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.NumberTheory.DirichletCharacter.Basic | {
"line": 264,
"column": 2
} | {
"line": 265,
"column": 68
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝ : CommMonoidWithZero R\nn : ℕ\nχ : DirichletCharacter R n\n⊢ n = 0 → χ.conductor = 0",
"usedConstants": [
"Iff.mpr",
"DirichletCharacter.conductor",
"Nat.sInf_eq_zero",
"Membership.mem",
"instOfNatNat",
"Nat.instInfSet",
"... | · rintro rfl
exact Nat.sInf_eq_zero.mpr <| Or.inl <| level_mem_conductorSet χ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 78,
"column": 2
} | {
"line": 85,
"column": 90
} | [
{
"pp": "n : ℕ\nhn : n % 2 = 1\n⊢ χ₄ ↑n = (-1) ^ (n / 2)",
"usedConstants": [
"one_pow",
"Eq.mpr",
"MulOne.toOne",
"False",
"Semigroup.toMul",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Nat.instIsOrderedAddMonoid",
"HMul.hMul",
"... | rw [χ₄_nat_eq_if_mod_four]
simp only [hn, Nat.one_ne_zero, if_false]
nth_rewrite 3 [← Nat.div_add_mod n 4]
nth_rewrite 3 [show 4 = 2 * 2 by lia]
rw [mul_assoc, add_comm, Nat.add_mul_div_left _ _ zero_lt_two, pow_add, pow_mul,
neg_one_sq, one_pow, mul_one]
have help : ∀ m : ℕ, m < 4 → m % 2 = 1 → ite (m = ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 78,
"column": 2
} | {
"line": 85,
"column": 90
} | [
{
"pp": "n : ℕ\nhn : n % 2 = 1\n⊢ χ₄ ↑n = (-1) ^ (n / 2)",
"usedConstants": [
"one_pow",
"Eq.mpr",
"MulOne.toOne",
"False",
"Semigroup.toMul",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Nat.instIsOrderedAddMonoid",
"HMul.hMul",
"... | rw [χ₄_nat_eq_if_mod_four]
simp only [hn, Nat.one_ne_zero, if_false]
nth_rewrite 3 [← Nat.div_add_mod n 4]
nth_rewrite 3 [show 4 = 2 * 2 by lia]
rw [mul_assoc, add_comm, Nat.add_mul_div_left _ _ zero_lt_two, pow_add, pow_mul,
neg_one_sq, one_pow, mul_one]
have help : ∀ m : ℕ, m < 4 → m % 2 = 1 → ite (m = ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 820,
"column": 6
} | {
"line": 820,
"column": 95
} | [
{
"pp": "case refine_3\nn : ℕ\ninst✝¹⁰ : NeZero n\nS T : Set ℕ\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst✝⁹ : CommRing A\ninst✝⁸ : CommRing B\ninst✝⁷ : Algebra A B\ninst✝⁶ : Field K\ninst✝⁵ : Field L\ninst✝⁴ : Algebra K L\ninst✝³ : Algebra A K\ninst✝² : IsFractionRing A K\ninst✝¹ : IsDomain A\ninst✝ ... | refine ⟨⟨a.1 * b.2 + b.1 * a.2, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 829,
"column": 33
} | {
"line": 829,
"column": 70
} | [
{
"pp": "n : ℕ\ninst✝¹⁰ : NeZero n\nS T : Set ℕ\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst✝⁹ : CommRing A\ninst✝⁸ : CommRing B\ninst✝⁷ : Algebra A B\ninst✝⁶ : Field K\ninst✝⁵ : Field L\ninst✝⁴ : Algebra K L\ninst✝³ : Algebra A K\ninst✝² : IsFractionRing A K\ninst✝¹ : IsDomain A\ninst✝ : NeZero ↑n\nx ... | rw [adjoin_algebra_injective n A K h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 829,
"column": 33
} | {
"line": 829,
"column": 70
} | [
{
"pp": "n : ℕ\ninst✝¹⁰ : NeZero n\nS T : Set ℕ\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst✝⁹ : CommRing A\ninst✝⁸ : CommRing B\ninst✝⁷ : Algebra A B\ninst✝⁶ : Field K\ninst✝⁵ : Field L\ninst✝⁴ : Algebra K L\ninst✝³ : Algebra A K\ninst✝² : IsFractionRing A K\ninst✝¹ : IsDomain A\ninst✝ : NeZero ↑n\nx ... | rw [adjoin_algebra_injective n A K h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 829,
"column": 33
} | {
"line": 829,
"column": 70
} | [
{
"pp": "n : ℕ\ninst✝¹⁰ : NeZero n\nS T : Set ℕ\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst✝⁹ : CommRing A\ninst✝⁸ : CommRing B\ninst✝⁷ : Algebra A B\ninst✝⁶ : Field K\ninst✝⁵ : Field L\ninst✝⁴ : Algebra K L\ninst✝³ : Algebra A K\ninst✝² : IsFractionRing A K\ninst✝¹ : IsDomain A\ninst✝ : NeZero ↑n\nx ... | rw [adjoin_algebra_injective n A K h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Fourier.ZMod | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 70
} | [
{
"pp": "N : ℕ\ninst✝² : NeZero N\nE : Type u_1\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℂ E\nΦ : ZMod N → E\n⊢ 𝓕 Φ 0 = ∑ j, Φ j",
"usedConstants": [
"Pi.Function.module",
"NegZeroClass.toNeg",
"MulOne.toOne",
"instHSMul",
"Pi.addCommMonoid",
"HMul.hMul",
"Finset.u... | simp only [dft_apply, mul_zero, neg_zero, map_zero_eq_one, one_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Fourier.ZMod | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 70
} | [
{
"pp": "N : ℕ\ninst✝² : NeZero N\nE : Type u_1\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℂ E\nΦ : ZMod N → E\n⊢ 𝓕 Φ 0 = ∑ j, Φ j",
"usedConstants": [
"Pi.Function.module",
"NegZeroClass.toNeg",
"MulOne.toOne",
"instHSMul",
"Pi.addCommMonoid",
"HMul.hMul",
"Finset.u... | simp only [dft_apply, mul_zero, neg_zero, map_zero_eq_one, one_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Fourier.ZMod | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 70
} | [
{
"pp": "N : ℕ\ninst✝² : NeZero N\nE : Type u_1\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℂ E\nΦ : ZMod N → E\n⊢ 𝓕 Φ 0 = ∑ j, Φ j",
"usedConstants": [
"Pi.Function.module",
"NegZeroClass.toNeg",
"MulOne.toOne",
"instHSMul",
"Pi.addCommMonoid",
"HMul.hMul",
"Finset.u... | simp only [dft_apply, mul_zero, neg_zero, map_zero_eq_one, one_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.GaussSum | {
"line": 113,
"column": 4
} | {
"line": 113,
"column": 34
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : Fintype R\nR' : Type v\ninst✝ : CommRing R'\nχ φ : MulChar R R'\nψ : AddChar R R'\nx : R\n⊢ ∑ y, χ x * φ y * ψ (x + y) = ∑ y, χ x * φ (y - x) * ψ y",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul"... | rw [sum_bij (fun a _ ↦ a + x)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 893,
"column": 4
} | {
"line": 899,
"column": 84
} | [
{
"pp": "n : ℕ\ninst✝⁴ : NeZero n\nA : Type u\nB : Type v\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsDomain B\nC : Subalgebra A B\nζ : B\nhζ : IsPrimitiveRoot ζ n\n⊢ C = adjoin A {x | ∃ n_1 ∈ {n}, n_1 ≠ 0 ∧ x ^ n_1 = 1} ↔ C = A[ζ]",
"usedConstants": [
