module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.Analysis.Fourier.RiemannLebesgueLemma | {
"line": 211,
"column": 2
} | {
"line": 211,
"column": 51
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℝ → E\n⊢ Tendsto (fun w ↦ ∫ (v : ℝ), 𝐞 (-(w * v)) • f v) (cocompact ℝ) (𝓝 0)",
"usedConstants": [
"Real",
"Real.instRCLike",
"FiniteDimensional.rclike_to_real",
"Real.normedAddCommGroup",
"Real... | exact tendsto_integral_exp_inner_smul_cocompact f | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 342,
"column": 4
} | {
"line": 342,
"column": 63
} | [
{
"pp": "case pos\nG : Type u_3\ninst✝ : DivisionCommMonoid G\nk : ℕ\nζ : G\nh : IsPrimitiveRoot ζ k\nl : ℤ\nh0 : 0 ≤ l\n⊢ ζ ^ l = 1 ↔ ↑k ∣ l",
"usedConstants": [
"zpow_natCast",
"Dvd.dvd",
"InvOneClass.toOne",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneCla... | lift l to ℕ using h0; exact_mod_cast h.pow_eq_one_iff_dvd l | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 342,
"column": 4
} | {
"line": 342,
"column": 63
} | [
{
"pp": "case pos\nG : Type u_3\ninst✝ : DivisionCommMonoid G\nk : ℕ\nζ : G\nh : IsPrimitiveRoot ζ k\nl : ℤ\nh0 : 0 ≤ l\n⊢ ζ ^ l = 1 ↔ ↑k ∣ l",
"usedConstants": [
"zpow_natCast",
"Dvd.dvd",
"InvOneClass.toOne",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneCla... | lift l to ℕ using h0; exact_mod_cast h.pow_eq_one_iff_dvd l | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.ZMod.ValMinAbs | {
"line": 67,
"column": 4
} | {
"line": 67,
"column": 47
} | [
{
"pp": "n : ℕ\na : ZMod (n + 1)\nh✝ : ¬0 ≤ a.valMinAbs\nh : ¬0 ≤ ↑(n + 1)\nhe : a.valMinAbs * 2 = ↑(n + 1)\n⊢ False",
"usedConstants": [
"Int.instIsStrictOrderedRing",
"Nat.cast_nonneg",
"SemilatticeInf.toPartialOrder",
"instOfNatNat",
"Int",
"instHAdd",
"HAdd.hAdd... | exacts [h (Nat.cast_nonneg _), zero_lt_two] | Batteries.Tactic._aux_Batteries_Tactic_Init___elabRules_Batteries_Tactic_exacts_1 | Batteries.Tactic.exacts |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 552,
"column": 24
} | {
"line": 552,
"column": 28
} | [
{
"pp": "case refine_2\nR : Type u_4\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\nk : ℕ\ninst✝ : NeZero k\nζ : Rˣ\nh : IsPrimitiveRoot ζ k\nn : ℤ\nhξ : ζ ^ n ∈ rootsOfUnity k R\nhk0 : 0 < ↑k\ni : ℤ := n % ↑k\ni₀ : ℕ\nhi₀ : ↑i₀ = i\n⊢ ζ ^ ↑i₀ = ζ ^ n",
"usedConstants": [
"Eq.mpr",
"congrArg",
... | hi₀, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.RootsOfUnity.Minpoly | {
"line": 129,
"column": 4
} | {
"line": 129,
"column": 37
} | [
{
"pp": "n : ℕ\nK : Type u_1\ninst✝² : CommRing K\nμ : K\nh : IsPrimitiveRoot μ n\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\np : ℕ\nhprime : Fact (Nat.Prime p)\nhdiv : ¬p ∣ n\nhn : ¬n = 0\nhpos : 0 < n\nP : ℤ[X] := minpoly ℤ μ\nQ : ℤ[X] := minpoly ℤ (μ ^ p)\nhdiff : ¬P = Q\nPmonic : P.Monic\nQmonic : Q.Monic\nPi... | refine IsCoprime.mul_dvd ?_ ?_ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Polynomial.Cyclotomic.Roots | {
"line": 101,
"column": 2
} | {
"line": 101,
"column": 67
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nn : ℕ\ninst✝¹ : IsDomain R\ninst✝ : NeZero ↑n\nμ : R\nhf : Function.Injective ⇑(algebraMap R (FractionRing R))\n⊢ (cyclotomic n R).IsRoot μ ↔ IsPrimitiveRoot μ n",
"usedConstants": [
"RingHom.instRingHomClass",
"OreLocalization.instAlgebra",
"Com... | haveI : NeZero (n : FractionRing R) := NeZero.nat_of_injective hf | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHaveI___1 | Lean.Parser.Tactic.tacticHaveI__ |
Mathlib.FieldTheory.KrullTopology | {
"line": 118,
"column": 4
} | {
"line": 118,
"column": 37
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\n⊢ ∀ (x₀ : Gal(L/K)) {U : Set Gal(L/K)},\n U ∈ (galBasis K L).sets → ∃ V ∈ (galBasis K L).sets, V ⊆ (fun x ↦ x₀ * x * x₀⁻¹) ⁻¹' U",
"usedConstants": [
"Field.toSemifield",
"Semifield.toDivisionSemirin... | rintro σ U ⟨H, ⟨E, hE, rfl⟩, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.FieldTheory.Finite.Basic | {
"line": 705,
"column": 42
} | {
"line": 705,
"column": 66
} | [
{
"pp": "case inr.inr\nn p k : ℕ\ninst✝ : Fact (Nat.Prime p)\nhn : n = p ^ k\nhn0 : n ≠ 0\nthis : NeZero n\na : ℕ\nha : (orderOf ↑a).Coprime n\nh : ¬IsUnit ↑(a - 1)\nha0 : 1 ≤ a\n⊢ a ^ p ^ k ≡ 1 [MOD p ^ k]",
"usedConstants": [
"ZMod.isUnit_iff_coprime",
"Nat.Coprime",
"ZMod.commRing",
... | ZMod.isUnit_iff_coprime, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Adjoin.PowerBasis | {
"line": 32,
"column": 2
} | {
"line": 35,
"column": 14
} | [
{
"pp": "K : Type u_1\nS : Type u_2\ninst✝² : Field K\ninst✝¹ : CommRing S\ninst✝ : Algebra K S\nx : S\nhx : IsIntegral K x\nhST : Function.Injective ⇑(algebraMap (↥K[x]) S)\n⊢ Basis (Fin (minpoly K x).natDegree) K ↥K[x]",
"usedConstants": [
"Subalgebra.instSetLike",
"Set.mem_singleton",
"... | have hx' :
IsIntegral K (⟨x, subset_adjoin (Set.mem_singleton x)⟩ : K[(x : S)]) := by
apply (isIntegral_algebraMap_iff hST).mp
convert hx | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.FieldTheory.Galois.Infinite | {
"line": 272,
"column": 8
} | {
"line": 275,
"column": 56
} | [
{
"pp": "case a\nk : Type u_1\nK : Type u_2\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\nL : IntermediateField k K\ninst✝ : IsGalois k K\nh : L.fixingSubgroup.Normal\ng : (x : K) → Subgroup Gal(↥(adjoin k {x}).toIntermediateField/k) := ⋯\nf : ↥L → IntermediateField k K := ⋯\nthis✝ : ∀ (x : K), (g ... | apply iSup_le
intro l
simpa only [f, g, ← restrict_fixedField L.fixingSubgroup (adjoin k {l.1}),
fixedField_fixingSubgroup L] using inf_le_left | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Galois.Infinite | {
"line": 272,
"column": 8
} | {
"line": 275,
"column": 56
} | [
{
"pp": "case a\nk : Type u_1\nK : Type u_2\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\nL : IntermediateField k K\ninst✝ : IsGalois k K\nh : L.fixingSubgroup.Normal\ng : (x : K) → Subgroup Gal(↥(adjoin k {x}).toIntermediateField/k) := ⋯\nf : ↥L → IntermediateField k K := ⋯\nthis✝ : ∀ (x : K), (g ... | apply iSup_le
