module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 217,
"column": 21
} | {
"line": 217,
"column": 32
} | [
{
"pp": "case add\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 s\na :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 218,
"column": 21
} | {
"line": 218,
"column": 32
} | [
{
"pp": "case mul\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 s\na :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic | {
"line": 135,
"column": 2
} | {
"line": 137,
"column": 38
} | [
{
"pp": "A : Type u_1\ninst✝⁹ : PartialOrder A\ninst✝⁸ : NonUnitalRing A\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : StarRing A\ninst✝⁵ : Module ℝ A\ninst✝⁴ : SMulCommClass ℝ A A\ninst✝³ : IsScalarTower ℝ A A\ninst✝² : StarOrderedRing A\ninst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : Nonne... | simp only [nnrpow_def, NNReal.nnrpow_def, NNReal.coe_ofNat, NNReal.rpow_ofNat, pow_two]
change cfcₙ (fun z : ℝ≥0 => id z * id z) a = a * a
rw [cfcₙ_mul id id a, cfcₙ_id ℝ≥0 a] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic | {
"line": 135,
"column": 2
} | {
"line": 137,
"column": 38
} | [
{
"pp": "A : Type u_1\ninst✝⁹ : PartialOrder A\ninst✝⁸ : NonUnitalRing A\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : StarRing A\ninst✝⁵ : Module ℝ A\ninst✝⁴ : SMulCommClass ℝ A A\ninst✝³ : IsScalarTower ℝ A A\ninst✝² : StarOrderedRing A\ninst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : Nonne... | simp only [nnrpow_def, NNReal.nnrpow_def, NNReal.coe_ofNat, NNReal.rpow_ofNat, pow_two]
change cfcₙ (fun z : ℝ≥0 => id z * id z) a = a * a
rw [cfcₙ_mul id id a, cfcₙ_id ℝ≥0 a] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 343,
"column": 8
} | {
"line": 343,
"column": 31
} | [
{
"pp": "X : Type u_1\n𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : NormedAlgebra 𝕜 A\ninst✝³ : IsometricContinuousFunctionalCalculus 𝕜 A p\ninst✝² : ContinuousStar A\ninst✝¹ : CompleteSpace A\ninst✝ : TopologicalSpace X\ns : Set 𝕜\nf : �... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 358,
"column": 53
} | {
"line": 358,
"column": 64
} | [
{
"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 : X → Set 𝕜\nf : 𝕜 → 𝕜\na : X → A\nha_... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 511,
"column": 8
} | {
"line": 511,
"column": 31
} | [
{
"pp": "X : Type u_1\nA : Type u_2\ninst✝¹¹ : NormedRing A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : NormedAlgebra ℝ A\ninst✝⁸ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁷ : ContinuousStar A\ninst✝⁶ : PartialOrder A\ninst✝⁵ : StarOrderedRing A\ninst✝⁴ : NonnegSpectrumClass ℝ A\ninst✝³ : T2Space A\ni... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic | {
"line": 486,
"column": 2
} | {
"line": 486,
"column": 13
} | [
{
"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 :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic | {
"line": 506,
"column": 2
} | {
"line": 506,
"column": 32
} | [
{
"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 : Aˣ\nha : 0 ≤ ↑a\n⊢ ↑a ^ (-1) * ↑a = 1",
"usedCons... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 526,
"column": 60
} | {
"line": 526,
"column": 71
} | [
{
"pp": "X : Type u_1\nA : Type u_2\ninst✝¹⁰ : NormedRing A\ninst✝⁹ : StarRing A\ninst✝⁸ : NormedAlgebra ℝ A\ninst✝⁷ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁶ : ContinuousStar A\ninst✝⁵ : PartialOrder A\ninst✝⁴ : StarOrderedRing A\ninst✝³ : NonnegSpectrumClass ℝ A\ninst✝² : T2Space A\nin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 602,
"column": 4
} | {
"line": 602,
"column": 50
} | [
{
"pp": "case neg\nX : Type u_1\nR : Type u_2\nA : Type u_3\np : A → Prop\ninst✝¹² : CommSemiring R\ninst✝¹¹ : StarRing R\ninst✝¹⁰ : MetricSpace R\ninst✝⁹ : Nontrivial R\ninst✝⁸ : IsTopologicalSemiring R\ninst✝⁷ : ContinuousStar R\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : TopologicalSpace A\ninst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 677,
"column": 4
} | {
"line": 677,
"column": 50
} | [
{
"pp": "case neg\nR : Type u_2\nA : Type u_3\np : A → Prop\ninst✝¹² : CommSemiring R\ninst✝¹¹ : StarRing R\ninst✝¹⁰ : MetricSpace R\ninst✝⁹ : Nontrivial R\ninst✝⁸ : IsTopologicalSemiring R\ninst✝⁷ : ContinuousStar R\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : MetricSpace A\ninst✝³ : Module R A\nin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 690,
"column": 2
} | {
"line": 690,
"column": 13
} | [
{
"pp": "R : Type u_2\nA : Type u_3\np : A → Prop\ninst✝¹² : CommSemiring R\ninst✝¹¹ : StarRing R\ninst✝¹⁰ : MetricSpace R\ninst✝⁹ : Nontrivial R\ninst✝⁸ : IsTopologicalSemiring R\ninst✝⁷ : ContinuousStar R\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : MetricSpace A\ninst✝³ : Module R A\ninst✝² : SMu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 713,
