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.LinearAlgebra.AffineSpace.Ceva | {
"line": 184,
"column": 68
} | {
"line": 184,
"column": 79
} | [
{
"pp": "k : 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 +... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Affine.Simplex | {
"line": 114,
"column": 82
} | {
"line": 132,
"column": 10
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : Module R V\ninst✝ : NormedAddTorsor V P\nn : ℕ\ns : Simplex R P n\nhr : s.Regular\n⊢ s.Equilateral",
"usedConstants": [
"Eq.mpr",
"instNeZeroNatHAdd_1",
... | by
refine ⟨dist (s.points 0) (s.points 1), fun i j hij ↦ ?_⟩
have hn : n ≠ 0 := by lia
by_cases hi : i = 1
· rw [hi, dist_comm]
rcases hr (Equiv.swap 0 j) with ⟨x, hx⟩
nth_rw 2 [← x.dist_eq]
simp_rw [← Function.comp_apply (f := x), ← hx]
simp only [comp_apply, Equiv.swap_apply_left]
convert!... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.AffineSpace.Ceva | {
"line": 186,
"column": 36
} | {
"line": 186,
"column": 54
} | [
{
"pp": "k : 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 +... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Ceva | {
"line": 187,
"column": 53
} | {
"line": 187,
"column": 83
} | [
{
"pp": "k : 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 +... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Ceva | {
"line": 188,
"column": 48
} | {
"line": 188,
"column": 78
} | [
{
"pp": "k : 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 +... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Ceva | {
"line": 190,
"column": 68
} | {
"line": 190,
"column": 79
} | [
{
"pp": "k : 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 +... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Polynomial.Factorization | {
"line": 49,
"column": 83
} | {
"line": 56,
"column": 82
} | [
{
"pp": "f : ℝ[X]\nn : ℕ\nhf : f.IsMonicOfDegree (n + 1)\n⊢ ∃ f₁ f₂, (f₁.IsMonicOfDegree 1 ∨ f₁.IsMonicOfDegree 2) ∧ f = f₁ * f₂",
"usedConstants": [
"Iff.mpr",
"NormedCommRing.toNormedRing",
"Irreducible.natDegree_le_two",
"Semigroup.toMul",
"Real",
"Polynomial.not_isUni... | by
obtain ⟨f₁, hm, hirr, f₂, hf₂⟩ :=
exists_monic_irreducible_factor f <| not_isUnit_of_natDegree_pos f <|
by grind [IsMonicOfDegree.natDegree_eq]
refine ⟨f₁, f₂, ?_, hf₂⟩
have help {P : ℕ → Prop} {m : ℕ} (hm₀ : 0 < m) (hm₂ : m ≤ 2) (h : P m) : P 1 ∨ P 2 := by
interval_cases m <;> tauto
exact help... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.AffineSpace.Ceva | {
"line": 192,
"column": 36
} | {
"line": 192,
"column": 54
} | [
{
"pp": "k : 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 +... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Ceva | {
"line": 173,
"column": 2
} | {
"line": 194,
"column": 66
} | [
{
"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 ... | · 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": 132,
"column": 37
} | {
"line": 132,
"column": 66
} | [
{
"pp": "X : Type u_1\nE : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : PreconnectedSpace X\ninst✝ : SeminormedAddCommGroup E\nf : X → E\nM : ℝ\nx : X\nhM : 0 < M\nhx : ‖f x‖ = M\nh : IsMinOn (fun x ↦ ‖f x‖) univ x\nhf : Continuous[inst✝², PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f\nH : ∀ {y : X}... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.GelfandMazur | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 47
} | [
{
"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‖ ≤ ‖... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.GelfandMazur | {
"line": 180,
"column": 4
} | {
"line": 181,
"column": 11
} | [
{
"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 : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Quaternion | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 41
} | [
{
"pp": "⊢ Continuous ⇑normSq",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.MatrixExponential | {
"line": 190,
"column": 10
} | {
"line": 190,
"column": 49
} | [
{
"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",
"usedConstants": []
}
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Quaternion | {
"line": 196,
"column": 2
} | {
"line": 196,
"column": 34
} | [
{
"pp": "⊢ Continuous fun q ↦ q.im",
"usedConstants": [
"Quaternion.coe",
"Eq.mpr",
"NegZeroClass.toNeg",
"Real",
"Continuous",
"Real.instZero",
"AddGroupWithOne.toAddGroup",
"congrArg",
"CommSemiring.toSemiring",
"AddGroupWithOne.toAddMonoidWithOn... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Quaternion | {
"line": 210,
"column": 4
} | {
"line": 210,
"column": 20
} | [
{
"pp": "α : Type u_1\nL : SummationFilter α\nf : α → ℝ\nr : ℝ\nh : HasSum (fun a ↦ ↑(f a)) (↑r) L\n⊢ HasSum f r L",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Quaternion | {
"line": 212,
"column": 16
} | {
"line": 212,
"column": 32
} | [
{
