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.RingTheory.Polynomial.Wronskian | {
"line": 102,
"column": 10
} | {
"line": 102,
"column": 47
} | [
{
"pp": "case hbc\nR : Type u_1\ninst✝ : CommRing R\na b : R[X]\nha : a ≠ 0\nhb : b ≠ 0\n⊢ (derivative a).degree + b.degree < a.degree + b.degree",
"usedConstants": [
"WithBot",
"congrArg",
"CommSemiring.toSemiring",
"Eq.mp",
"Ne",
"Bot.bot",
"Polynomial.degree",
... | rw [← Polynomial.degree_ne_bot] at hb | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Radical.Basic | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 74
} | [
{
"pp": "M : Type u_1\ninst✝² : CommMonoidWithZero M\ninst✝¹ : NormalizationMonoid M\ninst✝ : UniqueFactorizationMonoid M\na : M\nh : IsUnit a\n⊢ primeFactors a = ∅",
"usedConstants": [
"UniqueFactorizationMonoid.normalizedFactors",
"Multiset.toFinset",
"Eq.mpr",
"congrArg",
"F... | rw [primeFactors, normalizedFactors_of_isUnit h, Multiset.toFinset_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Radical.Basic | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 74
} | [
{
"pp": "M : Type u_1\ninst✝² : CommMonoidWithZero M\ninst✝¹ : NormalizationMonoid M\ninst✝ : UniqueFactorizationMonoid M\na : M\nh : IsUnit a\n⊢ primeFactors a = ∅",
"usedConstants": [
"UniqueFactorizationMonoid.normalizedFactors",
"Multiset.toFinset",
"Eq.mpr",
"congrArg",
"F... | rw [primeFactors, normalizedFactors_of_isUnit h, Multiset.toFinset_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Radical.Basic | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 74
} | [
{
"pp": "M : Type u_1\ninst✝² : CommMonoidWithZero M\ninst✝¹ : NormalizationMonoid M\ninst✝ : UniqueFactorizationMonoid M\na : M\nh : IsUnit a\n⊢ primeFactors a = ∅",
"usedConstants": [
"UniqueFactorizationMonoid.normalizedFactors",
"Multiset.toFinset",
"Eq.mpr",
"congrArg",
"F... | rw [primeFactors, normalizedFactors_of_isUnit h, Multiset.toFinset_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Radical.Basic | {
"line": 291,
"column": 4
} | {
"line": 291,
"column": 12
} | [
{
"pp": "case inl\nM : Type u_1\ninst✝² : CommMonoidWithZero M\ninst✝¹ : NormalizationMonoid M\ninst✝ : UniqueFactorizationMonoid M\nb : M\nhb₀ : b ≠ 0\nha : IsRadical 0\nhab : 0 ∣ b\n⊢ 0 ∣ radical b",
"usedConstants": [
"False",
"Dvd.dvd",
"eq_false",
"False.elim",
"semigroupD... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Radical.Basic | {
"line": 291,
"column": 4
} | {
"line": 291,
"column": 12
} | [
{
"pp": "case inl\nM : Type u_1\ninst✝² : CommMonoidWithZero M\ninst✝¹ : NormalizationMonoid M\ninst✝ : UniqueFactorizationMonoid M\nb : M\nhb₀ : b ≠ 0\nha : IsRadical 0\nhab : 0 ∣ b\n⊢ 0 ∣ radical b",
"usedConstants": [
"False",
"Dvd.dvd",
"eq_false",
"False.elim",
"semigroupD... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Radical.Basic | {
"line": 291,
"column": 4
} | {
"line": 291,
"column": 12
} | [
{
"pp": "case inl\nM : Type u_1\ninst✝² : CommMonoidWithZero M\ninst✝¹ : NormalizationMonoid M\ninst✝ : UniqueFactorizationMonoid M\nb : M\nhb₀ : b ≠ 0\nha : IsRadical 0\nhab : 0 ∣ b\n⊢ 0 ∣ radical b",
"usedConstants": [
"False",
"Dvd.dvd",
"eq_false",
"False.elim",
"semigroupD... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 125,
"column": 4
} | {
"line": 126,
"column": 18
} | [
{
"pp": "case inl.inr\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : x.gcd y = 1\nhx : x % 2 = 0\nhy : y % 2 = 1\n⊢ x % 2 = 0 ∧ y % 2 = 1 ∨ x % 2 = 1 ∧ y % 2 = 0",
"usedConstants": [
"instHMod",
"Int",
"HMod.hMod",
"And",
"instOfNat",
"And.intro",
"Int.instMod",
... | left
exact ⟨hx, hy⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 125,
"column": 4
} | {
"line": 126,
"column": 18
} | [
{
"pp": "case inl.inr\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : x.gcd y = 1\nhx : x % 2 = 0\nhy : y % 2 = 1\n⊢ x % 2 = 0 ∧ y % 2 = 1 ∨ x % 2 = 1 ∧ y % 2 = 0",
"usedConstants": [
"instHMod",
"Int",
"HMod.hMod",
"And",
"instOfNat",
"And.intro",
"Int.instMod",
... | left
exact ⟨hx, hy⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 177,
"column": 97
} | {
"line": 180,
"column": 5
} | [
{
"pp": "x y z : ℤ\nh : PythagoreanTriple x y z\nhp : h.IsPrimitiveClassified\n⊢ h.IsClassified",
"usedConstants": [
"Int.gcd",
"HMul.hMul",
"PythagoreanTriple.IsPrimitiveClassified",
"HSub.hSub",
"Exists",
"instHMod",
"instOfNatNat",
"Int",
"_private.Ma... | by
obtain ⟨m, n, H⟩ := hp
use 1, m, n
lia | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 201,
"column": 2
} | {
"line": 201,
"column": 67
} | [
{
"pp": "case inl\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : x.gcd y = 1\nhc' : x.gcd y ≠ 0\nhxz : x ≠ 0\n⊢ 0 < x ^ 2 + y ^ 2",
"usedConstants": [
"Int.instAddCommGroup",
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"Int.instIsStrictOrderedRing",
"IsOrderedRing.toPosMulM... | · apply lt_add_of_pos_of_le (sq_pos_of_ne_zero hxz) (sq_nonneg y) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 416,
"column": 29
} | {
"line": 416,
"column": 80
} | [
{
"pp": "case h\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : x.gcd y = 1\nhzpos : 0 < z\nm n : ℤ\nhm2n2 : 0 < m ^ 2 + n ^ 2\nhv2 : ↑x / ↑z = 2 * ↑m * ↑n / (↑m ^ 2 + ↑n ^ 2)\nhw2 : ↑y / ↑z = (↑m ^ 2 - ↑n ^ 2) / (↑m ^ 2 + ↑n ^ 2)\nH : (m ^ 2 - n ^ 2).gcd (m ^ 2 + n ^ 2) = 1\nco : m.gcd n = 1\npp : m % 2 = 0 ∧ n ... | ← div_left_inj' ((mt (Rat.coe_int_inj z 0).mp) hz), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 506,
"column": 2
} | {
"line": 506,
"column": 76
} | [
{
"pp": "case inl\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : x.gcd y = 1\nhzpos : 0 < z\nh1 : x % 2 = 0 ∧ y % 2 = 1\n⊢ h.IsPrimitiveClassified",
"usedConstants": [
"PythagoreanTriple.isPrimitiveClassified_of_coprime_of_odd_of_pos",
"instHMod",
"Int",
"HMod.hMod",
"And.right"... | · exact h.isPrimitiveClassified_of_coprime_of_odd_of_pos hc h1.right hzpos | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Instances.Complex | {
"line": 65,
"column": 4
} | {
"line": 65,
"column": 60
} | [
{
