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.Analysis.Normed.Algebra.TrivSqZeroExt | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 51
} | {
"line": 159,
"column": 0
} | [
{
"pp": "R : Type u_3\nM : Type u_4\ninst✝¹⁴ : CommRing R\ninst✝¹³ : AddCommGroup M\ninst✝¹² : Algebra ℚ R\ninst✝¹¹ : Module ℚ M\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module Rᵐᵒᵖ M\ninst✝⁸ : IsCentralScalar R M\ninst✝⁷ : TopologicalSpace R\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : IsTopologicalRing R\ninst✝⁴ : IsTopolog... | [] | rw [exp_def, fst_add, fst_inl, fst_inr, add_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Algebra.TrivSqZeroExt | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 51
} | {
"line": 159,
"column": 0
} | [
{
"pp": "R : Type u_3\nM : Type u_4\ninst✝¹⁴ : CommRing R\ninst✝¹³ : AddCommGroup M\ninst✝¹² : Algebra ℚ R\ninst✝¹¹ : Module ℚ M\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module Rᵐᵒᵖ M\ninst✝⁸ : IsCentralScalar R M\ninst✝⁷ : TopologicalSpace R\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : IsTopologicalRing R\ninst✝⁴ : IsTopolog... | [] | rw [exp_def, fst_add, fst_inl, fst_inr, add_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Algebra.TrivSqZeroExt | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 51
} | {
"line": 159,
"column": 0
} | [
{
"pp": "R : Type u_3\nM : Type u_4\ninst✝¹⁴ : CommRing R\ninst✝¹³ : AddCommGroup M\ninst✝¹² : Algebra ℚ R\ninst✝¹¹ : Module ℚ M\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module Rᵐᵒᵖ M\ninst✝⁸ : IsCentralScalar R M\ninst✝⁷ : TopologicalSpace R\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : IsTopologicalRing R\ninst✝⁴ : IsTopolog... | [] | rw [exp_def, fst_add, fst_inl, fst_inr, add_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.Ceva | {
"line": 175,
"column": 2
} | {
"line": 196,
"column": 66
} | {
"line": 197,
"column": 2
} | [
{
"pp": "case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : CommRing k\ninst✝³ : NoZeroDivisors k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nt : Triangle k P\nr : Fin 3 → k\nh✝ : Nontrivial k\nw : ↑Set.univ → Fin 3 → k := fun i ↦ Finset.affineCombinationLineMapWeights (↑i ... | [
"case neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : CommRing k\ninst✝³ : NoZeroDivisors k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nt : Triangle k P\nr : Fin 3 → k\nh✝ : Nontrivial k\nw : ↑Set.univ → Fin 3 → k := fun i ↦ Finset.affineCombinationLineMapWeights (↑i + 1) (↑i + 2... | · rw [Finset.prod_eq_zero_iff] at hc
obtain ⟨i, -, hi⟩ := hc
have hw'i1 : w' (i + 1) = 0 := by simpa [hi] using (hc1 i).symm
have hw'i2 : w' (i + 2) = 0 := by simpa [hi] using (hc2 i).symm
have hw'i0 : w' i = 1 := by
rw [← hw', Fin.sum_univ_three]
fin_cases i <;> grind
have hi1 : c (i + ... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Algebra.GelfandMazur | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 48
} | {
"line": 156,
"column": 49
} | [
{
"pp": "𝕜 : Type u_1\nF : Type u_2\ninst✝⁴ : NormedField 𝕜\ninst✝³ : ProperSpace 𝕜\ninst✝² : SeminormedRing F\ninst✝¹ : NormedAlgebra 𝕜 F\ninst✝ : NormOneClass F\nx : F\nthis : Tendsto (fun x_1 ↦ ‖x - (algebraMap 𝕜 F) x_1‖) (cobounded 𝕜) atTop\n⊢ Bornology.IsBounded {x_1 | ‖x - (algebraMap 𝕜 F) x_1‖ ≤ ‖... | [
"𝕜 : Type u_1\nF : Type u_2\ninst✝⁴ : NormedField 𝕜\ninst✝³ : ProperSpace 𝕜\ninst✝² : SeminormedRing F\ninst✝¹ : NormedAlgebra 𝕜 F\ninst✝ : NormOneClass F\nx : F\nthis : Tendsto (fun x_1 ↦ ‖x - (algebraMap 𝕜 F) x_1‖) (cobounded 𝕜) atTop\n⊢ {a | ‖x‖ < ‖x - (algebraMap 𝕜 F) a‖} ∈ cobounded 𝕜"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.GelfandMazur | {
"line": 180,
"column": 4
} | {
"line": 181,
"column": 11
} | {
"line": 181,
"column": 12
} | [
{
"pp": "case succ\nF : Type u_1\ninst✝³ : NormedRing F\ninst✝² : NormOneClass F\ninst✝¹ : NormMulClass F\ninst✝ : NormedAlgebra ℂ F\nx : F\nM : ℝ\nhM : 0 ≤ M\nh : ∀ (z' : ℂ), M ≤ ‖x - (algebraMap ℂ F) z'‖\nc : ℂ\nn : ℕ\nih : ∀ {p : ℂ[X]}, p.IsMonicOfDegree n → M ^ n ≤ ‖(aeval (x - (algebraMap ℂ F) c)) p‖\np : ... | [
"case succ\nF : Type u_1\ninst✝³ : NormedRing F\ninst✝² : NormOneClass F\ninst✝¹ : NormMulClass F\ninst✝ : NormedAlgebra ℂ F\nx : F\nM : ℝ\nhM : 0 ≤ M\nh : ∀ (z' : ℂ), M ≤ ‖x - (algebraMap ℂ F) z'‖\nc : ℂ\nn : ℕ\nih : ∀ {p : ℂ[X]}, p.IsMonicOfDegree n → M ^ n ≤ ‖(aeval (x - (algebraMap ℂ F) c)) p‖\np : ℂ[X]\nhp : p... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.MatrixExponential | {
"line": 190,
"column": 10
} | {
"line": 190,
"column": 49
} | {
"line": 190,
"column": 50
} | [
{
"pp": "m : Type u_1\n𝔸 : Type u_5\ninst✝⁴ : Fintype m\ninst✝³ : DecidableEq m\ninst✝² : NormedCommRing 𝔸\ninst✝¹ : NormedAlgebra ℚ 𝔸\ninst✝ : CompleteSpace 𝔸\nU A : Matrix m m 𝔸\nhy : IsUnit U\nu : (Matrix m m 𝔸)ˣ\nhu : ↑u = U\n⊢ exp (↑u * A * (↑u)⁻¹) = ↑u * exp A * (↑u)⁻¹",
"ppTerm": "?m.39",
"... | [
"m : Type u_1\n𝔸 : Type u_5\ninst✝⁴ : Fintype m\ninst✝³ : DecidableEq m\ninst✝² : NormedCommRing 𝔸\ninst✝¹ : NormedAlgebra ℚ 𝔸\ninst✝ : CompleteSpace 𝔸\nU A : Matrix m m 𝔸\nhy : IsUnit U\nu : (Matrix m m 𝔸)ˣ\nhu : ↑u = U\n⊢ exp (↑u * A * (↑u)⁻¹) = ↑u * exp A * (↑u)⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.MatrixExponential | {
"line": 195,
"column": 10
} | {
"line": 195,
"column": 49
} | {
"line": 195,
"column": 50
} | [
{
"pp": "m : Type u_1\n𝔸 : Type u_5\ninst✝⁴ : Fintype m\ninst✝³ : DecidableEq m\ninst✝² : NormedCommRing 𝔸\ninst✝¹ : NormedAlgebra ℚ 𝔸\ninst✝ : CompleteSpace 𝔸\nU A : Matrix m m 𝔸\nhy : IsUnit U\nu : (Matrix m m 𝔸)ˣ\nhu : ↑u = U\n⊢ exp ((↑u)⁻¹ * A * ↑u) = (↑u)⁻¹ * exp A * ↑u",
"ppTerm": "?m.39",
"... | [
"m : Type u_1\n𝔸 : Type u_5\ninst✝⁴ : Fintype m\ninst✝³ : DecidableEq m\ninst✝² : NormedCommRing 𝔸\ninst✝¹ : NormedAlgebra ℚ 𝔸\ninst✝ : CompleteSpace 𝔸\nU A : Matrix m m 𝔸\nhy : IsUnit U\nu : (Matrix m m 𝔸)ˣ\nhu : ↑u = U\n⊢ exp ((↑u)⁻¹ * A * ↑u) = (↑u)⁻¹ * exp A * ↑u"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Quaternion | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 41
} | {
"line": 176,
"column": 4
} | [
{
"pp": "⊢ Continuous ⇑normSq",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ Continuous ⇑normSq"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.Ultra | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 41
} | {
"line": 52,
"column": 42
} | [
{
"pp": "R : Type u_1\ninst✝² : SeminormedRing R\ninst✝¹ : NormOneClass R\ninst✝ : IsUltrametricDist R\nx : R\n⊢ ‖x + 1‖ ≤ max ‖x‖ 1",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"SeminormedRing.toNorm",
"Real.instLE",
"Real",
"P... | [
"R : Type u_1\ninst✝² : SeminormedRing R\ninst✝¹ : NormOneClass R\ninst✝ : IsUltrametricDist R\nx : R\n⊢ ‖x + 1‖ ≤ ‖x‖ ∨ ‖x + 1‖ ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.Ultra | {
"line": 64,
"column": 17
} | {
"line": 64,
"column": 80
} | {
"line": 65,
"column": 4
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝² : SeminormedRing R\ninst✝¹ : NormOneClass R\ninst✝ : IsUltrametricDist R\nn : ℕ\nhn : ‖↑n‖₊ ≤ 1\n⊢ ‖↑(n + 1)‖₊ ≤ 1",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AddMonoid.toAddSemigroup",
"congrArg",
"SeminormedA... | [
"case succ\nR : Type u_1\ninst✝² : SeminormedRing R\ninst✝¹ : NormOneClass R\ninst✝ : IsUltrametricDist R\nn : ℕ\nhn : ‖↑n‖₊ ≤ 1\n⊢ ‖↑n + 1‖₊ ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.Ultra | {
"line": 74,
"column": 2
} | {
"line": 75,
"column": 9
} | {
"line": 75,
"column": 10
} | [
{
"pp": "case ofNat\nR : Type u_1\ninst✝² : SeminormedRing R\ninst✝¹ : NormOneClass R\ninst✝ : IsUltrametricDist R\na✝ : ℕ\n⊢ ‖↑(Int.ofNat a✝)‖₊ ≤ 1",
"ppTerm": "?ofNat",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Int.cast_natCast",
"congrArg",
"Seminor... | [
"case ofNat\nR : Type u_1\ninst✝² : SeminormedRing R\ninst✝¹ : NormOneClass R\ninst✝ : IsUltrametricDist R\na✝ : ℕ\n⊢ ‖↑a✝‖₊ ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.Ultra | {
"line": 74,
"column": 2
} | {
"line": 75,
"column": 9
} | {
"line": 75,
"column": 10
} | [
{
"pp": "case negSucc\nR : Type u_1\ninst✝² : SeminormedRing R\ninst✝¹ : NormOneClass R\ninst✝ : IsUltrametricDist R\na✝ : ℕ\n⊢ ‖↑(Int.negSucc a✝)‖₊ ≤ 1",
"ppTerm": "?negSucc",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast",
"Eq.mpr",
"NegZero... | [