intro l
simpa only [f, g, ← restrict_fixedField L.fixingSubgroup (adjoin k {l.1}),
fixedField_fixingSubgroup L] using inf_le_left | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 238,
"column": 6
} | {
"line": 242,
"column": 37
} | [
{
"pp": "n : ℕ\nq : ℝ\nhn' : 3 ≤ n\nhq' : 1 < q\nhn : 0 < n\nhq : 0 < q\nhfor : ∀ ζ' ∈ primitiveRoots n ℂ, ‖↑q - ζ'‖ ≤ q + 1\nζ : ℂ := Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)\nhζ : IsPrimitiveRoot ζ n\nthis : ¬SameRay ℝ (↑q) (-ζ)\n⊢ ‖↑q - ζ‖ < q + 1",
"usedConstants": [
"instInnerProductSpaceRealC... | convert norm_add_lt_of_not_sameRay this using 2
· rw [Complex.norm_real]
symm
exact abs_eq_self.mpr hq.le
· simp [hζ.norm'_eq_one hn.ne'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 238,
"column": 6
} | {
"line": 242,
"column": 37
} | [
{
"pp": "n : ℕ\nq : ℝ\nhn' : 3 ≤ n\nhq' : 1 < q\nhn : 0 < n\nhq : 0 < q\nhfor : ∀ ζ' ∈ primitiveRoots n ℂ, ‖↑q - ζ'‖ ≤ q + 1\nζ : ℂ := Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)\nhζ : IsPrimitiveRoot ζ n\nthis : ¬SameRay ℝ (↑q) (-ζ)\n⊢ ‖↑q - ζ‖ < q + 1",
"usedConstants": [
"instInnerProductSpaceRealC... | convert norm_add_lt_of_not_sameRay this using 2
· rw [Complex.norm_real]
symm
exact abs_eq_self.mpr hq.le
· simp [hζ.norm'_eq_one hn.ne'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 297,
"column": 4
} | {
"line": 299,
"column": 96
} | [
{
"pp": "case neg\nn : ℕ\ninst✝⁵ : NeZero n\nK : Type u\nL : Type v\ninst✝⁴ : CommRing L\nζ : L\ninst✝³ : Field K\ninst✝² : Algebra K L\nhζ : IsPrimitiveRoot ζ n\ninst✝¹ : IsDomain L\ninst✝ : IsCyclotomicExtension {n} K L\nhn : n ≠ 2\nhirr : Irreducible (cyclotomic n K)\nthis : NeZero ↑n\nh1 : 2 ≤ n\n⊢ (Algebra... | rw [← hζ.powerBasis_gen K, PowerBasis.norm_gen_eq_coeff_zero_minpoly, hζ.powerBasis_gen K,
← hζ.minpoly_eq_cyclotomic_of_irreducible hirr, cyclotomic_coeff_zero K h1, mul_one,
hζ.powerBasis_dim K, ← hζ.minpoly_eq_cyclotomic_of_irreducible hirr, natDegree_cyclotomic] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 291,
"column": 60
} | {
"line": 300,
"column": 59
} | [
{
"pp": "n : ℕ\ninst✝⁵ : NeZero n\nK : Type u\nL : Type v\ninst✝⁴ : CommRing L\nζ : L\ninst✝³ : Field K\ninst✝² : Algebra K L\nhζ : IsPrimitiveRoot ζ n\ninst✝¹ : IsDomain L\ninst✝ : IsCyclotomicExtension {n} K L\nhn : n ≠ 2\nhirr : Irreducible (cyclotomic n K)\n⊢ (Algebra.norm K) ζ = 1",
"usedConstants": [
... | by
haveI := IsCyclotomicExtension.neZero' n K L
by_cases h1 : n = 1
· rw [h1, one_right_iff] at hζ
rw [hζ, show 1 = algebraMap K L 1 by simp, Algebra.norm_algebraMap, one_pow]
· replace h1 : 2 ≤ n := (two_le_iff n).mpr ⟨NeZero.ne _, h1⟩
rw [← hζ.powerBasis_gen K, PowerBasis.norm_gen_eq_coeff_zero_minpol... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.ZMod.Units | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 35
} | [
{
"pp": "N : ℕ\na : ZMod N\n⊢ ∃ d, d ∣ N ∧ ∃ u, IsUnit u ∧ a = u * ↑d",
"usedConstants": [
"instOfNatNat",
"Nat",
"eq_or_ne",
"OfNat.ofNat"
]
}
] | rcases eq_or_ne N 0 with rfl | hN | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 346,
"column": 24
} | {
"line": 346,
"column": 35
} | [
{
"pp": "case a\nn : ℕ\ninst✝⁴ : NeZero n\nK : Type u\nL : Type v\ninst✝³ : Field L\nζ : L\ninst✝² : Field K\ninst✝¹ : Algebra K L\nhζ : IsPrimitiveRoot ζ n\ninst✝ : IsCyclotomicExtension {n} K L\nh : 2 < n\nhirr : Irreducible (cyclotomic n K)\nthis✝¹ : NeZero ↑n\nE : Type v := AlgebraicClosure L\nz : E\nhz : (... | prod_const, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity | {
"line": 75,
"column": 4
} | {
"line": 75,
"column": 57
} | [
{
"pp": "case refine_2\nM : Type u_1\ninst✝² : CommMonoid M\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : HasEnoughRootsOfUnity M n\nζ : M\nh : IsPrimitiveRoot ζ n\n⊢ orderOf ζ ∣ Monoid.exponent ↥(rootsOfUnity n M)",
"usedConstants": [
"Units.val",
"Eq.mpr",
"Nat.instMulZeroClass",
"Dvd.dvd",
... | rw [← (h.isUnit NeZero.out).unit_spec, orderOf_units] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.LegendreSymbol.AddCharacter | {
"line": 222,
"column": 6
} | {
"line": 222,
"column": 65
} | [
{
"pp": "case inr\nF : Type u_1\nF' : Type u_2\ninst✝² : Field F\ninst✝¹ : Finite F\ninst✝ : Field F'\nh : ringChar F' ≠ ringChar F\np : ℕ := ringChar F\nhp : Fact (Nat.Prime p)\npp : ℕ+ := p.toPNat ⋯\nhq : ringChar F' = 0\n⊢ ¬0 ∣ p",
"usedConstants": [
"Nat.Prime",
"Dvd.dvd",
"Nat.instSem... | exact fun hf => Nat.Prime.ne_zero hp.1 (zero_dvd_iff.mp hf) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 236,
"column": 4
} | {
"line": 236,
"column": 44
} | [
{
"pp": "case refine_2\nn : ℕ\ninst✝³ : NeZero n\nS : Set ℕ\nA : Type u\nB : Type v\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nh : ∃ s ∈ S, s ≠ 0 ∧ n ∣ s\nH : IsCyclotomicExtension S A B\n⊢ ⊤ ≤ adjoin A {b | ∃ n_1 ∈ S ∪ {n}, n_1 ≠ 0 ∧ b ^ n_1 = 1}",
"usedConstants": [
"Eq.mpr",
... | rw [← ((iff_adjoin_eq_top S A B).1 H).2] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.DirichletCharacter.Basic | {
"line": 181,
"column": 2
} | {
"line": 182,
"column": 21
} | [
{
"pp": "R : Type u_1\ninst✝ : CommMonoidWithZero R\nχ : DirichletCharacter R 1\n⊢ χ = 1",
"usedConstants": [
"Units.val",
"MulOne.toOne",
"ZMod.commRing",
"Monoid.toMulOneClass",
"MulChar.hasOne",
"congrArg",
"ZMod.instUnique",
"MonoidHomClass.toOneHomClass",... | ext
simp [Units.eq_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.DirichletCharacter.Basic | {
"line": 181,
"column": 2
} | {
"line": 182,
"column": 21
} | [
{
"pp": "R : Type u_1\ninst✝ : CommMonoidWithZero R\nχ : DirichletCharacter R 1\n⊢ χ = 1",
"usedConstants": [
"Units.val",
"MulOne.toOne",
"ZMod.commRing",
"Monoid.toMulOneClass",
"MulChar.hasOne",
"congrArg",
"ZMod.instUnique",
"MonoidHomClass.toOneHomClass",... | ext
simp [Units.eq_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 537,
"column": 49
} | {
"line": 537,
"column": 85
} | [
{