"pp": "case refine_2\nK : Subfield ℂ\nψ : ↥K →+* ℂ\nhc : UniformContinuous ⇑ψ\nthis✝ : IsTopologicalDivisionRing ℂ :=\n { toIsTopologicalRing := NormedDivisionRing.to_isTopologicalDivisionRing.toIsTopologicalRing,\n toContinuousInv₀ := NormedDivisionRing.to_isTopologicalDivisionRing.toContinuousInv₀ }\nth... | let extψ := IsDenseInducing.extendRingHom ui di.dense hc | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 302,
"column": 2
} | {
"line": 302,
"column": 19
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw : InfinitePlace K\n⊢ #({w.embedding} ∪ {ComplexEmbedding.conjugate w.embedding}) = if w.IsReal then 1 else 2",
"usedConstants": [
"NumberField.ComplexEmbedding.conjugate",
"Finset.instUnion",
"Finset",
"Classical.propD... | split_ifs with hw | Mathlib.Tactic._aux_Mathlib_Tactic_SplitIfs___elabRules_Mathlib_Tactic_splitIfs_1 | Mathlib.Tactic.splitIfs |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 323,
"column": 58
} | {
"line": 328,
"column": 82
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ ∑ w, w.mult = finrank ℚ K",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"instHSMul",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"instSMulOfMul",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
... | by
classical
rw [← Embeddings.card K ℂ, Fintype.card, Finset.card_eq_sum_ones, ← Finset.univ.sum_fiberwise
(fun φ => InfinitePlace.mk φ)]
exact Finset.sum_congr rfl
(fun _ _ => by rw [Finset.sum_const, smul_eq_mul, mul_one, card_filter_mk_eq]) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 13
} | [
{
"pp": "case pos\nk : Type u_1\ninst✝¹ : Field k\nK : Type u_2\ninst✝ : Field K\nf : k →+* K\nw : InfinitePlace K\nh₁ : (w.comap f).IsReal\nh₂ : w.IsReal\n⊢ 1 ≤ 1",
"usedConstants": []
},
{
"pp": "case neg\nk : Type u_1\ninst✝¹ : Field k\nK : Type u_2\ninst✝ : Field K\nf : k →+* K\nw : InfinitePlac... | pick_goal 3 | Batteries.Tactic.«_aux_Batteries_Tactic_PermuteGoals___elabRules_Batteries_Tactic_tacticPick_goal-__1» | Batteries.Tactic.«tacticPick_goal-_» |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 391,
"column": 4
} | {
"line": 394,
"column": 68
} | [
{
"pp": "case inr\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : ℝ\nhB : 0 < B\n⊢ volume {x | convexBodySumFun x < 1} = ↑(convexBodySumFactor K)",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Iff.mpr",
"IsModuleTopology.toContinuousSMul",
"Eq.mpr",
"Pi.F... | rw [measure_lt_one_eq_integral_div_gamma (g := fun x : (mixedSpace K) => convexBodySumFun x)
volume ((convexBodySumFun_eq_zero_iff 0).mpr rfl) convexBodySumFun_neg convexBodySumFun_add_le
(fun hx => (convexBodySumFun_eq_zero_iff _).mp hx)
(fun r x => le_of_eq (convexBodySumFun_smul r x)) zero_lt_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 201,
"column": 20
} | {
"line": 201,
"column": 36
} | [
{
"pp": "k : Type u_1\ninst✝² : Field k\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra k K\nw : InfinitePlace K\n⊢ (w.comap (algebraMap k K)).mult = w.mult ↔ w.mult ≤ (w.comap (algebraMap k K)).mult",
"usedConstants": [
"Eq.mpr",
"Algebra.algebraMap",
"congrArg",
"PartialOrder.toPr... | le_antisymm_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 415,
"column": 88
} | {
"line": 418,
"column": 31
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : ℝ\nhB : 0 < B\n⊢ (∫ (x : ℝ), rexp (-|x|)) ^ nrRealPlaces K * (∫ (x : ℂ), rexp (-2 * ‖x‖)) ^ nrComplexPlaces K =\n (2 * Gamma (1 / 1 + 1)) ^ nrRealPlaces K * (π * 2 ^ (-2 / 1) * Gamma (2 / 1 + 1)) ^ nrComplexPlaces K",
"usedConstants": [
... | by
rw [integral_comp_abs (f := fun x => exp (-x)), ← integral_exp_neg_rpow zero_lt_one,
← Complex.integral_exp_neg_mul_rpow le_rfl zero_lt_two]
simp_rw [Real.rpow_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 449,
"column": 4
} | {
"line": 451,
"column": 27
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\n⊢ Free.ChooseBasisIndex ℤ ↥↑↑I ≃ Free.ChooseBasisIndex ℤ (𝓞 K)",
"usedConstants": [
"Units.val",
"Eq.mpr",
"Submodule",
"NumberField.instFreeIntSubtypeMemSubmoduleRingOfIntegersCoeToSub... | refine Fintype.equivOfCardEq ?_
rw [← finrank_eq_card_chooseBasisIndex, ← finrank_eq_card_chooseBasisIndex,
fractionalIdeal_rank] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 449,
"column": 4
} | {
"line": 451,
"column": 27
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\n⊢ Free.ChooseBasisIndex ℤ ↥↑↑I ≃ Free.ChooseBasisIndex ℤ (𝓞 K)",
"usedConstants": [
"Units.val",
"Eq.mpr",
"Submodule",
"NumberField.instFreeIntSubtypeMemSubmoduleRingOfIntegersCoeToSub... | refine Fintype.equivOfCardEq ?_
rw [← finrank_eq_card_chooseBasisIndex, ← finrank_eq_card_chooseBasisIndex,
fractionalIdeal_rank] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 335,
"column": 2
} | {
"line": 335,
"column": 21
} | [
{
"pp": "k : Type u_1\ninst✝³ : Field k\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra k K\ninst✝ : IsGalois k K\nφ : K →+* ℂ\nH : ∀ (σ : Gal(K/k)), ComplexEmbedding.IsConj φ σ → σ = 1\nhφ : ¬ComplexEmbedding.IsConj φ 1 ∧ ComplexEmbedding.IsReal (φ.comp (algebraMap k K))\nthis : Algebra k ℂ := (φ.comp (algeb... | letI := φ.toAlgebra | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLetI___1 | Lean.Parser.Tactic.tacticLetI__ |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 744,
"column": 2
} | {
"line": 744,
"column": 60
} | [
{
"pp": "K : Type u_4\nL : Type u_5\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nv : InfinitePlace K\n⊢ Set.SurjOn v.embeddingConjugateIte (unramifiedPlacesOver L v) (unmixedEmbeddingsOver L v.embedding)",
"usedConstants": [
"_private.Mathlib.NumberTheory.NumberField.InfinitePlace.Ramific... | refine fun ψ h ↦ ⟨mk ψ, mk_mem_unramifiedPlacesOver h, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.RamificationInertia.Galois | {
"line": 331,
"column": 2
} | {
"line": 332,
"column": 67
} | [
{
"pp": "case refine_2\nR : Type u_1\nS : Type u_2\nG : Type u_3\ninst✝¹³ : CommRing R\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Group G\ninst✝⁹ : MulSemiringAction G S\ninst✝⁸ : IsGaloisGroup G R S\ninst✝⁷ : Finite G\ninst✝⁶ : IsDedekindDomain R\ninst✝⁵ : IsDedekindDomain S\ninst✝⁴ : Module.Finit... | rw [H, mul_assoc, ← inertiaDeg_algebraMap, ← inertiaDegIn_eq_inertiaDeg p P G,