"case negSucc\nR : Type u_1\ninst✝² : SeminormedRing R\ninst✝¹ : NormOneClass R\ninst✝ : IsUltrametricDist R\na✝ : ℕ\n⊢ ‖↑(a✝ + 1)‖₊ ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Series | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 36
} | {
"line": 125,
"column": 37
} | [
{
"pp": "z : ℂ\n⊢ HasSum (fun n ↦ z ^ (2 * n) / ↑(2 * n)!) (cosh z)",
"ppTerm": "?m.42",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"z : ℂ\n⊢ HasSum (fun n ↦ z ^ (2 * n) / ↑(2 * n)!) (cosh z)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Series | {
"line": 129,
"column": 2
} | {
"line": 130,
"column": 9
} | {
"line": 130,
"column": 10
} | [
{
"pp": "z : ℂ\n⊢ HasSum (fun n ↦ z ^ (2 * n + 1) / ↑(2 * n + 1)!) (sinh z)",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Complex.sinh",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg"... | [
"z : ℂ\n⊢ HasSum (fun n ↦ (z ^ 2) ^ n * z / ↑(2 * n + 1)!) (sinh z)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Ultra | {
"line": 31,
"column": 4
} | {
"line": 31,
"column": 15
} | {
"line": 31,
"column": 16
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝³ : NormedField K\ninst✝² : SeminormedRing L\ninst✝¹ : NormOneClass L\ninst✝ : NormedAlgebra K L\nh : IsUltrametricDist L\nx y z : K\n⊢ dist x z ≤ max (dist x y) (dist y z)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Norm... | [
"K : Type u_1\nL : Type u_2\ninst✝³ : NormedField K\ninst✝² : SeminormedRing L\ninst✝¹ : NormOneClass L\ninst✝ : NormedAlgebra K L\nh : IsUltrametricDist L\nx y z : K\n⊢ dist x z ≤ dist x y ∨ dist x z ≤ dist y z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Ultra | {
"line": 49,
"column": 4
} | {
"line": 49,
"column": 31
} | {
"line": 49,
"column": 32
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝ : NormedDivisionRing R\nh : ∀ (x : R), ‖x + 1‖ ≤ max ‖x‖ 1\nx : R\n⊢ ‖x + 0‖ ≤ max ‖x‖ ‖0‖",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"Real.instLE",
"Real",
"S... | [
"case inl\nR : Type u_1\ninst✝ : NormedDivisionRing R\nh : ∀ (x : R), ‖x + 1‖ ≤ max ‖x‖ 1\nx : R\n⊢ ‖x‖ ≤ max ‖x‖ ‖0‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Ultra | {
"line": 51,
"column": 4
} | {
"line": 52,
"column": 36
} | {
"line": 52,
"column": 37
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝ : NormedDivisionRing R\nh : ∀ (x : R), ‖x + 1‖ ≤ max ‖x‖ 1\nx y : R\nhy : y ≠ 0\np : 0 < ‖y‖\n⊢ ‖x + y‖ ≤ max ‖x‖ ‖y‖",
"ppTerm": "?inr",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case inr\nR : Type u_1\ninst✝ : NormedDivisionRing R\nh : ∀ (x : R), ‖x + 1‖ ≤ max ‖x‖ 1\nx y : R\nhy : y ≠ 0\np : 0 < ‖y‖\n⊢ ‖x + y‖ ≤ max ‖x‖ ‖y‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Ultra | {
"line": 61,
"column": 19
} | {
"line": 61,
"column": 30
} | {
"line": 61,
"column": 31
} | [
{
"pp": "R : Type u_1\ninst✝ : NormedDivisionRing R\nh : ∀ (x : R), ‖x‖ ≤ 1 → ‖x + 1‖ ≤ 1\nx : R\nH : 1 < ‖x‖\n⊢ x ≠ 0",
"ppTerm": "?m.100",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"DivisionSemiring.toGroupWithZero",
"NormedDivisionRing.toDivisionRing... | [
"R : Type u_1\ninst✝ : NormedDivisionRing R\nh : ∀ (x : R), ‖x‖ ≤ 1 → ‖x + 1‖ ≤ 1\nx : R\nH : 1 < ‖x‖\n⊢ ¬x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Ultra | {
"line": 68,
"column": 4
} | {
"line": 68,
"column": 43
} | {
"line": 68,
"column": 44
} | [
{
"pp": "R : Type u_1\ninst✝ : NormedDivisionRing R\nh : ∀ (x : R), ‖x‖ ≤ 1 → ‖x - 1‖ ≤ 1\nx : R\nhx : ‖x‖ ≤ 1\n⊢ ‖x + 1‖ ≤ 1",
"ppTerm": "?m.39",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : NormedDivisionRing R\nh : ∀ (x : R), ‖x‖ ≤ 1 → ‖x - 1‖ ≤ 1\nx : R\nhx : ‖x‖ ≤ 1\n⊢ ‖x + 1‖ ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Ultra | {
"line": 114,
"column": 6
} | {
"line": 114,
"column": 50
} | {
"line": 114,
"column": 51
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝ : NormedDivisionRing R\nh : ∀ (n : ℕ), ‖↑n‖ ≤ 1\nx✝ : R\nm : ℕ\nx : R\nn : ℕ\nhx : 0 < ‖x‖\n⊢ ‖↑n‖ * ‖x‖ ≤ ‖x‖",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"... | [
"case inr\nR : Type u_1\ninst✝ : NormedDivisionRing R\nh : ∀ (n : ℕ), ‖↑n‖ ≤ 1\nx✝ : R\nm : ℕ\nx : R\nn : ℕ\nhx : 0 < ‖x‖\n⊢ ‖↑n‖ ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Ultra | {
"line": 118,
"column": 4
} | {
"line": 119,
"column": 26
} | {
"line": 119,
"column": 27
} | [
{
"pp": "R : Type u_1\ninst✝ : NormedDivisionRing R\nx : R\nm : ℕ\nh : ∀ (x : R) (n : ℕ), ‖n • x‖ ≤ ‖x‖\n⊢ ‖x + 1‖ ^ m ≤ ∑ k ∈ Finset.range (m + 1), ‖x‖ ^ k",
"ppTerm": "?m.86",
"assigned": true,
"usedConstants": [
"one_pow",
"Norm.norm",
"Eq.mpr",
"NonAssocSemiring.toAddComm... | [
"R : Type u_1\ninst✝ : NormedDivisionRing R\nx : R\nm : ℕ\nh : ∀ (x : R) (n : ℕ), ‖n • x‖ ≤ ‖x‖\n⊢ ‖∑ x_1 ∈ Finset.range (m + 1), x ^ x_1 * ↑(m.choose x_1)‖ ≤ ∑ x_1 ∈ Finset.range (m + 1), ‖x ^ x_1‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Ultra | {
"line": 129,
"column": 22
} | {
"line": 129,
"column": 89
} | {
"line": 129,
"column": 90
} | [
{
"pp": "R : Type u_1\ninst✝ : NormedDivisionRing R\nx : R\nh : ∀ (x : R) (n : ℕ), ‖n • x‖ ≤ ‖x‖\ni : ℕ\nhm : max 1 (‖x‖ ^ 0) = 1\nhx : ‖x‖ ^ 0 ≤ 1\nhi : i ∈ Finset.range (0 + 1)\n⊢ i = 0",
"ppTerm": "?m.244",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : NormedDivisionRing R\nx : R\nh : ∀ (x : R) (n : ℕ), ‖n • x‖ ≤ ‖x‖\ni : ℕ\nhm : max 1 (‖x‖ ^ 0) = 1\nhx : ‖x‖ ^ 0 ≤ 1\nhi : i ∈ Finset.range (0 + 1)\n⊢ i = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Ultra | {
"line": 134,
"column": 4
} | {
"line": 134,
"column": 33
} | {
"line": 134,
"column": 34
} | [
{
"pp": "case hmn\nR : Type u_1\ninst✝ : NormedDivisionRing R\nx : R\nm : ℕ\nh : ∀ (x : R) (n : ℕ), ‖n • x‖ ≤ ‖x‖\nhm : max 1 (‖x‖ ^ m) = ‖x‖ ^ m\nhx : 1 < ‖x‖ ^ m\ni : ℕ\nhi : i ∈ Finset.range (m + 1)\n⊢ i ≤ m",
"ppTerm": "?hmn",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"use... | [
"case hmn\nR : Type u_1\ninst✝ : NormedDivisionRing R\nx : R\nm : ℕ\nh : ∀ (x : R) (n : ℕ), ‖n • x‖ ≤ ‖x‖\nhm : max 1 (‖x‖ ^ m) = ‖x‖ ^ m\nhx : 1 < ‖x‖ ^ m\ni : ℕ\nhi : i ∈ Finset.range (m + 1)\n⊢ i ≤ m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.GelfandMazur | {
"line": 377,
"column": 2
} | {
"line": 378,
"column": 9
} | {
"line": 378,
"column": 10
} | [
{
"pp": "case inr\nF : Type u_1\ninst✝² : NormedRing F\ninst✝¹ : NormedAlgebra ℝ F\ninst✝ : NormOneClass F\nx : F\nu : ℝ\nhc₀ : 0 < ‖x - (algebraMap ℝ F) u‖\nhu : ∀ (x_1 : ℝ), ‖x - (algebraMap ℝ F) u‖ ≤ ‖x - (algebraMap ℝ F) x_1‖\n⊢ Bornology.IsBounded {x_1 | ‖φ x x_1‖ ≤ ‖φ x (0, 0)‖}",
"ppTerm": "?inr",
... | [
"case inr\nF : Type u_1\ninst✝² : NormedRing F\ninst✝¹ : NormedAlgebra ℝ F\ninst✝ : NormOneClass F\nx : F\nu : ℝ\nhc₀ : 0 < ‖x - (algebraMap ℝ F) u‖\nhu : ∀ (x_1 : ℝ), ‖x - (algebraMap ℝ F) u‖ ≤ ‖x - (algebraMap ℝ F) x_1‖\n⊢ {a | ‖φ x (0, 0)‖ < ‖φ x a‖} ∈ cobounded (ℝ × ℝ)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.RingSeminorm | {
"line": 152,
"column": 4
} | {
"line": 152,
"column": 45
} | {
"line": 152,
"column": 46
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝ : Ring R\np : RingSeminorm R\nhp : p 1 ≤ 1\nh : p ≠ 0\nhp0 : p 1 = 0\nx : R\n⊢ p x ≤ 0",
"ppTerm": "?inl",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case inl\nR : Type u_1\ninst✝ : Ring R\np : RingSeminorm R\nhp : p 1 ≤ 1\nh : p ≠ 0\nhp0 : p 1 = 0\nx : R\n⊢ p x ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.RingSeminorm | {
"line": 154,
"column": 4
} | {
"line": 154,
"column": 30
} | {
"line": 154,
"column": 31
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝ : Ring R\np : RingSeminorm R\nhp : p 1 ≤ 1\nh : p ≠ 0\nhp0 : 0 < p 1\n⊢ p 1 ≤ p 1 * p 1",
"ppTerm": "?inr",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case inr\nR : Type u_1\ninst✝ : Ring R\np : RingSeminorm R\nhp : p 1 ≤ 1\nh : p ≠ 0\nhp0 : 0 < p 1\n⊢ p 1 ≤ p 1 * p 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded | {
"line": 78,
"column": 23
} | {
"line": 78,
"column": 51
} | {
"line": 78,
"column": 52
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nx : R\nhx : IsUnit x\nhfx : f x ≠ 0\nn : ℕ\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nh1 : f 1 ≠ 0\nhxn : f (x ^ n) = 0\n⊢ f 1 ≤ 0",
"ppTerm": "?m.59",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nx : R\nhx : IsUnit x\nhfx : f x ≠ 0\nn : ℕ\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nh1 : f 1 ≠ 0\nhxn : f (x ^ n) = 0\n⊢ f 1 ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded | {
"line": 86,
"column": 38
} | {
"line": 86,
"column": 54
} | {
"line": 86,
"column": 55
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx : R\nhx : f x = 0\ny : R\n⊢ f (x * y) ≤ 0",
"ppTerm": "?m.38",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx : R\nhx : f x = 0\ny : R\n⊢ f (x * y) ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.RingSeminorm | {
"line": 412,
"column": 6
} | {
"line": 419,
"column": 30
} | {
"line": 420,
"column": 6
} | [
{
"pp": "R : Type u_1\nK : Type u_2\ninst✝ : Field K\nf : RingSeminorm K\nhnt : f ≠ 0\nx : K\nhx : f.toFun x = 0\nc : K\nhc : f c ≠ 0\nhn0 : ¬x = 0\n⊢ False",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
... | [
"R : Type u_1\nK : Type u_2\ninst✝ : Field K\nf : RingSeminorm K\nhnt : f ≠ 0\nx : K\nhx : f.toFun x = 0\nc : K\nhc : f c ≠ 0\nhn0 : ¬x = 0\nhc0 : f c = 0\n⊢ False"
] | have hc0 : f c = 0 := by
rw [← mul_one c, ← mul_inv_cancel₀ hn0, ← mul_assoc, mul_comm c, mul_assoc]
exact
le_antisymm
(le_trans (map_mul_le_mul f _ _)
(by rw [← RingSeminorm.toFun_eq_coe, ← AddGroupSeminorm.toFun_eq_coe, hx,
zero_mul]))
(a... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded | {
"line": 103,
"column": 38
} | {
"line": 103,
"column": 54
} | {
"line": 103,
"column": 55
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nx : R\nf_mul : 1 ≤ c * f 1\nhx : 0 < f x\nf_nonneg : 0 ≤ f 1\nh1 : f 1 = 0\n⊢ 1 ≤ 0",
"ppTerm": "?m.161",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nx : R\nf_mul : 1 ≤ c * f 1\nhx : 0 < f x\nf_nonneg : 0 ≤ f 1\nh1 : f 1 = 0\n⊢ 1 ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 21
} | {
"line": 116,
"column": 22
} | [
{
"pp": "case h.inl\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx y : R\nhy0 : f y = 0 y\n⊢ (fun y ↦ f (x * y) / f y) y ≤ c * f x",
"ppTerm": "?h.inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero... | [
"case h.inl\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx y : R\nhy0 : f y = 0 y\n⊢ 0 ≤ c * f x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 33
} | {
"line": 117,
"column": 34
} | [
{
"pp": "case h.inr\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx y : R\nhy0 : 0 y < f y\n⊢ (fun y ↦ f (x * y) / f y) y ≤ c * f x",
"ppTerm": "?h.inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero... | [
"case h.inr\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx y : R\nhy0 : 0 y < f y\n⊢ f (x * y) ≤ c * f x * f y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 20
} | {
"line": 126,
"column": 21
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx y : R\nhy : f y = 0 y\n⊢ f (x * y) / f y ≤ c * f x",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero... | [
"case inl\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx y : R\nhy : f y = 0 y\n⊢ 0 ≤ c * f x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Approximation | {
"line": 89,
"column": 6
} | {
"line": 89,
"column": 41
} | {
"line": 89,
"column": 42
} | [
{
"pp": "K : Type u_1\ninst✝ : NormedField K\nf g : K[X]\nε : ℝ\nhε : 0 < ε\na : K\nha : eval a f = 0\nhfm : f.Monic\nhgm : g.Monic\nhdeg : g.natDegree = f.natDegree\nhcoeff : ∀ (i : ℕ), ‖g.coeff i - f.coeff i‖ < ε\nhg : g.Splits\n⊢ (g - f).natDegree < g.natDegree + 1",
"ppTerm": "?m.596",
"assigned": t... | [
"K : Type u_1\ninst✝ : NormedField K\nf g : K[X]\nε : ℝ\nhε : 0 < ε\na : K\nha : eval a f = 0\nhfm : f.Monic\nhgm : g.Monic\nhdeg : g.natDegree = f.natDegree\nhcoeff : ∀ (i : ℕ), ‖g.coeff i - f.coeff i‖ < ε\nhg : g.Splits\n⊢ (g - f).natDegree ≤ f.natDegree"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Approximation | {
"line": 93,
"column": 4
} | {
"line": 93,
"column": 42
} | {
"line": 93,
"column": 43
} | [
{
"pp": "K : Type u_1\ninst✝ : NormedField K\nf g : K[X]\nε : ℝ\nhε : 0 < ε\na : K\nha : eval a f = 0\nhfm : f.Monic\nhgm : g.Monic\nhdeg : g.natDegree = f.natDegree\nhcoeff : ∀ (i : ℕ), ‖g.coeff i - f.coeff i‖ < ε\nhg : g.Splits\nthis :\n ‖∑ i ∈ Finset.range (g.natDegree + 1), eval a (C (g.coeff i - f.coeff i... | [
"K : Type u_1\ninst✝ : NormedField K\nf g : K[X]\nε : ℝ\nhε : 0 < ε\na : K\nha : eval a f = 0\nhfm : f.Monic\nhgm : g.Monic\nhdeg : g.natDegree = f.natDegree\nhcoeff : ∀ (i : ℕ), ‖g.coeff i - f.coeff i‖ < ε\nhg : g.Splits\nthis :\n ‖∑ i ∈ Finset.range (g.natDegree + 1), eval a (C (g.coeff i - f.coeff i) * X ^ i)‖ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Approximation | {
"line": 106,
"column": 8
} | {
"line": 106,
"column": 31
} | {
"line": 106,
"column": 32
} | [
{
"pp": "case convert_3.h₁\nK : Type u_1\ninst✝ : NormedField K\nf g : K[X]\nε : ℝ\nhε : 0 < ε\na : K\nha : eval a f = 0\nhfm : f.Monic\nhgm : g.Monic\nhdeg : g.natDegree = f.natDegree\nhcoeff : ∀ (i : ℕ), ‖g.coeff i - f.coeff i‖ < ε\nhg : g.Splits\ni : ℕ\nhi : i < f.natDegree + 1\n⊢ ‖g.coeff i - f.coeff i‖ < ε... | [
"case convert_3.h₁\nK : Type u_1\ninst✝ : NormedField K\nf g : K[X]\nε : ℝ\nhε : 0 < ε\na : K\nha : eval a f = 0\nhfm : f.Monic\nhgm : g.Monic\nhdeg : g.natDegree = f.natDegree\nhcoeff : ∀ (i : ℕ), ‖g.coeff i - f.coeff i‖ < ε\nhg : g.Splits\ni : ℕ\nhi : i < f.natDegree + 1\n⊢ ‖g.coeff i - f.coeff i‖ < ε"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Approximation | {
"line": 119,
"column": 40
} | {
"line": 119,
"column": 51
} | {
"line": 119,
"column": 52
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝² : NormedField K\ninst✝¹ : NormedField L\ninst✝ : NormedAlgebra K L\nf g : K[X]\nε : ℝ\nhε : 0 < ε\na : L\nha : (aeval a) f = 0\nhfm : f.Monic\nhgm : g.Monic\nhdeg : g.natDegree = f.natDegree\nhcoeff : ∀ (i : ℕ), ‖g.coeff i - f.coeff i‖ < ε\nhg : (map (algebraMap K L) ... | [
"K : Type u_1\nL : Type u_2\ninst✝² : NormedField K\ninst✝¹ : NormedField L\ninst✝ : NormedAlgebra K L\nf g : K[X]\nε : ℝ\nhε : 0 < ε\na : L\nha : (aeval a) f = 0\nhfm : f.Monic\nhgm : g.Monic\nhdeg : g.natDegree = f.natDegree\nhcoeff : ∀ (i : ℕ), ‖g.coeff i - f.coeff i‖ < ε\nhg : (map (algebraMap K L) g).Splits\n⊢... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Approximation | {
"line": 120,
"column": 10
} | {
"line": 120,
"column": 21
} | {
"line": 120,
"column": 22
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝² : NormedField K\ninst✝¹ : NormedField L\ninst✝ : NormedAlgebra K L\nf g : K[X]\nε : ℝ\nhε : 0 < ε\na : L\nha : (aeval a) f = 0\nhfm : f.Monic\nhgm : g.Monic\nhdeg : g.natDegree = f.natDegree\nhcoeff : ∀ (i : ℕ), ‖g.coeff i - f.coeff i‖ < ε\nhg : (map (algebraMap K L) ... | [
"K : Type u_1\nL : Type u_2\ninst✝² : NormedField K\ninst✝¹ : NormedField L\ninst✝ : NormedAlgebra K L\nf g : K[X]\nε : ℝ\nhε : 0 < ε\na : L\nha : (aeval a) f = 0\nhfm : f.Monic\nhgm : g.Monic\nhdeg : g.natDegree = f.natDegree\nhcoeff : ∀ (i : ℕ), ‖g.coeff i - f.coeff i‖ < ε\nhg : (map (algebraMap K L) g).Splits\n⊢... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Approximation | {
"line": 120,
"column": 32
} | {
"line": 120,
"column": 55
} | {
"line": 120,
"column": 56
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝² : NormedField K\ninst✝¹ : NormedField L\ninst✝ : NormedAlgebra K L\nf g : K[X]\nε : ℝ\nhε : 0 < ε\na : L\nha : (aeval a) f = 0\nhfm : f.Monic\nhgm : g.Monic\nhdeg : g.natDegree = f.natDegree\nhcoeff : ∀ (i : ℕ), ‖g.coeff i - f.coeff i‖ < ε\nhg : (map (algebraMap K L) ... | [
"K : Type u_1\nL : Type u_2\ninst✝² : NormedField K\ninst✝¹ : NormedField L\ninst✝ : NormedAlgebra K L\nf g : K[X]\nε : ℝ\nhε : 0 < ε\na : L\nha : (aeval a) f = 0\nhfm : f.Monic\nhgm : g.Monic\nhdeg : g.natDegree = f.natDegree\nhcoeff : ∀ (i : ℕ), ‖g.coeff i - f.coeff i‖ < ε\nhg : (map (algebraMap K L) g).Splits\n⊢... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Approximation | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 13
} | {
"line": 122,
"column": 14
} | [
{
"pp": "case right\nK : Type u_1\nL : Type u_2\ninst✝² : NormedField K\ninst✝¹ : NormedField L\ninst✝ : NormedAlgebra K L\nf g : K[X]\nε : ℝ\nhε : 0 < ε\na : L\nha : (aeval a) f = 0\nhfm : f.Monic\nhgm : g.Monic\nhdeg : g.natDegree = f.natDegree\nhcoeff : ∀ (i : ℕ), ‖g.coeff i - f.coeff i‖ < ε\nhg : (map (alge... | [