"pp": "α : Type u_1\nL : SummationFilter α\nf : α → ℝ\nr : ℝ\nh : HasSum f r L\n⊢ HasSum (fun a ↦ ↑(f a)) (↑r) L",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.MatrixExponential | {
"line": 195,
"column": 10
} | {
"line": 195,
"column": 49
} | [
{
"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",
"usedConstants": []
}
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Quaternion | {
"line": 216,
"column": 2
} | {
"line": 216,
"column": 18
} | [
{
"pp": "α : Type u_1\nL : SummationFilter α\nf : α → ℝ\n⊢ Summable (fun a ↦ ↑(f a)) L ↔ Summable f L",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Series | {
"line": 123,
"column": 2
} | {
"line": 123,
"column": 36
} | [
{
"pp": "z : ℂ\n⊢ HasSum (fun n ↦ z ^ (2 * n) / ↑(2 * n)!) (cosh z)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Series | {
"line": 127,
"column": 2
} | {
"line": 128,
"column": 9
} | [
{
"pp": "z : ℂ\n⊢ HasSum (fun n ↦ z ^ (2 * n + 1) / ↑(2 * n + 1)!) (sinh z)",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Complex.sinh",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"Monoid.toMulOneClass",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.Ultra | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 41
} | [
{
"pp": "R : Type u_1\ninst✝² : SeminormedRing R\ninst✝¹ : NormOneClass R\ninst✝ : IsUltrametricDist R\nx : R\n⊢ ‖x + 1‖ ≤ max ‖x‖ 1",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"SeminormedRing.toNorm",
"Real.instLE",
"Real",
"PartialOrder.toPreorder",
"AddGroupWithO... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.Ultra | {
"line": 64,
"column": 17
} | {
"line": 64,
"column": 80
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"AddMonoid.toAddSemigroup",
"congrArg",
"SeminormedAddGroup.toNNNorm",
"NNNorm.nnnorm",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.Ultra | {
"line": 74,
"column": 2
} | {
"line": 75,
"column": 9
} | [
{
"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",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Int.cast_natCast",
"congrArg",
"SeminormedAddGroup.toNNNorm",
"NNNorm.nnnorm",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.Ultra | {
"line": 74,
"column": 2
} | {
"line": 75,
"column": 9
} | [
{
"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",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast",
"Eq.mpr",
"NegZeroClass.toNeg",
"AddGroupWithOne.toAddGroup"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Ultra | {
"line": 49,
"column": 4
} | {
"line": 49,
"column": 31
} | [
{
"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‖",
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"Real.instLE",
"Real",
"SeminormedAddGroup.toAddGroup",
"Normed... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Ultra | {
"line": 51,
"column": 4
} | {
"line": 52,
"column": 36
} | [
{
"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‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Ultra | {
"line": 61,
"column": 19
} | {
"line": 61,
"column": 30
} | [
{
"pp": "R : Type u_1\ninst✝ : NormedDivisionRing R\nh : ∀ (x : R), ‖x‖ ≤ 1 → ‖x + 1‖ ≤ 1\nx : R\nH : 1 < ‖x‖\n⊢ x ≠ 0",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"DivisionSemiring.toGroupWithZero",
"NormedDivisionRing.toDivisionRing",
"DivisionRing.toDivisionSemiring",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Ultra | {
"line": 68,
"column": 4
} | {
"line": 68,
"column": 43
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.GelfandMazur | {
"line": 378,
"column": 2
} | {
"line": 379,
"column": 9
} | [
{
"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)‖}",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Ultra | {
"line": 31,
"column": 4
} | {
"line": 31,
"column": 15
} | [
{
"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)",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Ultra | {
"line": 114,
"column": 6
} | {
"line": 114,
"column": 50
} | [
{
"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‖",
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NonAssocSemiring.toAddCommMo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Ultra | {
"line": 118,
"column": 4
} | {
"line": 119,
"column": 26
} | [
{
"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",
"usedConstants": [
"one_pow",
"Norm.norm",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Ultra | {
"line": 129,
"column": 22
} | {
"line": 129,
"column": 89
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Ultra | {
"line": 134,
"column": 4
} | {
"line": 134,
"column": 33
} | [
{
"pp": "case inr.h.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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.TrivSqZeroExt | {
"line": 113,
"column": 4
} | {
"line": 113,
"column": 31
} | [
{