ncard_primesOver_mul_ramificationIdxIn_mul_inertiaDegIn hp S G] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 175,
"column": 39
} | {
"line": 175,
"column": 57
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nB : ℝ := (minkowskiBound K I * ↑(convexBodySumFactor K)⁻¹).toReal ^ (1 / ↑(finrank ℚ K))\nh_le : minkowskiBound K I ≤ volume (convexBodySum K B)\nx✝ : K\n⊢ ↑(FractionalIdeal.absNorm ↑I) *\n (2⁻¹ ^ nrComplexPla... | NNReal.coe_natCast | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 250,
"column": 67
} | {
"line": 250,
"column": 75
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw : { w // w.IsReal }\n⊢ ∀ {a b : ({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)},\n a ∈ {x | x.1 w = 0} → b ∈ {x | x.1 w = 0} → a + b ∈ {x | x.1 w = 0}",
"usedConstants": [
"Real",
"AddMonoid.toAddSemigroup",
"Real.i... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 250,
"column": 67
} | {
"line": 250,
"column": 75
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw : { w // w.IsReal }\n⊢ ∀ {a b : ({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)},\n a ∈ {x | x.1 w = 0} → b ∈ {x | x.1 w = 0} → a + b ∈ {x | x.1 w = 0}",
"usedConstants": [
"Real",
"AddMonoid.toAddSemigroup",
"Real.i... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 250,
"column": 67
} | {
"line": 250,
"column": 75
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw : { w // w.IsReal }\n⊢ ∀ {a b : ({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)},\n a ∈ {x | x.1 w = 0} → b ∈ {x | x.1 w = 0} → a + b ∈ {x | x.1 w = 0}",
"usedConstants": [
"Real",
"AddMonoid.toAddSemigroup",
"Real.i... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 250,
"column": 87
} | {
"line": 250,
"column": 95
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw : { w // w.IsReal }\n⊢ ∀ (c : ℝ) {x : ({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)},\n x ∈ { carrier := {x | x.1 w = 0}, add_mem' := ⋯, zero_mem' := ⋯ }.carrier →\n c • x ∈ { carrier := {x | x.1 w = 0}, add_mem' := ⋯, zero_mem' :=... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 250,
"column": 87
} | {
"line": 250,
"column": 95
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw : { w // w.IsReal }\n⊢ ∀ (c : ℝ) {x : ({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)},\n x ∈ { carrier := {x | x.1 w = 0}, add_mem' := ⋯, zero_mem' := ⋯ }.carrier →\n c • x ∈ { carrier := {x | x.1 w = 0}, add_mem' := ⋯, zero_mem' :=... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 250,
"column": 87
} | {
"line": 250,
"column": 95
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw : { w // w.IsReal }\n⊢ ∀ (c : ℝ) {x : ({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)},\n x ∈ { carrier := {x | x.1 w = 0}, add_mem' := ⋯, zero_mem' := ⋯ }.carrier →\n c • x ∈ { carrier := {x | x.1 w = 0}, add_mem' := ⋯, zero_mem' :=... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 295,
"column": 6
} | {
"line": 295,
"column": 19
} | [
{
"pp": "case h\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : (K →+* ℂ) → ℂ\nh_zero : (commMap K) x = 0\nh_mem : x ∈ Submodule.span ℝ (Set.range ⇑(canonicalEmbedding K))\nφ : K →+* ℂ\n⊢ x φ = 0 φ",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | Pi.zero_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 603,
"column": 2
} | {
"line": 603,
"column": 12
} | [
{
"pp": "case inr\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : (K →+* ℂ) → ℂ\nhx : ∀ (φ : K →+* ℂ), (starRingEnd ℂ) (x φ) = x (ComplexEmbedding.conjugate φ)\nc : { w // w.IsComplex } × Fin 2\n⊢ ↑(((stdBasis K).repr (fun w ↦ (x (↑w).embedding).re, fun w ↦ x (↑w).embedding)) (Sum.inr c)) =\n ∑ x... | | inr c => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.RingTheory.Ideal.Quotient.Index | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 24
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nI : Ideal R\nN : Submodule R M\ninst✝¹ : Finite (R ⧸ I)\ninst✝ : Finite (M ⧸ N)\nhN : N.FG\ne : (↥N ⧸ comap N.subtype (I • N)) ≃ₗ[R] (R ⧸ I) ⊗[R] ↥N :=\n (comap N.subtype (I • N)).quotEquivOfEq (I • ⊤) ⋯ ≪≫ₗ... | exact Nat.card_pos.ne' | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 750,
"column": 4
} | {
"line": 752,
"column": 27
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\n⊢ ChooseBasisIndex ℤ (𝓞 K) ≃ ChooseBasisIndex ℤ ↥↑↑I",
"usedConstants": [
"Units.val",
"Eq.mpr",
"Submodule",
"NumberField.instFreeIntSubtypeMemSubmoduleRingOfIntegersCoeToSubmodule",
... | refine Fintype.equivOfCardEq ?_
rw [← finrank_eq_card_chooseBasisIndex, ← finrank_eq_card_chooseBasisIndex,
fractionalIdeal_rank] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 750,
"column": 4
} | {
"line": 752,
"column": 27
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\n⊢ ChooseBasisIndex ℤ (𝓞 K) ≃ ChooseBasisIndex ℤ ↥↑↑I",
"usedConstants": [
"Units.val",
"Eq.mpr",
"Submodule",
"NumberField.instFreeIntSubtypeMemSubmoduleRingOfIntegersCoeToSubmodule",
... | refine Fintype.equivOfCardEq ?_
rw [← finrank_eq_card_chooseBasisIndex, ← finrank_eq_card_chooseBasisIndex,
fractionalIdeal_rank] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 427,
"column": 58
} | {
"line": 427,
"column": 76
} | [
{
"pp": "case refine_2.refine_2\nA : Type u_2\ninst✝¹ : Field A\ninst✝ : CharZero A\nN : ℕ\nD : ℕ := rankOfDiscrBdd N\nB : ℝ≥0 := boundOfDiscBdd N\nC : ℕ := ⌈max B 1 ^ D * ↑(D.choose (D / 2))⌉₊\nx✝¹ : { F // FiniteDimensional ℚ ↥F }\nK : IntermediateField ℚ A\nhK₀ : FiniteDimensional ℚ ↥K\nx✝ : ⟨K, hK₀⟩ ∈ {K | ... | NNReal.coe_natCast | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 1021,
"column": 4
} | {
"line": 1023,
"column": 69
} | [
{
"pp": "case fst.h\nK : Type u_1\ninst✝ : Field K\ns : Set { w // w.IsReal }\nA : Set (mixedSpace K)\nx : mixedSpace K\nhA : ∀ (x : mixedSpace K), x ∈ A ↔ (fun w ↦ ‖x.1 w‖, x.2) ∈ A\nhx₁ : x ∈ A\nhx₂ : ∀ (w : { w // w.IsReal }), x.1 w ≠ 0\nx✝ : (∀ w ∈ s, x.1 w < 0) ∧ ∀ w ∉ s, x.1 w > 0\nh₁ : ∀ w ∈ s, x.1 w < 0... | by_cases hw : w ∈ s
· simp [negAt_apply_isReal_and_mem _ hw, abs_of_neg (h₁ w hw)]
