,
"column": 38
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace X\nf : E → X\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : MulAction G E\nh :\n IsCoveringMap f ∧\n Function.Surjective f ∧\n ContinuousConstSMul G E ∧ IsCancelSMul G E ∧ ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.o... | [] | rintro ⟨_, ⟨_, ⟨x, hx, rfl⟩, rfl⟩, y, hy, eq⟩
have := h.2.2.2.1
apply IsCancelSMul.right_cancel _ _ x.1
simp_rw [← eq, one_smul]
refine congr($(H.injective <| Prod.ext (Subtype.ext ?_) <| hy.trans hx.symm))
simp_rw [hH]
exact h.2.2.2.2.mpr ⟨_, eq.symm⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Covering.Quotient | {
"line": 327,
"column": 6
} | {
"line": 333,
"column": 38
} | {
"line": 333,
"column": 38
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace X\nf : E → X\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : MulAction G E\nh :\n IsCoveringMap f ∧\n Function.Surjective f ∧\n ContinuousConstSMul G E ∧ IsCancelSMul G E ∧ ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.o... | [] | rintro ⟨_, ⟨_, ⟨x, hx, rfl⟩, rfl⟩, y, hy, eq⟩
have := h.2.2.2.1
apply IsCancelSMul.right_cancel _ _ x.1
simp_rw [← eq, one_smul]
refine congr($(H.injective <| Prod.ext (Subtype.ext ?_) <| hy.trans hx.symm))
simp_rw [hH]
exact h.2.2.2.2.mpr ⟨_, eq.symm⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Complex.Circle | {
"line": 107,
"column": 6
} | {
"line": 107,
"column": 36
} | {
"line": 107,
"column": 37
} | [
{
"pp": "s : Set ℝ\nhs : ∀ x ∈ s, ∀ y ∈ s, x - y ∈ Ioo (-2 * π) (2 * π)\nt₁ : ℝ\nht₁ : t₁ ∈ s\nt₂ : ℝ\nht₂ : t₂ ∈ s\nh1 : -(2 * π) < t₁ - t₂\nh2 : t₁ - t₂ < 2 * π\nheq : cexp (↑t₁ * I) = cexp (↑t₂ * I)\n⊢ (cexp (↑(t₁ - t₂) * I)).re = 1",
"ppTerm": "?m.88",
"assigned": true,
"usedConstants": [
... | [
"s : Set ℝ\nhs : ∀ x ∈ s, ∀ y ∈ s, x - y ∈ Ioo (-2 * π) (2 * π)\nt₁ : ℝ\nht₁ : t₁ ∈ s\nt₂ : ℝ\nht₂ : t₂ ∈ s\nh1 : -(2 * π) < t₁ - t₂\nh2 : t₁ - t₂ < 2 * π\nheq : cexp (↑t₁ * I - ↑t₂ * I) = 1\n⊢ (cexp (↑(t₁ - t₂) * I)).re = 1"
] | exp_eq_exp_iff_exp_sub_eq_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.Unitary.Span | {
"line": 66,
"column": 72
} | {
"line": 66,
"column": 79
} | {
"line": 66,
"column": 79
} | [
{
"pp": "A : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : ↥(selfAdjoint A)\nha_norm : ‖a‖ ≤ 1\n⊢ IsSelfAdjoint (CFC.sqrt (1 - ↑a ^ 2))",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"... | [] | cfc_tac | _aux_Mathlib_Tactic_ContinuousFunctionalCalculus___macroRules_cfcTac_1 | cfcTac |
Mathlib.Analysis.CStarAlgebra.Unitary.Span | {
"line": 66,
"column": 72
} | {
"line": 66,
"column": 79
} | {
"line": 66,
"column": 79
} | [
{
"pp": "A : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : ↥(selfAdjoint A)\nha_norm : ‖a‖ ≤ 1\n⊢ IsSelfAdjoint (CFC.sqrt (1 - ↑a ^ 2))",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"... | [] | cfc_tac | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.Unitary.Span | {
"line": 66,
"column": 72
} | {
"line": 66,
"column": 79
} | {
"line": 66,
"column": 79
} | [
{
"pp": "A : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : ↥(selfAdjoint A)\nha_norm : ‖a‖ ≤ 1\n⊢ IsSelfAdjoint (CFC.sqrt (1 - ↑a ^ 2))",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"... | [] | cfc_tac | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 181,
"column": 2
} | {
"line": 182,
"column": 23
} | {
"line": 184,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : CompleteSpace E\nU : Submodule 𝕜 E\ninst✝ : CompleteSpace ↥U\n⊢ adjoint U.subtypeL = U.orthogonalProjectionOnto",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": ... | [] | symm
simp [eq_adjoint_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 181,
"column": 2
} | {
"line": 182,
"column": 23
} | {
"line": 184,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : CompleteSpace E\nU : Submodule 𝕜 E\ninst✝ : CompleteSpace ↥U\n⊢ adjoint U.subtypeL = U.orthogonalProjectionOnto",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": ... | [] | symm
simp [eq_adjoint_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 267,
"column": 49
} | {
"line": 268,
"column": 50
} | {
"line": 269,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝ : CStarAlgebra A\nu : ↥(unitary A)\nhu : u ∈ ball 1 2\n⊢ ‖↑u - 1‖ < 2",
"ppTerm": "?m.270",
"assigned": true,
"usedConstants": [
"Norm.norm",
"CStarAlgebra.toNonUnitalCStarAlgebra",
"MulOne.toOne",
"Real",
"NonUnitalCStarAlgebra.toNonUnitalN... | [] | by
simpa [Subtype.dist_eq, dist_eq_norm] using hu | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 457,
"column": 35
} | {
"line": 457,
"column": 85
} | {