"column": 12
} | {
"line": 713,
"column": 34
} | [
{
"pp": "case zero\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 : NonUnitalIsometricContinuousFunctiona... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 716,
"column": 21
} | {
"line": 716,
"column": 47
} | [
{
"pp": "case add\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 : NonUnitalIsometricContinuousFunctional... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 717,
"column": 21
} | {
"line": 717,
"column": 47
} | [
{
"pp": "case mul\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 : NonUnitalIsometricContinuousFunctional... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Module.Defs | {
"line": 193,
"column": 15
} | {
"line": 193,
"column": 78
} | [
{
"pp": "A : Type u_1\nE : Type u_2\ninst✝⁶ : NonUnitalCStarAlgebra A\ninst✝⁵ : PartialOrder A\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module ℂ E\ninst✝² : SMul A E\ninst✝¹ : Norm E\ninst✝ : CStarModule A E\nx : E\nh : ‖x‖ = 0\n⊢ x = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 718,
"column": 19
} | {
"line": 718,
"column": 46
} | [
{
"pp": "case smul\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 : NonUnitalIsometricContinuousFunctiona... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order | {
"line": 117,
"column": 2
} | {
"line": 117,
"column": 13
} | [
{
"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\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ ↑(f (a • x + b • y)) ≤ ↑... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 13
} | [
{
"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\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ ↑(a • f x + b • f y) ≤ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Module.Defs | {
"line": 269,
"column": 22
} | {
"line": 269,
"column": 90
} | [
{
"pp": "A : Type u_1\nE : Type u_2\ninst✝⁷ : NonUnitalCStarAlgebra A\ninst✝⁶ : PartialOrder A\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module ℂ E\ninst✝³ : SMul A E\ninst✝² : Norm E\ninst✝¹ : CStarModule A E\ninst✝ : StarOrderedRing A\nc : ℂ\nx : E\n⊢ ‖c •> x‖ = ‖c‖ * ‖x‖",
"usedConstants": [
"Norm.norm",
... | simp [norm_eq_sqrt_norm_inner_self (A := A), norm_smul, ← mul_assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order | {
"line": 230,
"column": 2
} | {
"line": 230,
"column": 13
} | [
{
"pp": "A : Type u_1\ninst✝³ : CStarAlgebra A\ninst✝² : PartialOrder A\ninst✝¹ : StarOrderedRing A\ninst✝ : Nontrivial A\na : A\nha : 0 ≤ a\n⊢ ‖a‖ ∈ spectrum ℝ a",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Module.Defs | {
"line": 269,
"column": 22
} | {
"line": 269,
"column": 90
} | [
{
"pp": "A : Type u_1\nE : Type u_2\ninst✝⁷ : NonUnitalCStarAlgebra A\ninst✝⁶ : PartialOrder A\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module ℂ E\ninst✝³ : SMul A E\ninst✝² : Norm E\ninst✝¹ : CStarModule A E\ninst✝ : StarOrderedRing A\nc : ℂ\nx : E\n⊢ ‖c •> x‖ = ‖c‖ * ‖x‖",
"usedConstants": [
"Norm.norm",
... | simp [norm_eq_sqrt_norm_inner_self (A := A), norm_smul, ← mul_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.Module.Defs | {
"line": 269,
"column": 22
} | {
"line": 269,
"column": 90
} | [
{
"pp": "A : Type u_1\nE : Type u_2\ninst✝⁷ : NonUnitalCStarAlgebra A\ninst✝⁶ : PartialOrder A\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module ℂ E\ninst✝³ : SMul A E\ninst✝² : Norm E\ninst✝¹ : CStarModule A E\ninst✝ : StarOrderedRing A\nc : ℂ\nx : E\n⊢ ‖c •> x‖ = ‖c‖ * ‖x‖",
"usedConstants": [
"Norm.norm",
... | simp [norm_eq_sqrt_norm_inner_self (A := A), norm_smul, ← mul_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.Module.Defs | {
"line": 284,
"column": 4
} | {
"line": 284,
"column": 53
} | [
{
"pp": "case refine_1\nA : Type u_1\nE : Type u_2\ninst✝⁷ : NonUnitalCStarAlgebra A\ninst✝⁶ : PartialOrder A\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module ℂ E\ninst✝³ : SMul A E\ninst✝² : Norm E\ninst✝¹ : CStarModule A E\ninst✝ : StarOrderedRing A\nv : E\ninstNACG : NormedAddCommGroup E := NormedAddCommGroup.ofCor... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order | {
"line": 247,
"column": 2
} | {
"line": 247,
"column": 13
} | [
{
"pp": "A : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : A\nha : 0 ≤ a\n⊢ ‖a‖ ≤ 1 ↔ a ≤ 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order | {
"line": 256,
"column": 2
} | {
"line": 256,
"column": 13
} | [
{