· simp [negAt_apply_isReal_and_notMem _ hw, abs_of_pos (h₂ w hw)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 1021,
"column": 4
} | {
"line": 1023,
"column": 69
} | [
{
"pp": "case fst.h\nK : Type u_1\ninst✝ : Field K\ns : Set { w // w.IsReal }\nA : Set (mixedSpace K)\nx : mixedSpace K\nhA : ∀ (x : mixedSpace K), x ∈ A ↔ (fun w ↦ ‖x.1 w‖, x.2) ∈ A\nhx₁ : x ∈ A\nhx₂ : ∀ (w : { w // w.IsReal }), x.1 w ≠ 0\nx✝ : (∀ w ∈ s, x.1 w < 0) ∧ ∀ w ∉ s, x.1 w > 0\nh₁ : ∀ w ∈ s, x.1 w < 0... | by_cases hw : w ∈ s
· simp [negAt_apply_isReal_and_mem _ hw, abs_of_neg (h₁ w hw)]
· simp [negAt_apply_isReal_and_notMem _ hw, abs_of_pos (h₂ w hw)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.FractionalIdeal.Extended | {
"line": 256,
"column": 6
} | {
"line": 256,
"column": 22
} | [
{
"pp": "A : Type u_1\nK : Type u_2\nL : Type u_3\nB : Type u_4\ninst✝¹⁸ : CommRing A\ninst✝¹⁷ : IsDomain A\ninst✝¹⁶ : CommRing B\ninst✝¹⁵ : IsDomain B\ninst✝¹⁴ : Algebra A B\ninst✝¹³ : IsTorsionFree A B\ninst✝¹² : Field K\ninst✝¹¹ : Field L\ninst✝¹⁰ : Algebra A K\ninst✝⁹ : Algebra B L\ninst✝⁸ : IsFractionRing ... | le_antisymm_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 1110,
"column": 65
} | {
"line": 1110,
"column": 73
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw : InfinitePlace K\n⊢ ∀ {a b : InfinitePlace K → ℝ}, a ∈ {x | x w = 0} → b ∈ {x | x w = 0} → a + b ∈ {x | x w = 0}",
"usedConstants": [
"Real",
"AddMonoid.toAddSemigroup",
"Real.instZero",
"Real.instAddMonoid",
"c... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 1110,
"column": 65
} | {
"line": 1110,
"column": 73
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw : InfinitePlace K\n⊢ ∀ {a b : InfinitePlace K → ℝ}, a ∈ {x | x w = 0} → b ∈ {x | x w = 0} → a + b ∈ {x | x w = 0}",
"usedConstants": [
"Real",
"AddMonoid.toAddSemigroup",
"Real.instZero",
"Real.instAddMonoid",
"c... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 1110,
"column": 65
} | {
"line": 1110,
"column": 73
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw : InfinitePlace K\n⊢ ∀ {a b : InfinitePlace K → ℝ}, a ∈ {x | x w = 0} → b ∈ {x | x w = 0} → a + b ∈ {x | x w = 0}",
"usedConstants": [
"Real",
"AddMonoid.toAddSemigroup",
"Real.instZero",
"Real.instAddMonoid",
"c... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 1110,
"column": 85
} | {
"line": 1110,
"column": 93
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw : InfinitePlace K\n⊢ ∀ (c : ℝ) {x : InfinitePlace K → ℝ},\n x ∈ { carrier := {x | x w = 0}, add_mem' := ⋯, zero_mem' := ⋯ }.carrier →\n c • x ∈ { carrier := {x | x w = 0}, add_mem' := ⋯, zero_mem' := ⋯ }.carrier",
"usedConstants": [
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 1110,
"column": 85
} | {
"line": 1110,
"column": 93
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw : InfinitePlace K\n⊢ ∀ (c : ℝ) {x : InfinitePlace K → ℝ},\n x ∈ { carrier := {x | x w = 0}, add_mem' := ⋯, zero_mem' := ⋯ }.carrier →\n c • x ∈ { carrier := {x | x w = 0}, add_mem' := ⋯, zero_mem' := ⋯ }.carrier",
"usedConstants": [
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 1110,
"column": 85
} | {
"line": 1110,
"column": 93
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw : InfinitePlace K\n⊢ ∀ (c : ℝ) {x : InfinitePlace K → ℝ},\n x ∈ { carrier := {x | x w = 0}, add_mem' := ⋯, zero_mem' := ⋯ }.carrier →\n c • x ∈ { carrier := {x | x w = 0}, add_mem' := ⋯, zero_mem' := ⋯ }.carrier",
"usedConstants": [
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.PID | {
"line": 82,
"column": 4
} | {
"line": 82,
"column": 57
} | [
{
"pp": "case refine_1\nR : Type u_2\nA : Type u_3\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nI : (FractionalIdeal S A)ˣ\nv : A\nhv : v ∈ ↑I⁻¹\nhinv : ↑I * ↑I⁻¹ = 1\nJ : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * (R ∙ v))\nh... | grw [FractionalIdeal.spanSingleton_le_iff_mem.mpr hw] | Mathlib.Tactic._aux_Mathlib_Tactic_GRewrite_Elab___macroRules_Mathlib_Tactic_grwSeq_1 | Mathlib.Tactic.grwSeq |
Mathlib.RingTheory.DedekindDomain.PID | {
"line": 99,
"column": 13
} | {
"line": 99,
"column": 15
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nA : Type u_2\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nhS : S ≤ R⁰\nhf : {I | I.IsMaximal}.Finite\nI I' : FractionalIdeal S A\nhinv : ↑I * ↑I' = ↑1\nhinv' : I * I' = 1\ns : Finset (Ideal R) := hf.toFinset\nthis : Decidabl... | s, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 507,
"column": 2
} | {
"line": 507,
"column": 26
} | [
{
"pp": "A : Type u_2\ninst✝¹ : Field A\ninst✝ : CharZero A\nN : ℕ\nK : IntermediateField ℚ A\nhK₀ : FiniteDimensional ℚ ↥K\nhK₁ : ⟨K, hK₀⟩ ∈ {K | |discr ↥↑K| ≤ ↑N}\nthis✝ : CharZero ↥K\nthis : NumberField ↥K\nw₀ : InfinitePlace ↥K\n⊢ ⟨K, hK₀⟩ ∈ {K | {w | w.IsReal}.Nonempty ∧ |discr ↥↑K| ≤ ↑N} ∪ {K | {w | w.IsC... | by_cases hw₀ : IsReal w₀ | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.RingTheory.Ideal.Int | {
"line": 82,
"column": 6
} | {
"line": 82,
"column": 19
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\nI : Ideal R\nh : absNorm (under ℤ I) = 0\nx : ℕ\nhx₁ : x ≠ 0\nhx₂ : x = 0\n⊢ False",
"usedConstants": []
}
] | exact hx₁ hx₂ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.DedekindDomain.LinearDisjoint | {
"line": 134,
"column": 7
} | {
"line": 134,
"column": 65
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nK : Type u_3\nL : Type u_4\ninst✝⁵⁵ : CommRing A\ninst✝⁵⁴ : Field K\ninst✝⁵³ : Algebra A K\ninst✝⁵² : IsFractionRing A K\ninst✝⁵¹ : CommRing B\ninst✝⁵⁰ : Field L\ninst✝⁴⁹ : Algebra B L\ninst✝⁴⁸ : Algebra A L\ninst✝⁴⁷ : Algebra K L\ninst✝⁴⁶ : FiniteDimensional K L\ninst✝⁴⁵ : ... | ← differentIdeal_eq_map_differentIdeal A B R₁ R₂ h₁ h₂ h₃, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.LinearDisjoint | {
"line": 187,
"column": 2
} | {
"line": 188,
"column": 74
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nK : Type u_3\nL : Type u_4\ninst✝⁵⁵ : CommRing A\ninst✝⁵⁴ : Field K\ninst✝⁵³ : Algebra A K\ninst✝⁵² : IsFractionRing A K\ninst✝⁵¹ : CommRing B\ninst✝⁵⁰ : Field L\ninst✝⁴⁹ : Algebra B L\ninst✝⁴⁸ : Algebra A L\ninst✝⁴⁷ : Algebra K L\ninst✝⁴⁶ : FiniteDimensional K L\ninst✝⁴⁵ : ... | have h_main := congr_arg (Submodule.restrictScalars R₁) <|
