module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.LinearAlgebra.PiTensorProduct | {
"line": 399,
"column": 26
} | {
"line": 399,
"column": 64
} | [
{
"pp": "ι : Type u_1\nι₂ : Type u_2\nι₃ : Type u_3\nR : Type u_4\ninst✝⁸ : CommSemiring R\nR₁ : Type u_5\nR₂ : Type u_6\ns : ι → Type u_7\ninst✝⁷ : (i : ι) → AddCommMonoid (s i)\ninst✝⁶ : (i : ι) → Module R (s i)\nM : Type u_8\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nE : Type u_9\ninst✝³ : AddCommMonoid... | simp_rw [← smul_add, φ.map_update_add] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.PiTensorProduct | {
"line": 399,
"column": 26
} | {
"line": 399,
"column": 64
} | [
{
"pp": "ι : Type u_1\nι₂ : Type u_2\nι₃ : Type u_3\nR : Type u_4\ninst✝⁸ : CommSemiring R\nR₁ : Type u_5\nR₂ : Type u_6\ns : ι → Type u_7\ninst✝⁷ : (i : ι) → AddCommMonoid (s i)\ninst✝⁶ : (i : ι) → Module R (s i)\nM : Type u_8\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nE : Type u_9\ninst✝³ : AddCommMonoid... | simp_rw [← smul_add, φ.map_update_add] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.PiTensorProduct | {
"line": 400,
"column": 26
} | {
"line": 400,
"column": 36
} | [
{
"pp": "ι : Type u_1\nι₂ : Type u_2\nι₃ : Type u_3\nR : Type u_4\ninst✝⁷ : CommSemiring R\nR₁ : Type u_5\nR₂ : Type u_6\ns : ι → Type u_7\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type u_8\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_9\ninst✝² : AddCommMonoid... | ← add_smul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.ClassGroup | {
"line": 350,
"column": 2
} | {
"line": 352,
"column": 49
} | [
{
"pp": "R : Type u_1\nK : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\nhI : (↑↑I).IsPrincipal\nx : K\nhx : ↑↑I = R ∙ x\n⊢ ∃ x, spanSingleton R⁰ ↑x = ↑I",
"usedConstants": [
"Units.val",
"Subm... | have hx' : (I : FractionalIdeal R⁰ K) = spanSingleton R⁰ x := by
apply Subtype.coe_injective
simp only [val_eq_coe, hx, coe_spanSingleton] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.ClassGroup | {
"line": 360,
"column": 39
} | {
"line": 362,
"column": 74
} | [
{
"pp": "R : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : Field K\ninst✝³ : Algebra R K\ninst✝² : IsFractionRing R K\ninst✝¹ : IsDomain R\ninst✝ : Subsingleton (ClassGroup R)\nI : FractionalIdeal R⁰ K\nhI : IsUnit I\n⊢ (↑I).IsPrincipal",
"usedConstants": [
"Units.val",
"Eq.mpr",
"... | by
rcases hI with ⟨I, rfl⟩
rw [← ClassGroup.mk_eq_one_iff, Subsingleton.elim (ClassGroup.mk K I) 1] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.LocalProperties.Projective | {
"line": 179,
"column": 4
} | {
"line": 180,
"column": 85
} | [
{
"pp": "case H\nR : Type u_1\nM : Type uM\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\nRₚ : (P : Ideal R) → [P.IsMaximal] → Type u_4\ninst✝⁷ : (P : Ideal R) → [inst : P.IsMaximal] → CommRing (Rₚ P)\ninst✝⁶ : (P : Ideal R) → [inst : P.IsMaximal] → Algebra R (Rₚ P)\ninst✝⁵ : ∀ (P : Ideal ... | apply ((Module.End.isUnit_iff _).mp
(IsLocalizedModule.map_units (LocalizedModule.mkLinearMap P.primeCompl M) s)).1 | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.LocallyConstant.Basic | {
"line": 297,
"column": 6
} | {
"line": 300,
"column": 10
} | [
{
"pp": "case refine_2\nX✝ : Type u_1\nY : Type u_2\nZ : Type u_3\nα : Type u_4\ninst✝² : TopologicalSpace X✝\nX : Type u_5\ninst✝¹ : TopologicalSpace X\nU : Set X\ninst✝ : (x : X) → Decidable (x ∈ U)\nhU : IsClopen U\n⊢ IsOpen[inst✝¹] ((fun x ↦ if x ∈ U then 0 else 1) ⁻¹' {1})",
"usedConstants": [
"S... | rw [← isClosed_compl_iff]
convert! hU.1
ext
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.LocallyConstant.Basic | {
"line": 297,
"column": 6
} | {
"line": 300,
"column": 10
} | [
{
"pp": "case refine_2\nX✝ : Type u_1\nY : Type u_2\nZ : Type u_3\nα : Type u_4\ninst✝² : TopologicalSpace X✝\nX : Type u_5\ninst✝¹ : TopologicalSpace X\nU : Set X\ninst✝ : (x : X) → Decidable (x ∈ U)\nhU : IsClopen U\n⊢ IsOpen[inst✝¹] ((fun x ↦ if x ∈ U then 0 else 1) ⁻¹' {1})",
"usedConstants": [
"S... | rw [← isClosed_compl_iff]
convert! hU.1
ext
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.LocallyConstant.Basic | {
"line": 552,
"column": 2
} | {
"line": 552,
"column": 17
} | [
{
"pp": "X : Type u_1\nZ : Type u_3\ninst✝¹ : TopologicalSpace X\nC₁ C₂ : Set X\nh₁ : IsClosed[inst✝¹] C₁\nh₂ : IsClosed[inst✝¹] C₂\nh : C₁ ∪ C₂ = univ\nf : LocallyConstant (↑C₁) Z\ng : LocallyConstant (↑C₂) Z\nhfg : ∀ (x : X) (hx : x ∈ C₁ ∩ C₂), f ⟨x, ⋯⟩ = g ⟨x, ⋯⟩\ninst✝ : DecidablePred fun x ↦ x ∈ C₁\nx : X\... | rw [dif_pos hx] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.ZMod.ValMinAbs | {
"line": 127,
"column": 4
} | {
"line": 127,
"column": 36
} | [
{
"pp": "case inr.refine_1\nn : ℕ\na : ZMod n\nha : 2 * a.val ≠ n\nh : NeZero n\n⊢ -a = ↑(-a.valMinAbs)",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast_neg",
"Int.cast",
"Eq.mpr",
"NegZeroClass.toNeg",
"ZMod.commRing",
"AddGroupWithOne.toAddGroup",
... | rw [Int.cast_neg, coe_valMinAbs] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.ZMod.ValMinAbs | {
"line": 127,
"column": 4
} | {
"line": 127,
"column": 36
} | [
{
"pp": "case inr.refine_1\nn : ℕ\na : ZMod n\nha : 2 * a.val ≠ n\nh : NeZero n\n⊢ -a = ↑(-a.valMinAbs)",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast_neg",
"Int.cast",
"Eq.mpr",
"NegZeroClass.toNeg",
"ZMod.commRing",
"AddGroupWithOne.toAddGroup",
... | rw [Int.cast_neg, coe_valMinAbs] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.ZMod.ValMinAbs | {
"line": 127,
"column": 4
} | {
"line": 127,
"column": 36
} | [
{
"pp": "case inr.refine_1\nn : ℕ\na : ZMod n\nha : 2 * a.val ≠ n\nh : NeZero n\n⊢ -a = ↑(-a.valMinAbs)",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast_neg",
"Int.cast",
"Eq.mpr",
"NegZeroClass.toNeg",
"ZMod.commRing",