"pp": "A : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : A\nn : ℕ\nha : 0 ≤ a\n⊢ ‖a‖ ≤ ↑n ↔ a ≤ ↑n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order | {
"line": 260,
"column": 2
} | {
"line": 260,
"column": 13
} | [
{
"pp": "A : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : A\nn : ℕ\nha : 0 ≤ a\n⊢ ‖a‖₊ ≤ ↑n ↔ a ≤ ↑n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order | {
"line": 278,
"column": 2
} | {
"line": 278,
"column": 28
} | [
{
"pp": "A : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nx : A\n⊢ x ∈ Icc 0 1 ↔ 0 ≤ x ∧ ‖x‖ ≤ 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Module.Constructions | {
"line": 88,
"column": 4
} | {
"line": 88,
"column": 20
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nx : A\n⊢ ‖x‖ ^ 2 = √‖x * star x‖ ^ 2",
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NonUnitalNormedRing.toNorm",
"Real.instLE",
"Real",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Module.Constructions | {
"line": 125,
"column": 32
} | {
"line": 125,
"column": 43
} | [
{
"pp": "A : Type u_1\ninst✝¹⁰ : NonUnitalCStarAlgebra A\ninst✝⁹ : PartialOrder A\nE : Type u_2\nF : Type u_3\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : Module ℂ E\ninst✝⁶ : SMul A E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : Module ℂ F\ninst✝³ : SMul A F\ninst✝² : CStarModule A E\ninst✝¹ : CStarModule A F\ninst✝ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order | {
"line": 369,
"column": 2
} | {
"line": 369,
"column": 13
} | [
{
"pp": "A : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na b : Aˣ\nha : 0 ≤ ↑a\nhb : 0 ≤ ↑b\n⊢ ↑a⁻¹ ≤ ↑b ↔ ↑b⁻¹ ≤ ↑a",
"usedConstants": [
"Units.val_le_val._simp_2",
"Units.val",
"Eq.mpr",
"NormedRing.toRing",
"congrArg",
"Partia... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order | {
"line": 373,
"column": 2
} | {
"line": 373,
"column": 13
} | [
{
"pp": "A : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na b : Aˣ\nha : 0 ≤ ↑a\nhb : 0 ≤ ↑b\n⊢ ↑a ≤ ↑b⁻¹ ↔ ↑b ≤ ↑a⁻¹",
"usedConstants": [
"Units.val_le_val._simp_2",
"Units.val",
"Eq.mpr",
"NormedRing.toRing",
"congrArg",
"Partia... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order | {
"line": 377,
"column": 2
} | {
"line": 377,
"column": 13
} | [
{
"pp": "A : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : Aˣ\nha : 0 ≤ ↑a\n⊢ 1 ≤ ↑a⁻¹ ↔ a ≤ 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order | {
"line": 381,
"column": 2
} | {
"line": 381,
"column": 13
} | [
{
"pp": "A : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : Aˣ\nha : 0 ≤ ↑a\n⊢ ↑a⁻¹ ≤ 1 ↔ 1 ≤ a",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order | {
"line": 387,
"column": 2
} | {
"line": 387,
"column": 13
} | [
{
"pp": "A : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : Aˣ\nha : 1 ≤ ↑a⁻¹\n⊢ ↑a ≤ 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order | {
"line": 453,
"column": 4
} | {
"line": 453,
"column": 54
} | [
{
"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\nthis : ∀ (a b : Unitization ℂ A), 0 ≤ a → a ≤ b → ‖a‖ ≤ ‖b‖\nhb : 0 ≤ b\n⊢ ‖a‖ ≤ ‖b‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NonUnitalNormedRi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Module.Constructions | {
"line": 183,
"column": 4
} | {
"line": 183,
"column": 15
} | [
{
"pp": "A : Type u_1\ninst✝¹⁰ : NonUnitalCStarAlgebra A\ninst✝⁹ : PartialOrder A\nE : Type u_2\nF : Type u_3\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : Module ℂ E\ninst✝⁶ : SMul A E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : Module ℂ F\ninst✝³ : SMul A F\ninst✝² : CStarModule A E\ninst✝¹ : CStarModule A F\ninst✝ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 853,
"column": 8
} | {
"line": 853,
"column": 31
} | [
{
"pp": "X : Type u_1\n𝕜 : Type u_2\nA : Type u_3\np : A → Prop\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 : ContinuousStar A\ninst✝² : NonUnitalIsometricContinuousFunctionalCalculus �... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order | {
"line": 468,
"column": 4
} | {
"line": 468,
"column": 38
} | [
{
"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⊢ ↑(star a * b * a) ≤ ↑(‖b‖ • (star a * a))",
"usedConstants": [
"Unitiza... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order | {
"line": 477,
"column": 2
} | {
"line": 477,
"column": 13
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na b : A\nhb : IsSelfAdjoint b\n⊢ a * b * star a ≤ ‖b‖ • (a * star a)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Module.Constructions | {
"line": 221,
"column": 77
} | {
"line": 222,
"column": 16
} | [
{