"pp": "S : Set ℕ\nK : Type w\nL : Type z\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : IsCyclotomicExtension S K L\nthis : Algebra.IsIntegral K L\nh : (IntermediateField.adjoin K {b | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}).toSubalgebra = ⊤\n⊢ Algebra.IsSeparable K L",
"usedConstants": [
"... | ← IntermediateField.top_toSubalgebra | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 135,
"column": 6
} | {
"line": 135,
"column": 28
} | [
{
"pp": "n : ℕ\n⊢ χ₈ ↑n = χ₈ ↑(n % 8)",
"usedConstants": [
"Eq.mpr",
"ZMod.χ₈",
"ZMod.commRing",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne",
"CommSemiring.toCommMonoidWithZero",
"id",
"Nat.instMod",
"instHMod",
"MulChar",
"MulChar.instF... | ← ZMod.natCast_mod n 8 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 943,
"column": 79
} | {
"line": 955,
"column": 60
} | [
{
"pp": "A : Type u\nB : Type v\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Algebra A B\ninst✝¹ : IsDomain B\nn₁ n₂ : ℕ\nC₁ C₂ : Subalgebra A B\nh₁ : IsCyclotomicExtension {n₁} A ↥C₁\nh₂ : IsCyclotomicExtension {n₂} A ↥C₂\ninst✝ : NeZero n₂\nh : n₁ ∣ n₂\n⊢ C₁ ≤ C₂",
"usedConstants": [
"Subalge... | by
have : NeZero n₁ := by
constructor
rintro rfl
exact NeZero.ne n₂ <| eq_zero_of_zero_dvd h
obtain ⟨ζ₂, hζ₂⟩ := h₂.1 rfl (NeZero.ne n₂)
replace hζ₂ := hζ₂.map_of_injective (FaithfulSMul.algebraMap_injective C₂ B)
obtain ⟨d, hd⟩ := h
have hζ₁ := IsPrimitiveRoot.pow n₂.pos_of_neZero hζ₂ (by rwa [mu... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Grade | {
"line": 372,
"column": 4
} | {
"line": 374,
"column": 46
} | [
{
"pp": "α : Type u_3\ninst✝ : PartialOrder α\ns : Flag α\na : ↥s\nh : IsMin a\nb : α\nhba : b ≤ ↑a\n⊢ ↑a ≤ b",
"usedConstants": [
"Iff.mpr",
"Preorder.toLT",
"SemilatticeInf.instIsCodirectedOrder",
"IsMin.isBot",
"PartialOrder.toPreorder",
"Classical.propDecidable",
... | refine @h ⟨b, mem_iff_forall_le_or_ge.2 fun c hc ↦ ?_⟩ hba
classical
exact .inl <| hba.trans <| h.isBot ⟨c, hc⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Grade | {
"line": 372,
"column": 4
} | {
"line": 374,
"column": 46
} | [
{
"pp": "α : Type u_3\ninst✝ : PartialOrder α\ns : Flag α\na : ↥s\nh : IsMin a\nb : α\nhba : b ≤ ↑a\n⊢ ↑a ≤ b",
"usedConstants": [
"Iff.mpr",
"Preorder.toLT",
"SemilatticeInf.instIsCodirectedOrder",
"IsMin.isBot",
"PartialOrder.toPreorder",
"Classical.propDecidable",
... | refine @h ⟨b, mem_iff_forall_le_or_ge.2 fun c hc ↦ ?_⟩ hba
classical
exact .inl <| hba.trans <| h.isBot ⟨c, hc⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.NormPow | {
"line": 110,
"column": 6
} | {
"line": 110,
"column": 29
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\np : ℝ\nhp : 1 < p\n⊢ ContDiff ℝ 1 fun x ↦ ‖x‖ ^ p",
"usedConstants": [
"Differentiable",
"Norm.norm",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real.instPow",
"Real",
"Continuou... | contDiff_one_iff_fderiv | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Hofer | {
"line": 68,
"column": 12
} | {
"line": 70,
"column": 22
} | [
{
"pp": "X : Type u_1\ninst✝¹ : MetricSpace X\ninst✝ : CompleteSpace X\nx : X\nε : ℝ\nε_pos : 0 < ε\nϕ : X → ℝ\ncont : Continuous ϕ\nnonneg : ∀ (y : X), 0 ≤ ϕ y\nreformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x'\nthis : Nonempty X\nF : ℕ → X → X\nhF : ∀ (k : ℕ) (x' : X), d x' x... | rw [Finset.sum_mul]
simp
field_simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Hofer | {
"line": 68,
"column": 12
} | {
"line": 70,
"column": 22
} | [
{
"pp": "X : Type u_1\ninst✝¹ : MetricSpace X\ninst✝ : CompleteSpace X\nx : X\nε : ℝ\nε_pos : 0 < ε\nϕ : X → ℝ\ncont : Continuous ϕ\nnonneg : ∀ (y : X), 0 ≤ ϕ y\nreformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x'\nthis : Nonempty X\nF : ℕ → X → X\nhF : ∀ (k : ℕ) (x' : X), d x' x... | rw [Finset.sum_mul]
simp
field_simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 488,
"column": 4
} | {
"line": 489,
"column": 56
} | [
{
"pp": "n : Type u_2\ninst✝⁴ : Fintype n\nK : Type u_5\ninst✝³ : Field K\ninst✝² : PartialOrder K\ninst✝¹ : StarRing K\ninst✝ : DecidableEq n\nM : Matrix n n K\nhM : M.PosDef\nh : ¬IsUnit M\n⊢ ∃ a, a ≠ 0 ∧ M *ᵥ a = 0",
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"False",... | obtain ⟨a, b, ha⟩ := Function.not_injective_iff.mp <| mulVec_injective_iff_isUnit.not.mpr h
exact ⟨a - b, by simp [sub_eq_zero, ha, mulVec_sub]⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 488,
"column": 4
} | {
"line": 489,
"column": 56
} | [
{
"pp": "n : Type u_2\ninst✝⁴ : Fintype n\nK : Type u_5\ninst✝³ : Field K\ninst✝² : PartialOrder K\ninst✝¹ : StarRing K\ninst✝ : DecidableEq n\nM : Matrix n n K\nhM : M.PosDef\nh : ¬IsUnit M\n⊢ ∃ a, a ≠ 0 ∧ M *ᵥ a = 0",
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"False",... | obtain ⟨a, b, ha⟩ := Function.not_injective_iff.mp <| mulVec_injective_iff_isUnit.not.mpr h
exact ⟨a - b, by simp [sub_eq_zero, ha, mulVec_sub]⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Coalgebra | {
"line": 56,
"column": 2
} | {
"line": 57,
"column": 54
} | [
{
"pp": "case h.h\n𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nn : Type u_3\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\ni✝ : n\n⊢ (counit ∘ₗ LinearMap.single 𝕜 (fun i ↦ 𝕜) i✝) 1 =\n ((adjoint (↑(equiv n 𝕜).symm.toLinearEquiv ∘ₗ Algebra.linearMap 𝕜 (n → 𝕜)) ∘ₗ ↑(equiv n 𝕜).symm.toLinearEquiv) ∘ₗ\n LinearM... | simp [← toSpanSingleton_one_eq_algebraLinearMap, comp_toSpanSingleton,
adjoint_toSpanSingleton, inner_eq_star_dotProduct] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.InnerProductSpace.TensorProduct | {
"line": 112,
"column": 2
} | {
"line": 115,
"column": 25
} | [
{
"pp": "𝕜 : 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\nE' : Submodule 𝕜 E\nF' : Submodule 𝕜 F\niE' : Module.Finite 𝕜 ↥E'\niF' : Module.Finite 𝕜 ↥F'\ny : ↥E' ⊗[𝕜] ↥... | have (i) (j) : (e.toBasis.tensorProduct f.toBasis).repr y (i, j) = 0 := by
rw [inner_mapIncl_mapIncl, inner_self y e f, RCLike.ofReal_eq_zero,
Finset.sum_eq_zero_iff_of_nonneg fun _ _ => sq_nonneg _] at hx