"AddGroupWithOne.toAddGroup",
... | rw [Int.cast_neg, coe_valMinAbs] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Spectrum.Prime.FreeLocus | {
"line": 295,
"column": 45
} | {
"line": 300,
"column": 29
} | [
{
"pp": "R : Type uR\nM : Type uM\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Flat R M\ninst✝ : Module.Finite R M\np : PrimeSpectrum R\n⊢ rankAtStalk M p = 0 ↔ p ∉ support R M",
"usedConstants": [
"Module.Finite.instLocalizationLocalizedModule",
"CharP.cast_eq_ze... | by
rw [notMem_support_iff]
refine ⟨fun h ↦ ?_, fun h ↦ Module.finrank_zero_of_subsingleton⟩
apply subsingleton_of_rank_zero (R := Localization.AtPrime p.asIdeal)
dsimp [rankAtStalk] at h
simp [← finrank_eq_rank, h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Finite.Basic | {
"line": 203,
"column": 4
} | {
"line": 203,
"column": 56
} | [
{
"pp": "case e_s\nK : Type u_1\ninst✝² : CommRing K\ninst✝¹ : NoZeroDivisors K\nG : Subgroup Kˣ\ninst✝ : Fintype ↥G\nk : ℕ\nk_pos : k ≠ 0\nk_lt_card_G : k < Nat.card ↥G\na✝ : Nontrivial K\nthis : IsDomain K\na : ↥G\nha : a ^ k ≠ 1\nas_comp : (fun x ↦ (↑↑x * ↑↑a) ^ k) = (fun x ↦ ↑↑x ^ k) ∘ fun x ↦ x * a\n⊢ Mult... | exact Multiset.map_univ_val_equiv (Equiv.mulRight a) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.FieldTheory.Finite.Basic | {
"line": 251,
"column": 2
} | {
"line": 251,
"column": 27
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\ninst✝¹ : Fintype K\np : ℕ\ninst✝ : CharP K p\nhp : Fact (Nat.Prime p)\nthis : Module (ZMod p) K :=\n let __src := (ZMod.castHom ⋯ K).toModule;\n __src\nn : ℕ\nh : q = p ^ n\n⊢ ∃ n, Nat.Prime p ∧ q = p ^ ↑n",
"usedConstants": [
"PNat.val",
"Nat.Prime",
... | refine ⟨⟨n, ?_⟩, hp.1, h⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.FieldTheory.Finite.Basic | {
"line": 665,
"column": 6
} | {
"line": 665,
"column": 30
} | [
{
"pp": "p : ℕ\nhp : Prime p\nn : ℕ\nhpn : n.Coprime p\n⊢ n ^ (p - 1) ≡ 1 [MOD p]",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Nat.instMonoid",
"HSub.hSub",
"Int.natCast_modEq_iff",
"id",
"instSubNat",
"instOfNatNat",
"Int",
"Nat.cast",
"Monoi... | ← Int.natCast_modEq_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.FreeModule.Finite.CardQuotient | {
"line": 77,
"column": 10
} | {
"line": 77,
"column": 39
} | [
{
"pp": "case neg\nM : Type u_1\ninst✝⁴ : AddCommGroup M\ninst✝³ : Free ℤ M\ninst✝² : Module.Finite ℤ M\nN : Submodule ℤ M\nE : Type u_2\ninst✝¹ : EquivLike E M ↥N\ninst✝ : AddEquivClass E M ↥N\ne : E\nb : Basis (Free.ChooseBasisIndex ℤ M) ℤ M := Free.chooseBasis ℤ M\nh✝ : finrank ℤ ↥N = finrank ℤ M\na : Free.C... | Matrix.diagonal_apply_ne _ h, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.PicardGroup | {
"line": 777,
"column": 88
} | {
"line": 785,
"column": 77
} | [
{
"pp": "R : Type u\nM : Type v\nN : Type u_1\nP : Type u_2\nQ : Type u_3\nA : Type u_4\ninst✝¹² : CommSemiring R\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid N\ninst✝⁹ : AddCommMonoid P\ninst✝⁸ : AddCommMonoid Q\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\ninst✝⁵ : Module R P\ninst✝⁴ : Module R Q\ninst✝³ ... | by
convert! (AlgebraTensorModule.congr (.refl ..) (submoduleAlgebraEquiv e) ≪≫ₗ e).bijective
ext x
refine x.induction_on (by simp) ?_ (by simp +contextual)
intro a x
obtain ⟨m, rfl⟩ := (submoduleAlgebraEquiv e).symm.surjective x
suffices a * toAlgebra e m = e (a ⊗ₜ[R] m) by simpa using this
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative | {
"line": 123,
"column": 2
} | {
"line": 123,
"column": 36
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_3\ninst✝ : CommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, IsRel... | · rw [ha0, zero_mul, h0, zero_mul] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Norm.Basic | {
"line": 100,
"column": 4
} | {
"line": 100,
"column": 40
} | [
{
"pp": "case mp\nR : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : Ring S\ninst✝⁴ : Algebra R S\ninst✝³ : IsDomain R\ninst✝² : IsDomain S\ninst✝¹ : Free R S\ninst✝ : Module.Finite R S\nx : S\n⊢ (norm R) x = 0 → x = 0",
"usedConstants": [
"Ring.toNonAssocRing",
"CommSemiring.toSemiring",... | let b := Module.Free.chooseBasis R S | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.Norm.Basic | {
"line": 142,
"column": 41
} | {
"line": 148,
"column": 67
} | [
{
"pp": "K : Type u_4\nL : Type u_5\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\nhx : ¬IsIntegral K x\n⊢ (norm K) (AdjoinSimple.gen K x) = 1",
"usedConstants": [
"Iff.mpr",
"Set.mem_singleton",
"Eq.mpr",
"Submodule",
"MonoidHom.instFunLike",
"NonUnital... | by
rw [norm_eq_one_of_not_exists_basis]
contrapose hx
obtain ⟨s, ⟨b⟩⟩ := hx
refine .of_mem_of_fg K⟮x⟯.toSubalgebra ?_ x ?_
· exact (Submodule.fg_iff_finiteDimensional _).mpr (b.finiteDimensional_of_finite)
· exact IntermediateField.subset_adjoin K _ (Set.mem_singleton x) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.PrimitiveElement | {
"line": 116,
"column": 4
} | {
"line": 126,
"column": 64
} | [
{
"pp": "F : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Infinite F\nE : Type u_2\ninst✝² : Field E\nα β : E\ninst✝¹ : Algebra F E\ninst✝ : Algebra.IsSeparable F E\nhα : IsIntegral F α\nhβ : IsIntegral F β\nf : F[X] := minpoly F α\ng : F[X] := minpoly F β\nιFE : F →+* E := algebraMap F E\nιEE' : E →+* (Polynomial.map ... | use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel_right α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.PrimitiveElement | {
"line": 116,
"column": 4
} | {
"line": 126,
"column": 64
} | [
{
"pp": "F : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Infinite F\nE : Type u_2\ninst✝² : Field E\nα β : E\ninst✝¹ : Algebra F E\ninst✝ : Algebra.IsSeparable F E\nhα : IsIntegral F α\nhβ : IsIntegral F β\nf : F[X] := minpoly F α\ng : F[X] := minpoly F β\nιFE : F →+* E := algebraMap F E\nιEE' : E →+* (Polynomial.map ... | use γ