"pp": "A : Type u_1\ninst✝⁶ : NonUnitalCStarAlgebra A\ninst✝⁵ : PartialOrder A\nι : Type u_2\nE : ι → Type u_3\ninst✝⁴ : Fintype ι\ninst✝³ : (i : ι) → NormedAddCommGroup (E i)\ninst✝² : (i : ι) → Module ℂ (E i)\ninst✝¹ : (i : ι) → SMul A (E i)\ninst✝ : (i : ι) → CStarModule A (E i)\nx : C⋆ᵐᵒᵈ(A, (i : ι) → E i... | by
simp [pi_norm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.ApproximateUnit | {
"line": 115,
"column": 6
} | {
"line": 115,
"column": 57
} | [
{
"pp": "case refine_1\nA : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nf : ℝ≥0 → ℝ≥0 := fun x ↦ 1 - (1 + x)⁻¹\ng : ℝ≥0 → ℝ≥0 := fun x ↦ x * (1 - x)⁻¹\nthis : ∀ (a b : A), 0 ≤ a → 0 ≤ b → ‖a‖ < 1 → ‖b‖ < 1 → a ≤ cfcₙ f (cfcₙ g a + cfcₙ g b)\na : A\nha₁ : 0 ≤ a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 869,
"column": 54
} | {
"line": 869,
"column": 65
} | [
{
"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 �... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order | {
"line": 526,
"column": 2
} | {
"line": 526,
"column": 13
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nf : ℝ → ℝ\ns : Set A\nhf : ConcaveOn ℝ (inr '' s) (cfc f)\nthis : ConcaveOn ℝ s (- -cfcₙ f)\n⊢ ConcaveOn ℝ s (cfcₙ f)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order | {
"line": 559,
"column": 14
} | {
"line": 559,
"column": 64
} | [
{
"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\nthis : a * e = a\n⊢ e * a = a",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order | {
"line": 564,
"column": 4
} | {
"line": 565,
"column": 11
} | [
{
"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\nha : 0 ≤ a\nhae : a ≤ e\nthis : sqrt a * (1 - e) = 0\n⊢ a * e = a",
"usedConstants"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Module.Constructions | {
"line": 282,
"column": 2
} | {
"line": 282,
"column": 13
} | [
{
"pp": "A : Type u_1\ninst✝⁷ : NonUnitalCStarAlgebra A\ninst✝⁶ : PartialOrder A\nι : Type u_2\nE : ι → Type u_3\ninst✝⁵ : Fintype ι\ninst✝⁴ : (i : ι) → NormedAddCommGroup (E i)\ninst✝³ : (i : ι) → Module ℂ (E i)\ninst✝² : (i : ι) → SMul A (E i)\ninst✝¹ : (i : ι) → CStarModule A (E i)\ninst✝ : StarOrderedRing A... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ApproximateUnit | {
"line": 177,
"column": 59
} | {
"line": 177,
"column": 70
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nl : Filter A\nh : ∀ (m : A), 0 ≤ m → ‖m‖ < 1 → Tendsto (fun x ↦ x * m) l (𝓝 m)\nn : ℕ\nc : Fin n → ℂ\nx : Fin n → ↑({x | 0 ≤ x} ∩ ball 0 1)\ni : Fin n\nx✝ : i ∈ Finset.univ\n⊢ ‖↑(x i)‖ < 1",
"usedCo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Module.Constructions | {
"line": 311,
"column": 4
} | {
"line": 311,
"column": 15
} | [
{
"pp": "A : Type u_1\ninst✝⁷ : NonUnitalCStarAlgebra A\ninst✝⁶ : PartialOrder A\nι : Type u_2\nE : ι → Type u_3\ninst✝⁵ : Fintype ι\ninst✝⁴ : (i : ι) → NormedAddCommGroup (E i)\ninst✝³ : (i : ι) → Module ℂ (E i)\ninst✝² : (i : ι) → SMul A (E i)\ninst✝¹ : (i : ι) → CStarModule A (E i)\ninst✝ : StarOrderedRing A... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Module.Constructions | {
"line": 359,
"column": 4
} | {
"line": 359,
"column": 46
} | [
{
"pp": "A : Type u_1\ninst✝³ : NonUnitalCStarAlgebra A\ninst✝² : PartialOrder A\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℂ E\nx : E\n⊢ ‖x‖ = √‖⟪x, x⟫_ℂ‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"AddMonoid.toAddSemigroup",
"Inner.inner"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ApproximateUnit | {
"line": 198,
"column": 14
} | {
"line": 198,
"column": 25
} | [
{
"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\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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ApproximateUnit | {
"line": 229,
"column": 4
} | {
"line": 229,
"column": 15
} | [
{
"pp": "A : Type u_2\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nx y z : A\nhx₀ : 0 ≤ x\nhy : y ∈ Set.Icc x 1\nc : ℝ≥0\nh : ‖star z * (1 - x) * z‖₊ ≤ c ^ 2\nhy₀ : y ∈ Set.Icc 0 1\nhy' : 1 - y ∈ Set.Icc 0 1\n⊢ (1 - y) ^ 2 ≤ 1 - y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 1028,
"column": 8
} | {
"line": 1028,
"column": 31
} | [
{
"pp": "X : Type u_1\nA : Type u_2\ninst✝¹³ : NonUnitalNormedRing A\ninst✝¹² : StarRing A\ninst✝¹¹ : NormedSpace ℝ A\ninst✝¹⁰ : IsScalarTower ℝ A A\ninst✝⁹ : SMulCommClass ℝ A A\ninst✝⁸ : ContinuousStar A\ninst✝⁷ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁶ : PartialOrder A\ninst✝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 1044,
"column": 61
} | {
"line": 1044,
"column": 72
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ApproximateUnit | {
"line": 322,
"column": 29
} | {
"line": 322,
"column": 40
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\ni✝ : A\nhx : 0 ≤ i✝ ∧ ‖i✝‖ < 1\n⊢ i✝ ∈ closedBall 0 1",