"case right\nK : Type u_1\nL : Type u_2\ninst✝² : NormedField K\ninst✝¹ : NormedField L\ninst✝ : NormedAlgebra K L\nf g : K[X]\nε : ℝ\nhε : 0 < ε\na : L\nha : (aeval a) f = 0\nhfm : f.Monic\nhgm : g.Monic\nhdeg : g.natDegree = f.natDegree\nhcoeff : ∀ (i : ℕ), ‖g.coeff i - f.coeff i‖ < ε\nhg : (map (algebraMap K L) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Approximation | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 16
} | {
"line": 124,
"column": 0
} | [
{
"pp": "case right\nK : Type u_1\nL : Type u_2\ninst✝² : NormedField K\ninst✝¹ : NormedField L\ninst✝ : NormedAlgebra K L\nf g : K[X]\nε : ℝ\nhε : 0 < ε\na : L\nha : (aeval a) f = 0\nhfm : f.Monic\nhgm : g.Monic\nhdeg : g.natDegree = f.natDegree\nhcoeff : ∀ (i : ℕ), ‖g.coeff i - f.coeff i‖ < ε\nhg : (map (alge... | [] | simpa using h2 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Normed.Field.Approximation | {
"line": 152,
"column": 8
} | {
"line": 152,
"column": 66
} | {
"line": 152,
"column": 67
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : NormedField L\ninst✝ : Algebra K L\nhd : DenseRange ⇑(algebraMap K L)\nf : L[X]\nhf : f.Monic\nε : ℝ\nhε : ε > 0\nh : ¬f.natDegree = 0\nc : ℕ → K\nhc : ∀ (i : ℕ), dist (f.coeff i) ((algebraMap K L) (c i)) < ε\ni : ℕ\nhi : i ∈ Finset.Iio f.natDegree... | [
"K : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : NormedField L\ninst✝ : Algebra K L\nhd : DenseRange ⇑(algebraMap K L)\nf : L[X]\nhf : f.Monic\nε : ℝ\nhε : ε > 0\nh : ¬f.natDegree = 0\nc : ℕ → K\nhc : ∀ (i : ℕ), dist (f.coeff i) ((algebraMap K L) (c i)) < ε\ni : ℕ\nhi : i ∈ Finset.Iio f.natDegree\n⊢ i < f.na... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Approximation | {
"line": 160,
"column": 6
} | {
"line": 160,
"column": 54
} | {
"line": 160,
"column": 55
} | [
{
"pp": "case h.refine_2.inl\nK : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : NormedField L\ninst✝ : Algebra K L\nhd : DenseRange ⇑(algebraMap K L)\nf : L[X]\nhf : f.Monic\nε : ℝ\nhε : ε > 0\nh✝ : ¬f.natDegree = 0\nc : ℕ → K\nhc : ∀ (i : ℕ), dist (f.coeff i) ((algebraMap K L) (c i)) < ε\nhdeg : (C 1 * X ... | [
"case h.refine_2.inl\nK : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : NormedField L\ninst✝ : Algebra K L\nhd : DenseRange ⇑(algebraMap K L)\nf : L[X]\nhf : f.Monic\nε : ℝ\nhε : ε > 0\nh✝ : ¬f.natDegree = 0\nc : ℕ → K\nhc : ∀ (i : ℕ), dist (f.coeff i) ((algebraMap K L) (c i)) < ε\nhdeg : (C 1 * X ^ f.natDegre... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 116,
"column": 2
} | {
"line": 116,
"column": 24
} | {
"line": 116,
"column": 25
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nx : R\nhx : μ x = 0\nh0 : ∀ (n : ℕ), 1 ≤ n → μ (x ^ n) ^ (1 / ↑n) = 0\nhL0 : ⨅ n, μ (x ^ ↑n) ^ (1 / ↑↑n) = 0\n⊢ Tendsto (smoothingSeminormSeq μ x) atTop (𝓝 (smoothingFun μ x))",
"ppTerm": "?m.152",
"assigned": true,
"usedConstants": [
... | [
"R : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nx : R\nhx : μ x = 0\nh0 : ∀ (n : ℕ), 1 ≤ n → μ (x ^ n) ^ (1 / ↑n) = 0\nhL0 : ⨅ n, μ (x ^ ↑n) ^ (1 / ↑↑n) = 0\n⊢ Tendsto (smoothingSeminormSeq μ x) atTop (𝓝 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.IsConjRoot | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 44
} | {
"line": 110,
"column": 45
} | [
{
"pp": "K : Type u_2\nS : Type u_4\ninst✝² : CommRing S\ninst✝¹ : Field K\ninst✝ : Algebra K S\nx y : S\nr : K\nh : IsConjRoot K x y\n⊢ IsConjRoot K (x - (algebraMap K S) r) (y - (algebraMap K S) r)",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Algebra.algebraMap"... | [
"K : Type u_2\nS : Type u_4\ninst✝² : CommRing S\ninst✝¹ : Field K\ninst✝ : Algebra K S\nx y : S\nr : K\nh : IsConjRoot K x y\n⊢ IsConjRoot K (x + -(algebraMap K S) r) (y + -(algebraMap K S) r)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SeminormFromConst | {
"line": 232,
"column": 4
} | {
"line": 232,
"column": 22
} | {
"line": 232,
"column": 23
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nc : R\nf : RingSeminorm R\nhf1 : f 1 ≤ 1\nhc : f c ≠ 0\nhpm : IsPowMul ⇑f\nx : R\nhx : ∀ (y : R), f (x * y) = f x * f y\ny : R\nhseq : seminormFromConst_seq c f (x * y) = fun n ↦ f x * seminormFromConst_seq c f y n\n⊢ Tendsto (seminormFromConst_seq c f (x * y)) atTop (... | [
"R : Type u_1\ninst✝ : CommRing R\nc : R\nf : RingSeminorm R\nhf1 : f 1 ≤ 1\nhc : f c ≠ 0\nhpm : IsPowMul ⇑f\nx : R\nhx : ∀ (y : R), f (x * y) = f x * f y\ny : R\nhseq : seminormFromConst_seq c f (x * y) = fun n ↦ f x * seminormFromConst_seq c f y n\n⊢ Tendsto (fun n ↦ f x * seminormFromConst_seq c f y n) atTop (𝓝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.IsConjRoot | {
"line": 224,
"column": 53
} | {
"line": 227,
"column": 34
} | {
"line": 229,
"column": 0
} | [
{
"pp": "K : Type u_2\nS : Type u_4\ninst✝³ : CommRing S\ninst✝² : Field K\ninst✝¹ : Algebra K S\ninst✝ : IsDomain S\nx y : S\nh : IsIntegral K x\n⊢ IsConjRoot K x y ↔ y ∈ (minpoly K x).aroots S",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instIsTorsionFreeOfIsDom... | [] | by
rw [Polynomial.mem_aroots, isConjRoot_iff_aeval_eq_zero h]
simp only [iff_and_self]
exact fun _ => minpoly.ne_zero h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Minpoly.IsConjRoot | {
"line": 271,
"column": 2
} | {
"line": 271,
"column": 30
} | {
"line": 271,
"column": 31
} | [
{
"pp": "R : Type u_1\nS : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : IsDomain S\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R S\nr : R\nx : S\nh : X - C r = minpoly R x\nhf : Function.Injective ⇑(algebraMap R S)\nthis : x ∈ (X - C r).aroots S\n⊢ x = (algebraMap R S) r",
... | [
"R : Type u_1\nS : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : IsDomain S\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R S\nr : R\nx : S\nh : X - C r = minpoly R x\nhf : Function.Injective ⇑(algebraMap R S)\nthis : x ∈ (X - C r).aroots S\n⊢ x = (algebraMap R S) r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SeminormFromConst | {
"line": 262,
"column": 2
} | {
"line": 262,
"column": 21
} | {
"line": 262,
"column": 22
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nc : R\nf : RingSeminorm R\nhf1 : f 1 ≤ 1\nhc : f c ≠ 0\nhpm : IsPowMul ⇑f\nx : R\nhlim : Tendsto (fun n ↦ seminormFromConst_seq c f x (n + 1)) atTop (𝓝 (seminormFromConst' c f x))\nhterm : seminormFromConst_seq c f (c * x) = fun n ↦ f c * seminormFromConst_seq c f x (... | [
"R : Type u_1\ninst✝ : CommRing R\nc : R\nf : RingSeminorm R\nhf1 : f 1 ≤ 1\nhc : f c ≠ 0\nhpm : IsPowMul ⇑f\nx : R\nhlim : Tendsto (fun n ↦ seminormFromConst_seq c f x (n + 1)) atTop (𝓝 (seminormFromConst' c f x))\nhterm : seminormFromConst_seq c f (c * x) = fun n ↦ f c * seminormFromConst_seq c f x (n + 1)\n⊢ Te... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Krasner | {
"line": 82,
"column": 25
} | {
"line": 82,
"column": 44
} | {
"line": 82,
"column": 45
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁵ : NormedField L\ninst✝⁴ : NontriviallyNormedField K\ninst✝³ : CompleteSpace K\ninst✝² : IsUltrametricDist K\ninst✝¹ : NormedAlgebra K L\ninst✝ : Normal K L\nx y : L\nxsep : IsSeparable K x\nsp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits\nyint : IsIntegral... | [
"K : Type u_1\nL : Type u_2\ninst✝⁵ : NormedField L\ninst✝⁴ : NontriviallyNormedField K\ninst✝³ : CompleteSpace K\ninst✝² : IsUltrametricDist K\ninst✝¹ : NormedAlgebra K L\ninst✝ : Normal K L\nx y : L\nxsep : IsSeparable K x\nsp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits\nyint : IsIntegral K y\nkr : ∀... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Krasner | {
"line": 98,
"column": 10
} | {
"line": 98,
"column": 48
} | {
"line": 98,
"column": 49
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁵ : NormedField L\ninst✝⁴ : NontriviallyNormedField K\ninst✝³ : CompleteSpace K\ninst✝² : IsUltrametricDist K\ninst✝¹ : NormedAlgebra K L\ninst✝ : Normal K L\nx y : L\nxsep : IsSeparable K x\nsp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits\nyint : IsIntegral... | [
"K : Type u_1\nL : Type u_2\ninst✝⁵ : NormedField L\ninst✝⁴ : NontriviallyNormedField K\ninst✝³ : CompleteSpace K\ninst✝² : IsUltrametricDist K\ninst✝¹ : NormedAlgebra K L\ninst✝ : Normal K L\nx y : L\nxsep : IsSeparable K x\nsp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits\nyint : IsIntegral K y\nkr : ∀... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Krasner | {
"line": 107,
"column": 12
} | {
"line": 107,
"column": 31
} | {
"line": 107,
"column": 32
} | [
{