"pp": "𝕜 : Type u_1\nR : Type u_3\nM : Type u_4\ninst✝¹⁶ : Field 𝕜\ninst✝¹⁵ : CharZero 𝕜\ninst✝¹⁴ : Ring R\ninst✝¹³ : AddCommGroup M\ninst✝¹² : Algebra 𝕜 R\ninst✝¹¹ : Module 𝕜 M\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module Rᵐᵒᵖ M\ninst✝⁸ : SMulCommClass R Rᵐᵒᵖ M\ninst✝⁷ : IsScalarTower 𝕜 R M\ninst✝⁶ : IsScala... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.TrivSqZeroExt | {
"line": 114,
"column": 2
} | {
"line": 114,
"column": 43
} | [
{
"pp": "𝕜 : Type u_1\nR : Type u_3\nM : Type u_4\ninst✝¹⁶ : Field 𝕜\ninst✝¹⁵ : CharZero 𝕜\ninst✝¹⁴ : Ring R\ninst✝¹³ : AddCommGroup M\ninst✝¹² : Algebra 𝕜 R\ninst✝¹¹ : Module 𝕜 M\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module Rᵐᵒᵖ M\ninst✝⁸ : SMulCommClass R Rᵐᵒᵖ M\ninst✝⁷ : IsScalarTower 𝕜 R M\ninst✝⁶ : IsScala... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.TrivSqZeroExt | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 51
} | [
{
"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
} | [
{
"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
} | [
{
"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.Analysis.Normed.Unbundled.RingSeminorm | {
"line": 152,
"column": 4
} | {
"line": 152,
"column": 45
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.RingSeminorm | {
"line": 154,
"column": 4
} | {
"line": 154,
"column": 30
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.RingSeminorm | {
"line": 412,
"column": 6
} | {
"line": 419,
"column": 30
} | [
{
"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",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"... | 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": 77,
"column": 23
} | {
"line": 77,
"column": 51
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded | {
"line": 85,
"column": 38
} | {
"line": 85,
"column": 54
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded | {
"line": 102,
"column": 38
} | {
"line": 102,
"column": 54
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nx : R\nhx : 0 < f x\nf_mul : 1 ≤ c * f 1\nf_nonneg : 0 ≤ f 1\nh1 : f 1 = 0\n⊢ 1 ≤ 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded | {
"line": 115,
"column": 4
} | {
"line": 115,
"column": 21
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Real.instLE",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 33
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"div_le_iff₀",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded | {
"line": 125,
"column": 4
} | {
"line": 125,
"column": 20
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Real",
"instHDiv",
"HM... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Approximation | {
"line": 89,
"column": 6
} | {
"line": 89,
"column": 41
} | [
{
"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",
"usedConstants": [
"NormedCommR... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Approximation | {
"line": 93,
"column": 4
} | {
"line": 93,
"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\nthis :\n ‖∑ i ∈ Finset.range (g.natDegree + 1), eval a (C (g.coeff i - f.coeff i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Approximation | {
"line": 106,
"column": 8
} | {
"line": 106,
"column": 31
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Approximation | {
"line": 119,
"column": 40
} | {
"line": 119,
"column": 51
} | [
{
"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) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Approximation | {
"line": 120,
"column": 10
} | {
"line": 120,
"column": 21
} | [
{
"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) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Approximation | {
"line": 120,
"column": 32
} | {
"line": 120,
"column": 55
} | [
{
"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) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Approximation | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 13
} | [
{
"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 | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Approximation | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 16
} | [
{
"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
} | [
{
"pp": "case h.h\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\ni : ℕ\nhi : i ∈ Finset.Iio f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Approximation | {
"line": 160,
"column": 6
} | {
"line": 160,
"column": 54
} | [
{
"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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 24
} | [
{
"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))",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SeminormFromConst | {
"line": 135,
"column": 8
} | {
"line": 135,
"column": 63
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nc : R\nf : RingSeminorm R\nhf1 : f 1 ≤ 1\nhc : f c ≠ 0\nhpm : IsPowMul ⇑f\n⊢ Tendsto (seminormFromConst_seq c f 0) atTop (𝓝 0)",
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SeminormFromConst | {
"line": 232,
"column": 4
} | {
"line": 232,