congr_arg coeToSubmodule <| (1 : FractionalIdeal B⁰ L).dual_dual R₁ F₁ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 121,
"column": 4
} | {
"line": 121,
"column": 23
} | [
{
"pp": "case neg\nA : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Field K\ninst✝⁸ : CommRing B\ninst✝⁷ : Field L\ninst✝⁶ : Algebra A K\ninst✝⁵ : Algebra B L\ninst✝⁴ : Algebra A B\ninst✝³ : Algebra K L\ninst✝² : Algebra A L\ninst✝¹ : IsScalarTower A K L\ninst✝ : IsScalarTowe... | obtain ⟨b, hb⟩ := h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Prime | {
"line": 48,
"column": 6
} | {
"line": 50,
"column": 65
} | [
{
"pp": "case insert.inr\nR : Type u_1\ninst✝² : CommMonoidWithZero R\ninst✝¹ : IsCancelMulZero R\nα : Type u_2\ninst✝ : DecidableEq α\np : α → R\ni : α\ns : Finset α\nhis : i ∉ s\nih :\n ∀ {x y a : R},\n (∀ i ∈ s, Prime (p i)) →\n x * y = a * ∏ i ∈ s, p i →\n ∃ t u b c, t ∪ u = s ∧ Disjoint t u... | rw [← mul_assoc, mul_right_comm b, mul_left_inj' hpi.ne_zero] at hbc
exact ⟨t, insert i u, b, d, by rw [union_insert, htus], disjoint_insert_right.2 ⟨hit, htu⟩, by
simp [← hbc, prod_insert hiu, mul_comm, mul_left_comm]⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Prime | {
"line": 48,
"column": 6
} | {
"line": 50,
"column": 65
} | [
{
"pp": "case insert.inr\nR : Type u_1\ninst✝² : CommMonoidWithZero R\ninst✝¹ : IsCancelMulZero R\nα : Type u_2\ninst✝ : DecidableEq α\np : α → R\ni : α\ns : Finset α\nhis : i ∉ s\nih :\n ∀ {x y a : R},\n (∀ i ∈ s, Prime (p i)) →\n x * y = a * ∏ i ∈ s, p i →\n ∃ t u b c, t ∪ u = s ∧ Disjoint t u... | rw [← mul_assoc, mul_right_comm b, mul_left_inj' hpi.ne_zero] at hbc
exact ⟨t, insert i u, b, d, by rw [union_insert, htus], disjoint_insert_right.2 ⟨hit, htu⟩, by
simp [← hbc, prod_insert hiu, mul_comm, mul_left_comm]⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 205,
"column": 2
} | {
"line": 205,
"column": 86
} | [
{
"pp": "case h\nA : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝¹⁷ : CommRing A\ninst✝¹⁶ : Field K\ninst✝¹⁵ : CommRing B\ninst✝¹⁴ : Field L\ninst✝¹³ : Algebra A K\ninst✝¹² : Algebra B L\ninst✝¹¹ : Algebra A B\ninst✝¹⁰ : Algebra K L\ninst✝⁹ : Algebra A L\ninst✝⁸ : IsScalarTower A K L\ninst✝⁷ : IsScal... | rw [smul_mul_assoc, b.equivFun.map_smul, discr_def, mul_comm, ← H, Algebra.smul_def] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 357,
"column": 37
} | {
"line": 357,
"column": 56
} | [
{
"pp": "case neg.a\nA : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝¹⁸ : CommRing A\ninst✝¹⁷ : Field K\ninst✝¹⁶ : CommRing B\ninst✝¹⁵ : Field L\ninst✝¹⁴ : Algebra A K\ninst✝¹³ : Algebra B L\ninst✝¹² : Algebra A B\ninst✝¹¹ : Algebra K L\ninst✝¹⁰ : Algebra A L\ninst✝⁹ : IsScalarTower A K L\ninst✝⁸ : I... | inv_mul_cancel₀ hI, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 369,
"column": 34
} | {
"line": 369,
"column": 53
} | [
{
"pp": "A : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝¹⁸ : CommRing A\ninst✝¹⁷ : Field K\ninst✝¹⁶ : CommRing B\ninst✝¹⁵ : Field L\ninst✝¹⁴ : Algebra A K\ninst✝¹³ : Algebra B L\ninst✝¹² : Algebra A B\ninst✝¹¹ : Algebra K L\ninst✝¹⁰ : Algebra A L\ninst✝⁹ : IsScalarTower A K L\ninst✝⁸ : IsScalarTower... | inv_mul_cancel₀ hI, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral | {
"line": 197,
"column": 4
} | {
"line": 201,
"column": 77
} | [
{
"pp": "case h.a.calc_2.calc_1\nR : Type u\nK : Type v\nL : Type z\np : R\ninst✝⁹ : CommRing R\ninst✝⁸ : Field K\ninst✝⁷ : Field L\ninst✝⁶ : Algebra K L\ninst✝⁵ : Algebra R L\ninst✝⁴ : Algebra R K\ninst✝³ : IsScalarTower R K L\ninst✝² : IsDomain R\ninst✝¹ : IsFractionRing R K\ninst✝ : IsIntegrallyClosed R\nB :... | have : ∀ i ∈ (range (Q.natDegree + 1)).erase 0,
Q.coeff i • (B.gen ^ i * B.gen ^ n) = p • Q.coeff i • f (i + n) := by
intro i hi
rw [← pow_add, ← (hf _ (aux i hi)).2, ← Algebra.smul_def, smul_smul, mul_comm _ p, smul_smul]
simp only [add_mul, smul_mul_assoc, one_mul, sum_mul, sum_congr rfl this] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral | {
"line": 197,
"column": 4
} | {
"line": 201,
"column": 77
} | [
{
"pp": "case h.a.calc_2.calc_1\nR : Type u\nK : Type v\nL : Type z\np : R\ninst✝⁹ : CommRing R\ninst✝⁸ : Field K\ninst✝⁷ : Field L\ninst✝⁶ : Algebra K L\ninst✝⁵ : Algebra R L\ninst✝⁴ : Algebra R K\ninst✝³ : IsScalarTower R K L\ninst✝² : IsDomain R\ninst✝¹ : IsFractionRing R K\ninst✝ : IsIntegrallyClosed R\nB :... | have : ∀ i ∈ (range (Q.natDegree + 1)).erase 0,
Q.coeff i • (B.gen ^ i * B.gen ^ n) = p • Q.coeff i • f (i + n) := by
intro i hi
rw [← pow_add, ← (hf _ (aux i hi)).2, ← Algebra.smul_def, smul_smul, mul_comm _ p, smul_smul]
simp only [add_mul, smul_mul_assoc, one_mul, sum_mul, sum_congr rfl this] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral | {
"line": 242,
"column": 2
} | {
"line": 242,
"column": 23
} | [
{
"pp": "R : Type u\nK : Type v\nL : Type z\np : R\ninst✝⁹ : CommRing R\ninst✝⁸ : Field K\ninst✝⁷ : Field L\ninst✝⁶ : Algebra K L\ninst✝⁵ : Algebra R L\ninst✝⁴ : Algebra R K\ninst✝³ : IsScalarTower R K L\ninst✝² : IsDomain R\ninst✝¹ : IsFractionRing R K\ninst✝ : IsIntegrallyClosed R\nB : PowerBasis K L\nhp : _r... | obtain ⟨Q₁, hQ⟩ := hz | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.NumberTheory.NumberField.Units.DirichletTheorem | {
"line": 123,
"column": 65
} | {
"line": 123,
"column": 78
} | [
{
"pp": "case refine_2.h\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : (𝓞 K)ˣ\nh : ∀ (w : InfinitePlace K), w ((algebraMap (𝓞 K) K) ↑x) = 1\nw : { w // w ≠ w₀ }\n⊢ 0 = 0 w",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"congrArg",
"Pi.zero_apply",
... | Pi.zero_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 727,
"column": 4
} | {
"line": 728,
"column": 84
} | [
{
"pp": "A : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝²¹ : CommRing A\ninst✝²⁰ : Field K\ninst✝¹⁹ : CommRing B\ninst✝¹⁸ : Field L\ninst✝¹⁷ : Algebra A K\ninst✝¹⁶ : Algebra B L\ninst✝¹⁵ : Algebra A B\ninst✝¹⁴ : Algebra K L\ninst✝¹³ : Algebra A L\ninst✝¹² : IsScalarTower A K L\ninst✝¹¹ : IsScalarTow... | rwa [mul_comm, ← smul_eq_mul, ← map_smul, Algebra.smul_def, mul_comm,