apply le_antisymm
· rw [adjoin_le_iff]
have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel_right α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c)
rintro x (rfl | rfl) <;> assumption
· rw [adjoin_simple_le_iff]
have α_in_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.RamificationInertia.Inertia | {
"line": 102,
"column": 49
} | {
"line": 109,
"column": 52
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\nS : Type v\ninst✝³ : CommRing S\ninst✝² : Algebra R S\np : Ideal R\nS₁ : Type u_1\ninst✝¹ : CommRing S₁\ninst✝ : Algebra R S₁\ne : S ≃ₐ[R] S₁\nP : Ideal S₁\n⊢ p.inertiaDeg (comap e P) = p.inertiaDeg P",
"usedConstants": [
"AlgEquiv.toAlgHom_toRingHom",
"... | by
have he : (P.comap e).comap (algebraMap R S) = p ↔ P.comap (algebraMap R S₁) = p := by
rw [← comap_coe e, comap_comap, ← e.toAlgHom_toRingHom, AlgHom.comp_algebraMap]
by_cases h : P.LiesOver p
· rw [inertiaDeg_algebraMap, inertiaDeg_algebraMap]
exact (Quotient.algEquivOfEqComap p e rfl).toLinearEquiv.f... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.Torsion.PrimaryComponent | {
"line": 70,
"column": 2
} | {
"line": 70,
"column": 7
} | [
{
"pp": "A : Type u_1\nM₁ : Type u_3\nM₂ : Type u_4\ninst✝⁴ : CommRing A\nI : Ideal A\ninst✝³ : AddCommMonoid M₁\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module A M₁\ninst✝ : Module A M₂\nφ : M₁ →ₗ[A] M₂\nc : M₁\nn : ℕ\nhn : ∀ a ∈ I ^ n, a • c = 0\n⊢ ∃ n, ∀ a ∈ I ^ n, φ (a • c) = 0",
"usedConstants": [
"i... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Algebra.Module.Torsion.PrimaryComponent | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 96
} | [
{
"pp": "A : Type u_1\nM : Type u_2\ninst✝² : CommRing A\nI : Ideal A\ninst✝¹ : AddCommMonoid M\ninst✝ : Module A M\nJ : Ideal A\nhD : IsCoprime I J\nthis : ∀ (n : ℕ), Disjoint (torsionBySet A M ↑(I ^ n)) (torsionBySet A M ↑J)\n⊢ primaryComponent (↥(torsionBySet A M ↑J)) I = ⊥",
"usedConstants": [
"Su... | apply Submodule.map_injective_of_injective (Submodule.subtype_injective (torsionBySet A M ↑J)) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.NumberTheory.RamificationInertia.Basic | {
"line": 191,
"column": 35
} | {
"line": 218,
"column": 52
} | [
{
"pp": "R : Type u\ninst✝¹⁵ : CommRing R\nS : Type v\ninst✝¹⁴ : CommRing S\ninst✝¹³ : Algebra R S\nK : Type u_1\ninst✝¹² : Field K\ninst✝¹¹ : Algebra R K\nV : Type u_3\nV' : Type u_4\nV'' : Type u_5\ninst✝¹⁰ : AddCommGroup V\ninst✝⁹ : Module R V\ninst✝⁸ : Module K V\ninst✝⁷ : IsScalarTower R K V\ninst✝⁶ : AddC... | by
contrapose hb' with hb
-- Informally, if we have a nontrivial linear dependence with coefficients `g` in `K`,
-- then we can find a linear dependence with coefficients `I.Quotient.mk g'` in `R/I`,
-- where `I = ker (algebraMap R S)`.
-- We make use of the same principle but stay in `R` everywhere.
simp o... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 430,
"column": 9
} | {
"line": 430,
"column": 29
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : CauSeq β abv\nh : f ≈ g\nε : α\nε0 : 0 < ε\nx✝ : ℕ\nH : ∀ j ≥ x✝, abv (↑(f - g) j) < ε / 2 ∧ ∀ k ≥ j, abv (↑f k - ↑f j) < ε / 2\nj : ℕ\n... | sub_add_sub_cancel', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 663,
"column": 2
} | {
"line": 663,
"column": 48
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nJ I : Ideal R\nhIJ : J * I ≤ J\nhJ : ¬J = 0\nhI : ¬I = 0\ns : Finset (HeightOneSpectrum R) := ⋯.toFinset\nthis✝ : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \\ {p}, q.asIdeal\na : HeightOneSpectrum R → R\nha : ∀ p ∈ s, a p ∈ J * ∏ q ∈ s ... | have inst : Nonempty {x // x ∈ s} := ⟨_, hp's⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 804,
"column": 2
} | {
"line": 812,
"column": 24
} | [
{
"pp": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na b : CauSeq α abs\nh : b ≤ a\n⊢ a ⊓ b ≈ b",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"CauSeq.instLTAbs._proof_1",
"Eq.mpr",
"CauSeq.instLTAbs",
"Preorder.toLT",
"Non... | obtain ⟨ε, ε0 : _ < _, i, h⟩ | h := h
· intro _ _
refine ⟨i, fun j hj => ?_⟩
dsimp
rw [← min_sub_sub_right]
rwa [sub_self, min_eq_right, abs_zero]
exact ε0.le.trans (h _ hj)
· refine Setoid.trans (inf_equiv_inf (Setoid.symm h) (Setoid.refl _)) ?_
rw [CauSeq.inf_idem] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 804,
"column": 2
} | {
"line": 812,
"column": 24
} | [
{
"pp": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na b : CauSeq α abs\nh : b ≤ a\n⊢ a ⊓ b ≈ b",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"CauSeq.instLTAbs._proof_1",
"Eq.mpr",
"CauSeq.instLTAbs",
"Preorder.toLT",
"Non... | obtain ⟨ε, ε0 : _ < _, i, h⟩ | h := h
· intro _ _
refine ⟨i, fun j hj => ?_⟩
dsimp
rw [← min_sub_sub_right]
rwa [sub_self, min_eq_right, abs_zero]
exact ε0.le.trans (h _ hj)
· refine Setoid.trans (inf_equiv_inf (Setoid.symm h) (Setoid.refl _)) ?_
rw [CauSeq.inf_idem] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Real.Basic | {
"line": 362,
"column": 4
} | {
"line": 362,
"column": 31
} | [
{
"pp": "x : ℝ\n⊢ ∀ (a b : ℝ), a ≤ b → ∀ (c : ℝ), a + c ≤ b + c",
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"_private.Mathlib.Data.Real.Basic.0.Real.instIsOrderedAddMonoid._simp_1",
"PartialOrder.toPreorder",
"Preorder.toLE",
"... | simp only [le_iff_eq_or_lt] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Real.Basic | {
"line": 512,
"column": 8
} | {
"line": 512,
"column": 29
} | [
{
"pp": "x : ℝ\nq : ℚ≥0\n⊢ ↑q = ↑q.num / ↑q.den",
"usedConstants": [
"Semiring.toNatCast",
"Real.instNNRatCast",
"Eq.mpr",
"Real",
"instHDiv",
"CauSeq.Completion.instNNRatCast",
"abs",
"congrArg",
"Real.instDivInvMonoid",