"Subalgebra.instSe... | simp only [Set.mem_singleton_iff, exists_eq_left]
suffices adjoin A {b | n ≠ 0 ∧ b ^ n = 1} = adjoin A {ζ} by rw [this]
apply le_antisymm
· refine adjoin_le fun x ⟨_, hx⟩ ↦ ?_
obtain ⟨k, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx
exact Subalgebra.pow_mem _ (self_mem_adjoin_singleton A ζ) _
· exac... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 893,
"column": 4
} | {
"line": 899,
"column": 84
} | [
{
"pp": "n : ℕ\ninst✝⁴ : NeZero n\nA : Type u\nB : Type v\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsDomain B\nC : Subalgebra A B\nζ : B\nhζ : IsPrimitiveRoot ζ n\n⊢ C = adjoin A {x | ∃ n_1 ∈ {n}, n_1 ≠ 0 ∧ x ^ n_1 = 1} ↔ C = A[ζ]",
"usedConstants": [
"Subalgebra.instSe... | simp only [Set.mem_singleton_iff, exists_eq_left]
suffices adjoin A {b | n ≠ 0 ∧ b ^ n = 1} = adjoin A {ζ} by rw [this]
apply le_antisymm
· refine adjoin_le fun x ⟨_, hx⟩ ↦ ?_
obtain ⟨k, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx
exact Subalgebra.pow_mem _ (self_mem_adjoin_singleton A ζ) _
· exac... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.ZPow | {
"line": 280,
"column": 49
} | {
"line": 280,
"column": 75
} | [
{
"pp": "n' : Type u_1\ninst✝³ : DecidableEq n'\ninst✝² : Fintype n'\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : StarRing R\nA : M\nn : ℕ\n⊢ (A ^ (n + 1))⁻¹ᴴ = (Aᴴ ^ (n + 1))⁻¹",
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Comm... | conjTranspose_nonsing_inv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 138,
"column": 10
} | {
"line": 143,
"column": 57
} | [
{
"pp": "ι : Type u_1\nA : ι → Type u_2\ninst✝² : (i : ι) → MeasurableSpace (A i)\nμ : (i : ι) → Measure (A i)\ninst✝¹ : DecidableEq ι\np : ℝ\ninst✝ : ∀ (i : ι), SigmaFinite (μ i)\nhp₀ : 0 ≤ p\ns : Finset ι\nhp : ↑(#s) * p ≤ 1\ni : ι\nhi : i ∉ s\nf : ((i : ι) → A i) → ℝ≥0∞\nhf : Measurable f\n⊢ ∫⋯∫⁻_insert i s,... | rw [lmarginal_insert' _ _ hi]
· simp only [Pi.mul_apply, Pi.pow_apply, Finset.prod_apply]
· change Measurable (fun x ↦ _)
simp only [Pi.mul_apply, Pi.pow_apply, Finset.prod_apply]
refine (hf.pow_const _).mul <| Finset.measurable_fun_prod _ ?_
exact fun _ _ ↦ hf.lm... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 138,
"column": 10
} | {
"line": 143,
"column": 57
} | [
{
"pp": "ι : Type u_1\nA : ι → Type u_2\ninst✝² : (i : ι) → MeasurableSpace (A i)\nμ : (i : ι) → Measure (A i)\ninst✝¹ : DecidableEq ι\np : ℝ\ninst✝ : ∀ (i : ι), SigmaFinite (μ i)\nhp₀ : 0 ≤ p\ns : Finset ι\nhp : ↑(#s) * p ≤ 1\ni : ι\nhi : i ∉ s\nf : ((i : ι) → A i) → ℝ≥0∞\nhf : Measurable f\n⊢ ∫⋯∫⁻_insert i s,... | rw [lmarginal_insert' _ _ hi]
· simp only [Pi.mul_apply, Pi.pow_apply, Finset.prod_apply]
· change Measurable (fun x ↦ _)
simp only [Pi.mul_apply, Pi.pow_apply, Finset.prod_apply]
refine (hf.pow_const _).mul <| Finset.measurable_fun_prod _ ?_
exact fun _ _ ↦ hf.lm... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 186,
"column": 85
} | {
"line": 194,
"column": 44
} | [
{
"pp": "ι : Type u_1\nA : ι → Type u_2\ninst✝² : (i : ι) → MeasurableSpace (A i)\nμ : (i : ι) → Measure (A i)\ninst✝¹ : DecidableEq ι\np : ℝ\ninst✝ : ∀ (i : ι), SigmaFinite (μ i)\nhp₀ : 0 ≤ p\ns : Finset ι\nhp : ↑(#s) * p ≤ 1\ni : ι\nhi : i ∉ s\nf : ((i : ι) → A i) → ℝ≥0∞\nhf : Measurable f\nx : (i : ι) → A i\... | by
-- absorb the newly-created integrals into `∫⋯∫`
congr! 2
· rw [lmarginal_singleton]
refine prod_congr rfl fun j hj => ?_
have hi' : i ∉ ({j} : Finset ι) := by
simp only [Finset.mem_singleton] at hj ⊢
exact fun h ↦ ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 230,
"column": 6
} | {
"line": 230,
"column": 35
} | [
{
"pp": "case h.e'_4.h.e'_7\nι : Type u_1\nA : ι → Type u_2\ninst✝³ : (i : ι) → MeasurableSpace (A i)\nμ : (i : ι) → Measure (A i)\ninst✝² : DecidableEq ι\np : ℝ\ninst✝¹ : Fintype ι\ninst✝ : ∀ (i : ι), SigmaFinite (μ i)\nhp₀ : 0 ≤ p\nhp : (↑#ι - 1) * p ≤ 1\nf : ((i : ι) → A i) → ℝ≥0∞\nhf : Measurable f\ns : Fin... | rw [← insert_compl_insert hi] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.Rayleigh | {
"line": 114,
"column": 24
} | {
"line": 114,
"column": 34
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →L[𝕜] E\nx : E\n⊢ ‖T x‖ * ‖x‖ / ‖x‖ ^ 2 ≤ ‖T‖",
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.norm",
"InnerProductSpace.toNormedSpace",
"NormedCommRing... | le_opNorm, | Mathlib.Tactic.evalGRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Rayleigh | {
"line": 282,
"column": 2
} | {
"line": 282,
"column": 51
} | [
{
"pp": "case h.e'_7.h.e'_3\n𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nT : E →L[𝕜] E\nhT : IsSelfAdjoint T\nx₀ : E\nhx₀ : x₀ ≠ 0\nhextr : IsMaxOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀\nhx₀' : 0 < ‖x₀‖\nhx₀'' : x₀ ∈ ... | rw [T.iSup_rayleigh_eq_iSup_rayleigh_sphere hx₀'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.Rayleigh | {
"line": 339,
"column": 4
} | {
"line": 339,
"column": 30
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : RCLike 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\nT : E →ₗ[𝕜] E\ninst✝ : Nontrivial E\nhT : T.IsSymmetric\nthis✝¹ : ProperSpace E\nT' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint\nx : E\nhx : x ≠ 0\nH₁... | simpa [← norm_eq_zero, Ne] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.InnerProductSpace.Rayleigh | {
"line": 358,
"column": 4
} | {
"line": 358,
"column": 30
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : RCLike 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\nT : E →ₗ[𝕜] E\ninst✝ : Nontrivial E\nhT : T.IsSymmetric\nthis✝¹ : ProperSpace E\nT' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint\nx : E\nhx : x ≠ 0\nH₁... | simpa [← norm_eq_zero, Ne] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Data.Fin.Tuple.Sort | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 42
} | [
{
"pp": "n : ℕ\nα : Type u_1\ninst✝ : LinearOrder α\nf : Fin n → α\n⊢ Monotone graph.proj",
"usedConstants": [
"Lex",
"Finset",
"Membership.mem",
"Tuple.graph",
"Subtype",
"Finset.instSetLike",