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"Real.instLE",
"Real",
"NonUnitalCSt... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.PositiveLinearMap | {
"line": 105,
"column": 6
} | {
"line": 105,
"column": 29
} | [
{
"pp": "case refine_2\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\nhcontra : ∀ (C : ℝ≥0), ∃ a, 0 ≤ a ∧ ↑C * ‖a‖ < ‖f a‖\nn : ℕ\nx... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.PositiveLinearMap | {
"line": 124,
"column": 4
} | {
"line": 124,
"column": 27
} | [
{
"pp": "A₁ : 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\nhcontra : ∀ (C : ℝ≥0), ∃ a, 0 ≤ a ∧ ↑C * ‖a‖ < ‖f a‖\nx : ℕ → A₁\nhx_nonneg :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.Normed | {
"line": 548,
"column": 2
} | {
"line": 549,
"column": 86
} | [
{
"pp": "m : Type u_3\nn : Type u_4\nα : Type u_5\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : SeminormedAddCommGroup α\nA : Matrix m n α\n⊢ ‖A‖₊ = (∑ i, ∑ j, ‖A i j‖₊ ^ 2) ^ (1 / 2)",
"usedConstants": [
"WithLp",
"Eq.mpr",
"Real",
"instHDiv",
"fact_one_le_two_ennreal",
... | change ‖toLp 2 fun i => toLp 2 fun j => A i j‖₊ = _
simp_rw [PiLp.nnnorm_eq_of_L2, NNReal.sq_sqrt, NNReal.sqrt_eq_rpow, NNReal.rpow_two] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Matrix.Normed | {
"line": 548,
"column": 2
} | {
"line": 549,
"column": 86
} | [
{
"pp": "m : Type u_3\nn : Type u_4\nα : Type u_5\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : SeminormedAddCommGroup α\nA : Matrix m n α\n⊢ ‖A‖₊ = (∑ i, ∑ j, ‖A i j‖₊ ^ 2) ^ (1 / 2)",
"usedConstants": [
"WithLp",
"Eq.mpr",
"Real",
"instHDiv",
"fact_one_le_two_ennreal",
... | change ‖toLp 2 fun i => toLp 2 fun j => A i j‖₊ = _
simp_rw [PiLp.nnnorm_eq_of_L2, NNReal.sq_sqrt, NNReal.sqrt_eq_rpow, NNReal.rpow_two] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Matrix.Normed | {
"line": 642,
"column": 2
} | {
"line": 644,
"column": 26
} | [
{
"pp": "case h₁.h.h\nl : Type u_2\nm : Type u_3\nn : Type u_4\nα : Type u_5\ninst✝³ : Fintype l\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : RCLike α\nA : Matrix l m α\nB : Matrix m n α\ni : l\na✝¹ : i ∈ Finset.univ\nj : n\na✝ : j ∈ Finset.univ\n⊢ ‖∑ j_1, A i j_1 * B j_1 j‖₊ ≤ (∑ j, ‖A i j‖₊ ^ 2) ^ (1 / 2)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.SpecificCodomains.ContinuousMap | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 51
} | [
{
"pp": "case h\nX : Type u_1\nY : Type u_2\ninst✝³ : MeasurableSpace X\nμ : Measure X\ninst✝² : TopologicalSpace Y\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : CompactSpace Y\nf : X → Y → E\ng : C(Y, E)\nf_ae_cont : ∀ᵐ (x : X) ∂μ, Continuous (f x)\nbound : X → ℝ\nbound_int : HasFiniteIntegral bound μ\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.SpecificCodomains.ContinuousMapZero | {
"line": 58,
"column": 2
} | {
"line": 58,
"column": 58
} | [
{
"pp": "case h\nX : Type u_1\nY : Type u_2\ninst✝⁴ : MeasurableSpace X\nμ : Measure X\ninst✝³ : TopologicalSpace Y\nE : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompactSpace Y\ninst✝ : Zero Y\nf : X → Y → E\ng : C(Y, E)₀\nf_ae_cont : ∀ᵐ (x : X) ∂μ, Continuous (f x)\nf_ae_zero : ∀ᵐ (x : X) ∂μ, f x 0 = ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.CompletelyPositiveMap | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 13
} | [
{
"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 : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.CompletelyPositiveMap | {
"line": 139,
"column": 2
} | {
"line": 139,
"column": 29
} | [
{
"pp": "A₁ : Type u_1\nA₂ : Type u_2\ninst✝⁶ : NonUnitalCStarAlgebra A₁\ninst✝⁵ : NonUnitalCStarAlgebra A₂\ninst✝⁴ : PartialOrder A₁\ninst✝³ : PartialOrder A₂\ninst✝² : StarOrderedRing A₁\ninst✝¹ : StarOrderedRing A₂\nn : Type u_3\ninst✝ : Fintype n\nφ : A₁ →CP A₂\nM : CStarMatrix n n A₁\nhM : 0 ≤ M\nk : ℕ := ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Integral | {
"line": 122,
"column": 4
} | {
"line": 124,
"column": 74
} | [
{
"pp": "case refine_2\nX : Type u_1\n𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁹ : RCLike 𝕜\ninst✝⁸ : MeasurableSpace X\nμ : Measure X\ninst✝⁷ : NormedRing A\ninst✝⁶ : StarRing A\ninst✝⁵ : NormedAlgebra 𝕜 A\ninst✝⁴ : ContinuousFunctionalCalculus 𝕜 A p\ninst✝³ : CompleteSpace A\ninst✝² : TopologicalSpa... | refine hasFiniteIntegral_mkD_restrict_of_bound f _ ?_ bound bound_int bound_ge
exact ae_restrict_of_forall_mem hs fun x hx ↦
hf.comp (Continuous.prodMk_right x).continuousOn fun _ hz ↦ ⟨hx, hz⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Integral | {
"line": 122,
"column": 4
} | {
"line": 124,