simpa using hx (i, j) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.InnerProductSpace.TensorProduct | {
"line": 387,
"column": 28
} | {
"line": 387,
"column": 82
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : InnerProductSpace 𝕜 G\nx y : E ⊗[𝕜] F ⊗[𝕜] G\na : E ⊗[𝕜... | simp only [add_tmul, inner_add_right, map_add, h1, h2] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.InnerProductSpace.TensorProduct | {
"line": 387,
"column": 28
} | {
"line": 387,
"column": 82
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : InnerProductSpace 𝕜 G\nx y : E ⊗[𝕜] F ⊗[𝕜] G\na : E ⊗[𝕜... | simp only [add_tmul, inner_add_right, map_add, h1, h2] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.TensorProduct | {
"line": 387,
"column": 28
} | {
"line": 387,
"column": 82
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : InnerProductSpace 𝕜 G\nx y : E ⊗[𝕜] F ⊗[𝕜] G\na : E ⊗[𝕜... | simp only [add_tmul, inner_add_right, map_add, h1, h2] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Fin.Tuple.Sort | {
"line": 98,
"column": 6
} | {
"line": 98,
"column": 20
} | [
{
"pp": "n : ℕ\nα : Type u_1\ninst✝ : LinearOrder α\nf : Fin n → α\n⊢ Monotone (f ∘ ⇑(sort f))",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Equiv.instEquivLike",
"congrArg",
"Lex",
"Finset",
"PartialOrder.toPreorder",
"Monotone",
"Function.comp",
... | self_comp_sort | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 101,
"column": 2
} | {
"line": 107,
"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\n⊢ OrthogonalFamily 𝕜 (fun μ ↦ ↥(eigenspace T μ)) fun μ ↦ (eigenspace T μ).subtypeₗᵢ",
"usedConstants": [
"LinearIsometry",
"LinearMap.IsSy... | rintro μ ν hμν ⟨v, hv⟩ ⟨w, hw⟩
by_cases hv' : v = 0
· simp [hv']
have H := hT.conj_eigenvalue_eq_self (hasEigenvalue_of_hasEigenvector ⟨hv, hv'⟩)
rw [mem_eigenspace_iff] at hv hw
refine Or.resolve_left ?_ hμν.symm
simpa [inner_smul_left, inner_smul_right, hv, hw, H] using (hT v w).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 101,
"column": 2
} | {
"line": 107,
"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\n⊢ OrthogonalFamily 𝕜 (fun μ ↦ ↥(eigenspace T μ)) fun μ ↦ (eigenspace T μ).subtypeₗᵢ",
"usedConstants": [
"LinearIsometry",
"LinearMap.IsSy... | rintro μ ν hμν ⟨v, hv⟩ ⟨w, hw⟩
by_cases hv' : v = 0
· simp [hv']
have H := hT.conj_eigenvalue_eq_self (hasEigenvalue_of_hasEigenvector ⟨hv, hv'⟩)
rw [mem_eigenspace_iff] at hv hw
refine Or.resolve_left ?_ hμν.symm
simpa [inner_smul_left, inner_smul_right, hv, hw, H] using (hT v w).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 127,
"column": 2
} | {
"line": 127,
"column": 96
} | [
{
"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\nμ : 𝕜\np : Submodule 𝕜 E := (⨆ μ, eigenspace T μ)ᗮ\n⊢ Disjoint (eigenspace T μ) (⨆ μ, eigenspace T μ)ᗮ",
"usedConstants": [
"InnerProductSpace.... | have H₂ : eigenspace T μ ⟂ p := (Submodule.isOrtho_orthogonal_right _).mono_left (le_iSup _ _) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 573,
"column": 6
} | {
"line": 576,
"column": 27
} | [
{
"pp": "E : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : MeasurableSpace E\ninst✝⁴ : BorelSpace E\ninst✝³ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝² : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : InnerProductSpace ℝ F'\nu : E → F'\nhu : ContDiff ... | suffices (C : ℝ) * γ = eLpNormLESNormFDerivOfEqInnerConst μ p by
rw [eLpNorm_nnreal_eq_lintegral h0p]
congr
norm_cast at this ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Topology.Algebra.Module.LinearPMap | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 21
} | [
{
"pp": "R : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : AddCommGroup E\ninst✝⁹ : AddCommGroup F\ninst✝⁸ : Module R E\ninst✝⁷ : Module R F\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : TopologicalSpace F\ninst✝⁴ : ContinuousAdd E\ninst✝³ : ContinuousAdd F\ninst✝² : TopologicalSpace R\ninst✝¹ ... | rw [closure_def hf] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.LaxMilgram | {
"line": 90,
"column": 72
} | {
"line": 90,
"column": 81
} | [
{
"pp": "V : Type u\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : CompleteSpace V\nB : V →L[ℝ] V →L[ℝ] ℝ\nthis : CompleteSpace ↑↑(↑(continuousLinearMapOfBilin B)).range\nv w : V\nmem_w_orthogonal : w ∈ (↑(continuousLinearMapOfBilin B)).rangeᗮ\nC : ℝ\nC_pos : 0 < C\ncoercivity : ∀ (u : ... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.LinearPMap | {
"line": 307,
"column": 39
} | {
"line": 307,
"column": 54
} | [
{
"pp": "case h\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 F\nT : E →ₗ.[𝕜] F\ninst✝ : CompleteSpace E\nhT : Dense ↑T.domain\nx : F × E\nh : ∀ (a : E) (a_1 : a ∈ T.do... | inner_conj_symm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Eigenspace.Matrix | {
"line": 97,
"column": 39
} | {
"line": 97,
"column": 55
} | [
{
"pp": "R : Type u_1\nn : Type u_2\nM : Type u_3\ninst✝⁵ : DecidableEq n\ninst✝⁴ : Fintype n\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nd : n → R\nμ : R\nb : Basis n R M\ninst✝ : IsDomain R\nx : M\nhx : x ∈ maxGenEigenspace ((toLin b b) (diagonal d)) μ\nk : ℕ\nhk : (((toLin b b) (diago... | simp [one_eq_id] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.InnerProductSpace.OfNorm | {
"line": 184,
"column": 2
} | {
"line": 184,
"column": 74
} | [
{
"pp": "case neg\n𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : InnerProductSpaceable E\nhI : ¬I = 0\nx y : E\nhI' : I * I = -1\n⊢ inner_ 𝕜 (I • x) y = (starRingEnd 𝕜) I * inner_ 𝕜 x y",
"usedConstants": [
"NormedCommRing.toNorme... | rw [conj_I, inner_, inner_, mul_left_comm, smul_smul, hI', neg_one_smul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Matrix.HermitianFunctionalCalculus | {
"line": 55,
"column": 18
} | {
"line": 57,
"column": 7
} | [
{
"pp": "n : Type u_1\n𝕜 : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nA : Matrix n n 𝕜\nhA : A.IsHermitian\nf g : C(↑(spectrum ℝ A), ℝ)\n⊢ ((conjStarAlgAut 𝕜 (Matrix n n 𝕜)) hA.eigenvectorUnitary)\n (diagonal (RCLike.ofReal ∘ ⇑(f * g) ∘ fun i ↦ ⟨hA.eigenvalues i, ⋯⟩)) =\n ... | by