"Prod",
"Fin",
"SetLike.instMembership"
]
}
] | rintro ⟨⟨x, i⟩, hx⟩ ⟨⟨y, j⟩, hy⟩ (_ | h) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Analysis.Matrix.Hermitian | {
"line": 36,
"column": 2
} | {
"line": 36,
"column": 64
} | [
{
"pp": "𝕜 : Type u_1\nn : Type u_3\nA : Matrix n n 𝕜\ninst✝ : RCLike 𝕜\nh : A.IsHermitian\ni : n\n⊢ ↑(re (A i i)) = A i i",
"usedConstants": [
"Eq.mpr",
"RCLike.star_def",
"Real",
"Matrix.IsHermitian.eq",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"AddMonoid.toA... | rw [← conj_eq_iff_re, ← star_def, ← conjTranspose_apply, h.eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Matrix.Hermitian | {
"line": 36,
"column": 2
} | {
"line": 36,
"column": 64
} | [
{
"pp": "𝕜 : Type u_1\nn : Type u_3\nA : Matrix n n 𝕜\ninst✝ : RCLike 𝕜\nh : A.IsHermitian\ni : n\n⊢ ↑(re (A i i)) = A i i",
"usedConstants": [
"Eq.mpr",
"RCLike.star_def",
"Real",
"Matrix.IsHermitian.eq",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"AddMonoid.toA... | rw [← conj_eq_iff_re, ← star_def, ← conjTranspose_apply, h.eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Matrix.Hermitian | {
"line": 36,
"column": 2
} | {
"line": 36,
"column": 64
} | [
{
"pp": "𝕜 : Type u_1\nn : Type u_3\nA : Matrix n n 𝕜\ninst✝ : RCLike 𝕜\nh : A.IsHermitian\ni : n\n⊢ ↑(re (A i i)) = A i i",
"usedConstants": [
"Eq.mpr",
"RCLike.star_def",
"Real",
"Matrix.IsHermitian.eq",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"AddMonoid.toA... | rw [← conj_eq_iff_re, ← star_def, ← conjTranspose_apply, h.eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 119,
"column": 80
} | {
"line": 121,
"column": 74
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nv : E\nhv : v ∈ (⨆ μ, eigenspace T μ)ᗮ\n⊢ T v ∈ (⨆ μ, eigenspace T μ)ᗮ",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
... | by
rw [← Submodule.iInf_orthogonal] at hv ⊢
exact T.iInf_invariant hT.invariant_orthogonalComplement_eigenspace v hv | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Matrix.Order | {
"line": 200,
"column": 28
} | {
"line": 201,
"column": 52
} | [
{
"pp": "𝕜 : Type u_1\nn : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nA : Matrix n n 𝕜\nhA : A.PosSemidef\n⊢ A.PosDef ↔ A.det ≠ 0",
"usedConstants": [
"NormedCommRing.toNormedRing",
"GroupWithZero.toMonoidWithZero",
"NormedRing.toRing",
"congrArg",
... | by
simp [hA.posDef_iff_isUnit, isUnit_iff_isUnit_det] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Matrix.Order | {
"line": 269,
"column": 35
} | {
"line": 269,
"column": 66
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nι : Type u_3\nA B : Matrix ι ι 𝕜\nhA : A.PosDef\nhB : B.PosDef\nx : ι →₀ 𝕜\nhx : x ≠ 0\nhAB : ((A ⊙ B).submatrix Subtype.val Subtype.val).PosDef\n⊢ 0 <\n ∑ x_1 ∈ Finset.subtype (fun x_1 ↦ x_1 ∈ x.support) x.support,\n ∑ x_2 ∈ Finset.subtype (fun x_2 ↦ x_2 ∈ x... | ← Finsupp.support_subtypeDomain | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs | {
"line": 156,
"column": 11
} | {
"line": 156,
"column": 96
} | [
{
"pp": "A : Type u_2\ninst✝¹⁶ : NonUnitalRing A\ninst✝¹⁵ : StarRing A\ninst✝¹⁴ : TopologicalSpace A\ninst✝¹³ : Module ℝ A\ninst✝¹² : SMulCommClass ℝ A A\ninst✝¹¹ : IsScalarTower ℝ A A\ninst✝¹⁰ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁹ : PartialOrder A\ninst✝⁸ : StarOrderedRing A\ninst✝⁷... | sqrt_eq_iff _ _ (star_mul_self_nonneg _) (smul_nonneg (by positivity) (abs_nonneg _)) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs | {
"line": 196,
"column": 2
} | {
"line": 196,
"column": 28
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁶ : RCLike 𝕜\ninst✝¹⁵ : NonUnitalRing A\ninst✝¹⁴ : TopologicalSpace A\ninst✝¹³ : Module 𝕜 A\ninst✝¹² : StarRing A\ninst✝¹¹ : PartialOrder A\ninst✝¹⁰ : StarOrderedRing A\ninst✝⁹ : IsScalarTower 𝕜 A A\ninst✝⁸ : SMulCommClass 𝕜 A A\ninst✝⁷ : NonUnitalCo... | simp [RCLike.conj_mul, sq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.InnerProductSpace.MeanErgodic | {
"line": 54,
"column": 14
} | {
"line": 54,
"column": 31
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : E →ₗ[𝕜] E\nhf : LipschitzWith 1 ⇑f\ng : E →L[𝕜] ↥(eqLocus f 1)\nhg_proj : ∀ (x : ↥(eqLocus f 1)), g ↑x = x\nhg_ker : ↑(↑g).ker ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑... | by simp [hg_proj] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.LinearPMap | {
"line": 212,
"column": 49
} | {
"line": 218,
"column": 65
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nA : E →L[𝕜] F\np : Submodule 𝕜 E\nhp : Dense ↑p\n⊢ ((↑A).to... | by
ext x y hxy
· simp only [LinearMap.toPMap_domain, Submodule.mem_top, iff_true,
LinearPMap.mem_adjoint_domain_iff]
exact ((innerSL 𝕜 x).comp <| A.comp <| Submodule.subtypeL _).cont
refine LinearPMap.adjoint_apply_eq hp _ fun v => ?_
simp only [adjoint_inner_left, LinearMap.toPMap_apply, coe_coe] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.LinearPMap | {
"line": 280,
"column": 4
} | {
"line": 280,
"column": 46
} | [
{
"pp": "case mp\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\ng : Submodule 𝕜 (E × F)\nx : F × E\ny : WithLp 2 (F × E)\nh1 :\n ∀ (u : WithLp 2 (F × E)) (x : E) (x_1... | specialize h1 (toLp 2 (b, -a)) a b hab rfl | Lean.Elab.Tactic.evalSpecialize | Lean.Parser.Tactic.specialize |
Mathlib.Analysis.InnerProductSpace.SingularValues | {
"line": 101,
"column": 22
} | {
"line": 101,
"column": 46
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : RCLike 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : FiniteDimensional 𝕜 E\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 F\ninst✝ : FiniteDimensional 𝕜 F\nT : E →ₗ[𝕜] F\ni : ℕ\n⊢ 0 ≤ (Finsupp.embDomain... | Finsupp.embDomain_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.LocallyConvex.ContinuousOfBounded | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 58
} | [
{