"IsAbsoluteValue.abs_isAbs... | ← ofCauchy_nnratCast, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Real.Archimedean | {
"line": 334,
"column": 4
} | {
"line": 334,
"column": 78
} | [
{
"pp": "case refine_1\nf : CauSeq ℝ abs\ns : Set ℝ := {x | const abs x < f}\nlb : ∃ x, x ∈ s\nub' : ∀ (x : ℝ), f < const abs x → ∀ y ∈ s, y ≤ x\nub : ∃ x, ∀ y ∈ s, y ≤ x\nε : ℝ\nε0 : ε > 0\ni : ℕ\nih : ∀ j ≥ i, ε ≤ ↑(const abs (sSup s) - f) j\nj : ℕ\nij : j ≥ i\n⊢ ε / 2 ≤ ↑(const abs (sSup s - ε / 2) - f) j",
... | rw [sub_apply, const_apply, sub_right_comm, le_sub_iff_add_le, add_halves] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Real.Archimedean | {
"line": 410,
"column": 40
} | {
"line": 414,
"column": 28
} | [
{
"pp": "b : ℝ\nhb : 0 < b\n⊢ ∃ n, 0 < n ∧ (↑n)⁻¹ < b",
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"Real.instArchimedean",
... | by
refine (exists_nat_gt b⁻¹).imp fun k hk ↦ ?_
have := (inv_pos_of_pos hb).trans hk
refine ⟨Nat.cast_pos.mp this, ?_⟩
rwa [inv_lt_comm₀ this hb] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Group.Pointwise.CompleteLattice | {
"line": 96,
"column": 2
} | {
"line": 97,
"column": 42
} | [
{
"pp": "M : Type u_1\ninst✝³ : CompleteLattice M\ninst✝² : Group M\ninst✝¹ : MulLeftMono M\ninst✝ : MulRightMono M\ns : Set M\n⊢ sSup s⁻¹ = (sInf s)⁻¹",
"usedConstants": [
"Eq.mpr",
"iInf",
"DivInvOneMonoid.toInvOneClass",
"congrArg",
"iSup",
"OrderIso.inv",
"Invol... | rw [← image_inv_eq_inv, sSup_image]
exact ((OrderIso.inv M).map_sInf _).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Group.Pointwise.CompleteLattice | {
"line": 96,
"column": 2
} | {
"line": 97,
"column": 42
} | [
{
"pp": "M : Type u_1\ninst✝³ : CompleteLattice M\ninst✝² : Group M\ninst✝¹ : MulLeftMono M\ninst✝ : MulRightMono M\ns : Set M\n⊢ sSup s⁻¹ = (sInf s)⁻¹",
"usedConstants": [
"Eq.mpr",
"iInf",
"DivInvOneMonoid.toInvOneClass",
"congrArg",
"iSup",
"OrderIso.inv",
"Invol... | rw [← image_inv_eq_inv, sSup_image]
exact ((OrderIso.inv M).map_sInf _).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 929,
"column": 6
} | {
"line": 929,
"column": 43
} | [
{
"pp": "R : Type u_3\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\np : HeightOneSpectrum R\nI J : Ideal R\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\n⊢ count p.asIdeal (normalizedFactors (normalizedFactors I ∩ normalizedFactors J).prod) =\n min (count p.asIdeal (normalizedFactors I)) (count p.asIdeal (normalizedFactors J)... | normalizedFactors_prod_inter_eq_inter | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.NNReal.Defs | {
"line": 448,
"column": 2
} | {
"line": 459,
"column": 26
} | [
{
"pp": "s : Set ℝ≥0\n⊢ ↑(sSup s) = sSup (toReal '' s)",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.sSup_empty",
"Real.instLE",
"Real",
"Lattice.toSemilatticeSup",
"Set.Ici",
"Real.instZero",
"congrArg",
"OrderBot... | rcases Set.eq_empty_or_nonempty s with rfl | hs
· simp
by_cases H : BddAbove s
· have A : sSup (Subtype.val '' s) ∈ Set.Ici 0 := by
apply Real.sSup_nonneg
rintro - ⟨y, -, rfl⟩
exact y.2
exact (@subset_sSup_of_within ℝ (Set.Ici (0 : ℝ)) _ _ (_) s hs H A).symm
· simp only [csSup_of_not_bddAb... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.NNReal.Defs | {
"line": 448,
"column": 2
} | {
"line": 459,
"column": 26
} | [
{
"pp": "s : Set ℝ≥0\n⊢ ↑(sSup s) = sSup (toReal '' s)",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.sSup_empty",
"Real.instLE",
"Real",
"Lattice.toSemilatticeSup",
"Set.Ici",
"Real.instZero",
"congrArg",
"OrderBot... | rcases Set.eq_empty_or_nonempty s with rfl | hs
· simp
by_cases H : BddAbove s
· have A : sSup (Subtype.val '' s) ∈ Set.Ici 0 := by
apply Real.sSup_nonneg
rintro - ⟨y, -, rfl⟩
exact y.2
exact (@subset_sSup_of_within ℝ (Set.Ici (0 : ℝ)) _ _ (_) s hs H A).symm
· simp only [csSup_of_not_bddAb... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 970,
"column": 23
} | {
"line": 970,
"column": 34
} | [
{
"pp": "R : Type u_3\ninst✝³ : CommRing R\ninst✝² : IsDedekindDomain R\np : HeightOneSpectrum R\nι : Type u_4\ninst✝¹ : Finite ι\ninst✝ : Nonempty ι\nI : ι → Ideal R\nH : ∀ (i : ι), FiniteMultiplicity p.asIdeal (I i)\nhI : ⨆ i, I i = ⊥\n⊢ ∃ i, I i = ⊥",
"usedConstants": [
"Lattice.toSemilatticeSup",
... | iSup_eq_bot | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.NNReal.Defs | {
"line": 715,
"column": 26
} | {
"line": 717,
"column": 74
} | [
{
"pp": "a b : ℝ≥0\nha : 0 < a\nhb : b < 1\n⊢ ∃ n, b ^ n < a",
"usedConstants": [
"Iff.mpr",
"Real",
"Preorder.toLT",
"Real.instArchimedean",
"Real.instZero",
"congrArg",
"NNReal.coe_lt_coe._simp_1",
"PartialOrder.toPreorder",
"Real.instLT",
"Preor... | by
simpa only [← coe_pow, NNReal.coe_lt_coe] using
exists_pow_lt_of_lt_one (NNReal.coe_pos.2 ha) (NNReal.coe_lt_coe.2 hb) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.ENNReal.Operations | {
"line": 498,
"column": 76
} | {
"line": 498,
"column": 90
} | [
{
"pp": "x y : ℝ≥0\n⊢ ofNNReal '' uIoo x y = uIoo ↑x ↑y",
"usedConstants": [
"ENNReal.ofNNReal",
"Lattice.toSemilatticeSup",
"congrArg",
"PartialOrder.toPreorder",
"SemilatticeInf.toPartialOrder",
"SemilatticeSup.toMax",
"DistribLattice.toLattice",
"NNReal",
... | by simp [uIoo] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.ENNReal.Operations | {
"line": 638,
"column": 46
} | {
"line": 638,
"column": 61
} | [
{
"pp": "ι : Sort u_1\na : ℝ≥0∞\ninst✝ : Nonempty ι\nf : ι → ℝ≥0∞\nha : a ≠ ∞\ni : ι\n⊢ a + f i ≤ a + ⨆ i, f i",
"usedConstants": [
"ENNReal.instAdd",
"le_refl",
"ENNReal.instAddCommMonoid",
"iSup",
"CompletelyDistribLattice.toCompleteLattice",
"PartialOrder.toPreorder",
... | grw [← le_iSup] | Mathlib.Tactic._aux_Mathlib_Tactic_GRewrite_Elab___macroRules_Mathlib_Tactic_grwSeq_1 | Mathlib.Tactic.grwSeq |
Mathlib.Data.ENNReal.Operations | {
"line": 638,