"pp": "case a\nK : Type u_1\nL : Type u_2\ninst✝⁵ : NormedField L\ninst✝⁴ : NontriviallyNormedField K\ninst✝³ : CompleteSpace K\ninst✝² : IsUltrametricDist K\ninst✝¹ : NormedAlgebra K L\ninst✝ : Normal K L\nx y : L\nxsep : IsSeparable K x\nsp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits\nyint : Is... | [
"case a\nK : Type u_1\nL : Type u_2\ninst✝⁵ : NormedField L\ninst✝⁴ : NontriviallyNormedField K\ninst✝³ : CompleteSpace K\ninst✝² : IsUltrametricDist K\ninst✝¹ : NormedAlgebra K L\ninst✝ : Normal K L\nx y : L\nxsep : IsSeparable K x\nsp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits\nyint : IsIntegral K y... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Krasner | {
"line": 108,
"column": 12
} | {
"line": 108,
"column": 46
} | {
"line": 108,
"column": 47
} | [
{
"pp": "case a\nK : Type u_1\nL : Type u_2\ninst✝⁵ : NormedField L\ninst✝⁴ : NontriviallyNormedField K\ninst✝³ : CompleteSpace K\ninst✝² : IsUltrametricDist K\ninst✝¹ : NormedAlgebra K L\ninst✝ : Normal K L\nx y : L\nxsep : IsSeparable K x\nsp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits\nyint : Is... | [
"case a\nK : Type u_1\nL : Type u_2\ninst✝⁵ : NormedField L\ninst✝⁴ : NontriviallyNormedField K\ninst✝³ : CompleteSpace K\ninst✝² : IsUltrametricDist K\ninst✝¹ : NormedAlgebra K L\ninst✝ : Normal K L\nx y : L\nxsep : IsSeparable K x\nsp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits\nyint : IsIntegral K y... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 211,
"column": 8
} | {
"line": 211,
"column": 36
} | {
"line": 211,
"column": 37
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\nx : R\nhx : μ x ≠ 0\nL : ℝ := ⨅ n, μ (x ^ ↑n) ^ (1 / ↑↑n)\nhL0 : 0 ≤ L\nε : ℝ\nhε : ε > 0\nm1 : ℕ+\nhm1 : μ (x ^ ↑m1) ^ (1 / ↑↑m1) < (⨅ n, μ (x ^ ↑n) ^ (1 / ↑↑n)) + ε / 2\nm2 : ℕ\nhm2 : ∀ n ≥ m2, (L + ε / 2) ^ (-(↑(n % ↑m1) ... | [
"case pos\nR : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\nx : R\nhx : μ x ≠ 0\nL : ℝ := ⨅ n, μ (x ^ ↑n) ^ (1 / ↑↑n)\nhL0 : 0 ≤ L\nε : ℝ\nhε : ε > 0\nm1 : ℕ+\nhm1 : μ (x ^ ↑m1) ^ (1 / ↑↑m1) < (⨅ n, μ (x ^ ↑n) ^ (1 / ↑↑n)) + ε / 2\nm2 : ℕ\nhm2 : ∀ n ≥ m2, (L + ε / 2) ^ (-(↑(n % ↑m1) / ↑n)) * (μ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Instances | {
"line": 27,
"column": 59
} | {
"line": 27,
"column": 70
} | {
"line": 27,
"column": 71
} | [
{
"pp": "F : Type u_1\ninst✝ : NormedField F\nf : Filter F\nhc : Cauchy f\nhn : nhds 0 ⊓ f = ⊥\nδ : ℝ\nδ_pos : δ > 0\nhδ : ∀ᶠ (y : F) in f, δ ≤ ‖y‖\ny : F\nhy : δ ≤ ‖y‖\n⊢ δ ≤ ((fun x ↦ ‖x⁻¹‖) y)⁻¹",
"ppTerm": "?m.70",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Re... | [
"F : Type u_1\ninst✝ : NormedField F\nf : Filter F\nhc : Cauchy f\nhn : nhds 0 ⊓ f = ⊥\nδ : ℝ\nδ_pos : δ > 0\nhδ : ∀ᶠ (y : F) in f, δ ≤ ‖y‖\ny : F\nhy : δ ≤ ‖y‖\n⊢ δ ≤ ‖y‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Instances | {
"line": 28,
"column": 56
} | {
"line": 28,
"column": 67
} | {
"line": 28,
"column": 68
} | [
{
"pp": "F : Type u_1\ninst✝ : NormedField F\nf : Filter F\nhc : Cauchy f\nhn : nhds 0 ⊓ f = ⊥\nδ : ℝ\nδ_pos : δ > 0\nhδ : ∀ᶠ (y : F) in f, δ ≤ ‖y‖\nf_bdd : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f fun x ↦ ‖x⁻¹‖\ny : F\nhy : δ ≤ ‖y‖\n⊢ y ≠ 0",
"ppTerm": "?m.93",
"assigned": true,
"usedConstants": [
... | [
"F : Type u_1\ninst✝ : NormedField F\nf : Filter F\nhc : Cauchy f\nhn : nhds 0 ⊓ f = ⊥\nδ : ℝ\nδ_pos : δ > 0\nhδ : ∀ᶠ (y : F) in f, δ ≤ ‖y‖\nf_bdd : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f fun x ↦ ‖x⁻¹‖\ny : F\nhy : δ ≤ ‖y‖\n⊢ ¬y = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.ProperSpace | {
"line": 48,
"column": 26
} | {
"line": 48,
"column": 53
} | {
"line": 48,
"column": 54
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹ : NontriviallyNormedField 𝕜\ninst✝ : WeaklyLocallyCompactSpace 𝕜\nr : ℝ\nrpos : 0 < r\nhr : IsCompact (closedBall 0 r)\nc : 𝕜\nhc : 1 < ‖c‖\nn : ℕ\nthis : c ^ n ≠ 0\nx✝ : 𝕜\n⊢ 0 < ?m.159",
"ppTerm": "?m.160",
"assigned": false,
"usedConstants": [],
"usedFVars":... | [
"𝕜 : Type u_1\ninst✝¹ : NontriviallyNormedField 𝕜\ninst✝ : WeaklyLocallyCompactSpace 𝕜\nr : ℝ\nrpos : 0 < r\nhr : IsCompact (closedBall 0 r)\nc : 𝕜\nhc : 1 < ‖c‖\nn : ℕ\nthis : c ^ n ≠ 0\nx✝ : 𝕜\n⊢ 0 < ?m.159"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Dense | {
"line": 57,
"column": 10
} | {
"line": 57,
"column": 21
} | {
"line": 57,
"column": 22
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : NontriviallyNormedField L\ninst✝⁴ : CompleteSpace L\ninst✝³ : CharZero L\ninst✝² : IsUltrametricDist L\ninst✝¹ : Algebra K L\nhi : DenseRange ⇑(algebraMap K L)\ninst✝ : IsAlgClosed K\nf : L[X]\nfmon : f.Monic\nfirr : Irreducible f\nfnatdeg0 : f.nat... | [
"K : Type u_1\nL : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : NontriviallyNormedField L\ninst✝⁴ : CompleteSpace L\ninst✝³ : CharZero L\ninst✝² : IsUltrametricDist L\ninst✝¹ : Algebra K L\nhi : DenseRange ⇑(algebraMap K L)\ninst✝ : IsAlgClosed K\nf : L[X]\nfmon : f.Monic\nfirr : Irreducible f\nfnatdeg0 : f.natDegree ≠ 0\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Dense | {
"line": 61,
"column": 10
} | {
"line": 61,
"column": 21
} | {
"line": 61,
"column": 22
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : NontriviallyNormedField L\ninst✝⁴ : CompleteSpace L\ninst✝³ : CharZero L\ninst✝² : IsUltrametricDist L\ninst✝¹ : Algebra K L\nhi : DenseRange ⇑(algebraMap K L)\ninst✝ : IsAlgClosed K\nf : L[X]\nfmon : f.Monic\nfirr : Irreducible f\nfnatdeg0 : f.nat... | [
"K : Type u_1\nL : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : NontriviallyNormedField L\ninst✝⁴ : CompleteSpace L\ninst✝³ : CharZero L\ninst✝² : IsUltrametricDist L\ninst✝¹ : Algebra K L\nhi : DenseRange ⇑(algebraMap K L)\ninst✝ : IsAlgClosed K\nf : L[X]\nfmon : f.Monic\nfirr : Irreducible f\nfnatdeg0 : f.natDegree ≠ 0\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 252,
"column": 2
} | {
"line": 252,
"column": 43
} | {
"line": 252,
"column": 44
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\nx : R\n⊢ ∀ᶠ (c : ℕ) in atTop, 0 ≤ smoothingSeminormSeq μ x c",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"Filter.Eventually",
"instArchimedeanN... | [
"R : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\nx : R\n⊢ ∃ a, ∀ (b : ℕ), a ≤ b → 0 ≤ smoothingSeminormSeq μ x b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 294,
"column": 2
} | {
"line": 294,
"column": 18
} | {
"line": 294,
"column": 19
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nx y : R\nhn : ∀ (n : ℕ), ∃ m < n + 1, μ ((x + y) ^ n) ^ (1 / ↑n) ≤ (μ (x ^ m) * μ (y ^ (n - m))) ^ (1 / ↑n)\nn : ℕ\n⊢ mu μ hn n ≤ n",
"ppTerm": "?m.67",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Re... | [
"R : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nx y : R\nhn : ∀ (n : ℕ), ∃ m < n + 1, μ ((x + y) ^ n) ^ (1 / ↑n) ≤ (μ (x ^ m) * μ (y ^ (n - m))) ^ (1 / ↑n)\nn : ℕ\n⊢ Classical.choose ⋯ ≤ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Dense | {
"line": 98,
"column": 10
} | {
"line": 98,
"column": 42
} | {
"line": 98,
"column": 43
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : NontriviallyNormedField L\ninst✝⁴ : CompleteSpace L\ninst✝³ : CharZero L\ninst✝² : IsUltrametricDist L\ninst✝¹ : Algebra K L\nhi : DenseRange ⇑(algebraMap K L)\ninst✝ : IsAlgClosed K\nf : L[X]\nfmon : f.Monic\nfirr : Irreducible f\nfnatdeg0 : f.nat... | [
"K : Type u_1\nL : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : NontriviallyNormedField L\ninst✝⁴ : CompleteSpace L\ninst✝³ : CharZero L\ninst✝² : IsUltrametricDist L\ninst✝¹ : Algebra K L\nhi : DenseRange ⇑(algebraMap K L)\ninst✝ : IsAlgClosed K\nf : L[X]\nfmon : f.Monic\nfirr : Irreducible f\nfnatdeg0 : f.natDegree ≠ 0\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Dense | {
"line": 102,
"column": 6
} | {
"line": 102,
"column": 86
} | {
"line": 102,
"column": 87
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : NontriviallyNormedField L\ninst✝⁴ : CompleteSpace L\ninst✝³ : CharZero L\ninst✝² : IsUltrametricDist L\ninst✝¹ : Algebra K L\nhi : DenseRange ⇑(algebraMap K L)\ninst✝ : IsAlgClosed K\nf : L[X]\nfmon : f.Monic\nfirr : Irreducible f\nfnatdeg0 : f.nat... | [