"pp": "𝕜 : Type u_1\n𝕜' : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝¹¹ : AddCommGroup E\ninst✝¹⁰ : TopologicalSpace E\ninst✝⁹ : IsTopologicalAddGroup E\ninst✝⁸ : AddCommGroup F\ninst✝⁷ : TopologicalSpace F\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : ContinuousSMul 𝕜 E\ninst✝³ : No... | obtain ⟨c, hc0, hc1⟩ := NormedField.exists_norm_lt_one 𝕜 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.LocallyConvex.ContinuousOfBounded | {
"line": 90,
"column": 6
} | {
"line": 90,
"column": 11
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\n𝕜' : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝¹¹ : AddCommGroup E\ninst✝¹⁰ : TopologicalSpace E\ninst✝⁹ : IsTopologicalAddGroup E\ninst✝⁸ : AddCommGroup F\ninst✝⁷ : TopologicalSpace F\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : ContinuousSMul 𝕜... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Analysis.InnerProductSpace.Reproducing | {
"line": 203,
"column": 2
} | {
"line": 204,
"column": 88
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nX : Type u_2\nV : Type u_3\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace 𝕜 V\ninst✝ : CompleteSpace V\nK : Matrix X X (V →L[𝕜] V)\nthis : ∀ {h p1 p2 p3 : Prop}, (h → [p1, p2, p3].TFAE) → [h ∧ p1, h ∧ p2, h ∧ p3].TFAE\nhHerm : K.IsHermitia... | tfae_have 2 → 3 := fun h vv ↦ by
simpa [add_mul, Finsupp.sum_sum_index] using (h (vv.sum fun x v ↦ .single ⟨x, v⟩ 1)) | Mathlib.Tactic.TFAE._aux_Mathlib_Tactic_TFAE___macroRules_Mathlib_Tactic_TFAE_tfaeHave_1 | Mathlib.Tactic.TFAE.tfaeHave |
Mathlib.Analysis.InnerProductSpace.Reproducing | {
"line": 220,
"column": 25
} | {
"line": 225,
"column": 46
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁸ : RCLike 𝕜\nX : Type u_2\nV : Type u_3\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace 𝕜 V\nH : Type u_4\ninst✝⁵ : NormedAddCommGroup H\ninst✝⁴ : InnerProductSpace 𝕜 H\ninst✝³ : RKHS 𝕜 H X V\ninst✝² : CompleteSpace H\ninst✝¹ : CompleteSpace V\nK : Matrix X X (V →L[�... | by
rw [Finsupp.sum_comm]
simp only [map_finsuppSum]
congr! 6
rw [← (Fact.out : K.PosSemidef).isHermitian.apply]
simp [star, adjoint_inner_right, mul_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Matrix.LDL | {
"line": 116,
"column": 17
} | {
"line": 116,
"column": 43
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : RCLike 𝕜\nn : Type u_2\ninst✝³ : LinearOrder n\ninst✝² : WellFoundedLT n\ninst✝¹ : LocallyFiniteOrderBot n\nS : Matrix n n 𝕜\ninst✝ : Fintype n\nhS : S.PosDef\n⊢ (lowerInv hS)⁻¹ * diag hS * (lowerInv hS)⁻¹ᴴ = S",
"usedConstants": [
"LDL.lowerInv",
"Eq.mpr",
... | conjTranspose_nonsing_inv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.TwoDim | {
"line": 566,
"column": 2
} | {
"line": 566,
"column": 41
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\nf : E ≃ₗᵢ[ℝ] ℂ\nhf : (map (Fin 2) f.toLinearEquiv) o = Complex.orientation\nx y : E\n⊢ (o.kahler x) y = f y * (starRingEnd ℂ) (f x)",
"usedConstants": [
"i... | rw [← Complex.kahler, ← hf, kahler_map] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.LocallyConvex.WeakOperatorTopology | {
"line": 347,
"column": 2
} | {
"line": 347,
"column": 30
} | [
{
"pp": "𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝¹⁰ : NormedField 𝕜₁\ninst✝⁹ : NormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nE : Type u_3\nF : Type u_4\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : Module 𝕜₁ E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : TopologicalSpace F\ninst✝³ : Module 𝕜₂ F\ninst✝² : IsTopologi... | simpa [funext_iff] using hAB | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.MellinTransform | {
"line": 234,
"column": 2
} | {
"line": 258,
"column": 12
} | [
{
"pp": "b : ℝ\nf : ℝ → ℝ\nhfc : AEStronglyMeasurable f (volume.restrict (Ioi 0))\nhf : f =O[𝓝[>] 0] fun x ↦ x ^ (-b)\ns : ℝ\nhs : b < s\n⊢ ∃ c, 0 < c ∧ IntegrableOn (fun t ↦ t ^ (s - 1) * f t) (Ioc 0 c) volume",
"usedConstants": [
"MeasureTheory.ae",
"Iff.mpr",
"NormedCommRing.toNormedRi... | obtain ⟨d, _, hd'⟩ := hf.exists_pos
simp_rw [IsBigOWith, eventually_nhdsWithin_iff, Metric.eventually_nhds_iff, gt_iff_lt] at hd'
obtain ⟨ε, hε, hε'⟩ := hd'
refine ⟨ε, hε, Iff.mpr integrableOn_Ioc_iff_integrableOn_Ioo ⟨?_, ?_⟩⟩
· refine AEStronglyMeasurable.mul ?_ (hfc.mono_set Ioo_subset_Ioi_self)
refine (... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.MellinTransform | {
"line": 234,
"column": 2
} | {
"line": 258,
"column": 12
} | [
{
"pp": "b : ℝ\nf : ℝ → ℝ\nhfc : AEStronglyMeasurable f (volume.restrict (Ioi 0))\nhf : f =O[𝓝[>] 0] fun x ↦ x ^ (-b)\ns : ℝ\nhs : b < s\n⊢ ∃ c, 0 < c ∧ IntegrableOn (fun t ↦ t ^ (s - 1) * f t) (Ioc 0 c) volume",
"usedConstants": [
"MeasureTheory.ae",
"Iff.mpr",
"NormedCommRing.toNormedRi... | obtain ⟨d, _, hd'⟩ := hf.exists_pos
simp_rw [IsBigOWith, eventually_nhdsWithin_iff, Metric.eventually_nhds_iff, gt_iff_lt] at hd'
obtain ⟨ε, hε, hε'⟩ := hd'
refine ⟨ε, hε, Iff.mpr integrableOn_Ioc_iff_integrableOn_Ioo ⟨?_, ?_⟩⟩
· refine AEStronglyMeasurable.mul ?_ (hfc.mono_set Ioo_subset_Ioi_self)
refine (... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Complex.LogBounds | {
"line": 129,
"column": 10
} | {
"line": 129,
"column": 29
} | [
{
"pp": "t : ℝ\nht : 0 ≤ t ∧ t ≤ 1\nz : ℂ\nhz : ‖z‖ < 1\n⊢ 1 - ‖↑t * z‖ ≤ ‖1 + ↑t * z‖",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NegZeroClass.toNeg",
"NormedCommRing.toSeminormedCommRing",
"Real.instLE"... | ← norm_neg (t * z), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Complex.LogBounds | {
"line": 158,
"column": 4
} | {
"line": 158,
"column": 75
} | [
{
"pp": "n : ℕ\nz : ℂ\nhz : ‖z‖ < 1\nhelp : IntervalIntegrable (fun t ↦ t ^ n * (1 - ‖z‖)⁻¹) MeasureTheory.volume 0 1\nf : ℂ → ℂ := fun z ↦ log (1 + z) - logTaylor (n + 1) z\nf' : ℂ → ℂ := fun z ↦ (-z) ^ n * (1 + z)⁻¹\nhderiv : ∀ t ∈ Set.Icc 0 1, HasDerivAt f (f' (0 + ↑t * z)) (0 + ↑t * z)\nhcont : ContinuousOn... | convert! (integral_unitInterval_deriv_eq_sub hcont hderiv).symm using 1 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Topology.Algebra.AsymptoticCone | {
"line": 117,
"column": 2
} | {
"line": 117,
"column": 52
} | [
{