← IsScalarTower.algebraMap_apply, IsScalarTower.algebraMap_apply A B L, ← hz'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.NumberTheory.NumberField.Units.DirichletTheorem | {
"line": 126,
"column": 2
} | {
"line": 128,
"column": 6
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ (logEmbedding K).ker = Subgroup.toAddSubgroup (torsion K)",
"usedConstants": [
"Eq.mpr",
"Real",
"Equiv.instEquivLike",
"Subgroup.toAddSubgroup",
"NumberField.instCommRingRingOfIntegers",
"Real.instZero",
... | ext x
rw [AddMonoidHom.mem_ker, ← ofMul_toMul x, logEmbedding_eq_zero_iff]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.Units.DirichletTheorem | {
"line": 126,
"column": 2
} | {
"line": 128,
"column": 6
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ (logEmbedding K).ker = Subgroup.toAddSubgroup (torsion K)",
"usedConstants": [
"Eq.mpr",
"Real",
"Equiv.instEquivLike",
"Subgroup.toAddSubgroup",
"NumberField.instCommRingRingOfIntegers",
"Real.instZero",
... | ext x
rw [AddMonoidHom.mem_ker, ← ofMul_toMul x, logEmbedding_eq_zero_iff]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.Units.DirichletTheorem | {
"line": 132,
"column": 2
} | {
"line": 132,
"column": 69
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\ns : AddSubgroup (Additive (𝓞 K)ˣ)\n⊢ AddSubgroup.map (logEmbedding K) (s ⊔ Subgroup.toAddSubgroup (torsion K)) = AddSubgroup.map (logEmbedding K) s",
"usedConstants": [
"Eq.mpr",
"AddSubgroup.map_eq_map_iff",
"AddSubgroup.ins... | rw [← logEmbedding_ker, AddSubgroup.map_eq_map_iff, sup_right_idem] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.NumberField.Units.DirichletTheorem | {
"line": 132,
"column": 2
} | {
"line": 132,
"column": 69
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\ns : AddSubgroup (Additive (𝓞 K)ˣ)\n⊢ AddSubgroup.map (logEmbedding K) (s ⊔ Subgroup.toAddSubgroup (torsion K)) = AddSubgroup.map (logEmbedding K) s",
"usedConstants": [
"Eq.mpr",
"AddSubgroup.map_eq_map_iff",
"AddSubgroup.ins... | rw [← logEmbedding_ker, AddSubgroup.map_eq_map_iff, sup_right_idem] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.Units.DirichletTheorem | {
"line": 132,
"column": 2
} | {
"line": 132,
"column": 69
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\ns : AddSubgroup (Additive (𝓞 K)ˣ)\n⊢ AddSubgroup.map (logEmbedding K) (s ⊔ Subgroup.toAddSubgroup (torsion K)) = AddSubgroup.map (logEmbedding K) s",
"usedConstants": [
"Eq.mpr",
"AddSubgroup.map_eq_map_iff",
"AddSubgroup.ins... | rw [← logEmbedding_ker, AddSubgroup.map_eq_map_iff, sup_right_idem] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 80,
"column": 89
} | {
"line": 122,
"column": 85
} | [
{
"pp": "p k : ℕ\nK : Type u\ninst✝¹ : Field K\nζ : K\nhp : Fact (Nat.Prime p)\ninst✝ : CharZero K\nhcycl : IsCyclotomicExtension {p ^ k} ℚ K\nhζ : IsPrimitiveRoot ζ (p ^ k)\n⊢ IsIntegralClosure ↥ℤ[ζ] ℤ K",
"usedConstants": [
"Subalgebra.instSetLike",
"NormedCommRing.toNormedRing",
"PowerB... | by
refine ⟨Subtype.val_injective, @fun x => ⟨fun h => ⟨⟨x, ?_⟩, rfl⟩, ?_⟩⟩
swap
· rintro ⟨y, rfl⟩
exact
IsIntegral.algebraMap
((le_integralClosure_iff_isIntegral.1
(adjoin_le_integralClosure (hζ.isIntegral (NeZero.pos _)))).isIntegral _)
let B := hζ.subOnePowerBasis ℚ
have hint : I... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.NumberField.Units.DirichletTheorem | {
"line": 334,
"column": 4
} | {
"line": 334,
"column": 41
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : Basis { w // w ≠ w₀ } ℝ ({ w // w ≠ w₀ } → ℝ) := Pi.basisFun ℝ { w // w ≠ w₀ }\nv : { w // w ≠ w₀ } → logSpace K := fun w ↦ (logEmbedding K) (Additive.ofMul ⋯.choose)\nw : { w // w ≠ w₀ }\n⊢ 0 < |v w w| - v w w + ∑ x, v w x",
"usedConstants... | refine add_pos_of_nonneg_of_pos ?_ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 884,
"column": 4
} | {
"line": 885,
"column": 84
} | [
{
"pp": "A : Type u_1\nB : Type u_3\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : CommRing B\ninst✝⁹ : Algebra A B\ninst✝⁸ : IsDomain A\ninst✝⁷ : IsDedekindDomain A\ninst✝⁶ : IsDedekindDomain B\ninst✝⁵ : IsTorsionFree A B\ninst✝⁴ : Module.Finite A B\ninst✝³ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\np : Ideal ... | rwa [mul_comm, ← smul_eq_mul, ← map_smul, Algebra.smul_def, mul_comm,
← IsScalarTower.algebraMap_apply, IsScalarTower.algebraMap_apply A B L, ← hz'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.NumberTheory.NumberField.Units.DirichletTheorem | {
"line": 492,
"column": 46
} | {
"line": 493,
"column": 66
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx ζ : (𝓞 K)ˣ\nf : Fin (rank K) → ℤ\nhζ : ζ ∈ torsion K\nh : x = ζ * ∏ i, fundSystem K i ^ f i\n⊢ ∑ i, f i • Additive.ofMul ↑(fundSystem K i) = ∑ i, f i • (basisModTorsion K) i",
"usedConstants": [
"Eq.mpr",
"instHSMul",
"Equi... | by
simp_rw [fundSystem, QuotientGroup.out_eq', ofMul_toMul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic | {
"line": 161,
"column": 2
} | {
"line": 161,
"column": 87
} | [
{
"pp": "case pos\nF : Type u_1\ninst✝² : Field F\ninst✝¹ : Fintype F\ninst✝ : DecidableEq F\na : F\nha : a = 0\n⊢ (quadraticChar F) a = -1 ↔ ¬IsSquare a",
"usedConstants": [
"Int.instAddCommGroup",
"_private.Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic.0.quadraticChar_neg_one_iff_not... | · simp only [ha, MulChar.map_zero, zero_eq_neg, one_ne_zero, IsSquare.zero, not_true] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic | {
"line": 228,
"column": 17
} | {
"line": 228,
"column": 19
} | [
{
"pp": "case h\nF : Type u_1\ninst✝² : Field F\ninst✝¹ : Fintype F\ninst✝ : DecidableEq F\nhF : ringChar F ≠ 2\na : F\ns : Finset F := {x | x ^ 2 = a}.toFinset\nb : F\nh₀ : ¬b = 0\nh : a = b ^ 2\na✝ : F\n⊢ a✝ ∈ s ↔ a✝ ∈ [b, -b].toFinset",
"usedConstants": [
"NegZeroClass.toNeg",
"Finset",
... | s, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity | {
"line": 75,
"column": 2
} | {
"line": 75,