"K : Type u_1\nL : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : NontriviallyNormedField L\ninst✝⁴ : CompleteSpace L\ninst✝³ : CharZero L\ninst✝² : IsUltrametricDist L\ninst✝¹ : Algebra K L\nhi : DenseRange ⇑(algebraMap K L)\ninst✝ : IsAlgClosed K\nf : L[X]\nfmon : f.Monic\nfirr : Irreducible f\nfnatdeg0 : f.natDegree ≠ 0\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 335,
"column": 43
} | {
"line": 338,
"column": 60
} | {
"line": 339,
"column": 8
} | [
{
"pp": "K : Type u_2\ninst✝³ : NormedField K\nL : Type u_3\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : DecidableEq L\nf : AlgebraNorm K L\nhf_pm : IsPowMul ⇑f\nhf_na : IsNonarchimedean ⇑f\nhf1 : f 1 = 1\np : K[X]\ns : Multiset L\nhp : (mapAlg K L) p = (Multiset.map (fun a ↦ X - C a) s).prod\nh_le : 0 ≤ ⨆ ... | [] | by
have hs0 : 0 < s.card := hps ▸ hm.pos
obtain ⟨x, hx⟩ := card_pos_iff_exists_mem.mp hs0
exact Finset.card_pos.mpr ⟨x, mem_toFinset.mpr hx⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Field.Dense | {
"line": 116,
"column": 6
} | {
"line": 116,
"column": 68
} | {
"line": 117,
"column": 8
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : NontriviallyNormedField L\ninst✝⁴ : CompleteSpace L\ninst✝³ : CharZero L\ninst✝² : IsUltrametricDist L\ninst✝¹ : Algebra K L\nhi : DenseRange ⇑(algebraMap K L)\ninst✝ : IsAlgClosed K\nf : L[X]\nfmon : f.Monic\nfirr : Irreducible f\nfnatdeg0 : f.nat... | [
"K : Type u_1\nL : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : NontriviallyNormedField L\ninst✝⁴ : CompleteSpace L\ninst✝³ : CharZero L\ninst✝² : IsUltrametricDist L\ninst✝¹ : Algebra K L\nhi : DenseRange ⇑(algebraMap K L)\ninst✝ : IsAlgClosed K\nf : L[X]\nfmon : f.Monic\nfirr : Irreducible f\nfnatdeg0 : f.natDegree ≠ 0\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Dense | {
"line": 118,
"column": 4
} | {
"line": 118,
"column": 69
} | {
"line": 119,
"column": 6
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : NontriviallyNormedField L\ninst✝⁴ : CompleteSpace L\ninst✝³ : CharZero L\ninst✝² : IsUltrametricDist L\ninst✝¹ : Algebra K L\nhi : DenseRange ⇑(algebraMap K L)\ninst✝ : IsAlgClosed K\nf : L[X]\nfmon : f.Monic\nfirr : Irreducible f\nfnatdeg0 : f.nat... | [
"K : Type u_1\nL : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : NontriviallyNormedField L\ninst✝⁴ : CompleteSpace L\ninst✝³ : CharZero L\ninst✝² : IsUltrametricDist L\ninst✝¹ : Algebra K L\nhi : DenseRange ⇑(algebraMap K L)\ninst✝ : IsAlgClosed K\nf : L[X]\nfmon : f.Monic\nfirr : Irreducible f\nfnatdeg0 : f.natDegree ≠ 0\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.HomCompletion | {
"line": 147,
"column": 4
} | {
"line": 147,
"column": 15
} | {
"line": 147,
"column": 16
} | [
{
"pp": "G : Type u_1\ninst✝¹ : SeminormedAddCommGroup G\nH : Type u_2\ninst✝ : SeminormedAddCommGroup H\nf : NormedAddGroupHom G H\nx : G\n⊢ ‖f x‖ ≤ ‖f.completion‖ * ‖x‖",
"ppTerm": "?m.33",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\ninst✝¹ : SeminormedAddCommGroup G\nH : Type u_2\ninst✝ : SeminormedAddCommGroup H\nf : NormedAddGroupHom G H\nx : G\n⊢ ‖f x‖ ≤ ‖f.completion‖ * ‖x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.HomCompletion | {
"line": 162,
"column": 2
} | {
"line": 162,
"column": 67
} | {
"line": 163,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝¹ : SeminormedAddCommGroup G\nH : Type u_2\ninst✝ : SeminormedAddCommGroup H\nf : NormedAddGroupHom G H\nC : ℝ\nh : f.SurjectiveOnWith f.range C\nhatg : Completion G\nhatg_in : f.completion hatg = 0\nε : ℝ\nε_pos : 0 < ε\nC' : ℝ\nC'_pos : C' > 0\nhC' : f.SurjectiveOnWith f.range C'\n... | [
"G : Type u_1\ninst✝¹ : SeminormedAddCommGroup G\nH : Type u_2\ninst✝ : SeminormedAddCommGroup H\nf : NormedAddGroupHom G H\nC : ℝ\nh : f.SurjectiveOnWith f.range C\nhatg : Completion G\nhatg_in : f.completion hatg = 0\nε : ℝ\nε_pos : 0 < ε\nC' : ℝ\nC'_pos : C' > 0\nhC' : f.SurjectiveOnWith f.range C'\nδ : ℝ\nδ_pos... | rcases exists_pos_mul_lt ε_pos (1 + C' * ‖f‖) with ⟨δ, δ_pos, hδ⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 361,
"column": 6
} | {
"line": 362,
"column": 61
} | {
"line": 363,
"column": 4
} | [
{
"pp": "case pos\nK : Type u_2\ninst✝³ : NormedField K\nL : Type u_3\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : DecidableEq L\nf : AlgebraNorm K L\nhf_pm : IsPowMul ⇑f\nhf_na : IsNonarchimedean ⇑f\nhf1 : f 1 = 1\np : K[X]\ns : Multiset L\nhp : (mapAlg K L) p = (Multiset.map (fun a ↦ X - C a) s).prod\nh_l... | [] | exact le_trans this (pow_le_pow_left₀ (apply_nonneg _ _)
(le_trans (by rw [if_pos hyx]) (le_ciSup h_bdd y)) _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Group.HomCompletion | {
"line": 170,
"column": 6
} | {
"line": 170,
"column": 36
} | {
"line": 170,
"column": 37
} | [
{
"pp": "G : Type u_1\ninst✝¹ : SeminormedAddCommGroup G\nH : Type u_2\ninst✝ : SeminormedAddCommGroup H\nf : NormedAddGroupHom G H\nC : ℝ\nh : f.SurjectiveOnWith f.range C\nhatg : Completion G\nhatg_in : f.completion hatg = 0\nε : ℝ\nε_pos : 0 < ε\nC' : ℝ\nC'_pos : C' > 0\nhC' : f.SurjectiveOnWith f.range C'\n... | [
"G : Type u_1\ninst✝¹ : SeminormedAddCommGroup G\nH : Type u_2\ninst✝ : SeminormedAddCommGroup H\nf : NormedAddGroupHom G H\nC : ℝ\nh : f.SurjectiveOnWith f.range C\nhatg : Completion G\nhatg_in : f.completion hatg = 0\nε : ℝ\nε_pos : 0 < ε\nC' : ℝ\nC'_pos : C' > 0\nhC' : f.SurjectiveOnWith f.range C'\nδ : ℝ\nδ_pos... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 430,
"column": 2
} | {
"line": 430,
"column": 46
} | {
"line": 431,
"column": 2
} | [
{
"pp": "K : Type u_2\ninst✝² : NormedField K\nL : Type u_3\ninst✝¹ : Field L\ninst✝ : Algebra K L\ny : L\nhy : y ≠ 0\nhy_alg : IsAlgebraic K y\n⊢ 0 < spectralNorm K L y",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Real.partialOrder",
"Real",
"Real.instZero",
"s... | [
"K : Type u_2\ninst✝² : NormedField K\nL : Type u_3\ninst✝¹ : Field L\ninst✝ : Algebra K L\ny : L\nhy : y ≠ 0\nhy_alg : IsAlgebraic K y\n⊢ 0 ≠ spectralNorm K L y"
] | apply lt_of_le_of_ne (spectralNorm_nonneg _) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Normed.Group.HomCompletion | {
"line": 177,
"column": 16
} | {
"line": 177,
"column": 62
} | {
"line": 177,
"column": 63
} | [
{
"pp": "G : Type u_1\ninst✝¹ : SeminormedAddCommGroup G\nH : Type u_2\ninst✝ : SeminormedAddCommGroup H\nf : NormedAddGroupHom G H\nC : ℝ\nh : f.SurjectiveOnWith f.range C\nhatg : Completion G\nhatg_in : f.completion hatg = 0\nε : ℝ\nε_pos : 0 < ε\nC' : ℝ\nC'_pos : C' > 0\nhC' : f.SurjectiveOnWith f.range C'\n... | [
"G : Type u_1\ninst✝¹ : SeminormedAddCommGroup G\nH : Type u_2\ninst✝ : SeminormedAddCommGroup H\nf : NormedAddGroupHom G H\nC : ℝ\nh : f.SurjectiveOnWith f.range C\nhatg : Completion G\nhatg_in : f.completion hatg = 0\nε : ℝ\nε_pos : 0 < ε\nC' : ℝ\nC'_pos : C' > 0\nhC' : f.SurjectiveOnWith f.range C'\nδ : ℝ\nδ_pos... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.SemiNormedGrp.Kernels | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 74
} | {
"line": 105,
"column": 75
} | [
{
"pp": "V W : SemiNormedGrp\nf g : V ⟶ W\nv : { carrier := ↥(Hom.hom (f - g)).ker, str := AddSubgroup.seminormedAddCommGroup }.carrier\nthis : ↑v ∈ (Hom.hom (f - g)).ker\n⊢ (Hom.hom (ofHom (NormedAddGroupHom.incl (Hom.hom (f - g)).ker) ≫ f)) v =\n (Hom.hom (ofHom (NormedAddGroupHom.incl (Hom.hom (f - g)).ke... | [
"V W : SemiNormedGrp\nf g : V ⟶ W\nv : { carrier := ↥(Hom.hom (f - g)).ker, str := AddSubgroup.seminormedAddCommGroup }.carrier\nthis : ↑v ∈ (Hom.hom (f - g)).ker\n⊢ (Hom.hom f) ↑v = (Hom.hom g) ↑v"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.SeparationQuotient | {
"line": 141,
"column": 4
} | {
"line": 141,
"column": 32
} | {
"line": 141,
"column": 33
} | [
{
"pp": "case mpr\nM : Type u_1\ninst✝ : SeminormedAddCommGroup M\nh : ∀ (x : M), ‖x‖ = 0\nx : M\n⊢ normedMk x = 0 x",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
"Real",
"NormedAddGroupHom",
"Separ... | [
"case mpr\nM : Type u_1\ninst✝ : SeminormedAddCommGroup M\nh : ∀ (x : M), ‖x‖ = 0\nx : M\n⊢ ‖‖x‖‖ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 514,
"column": 2
} | {
"line": 514,
"column": 82
} | {
"line": 515,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\nhna : IsNonarchimedean ⇑μ\nx y : R\nhn : ∀ (n : ℕ), ∃ m < n + 1, μ ((x + y) ^ n) ^ (1 / ↑n) ≤ (μ (x ^ m) * μ (y ^ (n - m))) ^ (1 / ↑n)\nmu : ℕ → ℕ := ⋯\nnu : ℕ → ℕ := ⋯\nhnu : nu = fun n ↦ n - mu n\nhmu_le : ∀ (n : ℕ), mu n ≤ n\nhmu_b... | [