"pp": "k : Type u_1\nV : Type u_2\ninst✝⁹ : Field k\ninst✝⁸ : LinearOrder k\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module k V\ninst✝⁵ : TopologicalSpace V\ninst✝⁴ : TopologicalSpace k\ninst✝³ : OrderTopology k\ninst✝² : IsStrictOrderedRing k\ninst✝¹ : IsTopologicalAddGroup V\ninst✝ : ContinuousSMul k V\n⊢ asympto... | rw [← top_le_iff, ← iSup_pure_eq_top, iSup_le_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.AsymptoticCone | {
"line": 250,
"column": 2
} | {
"line": 250,
"column": 34
} | [
{
"pp": "case refine_2\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝¹⁰ : Field k\ninst✝⁹ : LinearOrder k\ninst✝⁸ : AddCommGroup V\ninst✝⁷ : Module k V\ninst✝⁶ : AddTorsor V P\ninst✝⁵ : TopologicalSpace V\ninst✝⁴ : TopologicalSpace k\ninst✝³ : OrderTopology k\ninst✝² : IsStrictOrderedRing k\ninst✝¹ : IsTopolo... | · simpa [mem_asymptoticCone_iff] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Affine.MazurUlam | {
"line": 151,
"column": 55
} | {
"line": 153,
"column": 5
} | [
{
"pp": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : MetricSpace PE\ninst✝⁴ : NormedAddTorsor E PE\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : MetricSpace PF\ninst✝ : NormedAddTorsor F PF\nf : PE ≃ᵢ PF\n⊢ f.to... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Algebra.QuaternionExponential | {
"line": 47,
"column": 65
} | {
"line": 47,
"column": 68
} | [
{
"pp": "q : ℍ\nhq : q.re = 0\nn : ℕ\nhq2 : q ^ 2 = -↑(normSq q)\nk : ℝ := ↑(2 * n)!\n⊢ k⁻¹ • (q ^ 2) ^ n = k⁻¹ • (-↑(normSq q)) ^ n",
"usedConstants": [
"Quaternion.coe",
"Eq.mpr",
"NegZeroClass.toNeg",
"Real",
"instHSMul",
"instSMulOfMul",
"Real.instZero",
"... | hq2 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Series | {
"line": 66,
"column": 33
} | {
"line": 66,
"column": 84
} | [
{
"pp": "case h.e'_6\nz : ℂ\nthis :\n HasSum\n (fun x ↦\n ((-z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)! - (z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)!) * I / 2)\n ((NormedSpace.exp (-z * I) - NormedSpace.exp (z * I)) * I / 2)\nk : ℕ\n⊢ (z ^ 2) ^ k * ((I ^ 2) ^ k * (z * I)) / ↑(2 * k + 1)! / I... | mul_div_cancel_left₀ _ (two_ne_zero : (2 : ℂ) ≠ 0), | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Normed.Algebra.QuaternionExponential | {
"line": 70,
"column": 85
} | {
"line": 70,
"column": 88
} | [
{
"pp": "q : ℍ\nhq : q.re = 0\nn : ℕ\nhq0 : q ≠ 0\nhq2 : q ^ 2 = -↑(normSq q)\nhqn : ‖q‖ ≠ 0\nk : ℝ := ↑(2 * n + 1)!\n⊢ k⁻¹ • ((q ^ 2) ^ n * q) = k⁻¹ • ((-↑(normSq q)) ^ n * q)",
"usedConstants": [
"Quaternion.coe",
"Eq.mpr",
"NegZeroClass.toNeg",
"Real",
"instHSMul",
"in... | hq2 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Series | {
"line": 123,
"column": 2
} | {
"line": 123,
"column": 56
} | [
{
"pp": "z : ℂ\n⊢ HasSum (fun n ↦ z ^ (2 * n) / ↑(2 * n)!) (cosh z)",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Complex.cos_mul_I",
"Semigroup.toMul",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"even_two._simp_1",
... | simpa [mul_assoc, cos_mul_I] using hasSum_cos' (z * I) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Series | {
"line": 123,
"column": 2
} | {
"line": 123,
"column": 56
} | [
{
"pp": "z : ℂ\n⊢ HasSum (fun n ↦ z ^ (2 * n) / ↑(2 * n)!) (cosh z)",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Complex.cos_mul_I",
"Semigroup.toMul",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"even_two._simp_1",
... | simpa [mul_assoc, cos_mul_I] using hasSum_cos' (z * I) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Series | {
"line": 123,
"column": 2
} | {
"line": 123,
"column": 56
} | [
{
"pp": "z : ℂ\n⊢ HasSum (fun n ↦ z ^ (2 * n) / ↑(2 * n)!) (cosh z)",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Complex.cos_mul_I",
"Semigroup.toMul",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"even_two._simp_1",
... | simpa [mul_assoc, cos_mul_I] using hasSum_cos' (z * I) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Algebra.TrivSqZeroExt | {
"line": 232,
"column": 4
} | {
"line": 232,
"column": 22
} | [
{
"pp": "𝕜 : Type u_1\nS : Type u_2\nR : Type u_3\nM : Type u_4\ninst✝¹¹ : SeminormedCommRing S\ninst✝¹⁰ : SeminormedRing R\ninst✝⁹ : SeminormedAddCommGroup M\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : IsBoundedSMul S R\ninst✝⁵ : IsBoundedSMul S M\ninst✝⁴ : Module R M\ninst✝³ : IsBoundedSMul R M\nins... | simp_rw [norm_def] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Topology.Algebra.Order.LiminfLimsup | {
"line": 228,
"column": 6
} | {
"line": 228,
"column": 46
} | [
{
"pp": "ι : Type u_1\nR : Type u_4\ninst✝⁶ : ConditionallyCompleteLinearOrder R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : OrderTopology R\nF : Filter ι\ninst✝³ : AddCommSemigroup R\ninst✝² : Sub R\ninst✝¹ : ContinuousSub R\ninst✝ : OrderedSub R\nf : ι → R\nc : R\nbdd_above : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) F... | exact fun _ _ h ↦ tsub_le_tsub_right h c | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Algebra.Order.LiminfLimsup | {
"line": 228,
"column": 6
} | {
"line": 228,
"column": 46
} | [
{
"pp": "ι : Type u_1\nR : Type u_4\ninst✝⁶ : ConditionallyCompleteLinearOrder R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : OrderTopology R\nF : Filter ι\ninst✝³ : AddCommSemigroup R\ninst✝² : Sub R\ninst✝¹ : ContinuousSub R\ninst✝ : OrderedSub R\nf : ι → R\nc : R\nbdd_above : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) F... | exact fun _ _ h ↦ tsub_le_tsub_right h c | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Order.LiminfLimsup | {
"line": 228,
"column": 6
} | {
"line": 228,
"column": 46
} | [
{
"pp": "ι : Type u_1\nR : Type u_4\ninst✝⁶ : ConditionallyCompleteLinearOrder R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : OrderTopology R\nF : Filter ι\ninst✝³ : AddCommSemigroup R\ninst✝² : Sub R\ninst✝¹ : ContinuousSub R\ninst✝ : OrderedSub R\nf : ι → R\nc : R\nbdd_above : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) F... | exact fun _ _ h ↦ tsub_le_tsub_right h c | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 159,