simp only [ContinuousMap.coe_mul, ← map_mul, diagonal_mul_diagonal, Function.comp_apply]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.OfNorm | {
"line": 194,
"column": 41
} | {
"line": 194,
"column": 55
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : InnerProductSpaceable E\nr : 𝕜\nx y : E\n⊢ inner_ 𝕜 (↑(re r) • x) y + inner_ 𝕜 ((↑(im r) * I) • x) y = (starRingEnd 𝕜) (↑(re r) + ↑(im r) * I) * inner_ 𝕜 x y",
"usedConstants": [
... | real_prop _ x, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Matrix.PosDef | {
"line": 36,
"column": 2
} | {
"line": 36,
"column": 38
} | [
{
"pp": "n : Type u_2\n𝕜 : Type u_3\ninst✝² : Fintype n\ninst✝¹ : RCLike 𝕜\nA : Matrix n n 𝕜\ninst✝ : DecidableEq n\nhA : A.IsHermitian\n⊢ A.PosSemidef ↔ 0 ≤ hA.eigenvalues",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"Matrix.smul",
"Real.instLE",
"Real",
... | conv_lhs => rw [hA.spectral_theorem] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convLHS_1 | Mathlib.Tactic.Conv.convLHS |
Mathlib.Analysis.Matrix.PosDef | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 38
} | [
{
"pp": "n : Type u_2\n𝕜 : Type u_3\ninst✝² : Fintype n\ninst✝¹ : RCLike 𝕜\nA : Matrix n n 𝕜\ninst✝ : DecidableEq n\nhA : A.IsHermitian\n⊢ A.PosDef ↔ ∀ (i : n), 0 < hA.eigenvalues i",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"Matrix.smul",
"Real",
"Algebra... | conv_lhs => rw [hA.spectral_theorem] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convLHS_1 | Mathlib.Tactic.Conv.convLHS |
Mathlib.Analysis.Matrix.Spectrum | {
"line": 208,
"column": 2
} | {
"line": 208,
"column": 38
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nn : Type u_2\ninst✝¹ : Fintype n\nA : Matrix n n 𝕜\ninst✝ : DecidableEq n\nhA : A.IsHermitian\nx : 𝕜\n⊢ x ∈ spectrum 𝕜 A ↔ x ∈ RCLike.ofReal '' Set.range hA.eigenvalues",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"Matrix.smul... | conv_lhs => rw [hA.spectral_theorem] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convLHS_1 | Mathlib.Tactic.Conv.convLHS |
Mathlib.Analysis.Matrix.Spectrum | {
"line": 221,
"column": 38
} | {
"line": 221,
"column": 62
} | [
{
"pp": "case h\n𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nn : Type u_2\ninst✝¹ : Fintype n\nA : Matrix n n 𝕜\ninst✝ : DecidableEq n\nhA : A.IsHermitian\nh : A = 0\nx✝ : n\n⊢ hA.eigenvalues x✝ = 0 x✝",
"usedConstants": [
"Pi.instStarForall",
"NormedCommRing.toSeminormedCommRing",
"Matrix.zero_mu... | simp [h, eigenvalues_eq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Matrix.Order | {
"line": 173,
"column": 4
} | {
"line": 175,
"column": 67
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\nn : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nA : Matrix n n 𝕜\nx✝ : A.IsHermitian ∧ spectrum 𝕜 A ⊆ {a | 0 ≤ a}\nh1 : A.IsHermitian\nh2 : spectrum 𝕜 A ⊆ {a | 0 ≤ a}\n⊢ A.PosSemidef",
"usedConstants": [
"NormedCommRing.toNormedRin... | rw [h1.posSemidef_iff_eigenvalues_nonneg]
intro i
simpa [h1.spectrum_eq_image_range] using @h2 (h1.eigenvalues i) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Matrix.Order | {
"line": 173,
"column": 4
} | {
"line": 175,
"column": 67
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\nn : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nA : Matrix n n 𝕜\nx✝ : A.IsHermitian ∧ spectrum 𝕜 A ⊆ {a | 0 ≤ a}\nh1 : A.IsHermitian\nh2 : spectrum 𝕜 A ⊆ {a | 0 ≤ a}\n⊢ A.PosSemidef",
"usedConstants": [
"NormedCommRing.toNormedRin... | rw [h1.posSemidef_iff_eigenvalues_nonneg]
intro i
simpa [h1.spectrum_eq_image_range] using @h2 (h1.eigenvalues i) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs | {
"line": 138,
"column": 2
} | {
"line": 139,
"column": 75
} | [
{
"pp": "A : Type u_2\ninst✝¹¹ : NonUnitalRing A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : TopologicalSpace A\ninst✝⁸ : Module ℝ A\ninst✝⁷ : SMulCommClass ℝ A A\ninst✝⁶ : IsScalarTower ℝ A A\ninst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁴ : PartialOrder A\ninst✝³ : StarOrderedRing A\ninst✝² : No... | simpa [two_smul] using
congr($(CFC.posPart_add_negPart a) - $(CFC.posPart_sub_negPart a)).symm | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs | {
"line": 138,
"column": 2
} | {
"line": 139,
"column": 75
} | [
{
"pp": "A : Type u_2\ninst✝¹¹ : NonUnitalRing A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : TopologicalSpace A\ninst✝⁸ : Module ℝ A\ninst✝⁷ : SMulCommClass ℝ A A\ninst✝⁶ : IsScalarTower ℝ A A\ninst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁴ : PartialOrder A\ninst✝³ : StarOrderedRing A\ninst✝² : No... | simpa [two_smul] using
congr($(CFC.posPart_add_negPart a) - $(CFC.posPart_sub_negPart a)).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs | {
"line": 138,
"column": 2
} | {
"line": 139,
"column": 75
} | [
{
"pp": "A : Type u_2\ninst✝¹¹ : NonUnitalRing A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : TopologicalSpace A\ninst✝⁸ : Module ℝ A\ninst✝⁷ : SMulCommClass ℝ A A\ninst✝⁶ : IsScalarTower ℝ A A\ninst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁴ : PartialOrder A\ninst✝³ : StarOrderedRing A\ninst✝² : No... | simpa [two_smul] using
congr($(CFC.posPart_add_negPart a) - $(CFC.posPart_sub_negPart a)).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 554,
"column": 19
} | {
"line": 554,
"column": 30
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\nT : E →L[𝕜] E\nhT : T.IsPositive\na : Fin (Module.finrank 𝕜 E) → E := fun i ↦ ↑√(⋯.eigenvalues ⋯ i) • (⋯.eigenvectorBasis ⋯) i\nx✝ : E\n⊢ T x✝ = ∑ x, (⟪(⋯.e... | smul_assoc, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.LocallyConvex.WeakSpace | {
"line": 46,
"column": 2
} | {
"line": 46,
"column": 61