"column": 43
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp : p ≠ 2\n⊢ IsSquare 2 ↔ p % 8 = 1 ∨ p % 8 = 7",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"FiniteField.isSquare_two_iff",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
"Nat.instOne",
"congrArg",
"ZMod.f... | rw [FiniteField.isSquare_two_iff, card p] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Ideal.NatInt | {
"line": 33,
"column": 4
} | {
"line": 33,
"column": 15
} | [
{
"pp": "a b : ℕ\nhab : a + b = 1\nh : a = 1 ∨ b = 1\n⊢ IsUnit a ∨ IsUnit b",
"usedConstants": [
"IsUnit",
"instOfNatNat",
"Nat",
"Semiring.toMonoid",
"Nat.instSemiring",
"Or.imp",
"OfNat.ofNat",
"Eq"
]
}
] | apply h.imp | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.NumberTheory.FLT.Three | {
"line": 343,
"column": 2
} | {
"line": 355,
"column": 42
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\nS' : Solution' hζ\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\n⊢ ∃ a' b', a' ^ 3 + b' ^ 3 = ↑S'.u * S'.c ^ 3 ∧ IsCoprime a' b' ∧ ¬λ ∣ a' ∧ ¬λ ∣ b' ∧ λ ^ 2 ∣ a' + b'",
"usedConstants": [
"IsPrimitiveRoot.toInte... | rcases lambda_sq_dvd_or_dvd_or_dvd S' with h | h | h
· exact ⟨S'.a, S'.b, S'.H, S'.coprime, S'.ha, S'.hb, h⟩
· refine ⟨S'.a, η * S'.b, ?_, ?_, S'.ha, fun ⟨x, hx⟩ ↦ S'.hb ⟨η ^ 2 * x, ?_⟩, h⟩
· simp [mul_pow, hζ.toInteger_cube_eq_one, one_mul, S'.H]
· refine (isCoprime_mul_unit_left_right (Units.isUnit η) _ _... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.FLT.Three | {
"line": 343,
"column": 2
} | {
"line": 355,
"column": 42
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\nS' : Solution' hζ\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\n⊢ ∃ a' b', a' ^ 3 + b' ^ 3 = ↑S'.u * S'.c ^ 3 ∧ IsCoprime a' b' ∧ ¬λ ∣ a' ∧ ¬λ ∣ b' ∧ λ ^ 2 ∣ a' + b'",
"usedConstants": [
"IsPrimitiveRoot.toInte... | rcases lambda_sq_dvd_or_dvd_or_dvd S' with h | h | h
· exact ⟨S'.a, S'.b, S'.H, S'.coprime, S'.ha, S'.hb, h⟩
· refine ⟨S'.a, η * S'.b, ?_, ?_, S'.ha, fun ⟨x, hx⟩ ↦ S'.hb ⟨η ^ 2 * x, ?_⟩, h⟩
· simp [mul_pow, hζ.toInteger_cube_eq_one, one_mul, S'.H]
· refine (isCoprime_mul_unit_left_right (Units.isUnit η) _ _... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.FLT.Three | {
"line": 359,
"column": 96
} | {
"line": 372,
"column": 22
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\nS' : Solution' hζ\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\n⊢ ∃ S₁, S₁.multiplicity = S'.multiplicity",
"usedConstants": [
"Units.val",
"Dvd.dvd",
"HMul.hMul",
"CommRing.toNonUnitalCommRin... | by
obtain ⟨a, b, H, coprime, ha, hb, hab⟩ := ex_cube_add_cube_eq_and_isCoprime_and_not_dvd_and_dvd S'
exact ⟨
{ a := a
b := b
c := S'.c
u := S'.u
ha := ha
hb := hb
hc := S'.hc
coprime := coprime
hcdvd := S'.hcdvd
H := H
hab := hab }, rfl⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Harmonic.Bounds | {
"line": 55,
"column": 4
} | {
"line": 55,
"column": 18
} | [
{
"pp": "case h.e'_3.h.e'_1\nn : ℕ\nhn0 : ¬n = 0\nhn : 1 ≤ n\n⊢ ↑n = ↑n / 1",
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"InvOneClass.toOne",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"congrArg",
"DivInvOneMonoid.toDivInvMon... | · rw [div_one] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.FLT.Three | {
"line": 453,
"column": 14
} | {
"line": 453,
"column": 40
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\nS : Solution hζ\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\np : 𝓞 K\nhp : Prime p\nhpaηb : p ∣ 1 * S.a + ↑η * S.b\nhpaηsqb : p ∣ 1 * S.a + ↑η ^ 2 * S.b\nthis : p ∣ ↑η ^ 2 - ↑η\n⊢ 2 * ↑η ^ 2 + 3 * ↑η + 1 = λ",
"use... | rw [eta_sq, coe_eta]; ring | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.FLT.Three | {
"line": 453,
"column": 14
} | {
"line": 453,
"column": 40
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\nS : Solution hζ\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\np : 𝓞 K\nhp : Prime p\nhpaηb : p ∣ 1 * S.a + ↑η * S.b\nhpaηsqb : p ∣ 1 * S.a + ↑η ^ 2 * S.b\nthis : p ∣ ↑η ^ 2 - ↑η\n⊢ 2 * ↑η ^ 2 + 3 * ↑η + 1 = λ",
"use... | rw [eta_sq, coe_eta]; ring | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.FrobeniusNumber | {
"line": 194,
"column": 6
} | {
"line": 194,
"column": 40
} | [
{
"pp": "s : AddSubmonoid ℕ\n⊢ s.FG",
"usedConstants": [
"Eq.mpr",
"Submodule",
"AddSubmonoid.toNatSubmodule",
"Submodule.toAddSubmonoid",
"congrArg",
"AddMonoid.toAddZeroClass",
"PartialOrder.toPreorder",
"AddSubmonoid.toNatSubmodule_toAddSubmonoid",
"P... | ← s.toNatSubmodule_toAddSubmonoid, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 107,
"column": 8
} | {
"line": 107,
"column": 56
} | [
{
"pp": "n : ℕ\nhn : 0 < n\nhv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x\n⊢ -1 < -2 ∨ -2 ≠ -1 ∧ 0 ∉ uIcc (↑n) (↑n + 1)",
"usedConstants": [
"False",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.lattice",
"Real.instZero",
"Real.instRCLike",
"congrArg",
... | refine Or.inr ⟨by simp, notMem_uIcc_of_lt ?_ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.FLT.Three | {
"line": 755,
"column": 2
} | {
"line": 755,
"column": 65
} | [
{
"pp": "K : Type := CyclotomicField 3 ℚ\nhζ : IsPrimitiveRoot (IsCyclotomicExtension.zeta 3 ℚ K) 3 := IsCyclotomicExtension.zeta_spec 3 ℚ K\nthis : NumberField K\n⊢ FermatLastTheoremFor 3",
"usedConstants": [
"Rat",
"Field.toDivisionRing",
"instOfNatNat",
"Field.toCommRing",
"... | apply FermatLastTheoremForThree_of_FermatLastTheoremThreeGen hζ | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.NumberTheory.Height.NumberField | {
"line": 259,
"column": 2
} | {
"line": 259,
"column": 59
} | [
{
"pp": "n : ℕ\ninst✝ : NeZero n\n⊢ logHeight₁ ↑n = Real.log ↑n",
"usedConstants": [
"Real",
"congrArg",
"Rat",
"Height.logHeight₁",
"NumberField.instAdmissibleAbsValues",
"Nat.cast",
"Real.log",
"Rat.numberField",
"Rat.instField",
"True",
"e... | simp [logHeight₁_eq_log_mulHeight₁, mulHeight₁_natCast n] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.Height.NumberField | {
"line": 259,
"column": 2
} | {
"line": 259,
"column": 59
} | [
{
"pp": "n : ℕ\ninst✝ : NeZero n\n⊢ logHeight₁ ↑n = Real.log ↑n",