"column": 2
} | {
"line": 159,
"column": 81
} | [
{
"pp": "case h\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) / ... | suffices h : smoothingSeminormSeq μ x n < L + ε by rwa [tsub_lt_iff_left hL_le] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Analysis.Normed.Unbundled.SeminormFromConst | {
"line": 157,
"column": 18
} | {
"line": 157,
"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 y : R\nn : ℕ\n⊢ ∃ a, n ≤ 2 * a",
"usedConstants": [
"HMul.hMul",
"Preorder.toLE",
"instMulNat",
"instOfNatNat",
"LE.le",
"Nat.instPreorder",
"Nat"... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Analysis.Normed.Unbundled.SeminormFromConst | {
"line": 191,
"column": 14
} | {
"line": 191,
"column": 19
} | [
{
"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\nm : ℕ\nhm : 1 ≤ m\nn : ℕ\n⊢ ∃ a, n ≤ m * a",
"usedConstants": [
"HMul.hMul",
"Preorder.toLE",
"instMulNat",
"LE.le",
"Nat.instPreorder",
"Nat",
... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Analysis.Normed.Unbundled.SeminormFromConst | {
"line": 253,
"column": 14
} | {
"line": 253,
"column": 19
} | [
{
"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\nn : ℕ\n⊢ ∃ a, n ≤ a + 1",
"usedConstants": [
"Preorder.toLE",
"instOfNatNat",
"LE.le",
"instHAdd",
"HAdd.hAdd",
"Nat.instPreorder",
"Nat",
... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 226,
"column": 80
} | {
"line": 226,
"column": 95
} | [
{
"pp": "R : 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)) * (... | by rw [pow_add] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Field.Dense | {
"line": 75,
"column": 10
} | {
"line": 75,
"column": 63
} | [
{
"pp": "case pos.h\nK : 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\nfnat... | ← (isConjRoot_iff_mem_minpoly_rootSet ⟨f, fmon, fa0⟩) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Group.ControlledClosure | {
"line": 37,
"column": 2
} | {
"line": 37,
"column": 50
} | [
{
"pp": "G : Type u_1\ninst✝² : NormedAddCommGroup G\ninst✝¹ : CompleteSpace G\nH : Type u_2\ninst✝ : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC ε : ℝ\nhC : 0 < C\nhε : 0 < ε\nhyp : f.SurjectiveOnWith K C\n⊢ f.SurjectiveOnWith K.topologicalClosure (C + ε)",
"usedConstants": []
}... | rintro (h : H) (h_in : h ∈ K.topologicalClosure) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 253,
"column": 49
} | {
"line": 253,
"column": 54
} | [
{
"pp": "K : Type u_2\ninst✝² : NormedField K\nL : Type u_3\ninst✝¹ : Field L\ninst✝ : Algebra K L\nf : AlgebraNorm K L\nhf_pm : IsPowMul ⇑f\nhf_na : IsNonarchimedean ⇑f\np : K[X]\nhp : p.Monic\nx : L\nhx : (aeval x) p = 0\nhx0 : ¬f x = 0\nh_ge : ∀ x_1 ∈ Set.range (spectralValueTerms p), x_1 < f x\nn : ℕ\nhn : ... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 258,
"column": 4
} | {
"line": 258,
"column": 81
} | [
{
"pp": "case neg\nK : Type u_2\ninst✝² : NormedField K\nL : Type u_3\ninst✝¹ : Field L\ninst✝ : Algebra K L\nf : AlgebraNorm K L\nhf_pm : IsPowMul ⇑f\nhf_na : IsNonarchimedean ⇑f\np : K[X]\nhp : p.Monic\nx : L\nhx : (aeval x) p = 0\nhx0 : ¬f x = 0\nh_ge : ¬f x ≤ spectralValue p\nhn_lt : ∀ n < p.natDegree, ‖p.c... | have h_deg : 0 < p.natDegree := natDegree_pos_of_monic_of_aeval_eq_zero hp hx | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Group.HomCompletion | {
"line": 147,
"column": 4
} | {
"line": 147,
"column": 40
} | [
{
"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": [
"Norm.norm",
"UniformSpace.Completion.coe'",
"Real.instLE",
"Real",
"NormedAddGroupHom... | simpa using f.completion.le_opNorm x | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Normed.Group.HomCompletion | {
"line": 147,
"column": 4
} | {
"line": 147,
"column": 40
} | [
{
"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": [
"Norm.norm",
"UniformSpace.Completion.coe'",
"Real.instLE",
"Real",
"NormedAddGroupHom... | simpa using f.completion.le_opNorm x | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Group.HomCompletion | {
"line": 147,
"column": 4
} | {
"line": 147,
"column": 40
} | [
{
"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": [
"Norm.norm",
"UniformSpace.Completion.coe'",
"Real.instLE",
"Real",
"NormedAddGroupHom... | simpa using f.completion.le_opNorm x | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 303,
"column": 6
} | {
"line": 306,
"column": 71
} | [
{
"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... | have hx0 : aeval x p = 0 := aeval_root_of_mapAlg_eq_multiset_prod_X_sub_C s hx hp
rw [if_pos hx]
exact norm_root_le_spectralValue hf_pm hf_na
(monic_of_monic_mapAlg (hp ▸ monic_multisetProd_X_sub_C s)) hx0 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 303,
"column": 6
} | {
"line": 306,
"column": 71
} | [
{
"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... | have hx0 : aeval x p = 0 := aeval_root_of_mapAlg_eq_multiset_prod_X_sub_C s hx hp
rw [if_pos hx]
exact norm_root_le_spectralValue hf_pm hf_na
(monic_of_monic_mapAlg (hp ▸ monic_multisetProd_X_sub_C s)) hx0 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 451,
"column": 2
} | {
"line": 451,
"column": 47
} | [
{
"pp": "K : Type u_2\ninst✝² : NormedField K\nL : Type u_3\ninst✝¹ : Field L\ninst✝ : Algebra K L\nσ : Gal(L/K)\nx : L\n⊢ spectralNorm K L x = spectralNorm K L (σ x)",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real",
"congrArg",
"CommSemiring.toSemiring",
"Nor... | simp only [spectralNorm, minpoly.algEquiv_eq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 451,
"column": 2
} | {
"line": 451,
"column": 47
} | [
{
"pp": "K : Type u_2\ninst✝² : NormedField K\nL : Type u_3\ninst✝¹ : Field L\ninst✝ : Algebra K L\nσ : Gal(L/K)\nx : L\n⊢ spectralNorm K L x = spectralNorm K L (σ x)",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real",
"congrArg",
"CommSemiring.toSemiring",
"Nor... | simp only [spectralNorm, minpoly.algEquiv_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 451,
"column": 2
} | {
"line": 451,