"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\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 (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SeminormFromConst | {
"line": 262,
"column": 2
} | {
"line": 262,
"column": 21
} | [
{
"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 (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 210,
"column": 8
} | {
"line": 210,
"column": 36
} | [
{
"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) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.IsConjRoot | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 44
} | [
{
"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)",
"usedConstants": [
"Eq.mpr",
"Algebra.algebraMap",
"AddGroupWithOne.toAddGroup",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.IsConjRoot | {
"line": 227,
"column": 53
} | {
"line": 230,
"column": 34
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"instIsTorsionFreeOfIsDomainOfNoZeroSMulDivisors",
"congrArg",
... | 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": 274,
"column": 2
} | {
"line": 274,
"column": 30
} | [
{
"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",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 251,
"column": 2
} | {
"line": 251,
"column": 43
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"congrArg",
"Filter.Eventually",
"instArchimedeanNat",
"Preorder.toLE",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 293,
"column": 2
} | {
"line": 293,
"column": 18
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real.instLE",
"Real",
"DivInvMonoid... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Krasner | {
"line": 82,
"column": 25
} | {
"line": 82,
"column": 44
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁶ : NormedField L\ninst✝⁵ : NontriviallyNormedField K\ninst✝⁴ : CompleteSpace K\ninst✝³ : IsUltrametricDist K\ninst✝² : NormedAlgebra K L\ninst✝¹ : Algebra.IsAlgebraic K L\ninst✝ : Normal K L\nx y : L\nxsep : IsSeparable K x\nsp : (Polynomial.map (algebraMap K L) (minpo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Krasner | {
"line": 98,
"column": 10
} | {
"line": 98,
"column": 48
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁶ : NormedField L\ninst✝⁵ : NontriviallyNormedField K\ninst✝⁴ : CompleteSpace K\ninst✝³ : IsUltrametricDist K\ninst✝² : NormedAlgebra K L\ninst✝¹ : Algebra.IsAlgebraic K L\ninst✝ : Normal K L\nx y : L\nxsep : IsSeparable K x\nsp : (Polynomial.map (algebraMap K L) (minpo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Krasner | {
"line": 107,
"column": 12
} | {
"line": 107,
"column": 31
} | [
{
"pp": "case a\nK : Type u_1\nL : Type u_2\ninst✝⁶ : NormedField L\ninst✝⁵ : NontriviallyNormedField K\ninst✝⁴ : CompleteSpace K\ninst✝³ : IsUltrametricDist K\ninst✝² : NormedAlgebra K L\ninst✝¹ : Algebra.IsAlgebraic K L\ninst✝ : Normal K L\nx y : L\nxsep : IsSeparable K x\nsp : (Polynomial.map (algebraMap K L... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Krasner | {
"line": 108,
"column": 12
} | {
"line": 108,
"column": 46
} | [
{
"pp": "case a\nK : Type u_1\nL : Type u_2\ninst✝⁶ : NormedField L\ninst✝⁵ : NontriviallyNormedField K\ninst✝⁴ : CompleteSpace K\ninst✝³ : IsUltrametricDist K\ninst✝² : NormedAlgebra K L\ninst✝¹ : Algebra.IsAlgebraic K L\ninst✝ : Normal K L\nx y : L\nxsep : IsSeparable K x\nsp : (Polynomial.map (algebraMap K L... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Krasner | {
"line": 131,
"column": 6
} | {
"line": 131,
"column": 73
} | [
{
"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✝ : Algebra.IsAlgebraic K L\nx y : L\nxsep : IsSeparable K x\nsp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits\nyint... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Krasner | {
"line": 135,
"column": 34
} | {
"line": 135,
"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✝ : Algebra.IsAlgebraic K L\nx y : L\nxsep : IsSeparable K x\nsp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits\nyint... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 390,
"column": 10
} | {
"line": 403,
"column": 74
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\ns : ℕ → ℕ\nhs_le : ∀ (n : ℕ), s n ≤ n\nx : R\na : ℝ\na_in : a ∈ Set.Icc 0 1\nψ : ℕ → ℕ\nhψ_mono : StrictMono ψ\nhψ_lim : Tendsto ((fun n ↦ ↑(s n) / ↑n) ∘ ψ) atTop (𝓝 0)\nha : a = 0\n⊢ limsup (fun n ↦ μ (x ^ s (ψ n)) ^ (1 / ↑(ψ n))) a... | apply csInf_le_csInf _ (μ_nonempty μ hs_le ψ)
· intro b hb
simp only [eventually_map, eventually_atTop, ge_iff_le, Set.mem_setOf_eq] at hb ⊢
obtain ⟨m, hm⟩ := hb
use m
intro k hkm
apply le_trans _ (hm k hkm)
rw [rpow_mul (apply_nonneg μ x... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 390,
"column": 10
} | {
"line": 403,
"column": 74