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜 E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : IsScalarTower ℝ 𝕜 E\ninst✝³ : TopologicalSpace E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\ns : Set E\nhs : Convex ℝ s\nx ... | have : ContinuousSMul ℝ E := IsScalarTower.continuousSMul 𝕜 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.InnerProductSpace.TwoDim | {
"line": 419,
"column": 2
} | {
"line": 419,
"column": 37
} | [
{
"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)\nx y : E\n⊢ ↑⟪x, y⟫ + (o.areaForm x) y • Complex.I = (starRingEnd ℂ) (↑⟪y, x⟫ + (o.areaForm y) x • Complex.I)",
"usedConstants": [
"Eq.mpr",
"InnerPro... | rw [real_inner_comm, areaForm_swap] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.MellinTransform | {
"line": 150,
"column": 4
} | {
"line": 150,
"column": 13
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℝ → E\ns : ℂ\na : ℝ\nha : 0 < a\nthis : EqOn (fun t ↦ ↑t ^ (s - 1) • f (a * t)) (fun t ↦ ↑a ^ (1 - s) • (fun u ↦ ↑u ^ (s - 1) • f u) (a * t)) (Ioi 0)\n⊢ ↑a ^ (1 - s) • a⁻¹ • ∫ (x : ℝ) in Ioi (a * 0), ↑x ^ (s - 1) • f x = ↑a ^ (-s... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd | {
"line": 76,
"column": 29
} | {
"line": 76,
"column": 38
} | [
{
"pp": "case h.e'_3\nz : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nder1 : ∀ x ∈ uIcc 0 (π / 2), HasDerivAt (fun y ↦ ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x\n⊢ ↑n / (2 * z) * ∫ (x : ℝ) in 0..π / 2, Complex.sin (2 * z * ↑x) * ↑(sin x) * ↑(cos x) ^ (n - 1) =\n ↑(cos (π / 2)) ^ n * (Complex.sin (2 * z * ↑... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd | {
"line": 76,
"column": 67
} | {
"line": 76,
"column": 76
} | [
{
"pp": "case h.e'_3\nz : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nder1 : ∀ x ∈ uIcc 0 (π / 2), HasDerivAt (fun y ↦ ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x\n⊢ ↑n / (2 * z) * ∫ (x : ℝ) in 0..π / 2, Complex.sin (2 * z * ↑x) * ↑(sin x) * ↑(cos x) ^ (n - 1) =\n ↑(cos (π / 2)) ^ n * (Complex.sin (2 * z * ↑... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd | {
"line": 114,
"column": 6
} | {
"line": 114,
"column": 15
} | [
{
"pp": "case h.e'_3\nz : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nder1 :\n ∀ x ∈ uIcc 0 (π / 2),\n HasDerivAt (fun y ↦ ↑(sin y) * ↑(cos y) ^ (n - 1)) (↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) x\n⊢ (↑n / (2 * z) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) -\n (↑n - 1)... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Complex.LogBounds | {
"line": 95,
"column": 6
} | {
"line": 95,
"column": 30
} | [
{
"pp": "z : ℂ\nn : ℕ\nih : HasDerivAt (logTaylor (n + 1)) (∑ j ∈ Finset.range n, (-1) ^ j * z ^ j) z\n⊢ HasDerivAt (fun x ↦ x ^ (n + 1) / (↑n + 1)) (z ^ n) z",
"usedConstants": [
"IsModuleTopology.toContinuousSMul",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"DivInvMonoid.toIn... | simp_rw [div_eq_mul_inv] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd | {
"line": 142,
"column": 65
} | {
"line": 142,
"column": 73
} | [
{
"pp": "z : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nnne : ↑n ≠ 0\nthis :\n ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n =\n ↑n / (2 * z) *\n (-((↑n - 1) / (2 * z) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (n - 2)) +\n ↑n / (2 * z) * ∫ (x : ℝ) in 0..π / 2, Comple... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup | {
"line": 111,
"column": 2
} | {
"line": 112,
"column": 99
} | [
{
"pp": "case h.e'_4\ns t a b : ℝ\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : ℝ → ℝ → ℝ → ℝ := fun c u x ↦ rexp (-c * x) * x ^ (c * (u - 1))\ne : (1 / a).HolderConjugate (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : ∀ (c u x : ℝ), x ∈ Ioi 0 → 0 ≤ f c u x\nposf' : ∀ (c u : ... | · rw [one_div_one_div, one_div_one_div]
congr 2 <;> exact setIntegral_congr_fun measurableSet_Ioi fun x hx => fpow (by assumption) _ hx | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup | {
"line": 166,
"column": 6
} | {
"line": 167,
"column": 10
} | [
{
"pp": "case succ\nf : ℝ → ℝ\nx : ℝ\nhf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y\nhx : 0 < x\nn : ℕ\nhn : f (x + ↑n) = f x + ∑ m ∈ Finset.range n, log (x + ↑m)\nthis : x + ↑n.succ = x + ↑n + 1\n⊢ f x + ∑ m ∈ Finset.range n, log (x + ↑m) + log (x + ↑n) = f x + ∑ m ∈ Finset.range (n + 1), log (x + ↑m)",
... | rw [Finset.range_add_one, Finset.sum_insert Finset.notMem_range_self]
abel | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup | {
"line": 166,
"column": 6
} | {
"line": 167,
"column": 10
} | [
{
"pp": "case succ\nf : ℝ → ℝ\nx : ℝ\nhf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y\nhx : 0 < x\nn : ℕ\nhn : f (x + ↑n) = f x + ∑ m ∈ Finset.range n, log (x + ↑m)\nthis : x + ↑n.succ = x + ↑n + 1\n⊢ f x + ∑ m ∈ Finset.range n, log (x + ↑m) + log (x + ↑n) = f x + ∑ m ∈ Finset.range (n + 1), log (x + ↑m)",
... | rw [Finset.range_add_one, Finset.sum_insert Finset.notMem_range_self]
abel | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd | {
"line": 195,
"column": 13
} | {
"line": 195,
"column": 22
} | [
{
"pp": "case inr.zero\nz : ℂ\nhz : z ≠ 0\n⊢ Complex.sin (↑π * z) =\n ((↑π * z * ∏ j ∈ Finset.range 0, (1 - z ^ 2 / (↑j + 1) ^ 2)) *\n ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * 0)) /\n ↑(∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * 0))",
"usedConstants": [
"instInnerProdu... | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd | {
"line": 198,
"column": 6
} | {
"line": 198,
"column": 15
} | [
{
"pp": "case inr.zero\nz : ℂ\nhz : z ≠ 0\n⊢ Complex.sin (↑π * z) =\n ↑π * z * (Complex.sin (2 * z * ↑(π / 2)) / (2 * z) - Complex.sin (2 * z * 0) / (2 * z)) / ↑(π / 2)",
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd | {
"line": 206,
"column": 70
} | {
"line": 206,
"column": 78
} | [
{
"pp": "z : ℂ\nhz : z ≠ 0\nn : ℕ\nA : ℂ := ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)\nB : ℂ := ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)\nC : ℝ := ∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * n)\nhn : Complex.sin (↑π * z) = ↑π * z * A * B / ↑C\n⊢ 2 * (n + 1) = 2 * n + 2",