"column": 47
} | [
{
"pp": "K : Type u_2\ninst✝² : NormedField K\nL : Type u_3\ninst✝¹ : Field L\ninst✝ : Algebra K L\nσ : Gal(L/K)\nx : L\n⊢ spectralNorm K L x = spectralNorm K L (σ x)",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real",
"congrArg",
"CommSemiring.toSemiring",
"Nor... | simp only [spectralNorm, minpoly.algEquiv_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 472,
"column": 2
} | {
"line": 487,
"column": 48
} | [
{
"pp": "K : Type u_2\ninst✝² : NormedField K\nL : Type u_3\ninst✝¹ : Field L\ninst✝ : Algebra K L\nh_fin : FiniteDimensional K L\nhn : Normal K L\nf : AlgebraNorm K L\nhf_pm : IsPowMul ⇑f\nhf_na : IsNonarchimedean ⇑f\nhf_ext : ∀ (x : K), f ((algebraMap K L) x) = ‖x‖\nx : L\nhf1 : f 1 = 1\n⊢ spectralNorm K L x ... | · set p := minpoly K x
have hp_sp : Splits ((minpoly K x).map (algebraMap K L)) := hn.splits x
obtain ⟨s, hs⟩ := splits_iff_exists_multiset.mp hp_sp
have h_lc : (algebraMap K L) (minpoly K x).leadingCoeff = 1 := by
rw [minpoly.monic (hn.isIntegral x), map_one]
rw [leadingCoeff_map, h_lc, map_one, ... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Group.SeparationQuotient | {
"line": 63,
"column": 4
} | {
"line": 63,
"column": 18
} | [
{
"pp": "M : Type u_1\nN : Type u_2\ninst✝³ : SeminormedAddCommGroup M\ninst✝² : SeminormedAddCommGroup N\nF : Type u_3\ninst✝¹ : FunLike F M N\ninst✝ : AddMonoidHomClass F M N\nf : F\nhf : ∀ (x : M), ‖x‖ = 0 → f x = 0\nx y : M\nh : Inseparable x y\n⊢ f x - f y = 0",
"usedConstants": [
"Eq.mpr",
... | rw [← map_sub] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Group.Tannery | {
"line": 65,
"column": 4
} | {
"line": 66,
"column": 61
} | [
{
"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... | rw [HasSum, Metric.tendsto_nhds] at hS
classical exact Eventually.exists <| hS _ (by positivity) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Group.Tannery | {
"line": 65,
"column": 4
} | {
"line": 66,
"column": 61
} | [
{
"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... | rw [HasSum, Metric.tendsto_nhds] at hS
classical exact Eventually.exists <| hS _ (by positivity) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 1001,
"column": 4
} | {
"line": 1001,
"column": 19
} | [
{
"pp": "K : 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\nE : Type v := ((mapAlg K L) (minpoly K x)).SplittingField\nhspl : ((mapAlg K E) (minpoly K x)).Splits\nthis✝ : ... | map_neg_eq_map, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.Bases | {
"line": 248,
"column": 2
} | {
"line": 248,
"column": 19
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\nX : Type u_2\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\nβ : Type u_3\nb : UnconditionalSchauderBasis β 𝕜 X\ninst✝ : CompleteSpace X\nC : ℝ\nhC : ∀ (A : Finset β), ‖GeneralSchauderBasis.proj b A‖ ≤ C\nhCpos : 0 ≤ C\n⊢ C.toNNReal ∈ uppe... | rintro _ ⟨A, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 154,
"column": 12
} | {
"line": 154,
"column": 32
} | [
{
"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\nx : X\n⊢ ‖P • Q • x‖ + ‖Q • x - P • Q • x + (x - Q • x)‖ = ‖(P * Q) • x‖ + ‖(1 - P * Q) • x‖",
"usedConstants": [
... | sub_add_sub_cancel', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 238,
"column": 2
} | {
"line": 238,
"column": 34
} | [
{
"pp": "X : Type u_1\ninst✝² : NormedAddCommGroup X\nM : Type u_2\ninst✝¹ : Ring M\ninst✝ : Module M X\nP : { P // IsLprojection X P }\nQ : M\n⊢ ↑Pᶜ * Q = Q - ↑P * Q",
"usedConstants": [
"Eq.mpr",
"IsLprojection",
"MulOne.toOne",
"HMul.hMul",
"Ring.toNonAssocRing",
"AddG... | rw [coe_compl, sub_mul, one_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 238,
"column": 2
} | {
"line": 238,
"column": 34
} | [
{
"pp": "X : Type u_1\ninst✝² : NormedAddCommGroup X\nM : Type u_2\ninst✝¹ : Ring M\ninst✝ : Module M X\nP : { P // IsLprojection X P }\nQ : M\n⊢ ↑Pᶜ * Q = Q - ↑P * Q",
"usedConstants": [
"Eq.mpr",
"IsLprojection",
"MulOne.toOne",
"HMul.hMul",
"Ring.toNonAssocRing",
"AddG... | rw [coe_compl, sub_mul, one_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 238,
"column": 2
} | {
"line": 238,
"column": 34
} | [
{
"pp": "X : Type u_1\ninst✝² : NormedAddCommGroup X\nM : Type u_2\ninst✝¹ : Ring M\ninst✝ : Module M X\nP : { P // IsLprojection X P }\nQ : M\n⊢ ↑Pᶜ * Q = Q - ↑P * Q",
"usedConstants": [
"Eq.mpr",
"IsLprojection",
"MulOne.toOne",
"HMul.hMul",
"Ring.toNonAssocRing",
"AddG... | rw [coe_compl, sub_mul, one_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 268,
"column": 4
} | {
"line": 268,
"column": 15
} | [
{
"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 R : { P // IsLprojection X P }\n⊢ ↑P = ↑P * ↑R → ↑Q = ↑Q * ↑R → ↑P + (↑Q - ↑P * ↑Q) = ↑P * ↑R + (↑Q * ↑R - ↑P * (↑Q * ↑R))",
"usedConstants": [
"IsLprojection",
... | intro h₁ h₂ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 276,
"column": 4
} | {
"line": 276,
"column": 15
} | [
{
"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 R : { P // IsLprojection X P }\n⊢ ↑P = ↑P * ↑Q → ↑P = ↑P * ↑R → ↑P = ↑P * ↑Q * ↑R",
"usedConstants": [
"IsLprojection",
"HMul.hMul",
"instDistribOfSemirin... | intro h₁ h₂ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Normed.Module.Bases | {
"line": 498,
"column": 6
} | {
"line": 498,
"column": 69
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nX : Type u_2\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\nβ : Type u_3\nL : SummationFilter β\nb : SchauderBasis 𝕜 X\nD : RankOneDecomposition 𝕜 X\ncoeff : ℕ → X → 𝕜 := D.basisCoeff\nhcoeff : ∀ (n : ℕ) (x : X), (succSub D.P n) x = coef... | simp only [mkContinuous_apply, LinearMap.coe_mk, AddHom.coe_mk] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Normed.Order.UpperLower | {
"line": 202,
"column": 2
} | {
"line": 202,
"column": 48
} | [
{
"pp": "case intro\nι : Type u_2\ninst✝ : Finite ι\ns : Set (ι → ℝ)\nhs : IsClosed s\nhs' : BddBelow s\nval✝ : Fintype ι\n⊢ IsClosed ↑(upperClosure s)",
"usedConstants": [
"Real",
"Pi.preorder",