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\ns : ℕ → ℕ\nhs_le : ∀ (n : ℕ), s n ≤ n\nx : R\na : ℝ\na_in : a ∈ Set.Icc 0 1\nψ : ℕ → ℕ\nhψ_mono : StrictMono ψ\nhψ_lim : Tendsto ((fun n ↦ ↑(s n) / ↑n) ∘ ψ) atTop (𝓝 0)\nha : a = 0\n⊢ limsup (fun n ↦ μ (x ^ s (ψ n)) ^ (1 / ↑(ψ n))) a... | apply csInf_le_csInf _ (μ_nonempty μ hs_le ψ)
· intro b hb
simp only [eventually_map, eventually_atTop, ge_iff_le, Set.mem_setOf_eq] at hb ⊢
obtain ⟨m, hm⟩ := hb
use m
intro k hkm
apply le_trans _ (hm k hkm)
rw [rpow_mul (apply_nonneg μ x... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Field.Krasner | {
"line": 143,
"column": 6
} | {
"line": 143,
"column": 38
} | [
{
"pp": "case refine_2\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✝ : Algebra.IsAlgebraic K L\nx y : L\nxsep : IsSeparable K x\nsp : (Polynomial.map (algebraMap K L) (minpoly K x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Instances | {
"line": 27,
"column": 59
} | {
"line": 27,
"column": 70
} | [
{
"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",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.partialOrder",
"Real",
"GroupW... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Instances | {
"line": 28,
"column": 56
} | {
"line": 28,
"column": 67
} | [
{
"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",
"usedConstants": [
"NormedField.toField",
"id",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.ProperSpace | {
"line": 48,
"column": 26
} | {
"line": 48,
"column": 53
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Completeness | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 35
} | [
{
"pp": "case a\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nh : ∀ (u : ℕ → E), (Summable fun x ↦ ‖u x‖) → ∃ a, Tendsto (fun n ↦ ∑ i ∈ range n, u i) atTop (𝓝 a)\nu : ℕ → E\nhu : CauchySeq u\nf : ℕ → ℕ\nhf₁ : StrictMono f\nhf₂ : Summable fun i ↦ ‖u (f (i + 1)) - u (f i)‖\nv : ℕ → E := fun n ↦ u (f (n + 1)) - u ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Dense | {
"line": 57,
"column": 10
} | {
"line": 57,
"column": 21
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Dense | {
"line": 61,
"column": 10
} | {
"line": 61,
"column": 21
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 513,
"column": 2
} | {
"line": 513,
"column": 82
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 570,
"column": 2
} | {
"line": 570,
"column": 20
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Dense | {
"line": 98,
"column": 10
} | {
"line": 98,
"column": 42
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Dense | {
"line": 102,
"column": 6
} | {
"line": 102,
"column": 86
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Dense | {
"line": 116,
"column": 6
} | {
"line": 116,
"column": 68
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Dense | {
"line": 118,
"column": 4
} | {
"line": 118,
"column": 69
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.HomCompletion | {
"line": 147,
"column": 4
} | {
"line": 147,
"column": 15
} | [
{
"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‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.HomCompletion | {
"line": 162,
"column": 2
} | {
"line": 162,
"column": 67
} | [
{
"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... | 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.Group.HomCompletion | {
"line": 170,
"column": 6
} | {
"line": 170,
"column": 36
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.HomCompletion | {
"line": 177,
"column": 16
} | {
"line": 177,
"column": 62
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 333,
"column": 43
} | {
"line": 336,
"column": 60
} | [
{
"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.Unbundled.SpectralNorm | {
"line": 348,
"column": 8
} | {
"line": 349,
"column": 60
} | [
{
"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 ≤ ⨆ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.SemiNormedGrp.Kernels | {
"line": 102,
"column": 4
} | {
"line": 102,
"column": 74
} | [
{
"pp": "case hf.H\nV 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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 359,
"column": 6
} | {
"line": 360,
"column": 61
} | [
{
"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.Unbundled.SpectralNorm | {
"line": 428,
"column": 2
} | {
"line": 428,
"column": 46
} | [
{
"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",
"usedConstants": [
"Real.partialOrder",
"Real",
"Real.instZero",
"spectralNorm",
"spectralNorm_nonneg",
... | apply lt_of_le_of_ne (spectralNorm_nonneg _) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Normed.Group.SeparationQuotient | {
"line": 124,
"column": 4
} | {
"line": 124,
"column": 46
} | [
{
"pp": "case h_above\nM : Type u_1\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : NontrivialTopology M\n⊢ ∀ (x : M), ‖normedMk x‖ ≤ 1 * ‖x‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"NormedAddGroupHom",
"HMul.hMul",
"SeparationQuotient.instNor... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.