"column": 46
} | {
"line": 638,
"column": 61
} | [
{
"pp": "ι : Sort u_1\na : ℝ≥0∞\ninst✝ : Nonempty ι\nf : ι → ℝ≥0∞\nha : a ≠ ∞\ni : ι\n⊢ a + f i ≤ a + ⨆ i, f i",
"usedConstants": [
"ENNReal.instAdd",
"le_refl",
"ENNReal.instAddCommMonoid",
"iSup",
"CompletelyDistribLattice.toCompleteLattice",
"PartialOrder.toPreorder",
... | grw [← le_iSup] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.ENNReal.Operations | {
"line": 638,
"column": 46
} | {
"line": 638,
"column": 61
} | [
{
"pp": "ι : Sort u_1\na : ℝ≥0∞\ninst✝ : Nonempty ι\nf : ι → ℝ≥0∞\nha : a ≠ ∞\ni : ι\n⊢ a + f i ≤ a + ⨆ i, f i",
"usedConstants": [
"ENNReal.instAdd",
"le_refl",
"ENNReal.instAddCommMonoid",
"iSup",
"CompletelyDistribLattice.toCompleteLattice",
"PartialOrder.toPreorder",
... | grw [← le_iSup] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.EReal.Basic | {
"line": 205,
"column": 29
} | {
"line": 205,
"column": 44
} | [
{
"pp": "x : EReal\n⊢ x * 1 = x",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"EReal",
"id",
"EReal.mul_comm",
"One.toOfNat1",
"OfNat.ofNat",
"Eq",
"instOneEReal",
"EReal.instMul",
"instHMul"
]
}
] | EReal.mul_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.EReal.Basic | {
"line": 207,
"column": 30
} | {
"line": 207,
"column": 45
} | [
{
"pp": "x : EReal\n⊢ x * 0 = 0",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"EReal",
"id",
"instZeroEReal",
"EReal.mul_comm",
"Zero.toOfNat0",
"OfNat.ofNat",
"Eq",
"EReal.instMul",
"instHMul"
]
}
] | EReal.mul_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.EReal.Basic | {
"line": 684,
"column": 39
} | {
"line": 684,
"column": 54
} | [
{
"pp": "x : ℝ≥0\n⊢ ⊤ * ↑↑x = ↑↑x * ↑∞",
"usedConstants": [
"Eq.mpr",
"ENNReal.ofNNReal",
"HMul.hMul",
"congrArg",
"EReal",
"instTopEReal",
"id",
"ENNReal.toEReal",
"EReal.mul_comm",
"ENNReal",
"ENNReal.instTop",
"Top.top",
"Eq",
... | EReal.mul_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.ENNReal.BigOperators | {
"line": 168,
"column": 4
} | {
"line": 168,
"column": 31
} | [
{
"pp": "case cons.h₁\nι : Type u_1\nα : Type u_2\nf : α → ι → ℝ≥0∞\nhf : ∀ (i j : ι), ∃ k, ∀ (a : α), f a i ≤ f a k ∧ f a j ≤ f a k\na : α\ns : Finset α\nha : a ∉ s\nihs : ∑ a ∈ s, ⨆ i, f a i = ⨆ i, ∑ a ∈ s, f a i\ni j k : ι\nhk : ∀ (a : α), f a i ≤ f a k ∧ f a j ≤ f a k\n⊢ f a i ≤ f a k",
"usedConstants":... | exacts [(hk a).1, (hk _).2] | Batteries.Tactic._aux_Batteries_Tactic_Init___elabRules_Batteries_Tactic_exacts_1 | Batteries.Tactic.exacts |
Mathlib.Topology.Order.MonotoneConvergence | {
"line": 235,
"column": 4
} | {
"line": 235,
"column": 61
} | [
{
"pp": "case mpr\nι₁ : Type u_3\nι₂ : Type u_4\nα : Type u_5\ninst✝⁶ : SemilatticeSup ι₁\ninst✝⁵ : Preorder ι₂\ninst✝⁴ : Nonempty ι₁\ninst✝³ : TopologicalSpace α\ninst✝² : ConditionallyCompleteLinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : NoMaxOrder α\nf : ι₂ → α\nφ : ι₁ → ι₂\nl : α\nhf : Monotone f\nhg : T... | rcases tendsto_atTop_of_monotone hf with (h' | ⟨l', hl'⟩) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Topology.Order.IsLUB | {
"line": 220,
"column": 34
} | {
"line": 220,
"column": 74
} | [
{
"pp": "γ : Type u_2\nα : Type u_3\ninst✝³ : TopologicalSpace α\ninst✝² : ConditionallyCompleteLinearOrder α\ninst✝¹ : ClosedIicTopology α\nf : γ → α\ninst✝ : TopologicalSpace γ\nS : Set γ\nhS : Dense S\nhf : Continuous[inst✝, inst✝³] f\nh : ¬BddAbove (range fun x ↦ f ↑x)\nthis : ¬BddAbove (range f)\n⊢ ⨆ s, f ... | by simp [ciSup_of_not_bddAbove, this, h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.ENNReal.Inv | {
"line": 266,
"column": 2
} | {
"line": 267,
"column": 43
} | [
{
"pp": "c a b : ℝ≥0∞\nhc : c ≠ 0\nhc' : c ≠ ∞\n⊢ a * c / (b * c) = a / b",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"mul_mul_mul_comm",
"Monoid.toMulOneClass",
"CommSemiring.toNonUnitalCommSemiring",
"c... | rw [div_eq_mul_inv, div_eq_mul_inv, ENNReal.mul_inv (Or.inr hc') (Or.inr hc), mul_mul_mul_comm,
ENNReal.mul_inv_cancel hc hc', mul_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.ENNReal.Inv | {
"line": 266,
"column": 2
} | {
"line": 267,
"column": 43
} | [
{
"pp": "c a b : ℝ≥0∞\nhc : c ≠ 0\nhc' : c ≠ ∞\n⊢ a * c / (b * c) = a / b",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"mul_mul_mul_comm",
"Monoid.toMulOneClass",
"CommSemiring.toNonUnitalCommSemiring",
"c... | rw [div_eq_mul_inv, div_eq_mul_inv, ENNReal.mul_inv (Or.inr hc') (Or.inr hc), mul_mul_mul_comm,
ENNReal.mul_inv_cancel hc hc', mul_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.ENNReal.Inv | {
"line": 266,
"column": 2
} | {
"line": 267,
"column": 43
} | [
{
"pp": "c a b : ℝ≥0∞\nhc : c ≠ 0\nhc' : c ≠ ∞\n⊢ a * c / (b * c) = a / b",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"mul_mul_mul_comm",
"Monoid.toMulOneClass",
"CommSemiring.toNonUnitalCommSemiring",
"c... | rw [div_eq_mul_inv, div_eq_mul_inv, ENNReal.mul_inv (Or.inr hc') (Or.inr hc), mul_mul_mul_comm,
ENNReal.mul_inv_cancel hc hc', mul_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.ENNReal.Inv | {
"line": 293,
"column": 49
} | {
"line": 294,
"column": 54
} | [
{
"pp": "a b : ℝ≥0∞\n⊢ a < b⁻¹ ↔ b < a⁻¹",
"usedConstants": [
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder",
"Eq.mp",
"Iff",
"Inv.inv",
"ENNReal.instInvolutiveInv",
"inv_inv",
"LT.lt",
"ENNReal",
"ENNReal.instPartialOrder",
"EN... | by
simpa only [inv_inv] using @ENNReal.inv_lt_inv a⁻¹ b | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.ENNReal.Inv | {
"line": 648,
"column": 4
} | {
"line": 648,
"column": 54
} | [
{
"pp": "a b : ℝ≥0∞\nha : a ≠ 0\nhb : b ≠ ∞\nn : ℕ\nhn : b / a < ↑n\n⊢ b < ↑n * a",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"instHDiv",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"PartialOrder.toPreorder",
"id",
"HDiv.hDiv",