"UpperSet",
"Pi.topologicalSpace",
"FrechetUrysohnSpace.to_sequentialSpace",
... | refine IsSeqClosed.isClosed fun f x hf hx ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Normed.Order.UpperLower | {
"line": 213,
"column": 2
} | {
"line": 213,
"column": 48
} | [
{
"pp": "case intro\nι : Type u_2\ninst✝ : Finite ι\ns : Set (ι → ℝ)\nhs : IsClosed s\nhs' : BddAbove s\nval✝ : Fintype ι\n⊢ IsClosed ↑(lowerClosure s)",
"usedConstants": [
"Real",
"Pi.preorder",
"Pi.topologicalSpace",
"FrechetUrysohnSpace.to_sequentialSpace",
"TopologicalSpace... | refine IsSeqClosed.isClosed fun f x hf hx ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.ODE.DiscreteGronwall | {
"line": 47,
"column": 56
} | {
"line": 63,
"column": 65
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\nu b c : ℕ → R\nn₀ : ℕ\nhu : ∀ n ≥ n₀, u (n + 1) ≤ c n * u n + b n\nhc : ∀ n ≥ n₀, 0 ≤ c n\nn : ℕ\nhn : n₀ ≤ n\n⊢ u n ≤ u n₀ * ∏ i ∈ Ico n₀ n, c i + ∑ k ∈ Ico n₀ n, b k * ∏ i ∈ Ico (k + 1) n, c i",
"usedConstant... | by
induction n, hn using Nat.le_induction with
| base => simp
| succ k hk ih =>
have hck : 0 ≤ c k := hc k hk
have heq : c k * ∑ j ∈ Ico n₀ k, b j * ∏ i ∈ Ico (j + 1) k, c i + b k =
∑ j ∈ Ico n₀ (k + 1), b j * ∏ i ∈ Ico (j + 1) (k + 1), c i := by
rw [sum_Ico_succ_top hk, mul_sum, Ico_self, p... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.ODE.PicardLindelof | {
"line": 244,
"column": 47
} | {
"line": 246,
"column": 44
} | [
{
"pp": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ntmin tmax : ℝ\nt₀ : ↑(Icc tmin tmax)\nx₀ : E\na r L : ℝ≥0\nα : FunSpace t₀ x₀ r L\nh : ↑L * max (tmax - ↑t₀) (↑t₀ - tmin) ≤ ↑a - ↑r\nt : ↑(Icc tmin tmax)\n⊢ ↑L * |↑t - ↑t₀| + ↑r ≤ ↑L * max (tmax - ↑t₀) (↑t₀ - tmin) + ↑r",
"usedConstants": [
"Real.in... | by
gcongr
exact abs_sub_le_max_sub t.2.1 t.2.2 _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Polynomial.Basic | {
"line": 50,
"column": 2
} | {
"line": 50,
"column": 20
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nP : 𝕜[X]\ninst✝ : OrderTopology 𝕜\n⊢ (fun x ↦ eval x P) ~[atTop] fun x ↦ P.leadingCoeff * x ^ P.natDegree",
"usedConstants": [
"Polynomial.eval",
"NormedCommRing.toSeminormedCommRing",
... | by_cases h : P = 0 | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Analysis.Polynomial.Basic | {
"line": 345,
"column": 2
} | {
"line": 345,
"column": 20
} | [
{
"pp": "R : Type u_2\ninst✝¹ : NormedRing R\ninst✝ : NormMulClass R\nP : R[X]\n⊢ (fun x ↦ eval x P) ~[cobounded R] fun x ↦ P.leadingCoeff * x ^ P.natDegree",
"usedConstants": [
"Polynomial.eval",
"NormedRing.toRing",
"HMul.hMul",
"PseudoMetricSpace.toBornology",
"Polynomial.in... | by_cases h : P = 0 | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Analysis.RCLike.BoundedContinuous | {
"line": 52,
"column": 20
} | {
"line": 52,
"column": 29
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝¹ : RCLike 𝕜\ninst✝ : PseudoEMetricSpace E\nA : StarSubalgebra 𝕜 (E →ᵇ 𝕜)\ng : C(E, ℝ)\nx : E →ᵇ 𝕜\nhxA : x ∈ A\nhxg : (toContinuousMapStarₐ 𝕜) x = (AlgHom.compLeftContinuous ℝ ofRealAm ⋯) g\nhg_apply : ∀ (a : E), x a = ↑(g a)\nh_comp_eq : (AlgHom.compLeftContinuo... | h_comp_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.GaussNorm | {
"line": 176,
"column": 2
} | {
"line": 176,
"column": 42
} | [
{
"pp": "case inr.inr\nR : Type u_1\nF : Type u_2\ninst✝³ : Semiring R\ninst✝² : FunLike F R ℝ\nv : F\ninst✝¹ : ZeroHomClass F R ℝ\ninst✝ : NonnegHomClass F R ℝ\nhna : IsNonarchimedean ⇑v\nc : ℝ\nhc : 0 ≤ c\np q : R[X]\nh✝¹ : p ≠ 0\nh✝ : q ≠ 0\n⊢ gaussNorm v c (p + q) ≤ max (gaussNorm v c p) (gaussNorm v c q)",... | rcases eq_or_ne (p + q) 0 with hpq | hpq | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Real.OfDigits | {
"line": 121,
"column": 4
} | {
"line": 126,
"column": 37
} | [
{
"pp": "case succ\nx : ℝ\nb : ℕ\ninst✝ : NeZero b\nhx : x ∈ Set.Ico 0 1\nthis : b ≠ 0\nn : ℕ\nih : ↑b ^ n * ∑ i ∈ Finset.range n, ofDigitsTerm (x.digits b) i = ↑⌊↑b ^ n * x⌋₊\n⊢ ↑b ^ (n + 1) * ∑ i ∈ Finset.range (n + 1), ofDigitsTerm (x.digits b) i = ↑⌊↑b ^ (n + 1) * x⌋₊",
"usedConstants": [
"Nat.cas... | rw [Finset.sum_range_succ, mul_add, pow_succ', mul_assoc, ih, ofDigitsTerm, digits, ← pow_succ',
mul_left_comm, mul_inv_cancel₀ (by positivity), mul_one, mul_comm x, pow_succ', mul_assoc]
set y := (b : ℝ) ^ n * x
norm_cast
rw [← Nat.cast_mul_floor_div_cancel (a := y) (show b ≠ 0 by lia),
Fin.val... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Real.OfDigits | {
"line": 121,
"column": 4
} | {
"line": 126,
"column": 37
} | [
{
"pp": "case succ\nx : ℝ\nb : ℕ\ninst✝ : NeZero b\nhx : x ∈ Set.Ico 0 1\nthis : b ≠ 0\nn : ℕ\nih : ↑b ^ n * ∑ i ∈ Finset.range n, ofDigitsTerm (x.digits b) i = ↑⌊↑b ^ n * x⌋₊\n⊢ ↑b ^ (n + 1) * ∑ i ∈ Finset.range (n + 1), ofDigitsTerm (x.digits b) i = ↑⌊↑b ^ (n + 1) * x⌋₊",
"usedConstants": [
"Nat.cas... | rw [Finset.sum_range_succ, mul_add, pow_succ', mul_assoc, ih, ofDigitsTerm, digits, ← pow_succ',
mul_left_comm, mul_inv_cancel₀ (by positivity), mul_one, mul_comm x, pow_succ', mul_assoc]
set y := (b : ℝ) ^ n * x
norm_cast
rw [← Nat.cast_mul_floor_div_cancel (a := y) (show b ≠ 0 by lia),
Fin.val... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Real.Pi.Bounds | {
"line": 71,
"column": 2
} | {
"line": 76,
"column": 18
} | [
{
"pp": "c d a b n : ℕ\nz : ℝ\nhz : (↑c / ↑d).sqrtTwoAddSeries n ≤ z\nhb : 0 < b\nhd : 0 < d\nh : (2 * b + a) * d ^ 2 ≤ c ^ 2 * b\n⊢ (↑a / ↑b).sqrtTwoAddSeries (n + 1) ≤ z",
"usedConstants": [
"Nat.cast_mul._simp_1",
"Iff.mpr",
"Real.instIsOrderedRing",
"Eq.mpr",
"GroupWithZero... | refine le_trans ?_ hz; rw [sqrtTwoAddSeries_succ]; apply sqrtTwoAddSeries_monotone_left
have hb' : 0 < (b : ℝ) := Nat.cast_pos.2 hb
have hd' : 0 < (d : ℝ) := Nat.cast_pos.2 hd
rw [sqrt_le_left (div_nonneg c.cast_nonneg d.cast_nonneg), div_pow,
add_div_eq_mul_add_div _ _ (ne_of_gt hb'), div_le_div_iff₀ hb' (po... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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