"R : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\nhna : IsNonarchimedean ⇑μ\nx y : R\nhn : ∀ (n : ℕ), ∃ m < n + 1, μ ((x + y) ^ n) ^ (1 / ↑n) ≤ (μ (x ^ m) * μ (y ^ (n - m))) ^ (1 / ↑n)\nmu : ℕ → ℕ := fun n ↦ _root_.mu μ hn n\nnu : ℕ → ℕ := fun n ↦ n - mu n\nhnu : nu = fun n ↦ n - mu n\nhmu_le : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Tannery | {
"line": 48,
"column": 4
} | {
"line": 48,
"column": 33
} | {
"line": 48,
"column": 34
} | [
{
"pp": "case inl\nα : Type u_1\nβ : Type u_2\nG : Type u_3\n𝓕 : Filter α\ninst✝¹ : NormedAddCommGroup G\ninst✝ : CompleteSpace G\nf : α → β → G\ng : β → G\nbound : β → ℝ\nh_sum : Summable bound\nhab : ∀ (k : β), Tendsto (fun x ↦ f x k) 𝓕 (𝓝 (g k))\nh_bound : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k\nh... | [
"case inl\nα : Type u_1\nβ : Type u_2\nG : Type u_3\n𝓕 : Filter α\ninst✝¹ : NormedAddCommGroup G\ninst✝ : CompleteSpace G\nf : α → β → G\ng : β → G\nbound : β → ℝ\nh_sum : Summable bound\nhab : ∀ (k : β), Tendsto (fun x ↦ f x k) 𝓕 (𝓝 (g k))\nh_bound : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k\nh✝ : IsEmpty ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 571,
"column": 2
} | {
"line": 571,
"column": 20
} | {
"line": 571,
"column": 21
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\nx : R\nm : ℕ\nhm : 1 ≤ m\nhlim : Tendsto (fun n ↦ smoothingSeminormSeq μ x (m * n)) atTop (𝓝 (smoothingFun μ x))\nh_eq : ∀ (n : ℕ), smoothingSeminormSeq μ x (m * n) ^ m = smoothingSeminormSeq μ (x ^ m) n\n⊢ Tendsto (fun x_1 ↦ smoothi... | [
"R : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\nx : R\nm : ℕ\nhm : 1 ≤ m\nhlim : Tendsto (fun n ↦ smoothingSeminormSeq μ x (m * n)) atTop (𝓝 (smoothingFun μ x))\nh_eq : ∀ (n : ℕ), smoothingSeminormSeq μ x (m * n) ^ m = smoothingSeminormSeq μ (x ^ m) n\n⊢ Tendsto (fun x_1 ↦ smoothingSeminormSe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Tannery | {
"line": 59,
"column": 38
} | {
"line": 59,
"column": 66
} | {
"line": 59,
"column": 67
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nG : Type u_3\n𝓕 : Filter α\ninst✝¹ : NormedAddCommGroup G\ninst✝ : CompleteSpace G\nf : α → β → G\ng : β → G\nbound : β → ℝ\nh_sum : Summable bound\nhab : ∀ (k : β), Tendsto (fun x ↦ f x k) 𝓕 (𝓝 (g k))\nh_bound : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k\nh✝¹ : Nonem... | [
"α : Type u_1\nβ : Type u_2\nG : Type u_3\n𝓕 : Filter α\ninst✝¹ : NormedAddCommGroup G\ninst✝ : CompleteSpace G\nf : α → β → G\ng : β → G\nbound : β → ℝ\nh_sum : Summable bound\nhab : ∀ (k : β), Tendsto (fun x ↦ f x k) 𝓕 (𝓝 (g k))\nh_bound : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k\nh✝¹ : Nonempty β\nh✝ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Tannery | {
"line": 71,
"column": 4
} | {
"line": 71,
"column": 67
} | {
"line": 71,
"column": 68
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nG : Type u_3\n𝓕 : Filter α\ninst✝¹ : NormedAddCommGroup G\ninst✝ : CompleteSpace G\nf : α → β → G\ng : β → G\nbound : β → ℝ\nh_sum : Summable bound\nhab : ∀ (k : β), Tendsto (fun x ↦ f x k) 𝓕 (𝓝 (g k))\nh_bound : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k\nh✝¹ : Nonem... | [
"α : Type u_1\nβ : Type u_2\nG : Type u_3\n𝓕 : Filter α\ninst✝¹ : NormedAddCommGroup G\ninst✝ : CompleteSpace G\nf : α → β → G\ng : β → G\nbound : β → ℝ\nh_sum : Summable bound\nhab : ∀ (k : β), Tendsto (fun x ↦ f x k) 𝓕 (𝓝 (g k))\nh_bound : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k\nh✝¹ : Nonempty β\nh✝ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Tannery | {
"line": 79,
"column": 4
} | {
"line": 79,
"column": 59
} | {
"line": 79,
"column": 60
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\nG : Type u_3\n𝓕 : Filter α\ninst✝¹ : NormedAddCommGroup G\ninst✝ : CompleteSpace G\nf : α → β → G\ng : β → G\nbound : β → ℝ\nh_sum : Summable bound\nhab : ∀ (k : β), Tendsto (fun x ↦ f x k) 𝓕 (𝓝 (g k))\nh_bound✝ : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ boun... | [
"case refine_1\nα : Type u_1\nβ : Type u_2\nG : Type u_3\n𝓕 : Filter α\ninst✝¹ : NormedAddCommGroup G\ninst✝ : CompleteSpace G\nf : α → β → G\ng : β → G\nbound : β → ℝ\nh_sum : Summable bound\nhab : ∀ (k : β), Tendsto (fun x ↦ f x k) 𝓕 (𝓝 (g k))\nh_bound✝ : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k\nh✝¹ : N... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 858,
"column": 22
} | {
"line": 858,
"column": 56
} | {
"line": 859,
"column": 4
} | [
{
"pp": "R : Type u_1\nK : Type u\ninst✝⁴ : NontriviallyNormedField K\nL : Type v\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : Algebra.IsAlgebraic K L\nhu : IsUltrametricDist K\ninst✝ : CompleteSpace K\nx : L\n⊢ spectralNorm K L (x - x) = 0",
"ppTerm": "?m.94",
"assigned": true,
"usedConstants"... | [] | simp [sub_self, spectralNorm_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 858,
"column": 22
} | {
"line": 858,
"column": 56
} | {
"line": 859,
"column": 4
} | [
{
"pp": "R : Type u_1\nK : Type u\ninst✝⁴ : NontriviallyNormedField K\nL : Type v\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : Algebra.IsAlgebraic K L\nhu : IsUltrametricDist K\ninst✝ : CompleteSpace K\nx : L\n⊢ spectralNorm K L (x - x) = 0",
"ppTerm": "?m.94",
"assigned": true,
"usedConstants"... | [] | simp [sub_self, spectralNorm_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 858,
"column": 22
} | {
"line": 858,
"column": 56
} | {
"line": 859,
"column": 4
} | [
{
"pp": "R : Type u_1\nK : Type u\ninst✝⁴ : NontriviallyNormedField K\nL : Type v\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : Algebra.IsAlgebraic K L\nhu : IsUltrametricDist K\ninst✝ : CompleteSpace K\nx : L\n⊢ spectralNorm K L (x - x) = 0",
"ppTerm": "?m.94",
"assigned": true,
"usedConstants"... | [] | simp [sub_self, spectralNorm_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Module.Ball.RadialEquiv | {
"line": 41,
"column": 31
} | {
"line": 41,
"column": 62
} | {
"line": 41,
"column": 63
} | [
{
"pp": "E✝ : Type u_1\ninst✝³ : NormedAddCommGroup E✝\ninst✝² : NormedSpace ℝ E✝\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nr : ℝ\nhr : 0 < r\nx : ↑{0}ᶜ\n⊢ 0 < ‖↑x‖",
"ppTerm": "?m.177",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"No... | [
"E✝ : Type u_1\ninst✝³ : NormedAddCommGroup E✝\ninst✝² : NormedSpace ℝ E✝\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nr : ℝ\nhr : 0 < r\nx : ↑{0}ᶜ\n⊢ ¬↑x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Ball.RadialEquiv | {
"line": 47,
"column": 25
} | {
"line": 47,
"column": 36
} | {
"line": 47,
"column": 37
} | [
{
"pp": "E✝ : Type u_1\ninst✝³ : NormedAddCommGroup E✝\ninst✝² : NormedSpace ℝ E✝\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nr : ℝ\nhr : 0 < r\nx : E\nhx : x ∈ {0}ᶜ\n⊢ 0 < ‖x‖",
"ppTerm": "?m.204",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",... | [
"E✝ : Type u_1\ninst✝³ : NormedAddCommGroup E✝\ninst✝² : NormedSpace ℝ E✝\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nr : ℝ\nhr : 0 < r\nx : E\nhx : x ∈ {0}ᶜ\n⊢ ¬x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.ContinuousInverse | {
"line": 292,
"column": 26
} | {
"line": 292,
"column": 37
} | {
"line": 292,
"column": 38
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : Semiring R\nE : Type u_2\nF : Type u_4\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommMonoid E\ninst✝³ : Module R E\ninst✝² : TopologicalSpace F\ninst✝¹ : AddCommMonoid F\ninst✝ : Module R F\nf : E ≃L[R] F\ny : F\n⊢ f (⋯.rightInverse y) = f (↑f.symm y)",
"ppTerm": "?m.109",
... | [
"R : Type u_1\ninst✝⁶ : Semiring R\nE : Type u_2\nF : Type u_4\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommMonoid E\ninst✝³ : Module R E\ninst✝² : TopologicalSpace F\ninst✝¹ : AddCommMonoid F\ninst✝ : Module R F\nf : E ≃L[R] F\ny : F\n⊢ f (⋯.rightInverse y) = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.ContinuousInverse | {
"line": 320,
"column": 34
} | {
"line": 320,
"column": 45
} | {
"line": 320,
"column": 46
} | [
{
"pp": "R : Type u_1\ninst✝⁹ : Semiring R\nE : Type u_2\nF : Type u_4\nG : Type u_6\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : AddCommMonoid E\ninst✝⁶ : Module R E\ninst✝⁵ : TopologicalSpace F\ninst✝⁴ : AddCommMonoid F\ninst✝³ : Module R F\nf : E →L[R] F\ninst✝² : TopologicalSpace G\ninst✝¹ : AddCommMonoid G\ninst... | [