"AddMono... | rwa [← ENNReal.div_lt_iff (Or.inl ha) (Or.inr hb)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Data.ENNReal.Inv | {
"line": 648,
"column": 4
} | {
"line": 648,
"column": 54
} | [
{
"pp": "a b : ℝ≥0∞\nha : a ≠ 0\nhb : b ≠ ∞\nn : ℕ\nhn : b / a < ↑n\n⊢ b < ↑n * a",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"instHDiv",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"PartialOrder.toPreorder",
"id",
"HDiv.hDiv",
"AddMono... | rwa [← ENNReal.div_lt_iff (Or.inl ha) (Or.inr hb)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.ENNReal.Inv | {
"line": 648,
"column": 4
} | {
"line": 648,
"column": 54
} | [
{
"pp": "a b : ℝ≥0∞\nha : a ≠ 0\nhb : b ≠ ∞\nn : ℕ\nhn : b / a < ↑n\n⊢ b < ↑n * a",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"instHDiv",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"PartialOrder.toPreorder",
"id",
"HDiv.hDiv",
"AddMono... | rwa [← ENNReal.div_lt_iff (Or.inl ha) (Or.inr hb)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Order.LiminfLimsup | {
"line": 211,
"column": 8
} | {
"line": 211,
"column": 58
} | [
{
"pp": "case pos\nα : Type u_2\ninst✝³ : ConditionallyCompleteLinearOrder α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\nf : Filter α\ninst✝ : f.NeBot\nhc : IsCobounded (fun x1 x2 ↦ x1 ≤ x2) f\nhb : IsBounded (fun x1 x2 ↦ x1 ≤ x2) f\nhn : Nontrivial α\nhtop : ∃ x, f.limsSup < x\nhbot : ∀ (x : α), f.... | exact lt_mem_sets_of_limsSup_lt hb h |>.frequently | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.ENNReal.Inv | {
"line": 744,
"column": 4
} | {
"line": 744,
"column": 71
} | [
{
"pp": "x : ℝ≥0∞\nhx : 1 ≤ x\na b : ℕ\nh : Int.ofNat a ≤ Int.negSucc b\n⊢ Int.negSucc b < Int.ofNat a",
"usedConstants": [
"PartialOrder.toPreorder",
"SemilatticeInf.toPartialOrder",
"Int.negSucc_lt_zero",
"Int.ofNat",
"Int",
"instOfNat",
"instLatticeInt",
"I... | exact lt_of_lt_of_le (Int.negSucc_lt_zero _) (Int.natCast_nonneg _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.EMetricSpace.Lipschitz | {
"line": 341,
"column": 85
} | {
"line": 348,
"column": 92
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\nf : α → β → γ\nK₁ K₂ : ℝ≥0\ns : Set α\nt : Set β\nhf₁ : ∀ b ∈ t, LipschitzOnWith K₁ (fun x ↦ f x b) s\nhf₂ : ∀ a ∈ s, LipschitzOnWith K₂ (f a) t\n⊢ Metric.ediam (image2 f s t)... | by
simp only [Metric.ediam_le_iff, forall_mem_image2]
intro a₁ ha₁ b₁ hb₁ a₂ ha₂ b₂ hb₂
refine (edist_triangle _ (f a₂ b₁) _).trans ?_
exact
add_le_add
((hf₁ b₁ hb₁ ha₁ ha₂).trans <| mul_right_mono <| Metric.edist_le_ediam_of_mem ha₁ ha₂)
((hf₂ a₂ ha₂ hb₁ hb₂).trans <| mul_right_mono <| Metric.e... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.InfiniteSum.Group | {
"line": 318,
"column": 2
} | {
"line": 320,
"column": 5
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : UniformSpace α\ninst✝³ : CommGroup α\ninst✝² : IsUniformGroup α\ninst✝¹ : CompleteSpace α\ninst✝ : T2Space α\nf : β → α\nhf : Multipliable f\ns : Finset β\n⊢ (∏ x ∈ s, f x) * ∏' (x : { x // x ∉ s }), f ↑x = ∏' (x : β), f x",
"usedConstants": [
"Eq.mpr",
... | rw [← hf.tprod_subtype_mul_tprod_subtype_compl s]
simp only [Finset.tprod_subtype', mul_right_inj]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.InfiniteSum.Group | {
"line": 318,
"column": 2
} | {
"line": 320,
"column": 5
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : UniformSpace α\ninst✝³ : CommGroup α\ninst✝² : IsUniformGroup α\ninst✝¹ : CompleteSpace α\ninst✝ : T2Space α\nf : β → α\nhf : Multipliable f\ns : Finset β\n⊢ (∏ x ∈ s, f x) * ∏' (x : { x // x ∉ s }), f ↑x = ∏' (x : β), f x",
"usedConstants": [
"Eq.mpr",
... | rw [← hf.tprod_subtype_mul_tprod_subtype_compl s]
simp only [Finset.tprod_subtype', mul_right_inj]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.InfiniteSum.Order | {
"line": 158,
"column": 11
} | {
"line": 158,
"column": 24
} | [
{
"pp": "case neg\nι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝³ : CommMonoid α\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf : ι → α\na₂ : α\nha₂ : 1 ≤ a₂\nh : ∀ (s : Finset ι), ∏ i ∈ s, f i ≤ a₂\nhL : ¬L.NeBot\nhf : ¬(mulSupport f).Finite\n⊢ ∏'[L] (i : ι), f i... | tprod_bot hL, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.EReal.Operations | {
"line": 577,
"column": 6
} | {
"line": 577,
"column": 21
} | [
{
"pp": "x : ℝ≥0∞\nhx : x ≠ 0\n⊢ ↑x * ⊤ = ⊤",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"EReal",
"instTopEReal",
"id",
"ENNReal.toEReal",
"EReal.mul_comm",
"Top.top",
"Eq",
"EReal.instMul",
"instHMul"
]
}
] | EReal.mul_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.EReal.Operations | {
"line": 746,
"column": 53
} | {
"line": 746,
"column": 68
} | [
{
"pp": "a b : EReal\n⊢ -a = ⊥ ∧ b < 0 ∨ -a < 0 ∧ b = ⊥ ∨ -a = ⊤ ∧ 0 < b ∨ 0 < -a ∧ b = ⊤ ↔\n a = ⊥ ∧ 0 < b ∨ 0 < a ∧ b = ⊥ ∨ a = ⊤ ∧ b < 0 ∨ a < 0 ∧ b = ⊤",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder",
"EReal.instNeg",
"EReal",
... | neg_eq_bot_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order.IntermediateValue | {
"line": 396,
"column": 4
} | {
"line": 396,
"column": 42
} | [
{
"pp": "α : Type u\ninst✝³ : TopologicalSpace α\ninst✝² : ConditionallyCompleteLinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b : α\ns : Set α\nhs : IsClosed[inst✝³] (s ∩ Icc a b)\nha : a ∈ s\nh : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t\nhab : a ≤ b\nA : Set α := {t | t ∈ Icc a b ∧ Icc a... | rcases le_or_gt t' t₁ with h't' | h't' | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Data.EReal.Operations | {
"line": 775,
"column": 6
} | {
"line": 775,
"column": 21
} | [
{
"pp": "x y : EReal\nhy : 0 ≤ y\n⊢ (x * y).toENNReal = x.toENNReal * y.toENNReal",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"EReal.toENNReal",
"EReal",
"id",