"line": 458,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nT : E →L[𝕜] E\ninst✝ : CompleteSpace E\nhT : IsIdempotentElem T\nh : IsStarNormal T\n⊢ T - T * T = 0 ↔ IsIdempotentElem T",
"ppTerm": "?m.193",
"assigned": true,
"usedConstants"... | [] | simp only [sub_eq_zero, IsIdempotentElem, eq_comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 457,
"column": 35
} | {
"line": 457,
"column": 85
} | {
"line": 458,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nT : E →L[𝕜] E\ninst✝ : CompleteSpace E\nhT : IsIdempotentElem T\nh : IsStarNormal T\n⊢ T - T * T = 0 ↔ IsIdempotentElem T",
"ppTerm": "?m.193",
"assigned": true,
"usedConstants"... | [] | simp only [sub_eq_zero, IsIdempotentElem, eq_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 457,
"column": 35
} | {
"line": 457,
"column": 85
} | {
"line": 458,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nT : E →L[𝕜] E\ninst✝ : CompleteSpace E\nhT : IsIdempotentElem T\nh : IsStarNormal T\n⊢ T - T * T = 0 ↔ IsIdempotentElem T",
"ppTerm": "?m.193",
"assigned": true,
"usedConstants"... | [] | simp only [sub_eq_zero, IsIdempotentElem, eq_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 388,
"column": 2
} | {
"line": 388,
"column": 83
} | {
"line": 389,
"column": 2
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ns : Set P\ninst✝ : FiniteDimensional k ↥(vectorSpan k s)\np : P\nthis : FiniteDimensional k ↥(affineSpan k s).direction\n⊢ FiniteDimensional k ↥(vecto... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ns : Set P\ninst✝ : FiniteDimensional k ↥(vectorSpan k s)\np : P\nthis : FiniteDimensional k ↥(affineSpan k s).direction\n⊢ FiniteDimensional k ↥(vectorSpan k (ins... | rw [← direction_affineSpan, ← affineSpan_insert_affineSpan, direction_affineSpan] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 469,
"column": 4
} | {
"line": 469,
"column": 14
} | {
"line": 470,
"column": 4
} | [
{
"pp": "case h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\nv : V\nhp₀v : ∀ p ∈ s, ∃ r, p = r • v +ᵥ p₀\n⊢ ∀ v_1 ∈ vectorSpan k s, ∃ r, r • v = v_1",
"ppTerm": "?h",
"assigned": ... | [
"case h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₀ : P\nh : p₀ ∈ s\nv : V\nhp₀v : ∀ p ∈ s, ∃ r, p = r • v +ᵥ p₀\nw : V\nhw : w ∈ vectorSpan k s\n⊢ ∃ r, r • v = w"
] | intro w hw | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.MeasureTheory.Group.Integral | {
"line": 165,
"column": 76
} | {
"line": 167,
"column": 35
} | {
"line": 169,
"column": 0
} | [
{
"pp": "G : Type u_4\nE : Type u_5\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul G\ninst✝² : MeasurableInv G\nf : G → E\nμ : Measure G\ninst✝¹ : μ.IsInvInvariant\ninst✝ : μ.IsMulLeftInvariant\nx' : G\n⊢ ∫ (x : G), f (x' / x) ∂μ = ... | [] | by
simp_rw [div_eq_mul_inv, integral_inv_eq_self (fun x => f (x' * x)) μ,
integral_mul_left_eq_self f x'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.UniformLimitsDeriv | {
"line": 131,
"column": 8
} | {
"line": 131,
"column": 43
} | {
"line": 131,
"column": 43
} | [
{
"pp": "case left\nι : Type u_1\nl : Filter ι\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : IsRCLikeNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : ι → E → G\nf' : ι → E → E →L[𝕜]... | [
"case left\nι : Type u_1\nl : Filter ι\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : IsRCLikeNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : ι → E → G\nf' : ι → E → E →L[𝕜] G\nx : E\nh... | Metric.tendstoUniformlyOnFilter_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.Ball.Homeomorph | {
"line": 130,
"column": 74
} | {
"line": 131,
"column": 36
} | {
"line": 133,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nP : Type u_2\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor E P\nc : P\nr : ℝ\n⊢ (univBall c r).source = univ",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"OpenPartialHomeomorph.univBall",
... | [] | by
unfold univBall; split_ifs <;> rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Module.Ball.Homeomorph | {
"line": 138,
"column": 8
} | {
"line": 138,
"column": 28
} | {
"line": 138,
"column": 28
} | [
{
"pp": "case pos\nE : Type u_1\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nP : Type u_2\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor E P\nc : P\nr : ℝ\nhr : 0 < r\n⊢ ball c r ⊆ (univBall c r).target",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"OpenPar... | [
"case pos\nE : Type u_1\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nP : Type u_2\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor E P\nc : P\nr : ℝ\nhr : 0 < r\n⊢ ball c r ⊆ ball c r"
] | univBall_target c hr | Lean.Elab.Tactic.evalRewriteSeq | null |
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