"R : Type u_1\ninst✝⁹ : Semiring R\nE : Type u_2\nF : Type u_4\nG : Type u_6\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : AddCommMonoid E\ninst✝⁶ : Module R E\ninst✝⁵ : TopologicalSpace F\ninst✝⁴ : AddCommMonoid F\ninst✝³ : Module R F\nf : E →L[R] F\ninst✝² : TopologicalSpace G\ninst✝¹ : AddCommMonoid G\ninst✝ : Module R... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.DoubleDual | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 13
} | {
"line": 74,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : E\n⊢ ‖(inclusionInDoubleDual 𝕜 E) x‖ ≤ ‖x‖",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : E\n⊢ ‖(inclusionInDoubleDual 𝕜 E) x‖ ≤ ‖x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 123,
"column": 12
} | {
"line": 123,
"column": 59
} | {
"line": 123,
"column": 60
} | [
{
"pp": "X : Type u_1\ninst✝³ : NormedAddCommGroup X\nM : Type u_2\ninst✝² : Ring M\ninst✝¹ : Module M X\ninst✝ : FaithfulSMul M X\nP Q : M\nh₁ : IsLprojection X P\nh₂ : IsLprojection X Q\nR : M\nh₃ : IsLprojection X R\nx : X\nthis : ‖R • x‖ + 2 • ‖(1 - R) • P • R • x‖ ≤ ‖R • P • R • x‖ + ‖R • x - R • P • R • x... | [
"X : Type u_1\ninst✝³ : NormedAddCommGroup X\nM : Type u_2\ninst✝² : Ring M\ninst✝¹ : Module M X\ninst✝ : FaithfulSMul M X\nP Q : M\nh₁ : IsLprojection X P\nh₂ : IsLprojection X Q\nR : M\nh₃ : IsLprojection X R\nx : X\nthis : ‖R • x‖ + 2 • ‖(1 - R) • P • R • x‖ ≤ ‖R • P • R • x‖ + ‖R • x - R • P • R • x‖ + 2 • ‖(1 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 131,
"column": 4
} | {
"line": 131,
"column": 29
} | {
"line": 131,
"column": 30
} | [
{
"pp": "X : Type u_1\ninst✝³ : NormedAddCommGroup X\nM : Type u_2\ninst✝² : Ring M\ninst✝¹ : Module M X\ninst✝ : FaithfulSMul M X\nP Q : M\nh₁ : IsLprojection X P\nh₂ : IsLprojection X Q\nPR_eq_RPR : ∀ (R : M), IsLprojection X R → P * R = R * P * R\ne1 : Q * P - Q * P * Q = 0\n⊢ Q * P = Q * P * Q",
"ppTerm... | [
"X : Type u_1\ninst✝³ : NormedAddCommGroup X\nM : Type u_2\ninst✝² : Ring M\ninst✝¹ : Module M X\ninst✝ : FaithfulSMul M X\nP Q : M\nh₁ : IsLprojection X P\nh₂ : IsLprojection X Q\nPR_eq_RPR : ∀ (R : M), IsLprojection X R → P * R = R * P * R\ne1 : Q * P - Q * P * Q = 0\n⊢ Q * P = Q * P * Q"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 191,
"column": 18
} | {
"line": 191,
"column": 50
} | {
"line": 191,
"column": 51
} | [
{
"pp": "X : Type u_1\ninst✝³ : NormedAddCommGroup X\nM : Type u_2\ninst✝² : Ring M\ninst✝¹ : Module M X\ninst✝ : FaithfulSMul M X\nP : { P // IsLprojection X P }\n⊢ ↑P = ↑(P ⊓ P)",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsLprojection",
"congrArg",
... | [
"X : Type u_1\ninst✝³ : NormedAddCommGroup X\nM : Type u_2\ninst✝² : Ring M\ninst✝¹ : Module M X\ninst✝ : FaithfulSMul M X\nP : { P // IsLprojection X P }\n⊢ ↑P = ↑P ^ 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Ball.RadialEquiv | {
"line": 85,
"column": 4
} | {
"line": 85,
"column": 15
} | {
"line": 85,
"column": 16
} | [
{
"pp": "case inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nr : ℝ\nhr : r ≠ 0\nU : Set ℝ\nV : Set ↑(sphere 0 r)\nhU : IsOpen U\nhU₀ : 0 ∉ U\nhV : IsOpen[instTopologicalSpaceSubtype] V\nx : ℝ\nhxU : x ∈ U\ny : { x // x ∈ sphere 0 r }\nhyV : y ∈ V\nhx₀ : 0 < -x\nthis : Neg.neg ⁻¹' (-(... | [
"case inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nr : ℝ\nhr : r ≠ 0\nU : Set ℝ\nV : Set ↑(sphere 0 r)\nhU : IsOpen U\nhU₀ : 0 ∉ U\nhV : IsOpen[instTopologicalSpaceSubtype] V\nx : ℝ\nhxU : x ∈ U\ny : { x // x ∈ sphere 0 r }\nhyV : y ∈ V\nhx₀ : 0 < -x\nthis : Neg.neg ⁻¹' (-(U • Subtype.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Ball.RadialEquiv | {
"line": 86,
"column": 41
} | {
"line": 86,
"column": 52
} | {
"line": 86,
"column": 53
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nr : ℝ\nhr : r ≠ 0\nV : Set ↑(sphere 0 r)\nhV : IsOpen[instTopologicalSpaceSubtype] V\ny : { x // x ∈ sphere 0 r }\nhyV : y ∈ V\nU : Set ℝ\nhU : IsOpen U\nhU₀ : 0 ∉ U\nx : ℝ\nhxU : x ∈ U\nhx₀ : 0 < x\n⊢ 0 ≤ r",
"ppTerm": "?m.207",... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nr : ℝ\nhr : r ≠ 0\nV : Set ↑(sphere 0 r)\nhV : IsOpen[instTopologicalSpaceSubtype] V\ny : { x // x ∈ sphere 0 r }\nhyV : y ∈ V\nU : Set ℝ\nhU : IsOpen U\nhU₀ : 0 ∉ U\nx : ℝ\nhxU : x ∈ U\nhx₀ : 0 < x\n⊢ 0 ≤ r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Summable | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 13
} | {
"line": 138,
"column": 14
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\ninst✝¹ : NormedCommRing R\ninst✝ : NormOneClass R\nf : ι → R\nhf : Summable fun i ↦ ‖f i‖\ni : ι\n⊢ ‖‖1 + f i‖ - 1‖ ≤ ‖f i‖",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"NormedCommRing.toS... | [
"ι : Type u_1\nR : Type u_2\ninst✝¹ : NormedCommRing R\ninst✝ : NormOneClass R\nf : ι → R\nhf : Summable fun i ↦ ‖f i‖\ni : ι\n⊢ |‖1 + f i‖ - 1| ≤ ‖f i‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Summable | {
"line": 163,
"column": 38
} | {
"line": 163,
"column": 53
} | {
"line": 163,
"column": 54
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\ninst✝¹ : NormedCommRing R\ninst✝ : NormOneClass R\nf✝ : ι → R\nhf : Summable fun i ↦ ‖f✝ i‖\nε : ℝ\nhε : 0 < ε\nf : ℝ → ℝ := fun x ↦ Real.exp x - 1\n⊢ Set.Iio ε ∈ 𝓝 (f 0)",
"ppTerm": "?m.144",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
... | [
"ι : Type u_1\nR : Type u_2\ninst✝¹ : NormedCommRing R\ninst✝ : NormOneClass R\nf✝ : ι → R\nhf : Summable fun i ↦ ‖f✝ i‖\nε : ℝ\nhε : 0 < ε\nf : ℝ → ℝ := fun x ↦ Real.exp x - 1\n⊢ Set.Iio ε ∈ 𝓝 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Summable | {
"line": 212,
"column": 4
} | {
"line": 212,
"column": 15
} | {
"line": 212,
"column": 16
} | [
{
"pp": "case left\nι : Type u_1\nR : Type u_2\ninst✝¹ : NormedCommRing R\ninst✝ : NormOneClass R\nf : ι → R\nhu : Summable fun n ↦ ‖f n‖\ni : ι\n⊢ -‖f i‖ ≤ ‖1 + f i‖ - 1",
"ppTerm": "?left",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Eq.mpr",... | [
"case left\nι : Type u_1\nR : Type u_2\ninst✝¹ : NormedCommRing R\ninst✝ : NormOneClass R\nf : ι → R\nhu : Summable fun n ↦ ‖f n‖\ni : ι\n⊢ 1 ≤ ‖1 + f i‖ + ‖f i‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Summable | {
"line": 213,
"column": 4
} | {
"line": 213,
"column": 26
} | {
"line": 213,
"column": 27
} | [
{
"pp": "case right\nι : Type u_1\nR : Type u_2\ninst✝¹ : NormedCommRing R\ninst✝ : NormOneClass R\nf : ι → R\nhu : Summable fun n ↦ ‖f n‖\ni : ι\n⊢ ‖1 + f i‖ - 1 ≤ ‖f i‖",
"ppTerm": "?right",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Eq.mpr"... | [
"case right\nι : Type u_1\nR : Type u_2\ninst✝¹ : NormedCommRing R\ninst✝ : NormOneClass R\nf : ι → R\nhu : Summable fun n ↦ ‖f n‖\ni : ι\n⊢ ‖f i + 1‖ ≤ ‖f i‖ + 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.MultipliableUniformlyOn | {
"line": 44,
"column": 75
} | {
"line": 44,
"column": 86
} | {
"line": 44,
"column": 87
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nK : Set α\nu : ι → ℝ\nf : ι → α → ℂ\nhu : Summable u\nh : ∀ᶠ (i : ι) in cofinite, ∀ x ∈ K, ‖f i x‖ ≤ u i\ni : ι\nhi : ∀ x ∈ K, ‖f i x‖ ≤ u i\nhi' : u i ≤ 1 / 2\nx : α\nhx : x ∈ K\n⊢ 3 / 2 * ‖f i x‖ ≤ 3 / 2 * u i",
"ppTerm": "?m.133",
"assigned": true,
"usedConsta... | [
"α : Type u_1\nι : Type u_2\nK : Set α\nu : ι → ℝ\nf : ι → α → ℂ\nhu : Summable u\nh : ∀ᶠ (i : ι) in cofinite, ∀ x ∈ K, ‖f i x‖ ≤ u i\ni : ι\nhi : ∀ x ∈ K, ‖f i x‖ ≤ u i\nhi' : u i ≤ 1 / 2\nx : α\nhx : x ∈ K\n⊢ ‖f i x‖ ≤ u i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.MultipliableUniformlyOn | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 46
} | {
"line": 69,
"column": 47
} | [
{
"pp": "case refine_1\nα : Type u_1\nι : Type u_2\ns : Set α\nf : ι → α → ℂ\nhf : SummableUniformlyOn (fun i x ↦ log (f i x)) s\nhfn : ∀ x ∈ s, ∀ (i : ι), f i x ≠ 0\nhg : BddAbove ((fun x ↦ (∑' (i : ι), log (f i x)).re) '' s)\nr : α → ℂ\nhr : HasSumUniformlyOn (fun i x ↦ log (f i x)) r s\n⊢ BddAbove ((fun x ↦ ... | [
"case refine_1\nα : Type u_1\nι : Type u_2\ns : Set α\nf : ι → α → ℂ\nhf : SummableUniformlyOn (fun i x ↦ log (f i x)) s\nhfn : ∀ x ∈ s, ∀ (i : ι), f i x ≠ 0\nhg : BddAbove ((fun x ↦ (∑' (i : ι), log (f i x)).re) '' s)\nr : α → ℂ\nhr : HasSumUniformlyOn (fun i x ↦ log (f i x)) r s\n⊢ BddAbove ((fun a ↦ (∑' (b : ι),... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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