"ENNReal.instCommSemiring",
"instDistribOfSemiring",
... | EReal.mul_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 590,
"column": 6
} | {
"line": 590,
"column": 28
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nx y : α\nε₁ ε₂ : ℝ\nh : dist x y ≤ ε₂ - ε₁\nz : α\nzx : z ∈ ball x ε₁\n⊢ z ∈ ball y ε₂",
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"AddMonoid.toAddZeroClass",
"HSub.hSub",
"AddCommGroup.toAddGroup",
"Membershi... | ← add_sub_cancel ε₁ ε₂ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 1036,
"column": 85
} | {
"line": 1038,
"column": 5
} | [
{
"pp": "α : Type u_3\nU : UniformSpace α\nm : PseudoMetricSpace α\nH : 𝓤 α = 𝓤 α\n⊢ m.replaceUniformity H = m",
"usedConstants": [
"Real",
"PseudoMetricSpace.ext",
"funext",
"Dist.ext",
"Eq.refl",
"Dist.dist",
"PseudoMetricSpace.toDist",
"PseudoMetricSpace.... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 1056,
"column": 79
} | {
"line": 1058,
"column": 5
} | [
{
"pp": "γ : Type u_3\nU : TopologicalSpace γ\nm : PseudoMetricSpace γ\nH : U = toUniformSpace.toTopologicalSpace\n⊢ m.replaceTopology H = m",
"usedConstants": [
"Real",
"PseudoMetricSpace.ext",
"funext",
"Dist.ext",
"Eq.refl",
"Dist.dist",
"PseudoMetricSpace.toDist... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 1102,
"column": 50
} | {
"line": 1104,
"column": 5
} | [
{
"pp": "α : Type u_3\nm : PseudoMetricSpace α\nB : Bornology α\nH : ∀ (s : Set α), Bornology.IsBounded s ↔ Bornology.IsBounded s\n⊢ m.replaceBornology H = m",
"usedConstants": [
"Real",
"PseudoMetricSpace.ext",
"PseudoMetricSpace.replaceBornology",
"funext",
"Dist.ext",
... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.Defs | {
"line": 188,
"column": 44
} | {
"line": 190,
"column": 5
} | [
{
"pp": "α : Type u_3\nm : MetricSpace α\nB : Bornology α\nH : ∀ (s : Set α), Bornology.IsBounded s ↔ Bornology.IsBounded s\n⊢ m.replaceBornology H = m",
"usedConstants": [
"Real",
"MetricSpace.ext",
"funext",
"Dist.ext",
"MetricSpace.replaceBornology",
"Eq.refl",
"... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.Bounded | {
"line": 161,
"column": 4
} | {
"line": 161,
"column": 82
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nc : α\n⊢ (cobounded α).HasBasis (fun x ↦ True) fun i ↦ (fun x ↦ dist x c) ⁻¹' Ici i",
"usedConstants": [
"Real",
"PseudoMetricSpace.toBornology",
"congrArg",
"Compl.compl",
"PartialOrder.toPreorder",
"setOf",
"Real.i... | simpa only [compl_def, mem_ball, not_lt] using hasBasis_cobounded_compl_ball c | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.MetricSpace.Bounded | {
"line": 161,
"column": 4
} | {
"line": 161,
"column": 82
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nc : α\n⊢ (cobounded α).HasBasis (fun x ↦ True) fun i ↦ (fun x ↦ dist x c) ⁻¹' Ici i",
"usedConstants": [
"Real",
"PseudoMetricSpace.toBornology",
"congrArg",
"Compl.compl",
"PartialOrder.toPreorder",
"setOf",
"Real.i... | simpa only [compl_def, mem_ball, not_lt] using hasBasis_cobounded_compl_ball c | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Bounded | {
"line": 161,
"column": 4
} | {
"line": 161,
"column": 82
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nc : α\n⊢ (cobounded α).HasBasis (fun x ↦ True) fun i ↦ (fun x ↦ dist x c) ⁻¹' Ici i",
"usedConstants": [
"Real",
"PseudoMetricSpace.toBornology",
"congrArg",
"Compl.compl",
"PartialOrder.toPreorder",
"setOf",
"Real.i... | simpa only [compl_def, mem_ball, not_lt] using hasBasis_cobounded_compl_ball c | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Isometry | {
"line": 640,
"column": 53
} | {
"line": 640,
"column": 83
} | [
{
"pp": "F : Type u_1\nι : Type u_2\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\nm n : ℕ\nx✝¹ x✝ : (Fin m → α) × (Fin n → α)\n⊢ edist (Fin.append x✝¹.1 x✝¹.2) (Fin.append x✝.1 x✝.2) = edist x✝¹ x✝",
"usedConstants": [
... | Fin.edist_append_eq_max_edist, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.OuterMeasure.Basic | {
"line": 130,
"column": 37
} | {
"line": 130,
"column": 57
} | [
{
"pp": "α : Type u_1\nF : Type u_3\ninst✝¹ : FunLike F (Set α) ℝ≥0∞\ninst✝ : OuterMeasureClass F α\nμ : F\ns : Set α\nhs : s.Countable\n⊢ μ s = 0 ↔ μ (⋃ i ∈ s, {i}) = 0",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Membership.mem",
"Set.biUnion_of_singleton",
"Set.instSingletonS... | biUnion_of_singleton | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 153,
"column": 67
} | {
"line": 154,
"column": 75
} | [
{
"pp": "α : Type u_1\nf : α → ℝ≥0\nl : Filter α\n⊢ Tendsto (fun x ↦ ↑(f x)) l (𝓝 ∞) ↔ Tendsto f l atTop",
"usedConstants": [
"Eq.mpr",
"ENNReal.ofNNReal",
"Set.Ioi",
"Preorder.toLT",
"NNReal.instInhabited",
"ENNReal.tendsto_nhds_top_iff_nnreal",
"congrArg",
... | by
rw [tendsto_nhds_top_iff_nnreal, atTop_basis_Ioi.tendsto_right_iff]; simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 711,
"column": 8
} | {
"line": 711,
"column": 42
} | [
{
"pp": "case inr\na b : ℝ\nh : a < b\n⊢ ediam (Ioo a b) = ENNReal.ofReal (b - a)",
"usedConstants": [
"Eq.mpr",
"ConditionallyCompleteLinearOrder.toCompactIccSpace",
"Real",
"Real.ediam_eq",
"ENNReal.ofReal",
"congrArg",
"Real.instSub",
"Metric.isBounded_Ioo"... | Real.ediam_eq (isBounded_Ioo _ _), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Sign.Defs | {
"line": 310,
"column": 49
} | {
"line": 315,
"column": 43
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Zero α\ninst✝ : LinearOrder α\na : α\n⊢ sign a = 0 ↔ a = 0",
"usedConstants": [
"SignType.ctorIdx",
"False",
"Preorder.toLT",
"SignType.instOne",
"congrArg",
"HEq.refl",
"False.elim",
"PartialOrder.toPreorder",
"SignType.i... | by
refine ⟨fun h => ?_, fun h => h.symm ▸ sign_zero⟩
rw [sign_apply] at h
split_ifs at h with h_1 h_2
cases h