"column": 74
} | [
{
"pp": "case refine_2\nX : Type u_1\n𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁹ : RCLike 𝕜\ninst✝⁸ : MeasurableSpace X\nμ : Measure X\ninst✝⁷ : NormedRing A\ninst✝⁶ : StarRing A\ninst✝⁵ : NormedAlgebra 𝕜 A\ninst✝⁴ : ContinuousFunctionalCalculus 𝕜 A p\ninst✝³ : CompleteSpace A\ninst✝² : TopologicalSpa... | refine hasFiniteIntegral_mkD_restrict_of_bound f _ ?_ bound bound_int bound_ge
exact ae_restrict_of_forall_mem hs fun x hx ↦
hf.comp (Continuous.prodMk_right x).continuousOn fun _ hz ↦ ⟨hx, hz⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Group.NullSubmodule | {
"line": 42,
"column": 4
} | {
"line": 42,
"column": 41
} | [
{
"pp": "M : Type u_1\ninst✝ : SeminormedCommGroup M\nx y : M\nhx : ‖x‖ = 0\nhy : ‖y‖ = 0\n⊢ ‖x * y‖ ≤ 0",
"usedConstants": [
"Norm.norm",
"Real.instLE",
"Real",
"HMul.hMul",
"LE.le.trans_eq",
"Real.instZero",
"Monoid.toMulOneClass",
"SeminormedGroup.toGroup",... | refine (norm_mul_le' x y).trans_eq ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Normed.Group.NullSubmodule | {
"line": 45,
"column": 36
} | {
"line": 45,
"column": 82
} | [
{
"pp": "M : Type u_1\ninst✝ : SeminormedCommGroup M\nx : M\nhx : ‖x‖ = 0\n⊢ x⁻¹ ∈ {x | ‖x‖ = 0}",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"DivInvOneMonoid.toInvOneClass",
"Real.instZero",
"congrArg",
"setOf",
"Group.toDivisionMonoid",
"Members... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart | {
"line": 80,
"column": 4
} | {
"line": 80,
"column": 57
} | [
{
"pp": "A : Type u_1\ninst✝⁹ : TopologicalSpace A\ninst✝⁸ : NonUnitalRing A\ninst✝⁷ : StarRing A\ninst✝⁶ : Module ℂ A\ninst✝⁵ : IsScalarTower ℂ A A\ninst✝⁴ : SMulCommClass ℂ A A\ninst✝³ : StarModule ℂ A\ninst✝² : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal\ninst✝¹ : ContinuousMapZero.UniqueHom ℂ A\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart | {
"line": 92,
"column": 4
} | {
"line": 92,
"column": 57
} | [
{
"pp": "A : Type u_1\ninst✝⁹ : TopologicalSpace A\ninst✝⁸ : NonUnitalRing A\ninst✝⁷ : StarRing A\ninst✝⁶ : Module ℂ A\ninst✝⁵ : IsScalarTower ℂ A A\ninst✝⁴ : SMulCommClass ℂ A A\ninst✝³ : StarModule ℂ A\ninst✝² : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal\ninst✝¹ : ContinuousMapZero.UniqueHom ℂ A\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.CStarMatrix | {
"line": 569,
"column": 2
} | {
"line": 569,
"column": 13
} | [
{
"pp": "case h\nm : Type u_1\nn : Type u_2\nA : Type u_5\ninst✝⁴ : Fintype m\ninst✝³ : NonUnitalCStarAlgebra A\ninst✝² : PartialOrder A\ninst✝¹ : StarOrderedRing A\ninst✝ : Fintype n\nM : CStarMatrix m n A\nv : C⋆ᵐᵒᵈ(A, m → A)\nw : C⋆ᵐᵒᵈ(A, n → A)\n⊢ ⟪(toCLM M) v, w⟫_A = ⟪v, (toCLM Mᴴ) w⟫_A",
"usedConstant... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.CStarMatrix | {
"line": 582,
"column": 28
} | {
"line": 582,
"column": 51
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nA : Type u_5\ninst✝⁴ : Fintype m\ninst✝³ : NonUnitalCStarAlgebra A\ninst✝² : PartialOrder A\ninst✝¹ : StarOrderedRing A\ninst✝ : Fintype n\nM₁ M₂ : CStarMatrix m n A\n⊢ ‖M₁ + M₂‖ ≤ ‖M₁‖ + ‖M₂‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.CStarMatrix | {
"line": 584,
"column": 4
} | {
"line": 584,
"column": 75
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nA : Type u_5\ninst✝⁴ : Fintype m\ninst✝³ : NonUnitalCStarAlgebra A\ninst✝² : PartialOrder A\ninst✝¹ : StarOrderedRing A\ninst✝ : Fintype n\n⊢ ∀ (x : CStarMatrix m n A), ‖x‖ = 0 ↔ x = 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Pi.uniformSpace",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart | {
"line": 158,
"column": 4
} | {
"line": 158,
"column": 57
} | [
{
"pp": "A : Type u_1\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : Ring A\ninst✝⁵ : StarRing A\ninst✝⁴ : Algebra ℂ A\ninst✝³ : StarModule ℂ A\ninst✝² : ContinuousFunctionalCalculus ℂ A IsStarNormal\ninst✝¹ : ContinuousMap.UniqueHom ℂ A\ninst✝ : T2Space A\nf : ℝ → ℝ\na : A\nhf : ContinuousOn f (re '' spectrum ℂ a)\nha... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.CStarMatrix | {
"line": 607,
"column": 4
} | {
"line": 607,
"column": 36
} | [
{
"pp": "case inr\nm : Type u_1\nn : Type u_2\nA : Type u_5\ninst✝⁴ : Fintype m\ninst✝³ : NonUnitalCStarAlgebra A\ninst✝² : PartialOrder A\ninst✝¹ : StarOrderedRing A\ninst✝ : Fintype n\nM : CStarMatrix m n A\nC : ℝ≥0\nh : ∀ (v : C⋆ᵐᵒᵈ(A, m → A)) (w : C⋆ᵐᵒᵈ(A, n → A)), ‖⟪w, (toCLM M) v⟫_A‖ ≤ ↑C * ‖v‖ * ‖w‖\nv :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart | {
"line": 169,
"column": 4
} | {
"line": 169,
"column": 57
} | [
{