"usedCons... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Complex.LogBounds | {
"line": 190,
"column": 2
} | {
"line": 190,
"column": 29
} | [
{
"pp": "n : ℕ\n⊢ (fun z ↦ log (1 + z) - logTaylor (n + 1) z) =O[𝓝 0] fun z ↦ z ^ (n + 1)",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Complex.log",
"Real.instLE",
"Real",
"HMul.hMul",
"congrArg",
"Asymptotics.isBi... | rw [Asymptotics.isBigO_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup | {
"line": 261,
"column": 8
} | {
"line": 261,
"column": 22
} | [
{
"pp": "case succ\nf : ℝ → ℝ\nhf_conv : ConvexOn ℝ (Ioi 0) f\nhf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y\nm : ℕ\nhm : ∀ {x : ℝ}, 0 < x → ↑m < x → x ≤ ↑m + 1 → Tendsto (logGammaSeq x) atTop (𝓝 (f x - f 1))\nx : ℝ\nhx : 0 < x\nhy : ↑(m + 1) < x\nhy' : x ≤ ↑(m + 1) + 1\n⊢ Tendsto (logGammaSeq x) atTop (... | Nat.cast_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup | {
"line": 262,
"column": 8
} | {
"line": 262,
"column": 22
} | [
{
"pp": "case succ\nf : ℝ → ℝ\nhf_conv : ConvexOn ℝ (Ioi 0) f\nhf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y\nm : ℕ\nhm : ∀ {x : ℝ}, 0 < x → ↑m < x → x ≤ ↑m + 1 → Tendsto (logGammaSeq x) atTop (𝓝 (f x - f 1))\nx : ℝ\nhx : 0 < x\nhy : ↑(m + 1) < x\nhy' : x - 1 ≤ ↑m + 1\n⊢ Tendsto (logGammaSeq x) atTop (𝓝... | Nat.cast_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd | {
"line": 263,
"column": 36
} | {
"line": 263,
"column": 49
} | [
{
"pp": "f : ℝ → ℂ\nhf : ContinuousOn f (Icc 0 (π / 2))\nc_lt : ∀ y ∈ Icc 0 (π / 2), y ≠ 0 → cos y < cos 0\nc_nonneg : ∀ x ∈ Icc 0 (π / 2), 0 ≤ cos x\n⊢ 0 < cos 0",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"Real.cos",
"congrArg",
"Real.instLT",
"id",
... | rw [cos_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd | {
"line": 293,
"column": 29
} | {
"line": 293,
"column": 38
} | [
{
"pp": "case h.e'_5\nz : ℂ\nthis✝ : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)\nA :\n Tendsto\n (fun n ↦\n (↑π * z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *\n ((∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /\n ↑(∫ (x : ℝ) in 0..π / 2,... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup | {
"line": 452,
"column": 6
} | {
"line": 452,
"column": 52
} | [
{
"pp": "s : ℝ\nhs : 0 < s\n⊢ Γ s * Γ (s + 1 / 2) = Γ (2 * s) * 2 ^ (1 - 2 * s) * √π",
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real.partialOrder",
"Real",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.pi",
"HMul.hMul",
"MulZero... | ← doublingGamma_eq_Gamma (mul_pos two_pos hs), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.AsymptoticCone | {
"line": 294,
"column": 57
} | {
"line": 297,
"column": 29
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝¹² : Field k\ninst✝¹¹ : LinearOrder k\ninst✝¹⁰ : AddCommGroup V\ninst✝⁹ : Module k V\ninst✝⁸ : AddTorsor V P\ninst✝⁷ : TopologicalSpace V\ninst✝⁶ : TopologicalSpace k\ninst✝⁵ : OrderTopology k\ninst✝⁴ : IsStrictOrderedRing k\ninst✝³ : IsTopologicalAddGroup... | by
ext
simp_rw [mem_asymptoticCone_iff, mem_closure_iff_frequently, ← frequently_bind,
asymptoticNhds_bind_nhds] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Gamma.Beta | {
"line": 213,
"column": 39
} | {
"line": 213,
"column": 53
} | [
{
"pp": "case succ.refine_2\nn : ℕ\nIH : ∀ {u : ℂ}, 0 < u.re → u.betaIntegral (↑n + 1) = ↑n ! / ∏ j ∈ Finset.range (n + 1), (u + ↑j)\nu : ℂ\nhu : 0 < u.re\nthis : u.betaIntegral (↑n.succ + 1) = ↑n.succ * (u + 1).betaIntegral ↑n.succ / u\n⊢ ↑n.succ * (u + 1).betaIntegral ↑n.succ / u = ↑(n + 1)! / ((∏ k ∈ Finset.... | Nat.cast_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Beta | {
"line": 357,
"column": 8
} | {
"line": 357,
"column": 22
} | [
{
"pp": "case succ\nm : ℕ\nIH : ∀ (s : ℂ), -↑m < s.re → Tendsto s.GammaSeq atTop (𝓝 (GammaAux m s))\ns : ℂ\nhs : -↑(m + 1) < s.re\n⊢ Tendsto s.GammaSeq atTop (𝓝 (GammaAux (m + 1) s))",
"usedConstants": [
"Nat.cast_succ",
"Real",
"AddMonoid.toAddSemigroup",
"congrArg",
"AddGro... | Nat.cast_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Beta | {
"line": 385,
"column": 4
} | {
"line": 385,
"column": 75
} | [
{
"pp": "z : ℂ\nn : ℕ\nhn : n ≠ 0\naux : ∀ (a b c d : ℂ), a * b * (c * d) = a * c * (b * d)\n⊢ ↑n ^ z * ↑n ^ (1 - z) = ↑n",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"AddMonoid.toAddZeroClass",
"AddGroupWithOne.toAddMonoidWithOne",
"HSub.hSub"... | rw [← cpow_add _ _ (Nat.cast_ne_zero.mpr hn), add_sub_cancel, cpow_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Gamma.Beta | {
"line": 407,
"column": 42
} | {
"line": 407,
"column": 52
} | [
{
"pp": "case pos\nz : ℂ\npi_ne : ↑π ≠ 0\nhs : sin (-(↑π * z)) = 0\n⊢ Gamma z * Gamma (1 - z) = 0",
"usedConstants": [
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.pi",
"HMul.hMul",
"congrArg",
"Complex.sin",
"Complex.instNormedField",
"Complex.instMul",
... | ← mul_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Beta | {
"line": 575,
"column": 16
} | {
"line": 575,
"column": 51
} | [
{
"pp": "s : ℂ\nh1 : AnalyticOnNhd ℂ (fun z ↦ (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) univ\nh2 : AnalyticOnNhd ℂ (fun z ↦ (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1) / ↑√π) univ\nh3 : Tendsto ofReal (𝓝[≠] 1) (𝓝[≠] 1)\nt : ℝ\nht : 0 < t\n⊢ (↑(Real.Gamma t * Real.Gamma (t + 1 / 2)))⁻¹ = (Gamma (2 * ↑t))⁻¹ * 2 ^ (2 * ↑t - ... | Gamma_mul_Gamma_add_half_of_pos ht, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.AffineSpace.Ceva | {
"line": 186,
"column": 10
} | {
"line": 186,
"column": 18
} | [
{
"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 +... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.AffineSpace.Ceva | {
"line": 192,
"column": 10
} | {
"line": 192,
"column": 18
} | [
{
"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 +... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.AffineSpace.Ceva | {
"line": 209,
"column": 2
} | {
"line": 209,
"column": 94