"usedConstants": [
"Real",
"congrArg",
"Rat",
"Height.logHeight₁",
"NumberField.instAdmissibleAbsValues",
"Nat.cast",
"Real.log",
"Rat.numberField",
"Rat.instField",
"True",
"e... | simp [logHeight₁_eq_log_mulHeight₁, mulHeight₁_natCast n] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Height.NumberField | {
"line": 259,
"column": 2
} | {
"line": 259,
"column": 59
} | [
{
"pp": "n : ℕ\ninst✝ : NeZero n\n⊢ logHeight₁ ↑n = Real.log ↑n",
"usedConstants": [
"Real",
"congrArg",
"Rat",
"Height.logHeight₁",
"NumberField.instAdmissibleAbsValues",
"Nat.cast",
"Real.log",
"Rat.numberField",
"Rat.instField",
"True",
"e... | simp [logHeight₁_eq_log_mulHeight₁, mulHeight₁_natCast n] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 295,
"column": 4
} | {
"line": 295,
"column": 50
} | [
{
"pp": "this : Tendsto (fun s ↦ ∑' (n : ℕ), 1 / (↑n + 1) ^ s - 1 / (s - 1)) (𝓝[>] 1) (𝓝 γ)\n⊢ (Complex.ofReal ∘ fun s ↦ ∑' (n : ℕ), 1 / (↑n + 1) ^ s - 1 / (s - 1)) =ᶠ[𝓝[>] 1] fun s ↦ riemannZeta ↑s - 1 / (↑s - 1)",
"usedConstants": [
"Real.instPow",
"Real",
"Set.Ioi",
"instHDiv",... | filter_upwards [self_mem_nhdsWithin] with s hs | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 331,
"column": 4
} | {
"line": 331,
"column": 50
} | [
{
"pp": "case refine_1\nf : ℂ → ℂ := fun s ↦ riemannZeta s - 1 / (s - 1)\n⊢ ∀ᶠ (z : ℂ) in 𝓝[≠] 1, DifferentiableAt ℂ f z",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"Semiring.toModule",
"Complex.instNormedAddCommGroup",
"Compl... | filter_upwards [self_mem_nhdsWithin] with s hs | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 371,
"column": 20
} | {
"line": 371,
"column": 45
} | [
{
"pp": "h : Tendsto (fun b ↦ (b.Gammaℝ - 1) / (b - 1)) (𝓝[≠] 1) (𝓝 (-(↑γ + Complex.log (4 * ↑π)) / 2))\nthis✝ : Tendsto ((fun b ↦ (b.Gammaℝ - 1) / (b - 1)) / Gammaℝ) (𝓝[≠] 1) (𝓝 (-(↑γ + Complex.log (4 * ↑π)) / 2))\nthis : {z | 0 < z.re} ∈ 𝓝 1\n⊢ ∀ᶠ (x : ℂ) in 𝓝[≠] 1, ((fun b ↦ (b.Gammaℝ - 1) / (b - 1)) /... | eventually_nhdsWithin_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 379,
"column": 11
} | {
"line": 379,
"column": 29
} | [
{
"pp": "this :\n Tendsto (fun x ↦ riemannZeta x - 1 / (x - 1) + (1 / (x - 1) - 1 / x.Gammaℝ / (x - 1))) (𝓝[≠] 1)\n (𝓝 (↑γ + -(↑γ + Complex.log (4 * ↑π)) / 2))\n⊢ Tendsto (fun s ↦ riemannZeta s - 1 / s.Gammaℝ / (s - 1)) (𝓝[≠] 1) (𝓝 ((↑γ - Complex.log (4 * ↑π)) / 2))",
"usedConstants": [
"Norme... | sub_add_sub_cancel | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.JacobiSum.Basic | {
"line": 74,
"column": 59
} | {
"line": 79,
"column": 71
} | [
{
"pp": "F : Type u_1\nR : Type u_2\ninst✝⁴ : CommRing F\ninst✝³ : Nontrivial F\ninst✝² : Fintype F\ninst✝¹ : DecidableEq F\ninst✝ : CommRing R\nχ ψ : MulChar F R\n⊢ jacobiSum χ ψ = ∑ x ∈ univ \\ {0, 1}, χ x * ψ (1 - x)",
"usedConstants": [
"Eq.mpr",
"add_neg_cancel",
"NonUnitalCommRing.to... | by
simp only [jacobiSum, subset_univ, sum_sdiff_eq_sub, sub_eq_add_neg, left_eq_add,
neg_eq_zero]
apply sum_eq_zero
simp only [mem_insert, mem_singleton, forall_eq_or_imp, χ.map_zero, neg_zero, add_zero, map_one,
mul_one, forall_eq, add_neg_cancel, ψ.map_zero, mul_zero, and_self] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Height.MvPolynomial | {
"line": 193,
"column": 15
} | {
"line": 193,
"column": 23
} | [
{
"pp": "K : Type u_4\ninst✝¹ : Field K\nι : Type u_5\ninst✝ : Finite ι\nv : AbsoluteValue K ℝ\np : MvPolynomial ι K\nN : ℕ\nhp : p.IsHomogeneous N\nx : ι → K\n⊢ v (∑ d ∈ p.support, coeff d p * ∏ i ∈ d.support, x i ^ d i) ≤ (Finsupp.sum p fun x c ↦ v c) * (⨆ i, v (x i)) ^ N",
"usedConstants": [
"Finsu... | sum_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.JacobiSum.Basic | {
"line": 210,
"column": 2
} | {
"line": 210,
"column": 35
} | [
{
"pp": "F : Type u_1\nF' : Type u_2\ninst✝² : Fintype F\ninst✝¹ : Field F\ninst✝ : Field F'\nh : ringChar F' ≠ ringChar F\nχ φ : MulChar F F'\nhχ : χ ≠ 1\nhφ : φ ≠ 1\nhχφ : χ * φ ≠ 1\nn : ℕ+\nhp : Nat.Prime (ringChar F)\nhc : Fintype.card F = ringChar F ^ ↑n\nψ : PrimitiveAddChar F F' := FiniteField.primitiveC... | let FF' := CyclotomicField ψ.n F' | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.NumberTheory.Height.MvPolynomial | {
"line": 207,
"column": 4
} | {
"line": 207,
"column": 12
} | [
{
"pp": "case inl\nK : Type u_4\ninst✝¹ : Field K\nι : Type u_5\ninst✝ : Finite ι\nv : AbsoluteValue K ℝ\nhv : IsNonarchimedean ⇑v\nN : ℕ\nx : ι → K\nhp : IsHomogeneous 0 N\n⊢ v ((eval x) 0) ≤ (⨆ s, v (coeff (↑s) 0)) * (⨆ i, v (x i)) ^ N",
"usedConstants": [
"Finsupp.instAddZeroClass",
"RingHom.... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.NumberTheory.Height.MvPolynomial | {
"line": 207,
"column": 4
} | {
"line": 207,
"column": 12
} | [
{
"pp": "case inl\nK : Type u_4\ninst✝¹ : Field K\nι : Type u_5\ninst✝ : Finite ι\nv : AbsoluteValue K ℝ\nhv : IsNonarchimedean ⇑v\nN : ℕ\nx : ι → K\nhp : IsHomogeneous 0 N\n⊢ v ((eval x) 0) ≤ (⨆ s, v (coeff (↑s) 0)) * (⨆ i, v (x i)) ^ N",
"usedConstants": [
"Finsupp.instAddZeroClass",
"RingHom.... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Height.MvPolynomial | {
"line": 207,
"column": 4
} | {
"line": 207,
"column": 12
} | [
{
"pp": "case inl\nK : Type u_4\ninst✝¹ : Field K\nι : Type u_5\ninst✝ : Finite ι\nv : AbsoluteValue K ℝ\nhv : IsNonarchimedean ⇑v\nN : ℕ\nx : ι → K\nhp : IsHomogeneous 0 N\n⊢ v ((eval x) 0) ≤ (⨆ s, v (coeff (↑s) 0)) * (⨆ i, v (x i)) ^ N",
"usedConstants": [
"Finsupp.instAddZeroClass",
"RingHom.... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.JacobiSum.Basic | {
"line": 264,
"column": 2
} | {
"line": 264,
"column": 85
} | [
{
"pp": "case inr.inl\nF : Type u_1\nR : Type u_2\ninst✝⁴ : Field F\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : Finite F\nn : ℕ\ninst✝ : NeZero n\nχ ψ : MulChar F R\nμ : R\nhχ : χ ^ n = 1\nhψ : ψ ^ n = 1\nhμ : IsPrimitiveRoot μ n\nhx₀ : 1 ≠ 0\n⊢ ∃ z ∈ ℤ[μ], (χ 1 - 1) * (ψ (1 - 1) - 1) = z * (μ - 1) ^ 2"... | · exact ⟨0, Subalgebra.zero_mem _, by rw [χ.map_one, sub_self, zero_mul, zero_mul]⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
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