"pp": "A : Type u_1\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : Ring A\ninst✝⁵ : StarRing A\ninst✝⁴ : Algebra ℂ A\ninst✝³ : StarModule ℂ A\ninst✝² : ContinuousFunctionalCalculus ℂ A IsStarNormal\ninst✝¹ : ContinuousMap.UniqueHom ℂ A\ninst✝ : T2Space A\nf : ℝ → ℝ\na : A\nhf : ContinuousOn f (im '' spectrum ℂ a)\nha... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.CStarMatrix | {
"line": 757,
"column": 24
} | {
"line": 757,
"column": 61
} | [
{
"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\nx✝¹ x✝ : CStarMatrix n n A\n⊢ ‖x✝¹ * x✝‖ ≤ ‖x✝¹‖ * ‖x✝‖",
"usedConstants": [
"Pi.uniformSpace",
"Norm.norm",
"Eq.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.CStarMatrix | {
"line": 771,
"column": 20
} | {
"line": 774,
"column": 65
} | [
{
"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⊢ ‖v‖ * ‖(toCLM Mᴴ) ((toCLM M) v)‖ ≤ ‖v‖ * ‖toCLM Mᴴ ∘SL toCLM M‖ * ‖v‖",
"usedConstant... | rw [mul_assoc]
gcongr
rw [← ContinuousLinearMap.comp_apply]
exact le_opNorm ((toCLM Mᴴ).comp (toCLM M)) v | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.CStarMatrix | {
"line": 771,
"column": 20
} | {
"line": 774,
"column": 65
} | [
{
"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⊢ ‖v‖ * ‖(toCLM Mᴴ) ((toCLM M) v)‖ ≤ ‖v‖ * ‖toCLM Mᴴ ∘SL toCLM M‖ * ‖v‖",
"usedConstant... | rw [mul_assoc]
gcongr
rw [← ContinuousLinearMap.comp_apply]
exact le_opNorm ((toCLM Mᴴ).comp (toCLM M)) v | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.CStarMatrix | {
"line": 812,
"column": 25
} | {
"line": 812,
"column": 62
} | [
{
"pp": "A : Type u_1\ninst✝⁴ : CStarAlgebra A\ninst✝³ : PartialOrder A\ninst✝² : StarOrderedRing A\nn : Type u_2\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nr : ℂ\nM : CStarMatrix n n A\n⊢ ‖r • M‖ ≤ ‖r‖ * ‖M‖",
"usedConstants": [
"WithCStarModule.instZero",
"Pi.uniformSpace",
"CStarMatrix... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Extreme | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 35
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : Semiring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMul 𝕜 E\nA : Set E\nF : Set (Set E)\nhF : F.Nonempty\nhA : ∀ B ∈ F, IsExtreme 𝕜 A B\nthis : Nonempty ↑F\n⊢ IsExtreme 𝕜 A (⋂ B ∈ F, B)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Extreme | {
"line": 124,
"column": 31
} | {
"line": 124,
"column": 62
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : Semiring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMul 𝕜 E\nA : Set E\nF : Set (Set E)\nhF : F.Nonempty\nhAF : ∀ B ∈ F, IsExtreme 𝕜 A B\n⊢ IsExtreme 𝕜 A (⋂₀ F)",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.iInter",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Extreme | {
"line": 170,
"column": 13
} | {
"line": 170,
"column": 53
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : Semiring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMul 𝕜 E\nA B : Set E\nhAB : IsExtreme 𝕜 A B\nx✝ : E\n⊢ x✝ ∈ extremePoints 𝕜 B → x✝ ∈ extremePoints 𝕜 A",
"usedConstants": [
"Eq.mpr",
"Set.extremePoints",
"IsExtreme... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Range | {
"line": 59,
"column": 2
} | {
"line": 59,
"column": 13
} | [
{
"pp": "case e_s.e_s\n𝕜 : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : Ring A\ninst✝⁶ : StarRing A\ninst✝⁵ : Algebra 𝕜 A\ninst✝⁴ : TopologicalSpace A\ninst✝³ : StarModule 𝕜 A\ninst✝² : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 A p\ninst✝¹ : IsTopologicalRing A\ninst✝ : ContinuousS... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Range | {
"line": 129,
"column": 43
} | {
"line": 129,
"column": 58
} | [
{
"pp": "A : Type u_1\ninst✝¹¹ : Ring A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : Algebra ℝ A\ninst✝⁸ : TopologicalSpace A\ninst✝⁷ : IsTopologicalRing A\ninst✝⁶ : T2Space A\ninst✝⁵ : PartialOrder A\ninst✝⁴ : NonnegSpectrumClass ℝ A\ninst✝³ : StarOrderedRing A\ninst✝² : ContinuousStar A\ninst✝¹ : StarModule ℝ A\ninst✝ : C... | Set.inter_comm, | Mathlib.Tactic.evalGRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Range | {
"line": 137,
"column": 36
} | {
"line": 137,
"column": 51
} | [
{
"pp": "A : Type u_1\ninst✝¹¹ : Ring A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : Algebra ℝ A\ninst✝⁸ : TopologicalSpace A\ninst✝⁷ : IsTopologicalRing A\ninst✝⁶ : T2Space A\ninst✝⁵ : PartialOrder A\ninst✝⁴ : NonnegSpectrumClass ℝ A\ninst✝³ : StarOrderedRing A\ninst✝² : ContinuousStar A\ninst✝¹ : StarModule ℝ A\ninst✝ : C... | Set.inter_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Range | {
"line": 166,
"column": 2
} | {
"line": 166,
"column": 13
} | [
{