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Field k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nt : Triangle k P\nr : Fin 3 → k\nhr0 : ∀ (i : Fin 3), r i ≠ 0\np' : P\nhp' : ∀ (i : Fin 3), p' ∈ affineSpan k {t.points i, (AffineMap.lineMap (t.points (i + 1)) (t.points (... | rw [Finset.prod_div_distrib, ← prod_eq_prod_one_sub_of_mem_line_point_lineMap hp', div_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Algebra.QuaternionExponential | {
"line": 132,
"column": 30
} | {
"line": 132,
"column": 39
} | [
{
"pp": "case e_a.inr\nq : ℍ\nhv : ‖q.im‖ ≠ 0\n⊢ normSq ↑(Real.cos ‖q.im‖) + (Real.sin ‖q.im‖ / ‖q.im‖) ^ 2 * normSq q.im + 2 * 0 =\n Real.cos ‖q.im‖ ^ 2 + Real.sin ‖q.im‖ ^ 2",
"usedConstants": [
"Quaternion.coe",
"Norm.norm",
"Eq.mpr",
"NegZeroClass.toNeg",
"Real",
"... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Algebra.GelfandMazur | {
"line": 217,
"column": 2
} | {
"line": 217,
"column": 52
} | [
{
"pp": "F : Type u_1\ninst✝³ : NormedRing F\ninst✝² : NormOneClass F\ninst✝¹ : NormMulClass F\ninst✝ : NormedAlgebra ℂ F\nx : F\n⊢ ∃ z, ‖x - (algebraMap ℂ F) z‖ = 0",
"usedConstants": [
"Complex.instNormedField",
"NormedAlgebra.exists_isMinOn_norm_sub_smul",
"NormedRing.toSeminormedRing",... | obtain ⟨z, hz⟩ := exists_isMinOn_norm_sub_smul ℂ x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Normed.Algebra.GelfandMazur | {
"line": 284,
"column": 8
} | {
"line": 284,
"column": 16
} | [
{
"pp": "case succ\nF : Type u_1\ninst✝³ : NormedRing F\ninst✝² : NormedAlgebra ℝ F\ninst✝¹ : NormOneClass F\ninst✝ : NormMulClass F\nx : F\nM : ℝ\nhM : 0 ≤ M\nh : ∀ (z : ℝ × ℝ), M ≤ ‖φ x z‖\nn : ℕ\nih : ∀ {p : ℝ[X]}, p.IsMonicOfDegree (2 * n) → M ^ n ≤ ‖(aeval x) p‖\np : ℝ[X]\nhp : p.IsMonicOfDegree (2 * (n + ... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Algebra.GelfandMazur | {
"line": 343,
"column": 82
} | {
"line": 350,
"column": 12
} | [
{
"pp": "F : Type u_1\ninst✝² : NormedRing F\ninst✝¹ : NormedAlgebra ℝ F\ninst✝ : NormOneClass F\nx : F\nc : ℝ\nhc₀ : 0 < c\nhbd : ∀ (r : ℝ), c ≤ ‖x - (algebraMap ℝ F) r‖\ns : Set ℝ\nhs : Bornology.IsBounded sᶜ\nM : ℝ\nhM_pos : M > 0\nhM : ∀ y ∈ sᶜ, ‖y‖ ≤ M\nthis : Tendsto (fun x_1 ↦ ‖(algebraMap ℝ F) x_1.2‖ - ... | by
refine tendsto_atTop_mono' _ ?_ this
filter_upwards [prod_mem_prod (mem_principal_self sᶜ) univ_mem] with w hw
rw [norm_sub_rev]
refine le_trans ?_ (norm_sub_norm_le ..)
specialize hM _ (Set.mem_prod.mp hw).1
simp only [norm_algebraMap', norm_smul]
gcongr | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Algebra.GelfandMazur | {
"line": 359,
"column": 46
} | {
"line": 361,
"column": 18
} | [
{
"pp": "F : Type u_1\ninst✝² : NormedRing F\ninst✝¹ : NormedAlgebra ℝ F\ninst✝ : NormOneClass F\nx : F\nc : ℝ\nhc₀ : 0 < c\nhbd : ∀ (r : ℝ), c ≤ ‖x - (algebraMap ℝ F) r‖\nthis : Tendsto (fun y ↦ ‖y.1‖ * c) (cobounded ℝ ×ˢ ⊤) atTop\ny : ℝ × ℝ\nhy : y ∈ {0}ᶜ ×ˢ Set.univ\n⊢ ‖y.1‖ * ‖x - (algebraMap ℝ F) (y.1⁻¹ * ... | by
simp only [← norm_smul, smul_sub, smul_smul, Algebra.algebraMap_eq_smul_one]
simp_all | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Algebra.TrivSqZeroExt | {
"line": 89,
"column": 51
} | {
"line": 89,
"column": 62
} | [
{
"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... | smul_assoc, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Normed.Unbundled.RingSeminorm | {
"line": 411,
"column": 22
} | {
"line": 411,
"column": 50
} | [
{
"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⊢ f x * f (c * x⁻¹) ≤ 0",
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"Real",
"DivInvMonoid.toInv",
"NonUnitalCommRing.toNonU... | ← RingSeminorm.toFun_eq_coe, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded | {
"line": 67,
"column": 24
} | {
"line": 67,
"column": 33
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_ne_zero : f ≠ 0\nf_nonneg : 0 ≤ f\nh1 : f 1 = 0\nf_mul : ∀ (y : R), f y ≤ c * 0 * f y\n⊢ False",
"usedConstants": [
"Real.instLE",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"Real.instZero... | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded | {
"line": 285,
"column": 2
} | {
"line": 288,
"column": 31
} | [
{
"pp": "case a\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 : R\nhx : ∀ (y : R), f (x * y) = f x * f y\n⊢ ⨆ y, f x * (f y / f y) ≤ f x",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"MulOne.toOne... | · refine ciSup_le (fun x ↦ ?_)
by_cases hx : f x = 0
· rw [hx, div_zero, mul_zero]; exact f_nonneg _
· rw [div_self hx, mul_one] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 76,
"column": 4
} | {
"line": 82,
"column": 55
} | [
{
"pp": "case hf\nR : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nL : ℝ\nhL : 0 ≤ L\nε : ℝ\nhε : 0 < ε\nm1 : ℕ\nhm1 : 0 < m1\nx : R\nhx : μ x ≠ 0\nh_exp : Tendsto (fun n ↦ ↑(n % m1) / ↑n) atTop (𝓝 0)\n⊢ Tendsto (fun x ↦ (L + ε) ^ (-(↑(x % m1) / ↑x))) atTop (𝓝 1)",
"usedConstants": [
"Filter.Te... | have h0 : Tendsto (fun t : ℕ => -(((t % m1 : ℕ) : ℝ) / (t : ℝ))) atTop (𝓝 0) := by
rw [← neg_zero]
exact Tendsto.neg h_exp
rw [← rpow_zero (L + ε)]
apply Tendsto.rpow tendsto_const_nhds h0
rw [ne_eq, add_eq_zero_iff_of_nonneg hL (le_of_lt hε)]
exact Or.inl (not_and_of_not_right _ (ne_of_gt ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 76,
"column": 4
} | {
"line": 82,
"column": 55
} | [
{
"pp": "case hf\nR : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nL : ℝ\nhL : 0 ≤ L\nε : ℝ\nhε : 0 < ε\nm1 : ℕ\nhm1 : 0 < m1\nx : R\nhx : μ x ≠ 0\nh_exp : Tendsto (fun n ↦ ↑(n % m1) / ↑n) atTop (𝓝 0)\n⊢ Tendsto (fun x ↦ (L + ε) ^ (-(↑(x % m1) / ↑x))) atTop (𝓝 1)",
"usedConstants": [
"Filter.Te... | have h0 : Tendsto (fun t : ℕ => -(((t % m1 : ℕ) : ℝ) / (t : ℝ))) atTop (𝓝 0) := by
rw [← neg_zero]
exact Tendsto.neg h_exp
rw [← rpow_zero (L + ε)]
apply Tendsto.rpow tendsto_const_nhds h0
rw [ne_eq, add_eq_zero_iff_of_nonneg hL (le_of_lt hε)]
exact Or.inl (not_and_of_not_right _ (ne_of_gt ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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