"pp": "case e_s.e_s\n𝕜 : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹¹ : RCLike 𝕜\ninst✝¹⁰ : NonUnitalRing A\ninst✝⁹ : StarRing A\ninst✝⁸ : Module 𝕜 A\ninst✝⁷ : IsScalarTower 𝕜 A A\ninst✝⁶ : SMulCommClass 𝕜 A A\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ : ContinuousConstSMul 𝕜 A\ninst✝³ : StarModule 𝕜 A\nins... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Range | {
"line": 239,
"column": 44
} | {
"line": 239,
"column": 59
} | [
{
"pp": "A : Type u_1\ninst✝¹⁴ : NonUnitalRing A\ninst✝¹³ : StarRing A\ninst✝¹² : Module ℝ A\ninst✝¹¹ : IsScalarTower ℝ A A\ninst✝¹⁰ : SMulCommClass ℝ A A\ninst✝⁹ : TopologicalSpace A\ninst✝⁸ : IsTopologicalRing A\ninst✝⁷ : T2Space A\ninst✝⁶ : PartialOrder A\ninst✝⁵ : NonnegSpectrumClass ℝ A\ninst✝⁴ : StarOrder... | Set.inter_comm, | Mathlib.Tactic.evalGRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Range | {
"line": 246,
"column": 37
} | {
"line": 246,
"column": 52
} | [
{
"pp": "A : Type u_1\ninst✝¹⁴ : NonUnitalRing A\ninst✝¹³ : StarRing A\ninst✝¹² : Module ℝ A\ninst✝¹¹ : IsScalarTower ℝ A A\ninst✝¹⁰ : SMulCommClass ℝ A A\ninst✝⁹ : TopologicalSpace A\ninst✝⁸ : IsTopologicalRing A\ninst✝⁷ : T2Space A\ninst✝⁶ : PartialOrder A\ninst✝⁵ : NonnegSpectrumClass ℝ A\ninst✝⁴ : StarOrder... | Set.inter_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Range | {
"line": 251,
"column": 4
} | {
"line": 251,
"column": 33
} | [
{
"pp": "A : Type u_1\ninst✝¹⁴ : NonUnitalRing A\ninst✝¹³ : StarRing A\ninst✝¹² : Module ℝ A\ninst✝¹¹ : IsScalarTower ℝ A A\ninst✝¹⁰ : SMulCommClass ℝ A A\ninst✝⁹ : TopologicalSpace A\ninst✝⁸ : IsTopologicalRing A\ninst✝⁷ : T2Space A\ninst✝⁶ : PartialOrder A\ninst✝⁵ : NonnegSpectrumClass ℝ A\ninst✝⁴ : StarOrder... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Hom | {
"line": 60,
"column": 4
} | {
"line": 60,
"column": 26
} | [
{
"pp": "F : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : NonUnitalCStarAlgebra A\ninst✝³ : NonUnitalCStarAlgebra B\ninst✝² : FunLike F A B\ninst✝¹ : NonUnitalAlgHomClass F ℂ A B\ninst✝ : StarHomClass F A B\nφ : F\nhφ : Function.Injective ⇑φ\na : A\nthis : ∀ {ψ : Unitization ℂ A →⋆ₐ[ℂ] Unitization ℂ B}, Funct... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Extreme | {
"line": 51,
"column": 18
} | {
"line": 51,
"column": 36
} | [
{
"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 = ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Extreme | {
"line": 56,
"column": 6
} | {
"line": 56,
"column": 17
} | [
{
"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 = ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Extreme | {
"line": 60,
"column": 8
} | {
"line": 60,
"column": 37
} | [
{
"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 = ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.GelfandNaimarkSegal | {
"line": 124,
"column": 8
} | {
"line": 124,
"column": 19
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\nf : A →ₚ[ℂ] ℂ\ninst✝ : StarOrderedRing A\na : A\nx : f.PreGNS\nthis : star (f.ofPreGNS x) * star a * (a * f.ofPreGNS x) ≤ ‖a‖ ^ 2 • star (f.ofPreGNS x) * f.ofPreGNS x\n⊢ f\n (star (f.ofPreGNS ((↑f.toPreGNS ∘ₗ ↑((ContinuousLin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.GelfandNaimarkSegal | {
"line": 152,
"column": 18
} | {
"line": 152,
"column": 29
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\nf : A →ₚ[ℂ] ℂ\ninst✝ : StarOrderedRing A\n⊢ (f.leftMulMapPreGNS 0).completion = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Projection | {
"line": 102,
"column": 4
} | {
"line": 102,
"column": 68
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\np q : A\nhp : IsStarProjection p\nhq : IsStarProjection q\ntfae_1_to_2 : p ≤ q → q * p = p\nh : q * p = p\n⊢ p * q = p",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Projection | {
"line": 104,
"column": 32
} | {
"line": 104,
"column": 43
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\np q : A\nhp : IsStarProjection p\nhq : IsStarProjection q\ntfae_1_to_2 : p ≤ q → q * p = p\ntfae_2_to_3 : q * p = p → p * q = p\ntfae_3_to_4 : p * q = p → IsStarProjection (q - p)\nh : IsStarProjection (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Projection | {
"line": 104,
"column": 2
} | {
"line": 104,
"column": 52
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\np q : A\nhp : IsStarProjection p\nhq : IsStarProjection q\ntfae_1_to_2 : p ≤ q → q * p = p\ntfae_2_to_3 : q * p = p → p * q = p\ntfae_3_to_4 : p * q = p → IsStarProjection (q - p)\n⊢ [p ≤ q, q * p = p, p... | tfae_have 4 → 1 := fun h ↦ by simpa using h.nonneg | Mathlib.Tactic.TFAE._aux_Mathlib_Tactic_TFAE___macroRules_Mathlib_Tactic_TFAE_tfaeHave_1 | Mathlib.Tactic.TFAE.tfaeHave |
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