exact (le_of_not_gt h_1).eq_of_not_lt h_2 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Nat.Cast.Order.Field | {
"line": 36,
"column": 29
} | {
"line": 36,
"column": 42
} | [
{
"pp": "case zero\nα : Type u_1\ninst✝² : Semifield α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\nm : ℕ\n⊢ ↑(m / 0) ≤ 0",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHDiv",
"congrArg",
"Partia... | Nat.div_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.EReal.Inv | {
"line": 103,
"column": 67
} | {
"line": 111,
"column": 62
} | [
{
"pp": "x y : EReal\n⊢ sign (x * y) = sign x * sign y",
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Iff.mpr",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"MulOne.toOne",
"Real",
"SignType.instHasDistribNeg",
"Preorder.toLT",
"HMul.hMul",
... | by
induction x, y using induction₂_symm_neg with
| top_zero => simp only [mul_zero, sign_zero]
| top_top => rfl
| symm h => rwa [mul_comm, EReal.mul_comm]
| coe_coe => simp only [← coe_mul, sign_coe, _root_.sign_mul]
| top_pos _ h =>
rw [top_mul_coe_of_pos h, sign_top, one_mul, sign_pos (EReal.coe_pos.2... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.EReal.Inv | {
"line": 119,
"column": 6
} | {
"line": 119,
"column": 21
} | [
{
"pp": "x : EReal\n⊢ ↑x.abs * ↑(sign x) = x",
"usedConstants": [
"SignType.cast",
"Eq.mpr",
"EReal.abs",
"HMul.hMul",
"congrArg",
"PartialOrder.toPreorder",
"SignType.instLinearOrder",
"EReal.instNeg",
"EReal",
"SemilatticeInf.toPartialOrder",
... | EReal.mul_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.EReal.Inv | {
"line": 255,
"column": 6
} | {
"line": 258,
"column": 28
} | [
{
"pp": "case coe.inr.inr\na : ℝ\na_pos : 0 < a\n⊢ ↑(sign ↑a) * (↑(↑a).abs)⁻¹ = (↑a)⁻¹",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"SignType.cast",
"Eq.mpr",
"EReal.abs",
"MulOne.toOne",
"SignType.coe_one",
"Real",
"Inv",
"HMul.hMul",
"ERe... | rw [sign_coe, _root_.sign_pos a_pos, SignType.coe_one, one_mul]
simp only [abs_def a, coe_ennreal_ofReal, abs_nonneg, max_eq_left]
congr
exact abs_of_pos a_pos | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.EReal.Inv | {
"line": 255,
"column": 6
} | {
"line": 258,
"column": 28
} | [
{
"pp": "case coe.inr.inr\na : ℝ\na_pos : 0 < a\n⊢ ↑(sign ↑a) * (↑(↑a).abs)⁻¹ = (↑a)⁻¹",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"SignType.cast",
"Eq.mpr",
"EReal.abs",
"MulOne.toOne",
"SignType.coe_one",
"Real",
"Inv",
"HMul.hMul",
"ERe... | rw [sign_coe, _root_.sign_pos a_pos, SignType.coe_one, one_mul]
simp only [abs_def a, coe_ennreal_ofReal, abs_nonneg, max_eq_left]
congr
exact abs_of_pos a_pos | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 513,
"column": 4
} | {
"line": 513,
"column": 28
} | [
{
"pp": "case neg\nα : Type u_1\nsf sg : ℝ≥0∞\ni : α\nhsf : sf ≠ ∞\nhsg : ¬sg = ∞\nf : α → ℝ≥0\nhf : HasSum (fun i ↦ ↑(f i)) sf\ng : α → ℝ≥0\nhg : HasSum (fun i ↦ ↑(g i)) sg\nh : ∀ (a : α), (fun i ↦ ↑(f i)) a ≤ (fun i ↦ ↑(g i)) a\nhi : (fun i ↦ ↑(f i)) i < (fun i ↦ ↑(g i)) i\n⊢ sf < sg",
"usedConstants": [
... | lift sf to ℝ≥0 using hsf | Mathlib.Tactic._aux_Mathlib_Tactic_Lift___elabRules_Mathlib_Tactic_lift_1 | Mathlib.Tactic.lift |
Mathlib.GroupTheory.Archimedean | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 34
} | [
{
"pp": "G : Type u_1\ninst✝³ : CommGroup G\ninst✝² : LinearOrder G\ninst✝¹ : IsOrderedMonoid G\ninst✝ : MulArchimedean G\nH : Subgroup G\nhbot : H ≠ ⊥\na : G\nh₀ : 1 < a\nhd : Disjoint (↑H) (Ioo 1 a)\nhex : ∀ g > 1, ∃ n, g ∈ Ioc (a ^ n) (a ^ (n + 1))\nthis : ∃ n, (↑H ∩ Ioc (a ^ n) (a ^ (n + 1))).Nonempty\nn : ... | obtain ⟨m, hm, hya⟩ := hex y hy₀ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 187,
"column": 2
} | {
"line": 188,
"column": 18
} | [
{
"pp": "⊢ Tendsto toReal (𝓝[≠] ⊤) atTop",
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"Filter.tendsto_id",
"Filter.map",
"Compl.compl",
"EReal.instTopologicalSpace",
"nhdsWithin",
"EReal",
"Filter.tendsto_map'_iff",
"Function.comp",
... | rw [nhdsWithin_top, tendsto_map'_iff]
exact tendsto_id | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 187,
"column": 2
} | {
"line": 188,
"column": 18
} | [
{
"pp": "⊢ Tendsto toReal (𝓝[≠] ⊤) atTop",
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"Filter.tendsto_id",
"Filter.map",
"Compl.compl",
"EReal.instTopologicalSpace",
"nhdsWithin",
"EReal",
"Filter.tendsto_map'_iff",
"Function.comp",
... | rw [nhdsWithin_top, tendsto_map'_iff]
exact tendsto_id | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 128,
"column": 2
} | {
"line": 128,
"column": 39
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\ns : Set α\ny : β\ninst✝¹ : Zero β\ninst✝ : Preorder β\nhs : IsOpen[inst✝²] s\nhy : 0 ≤ y\nx : α\nz : β\nhz : s.indicator (fun _x ↦ y) x > z\n⊢ ∀ᶠ (x' : α) in 𝓝 x, (fun x1 x2 ↦ s.indicator (fun _x ↦ y) x1 > x2) x' z",
"usedConstants": [
... | by_cases h : x ∈ s <;> simp [h] at hz | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 39
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\ns : Set α\ny : β\ninst✝¹ : Zero β\ninst✝ : Preorder β\nhs : IsClosed[inst✝²] s\nhy : y ≤ 0\nx : α\nz : β\nhz : s.indicator (fun _x ↦ y) x > z\n⊢ ∀ᶠ (x' : α) in 𝓝 x, (fun x1 x2 ↦ s.indicator (fun _x ↦ y) x1 > x2) x' z",
"usedConstants": [
... | by_cases h : x ∈ s <;> simp [h] at hz | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 343,
"column": 6
} | {
"line": 344,
"column": 51
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝³ : TopologicalSpace α\nγ : Type u_4\ninst✝² : LinearOrder γ\ninst✝¹ : TopologicalSpace γ\ninst✝ : ClosedIciTopology γ\nf : α → γ\ns : Set α\nhs : IsClosed[inst✝³] s\nhf : ∀ x ∈ s, ∀ y < f x, ∀ᶠ (x : α) in 𝓝 x, x ∈ s → y < f x\nx : α\ny : γ\nh : (x, y).1 ∈ s → (x, y).2 < f... | filter_upwards [(hf x hx z hz).prodMk_nhds (eventually_lt_nhds hy')]
with _ ⟨h₂, h₃⟩ h₄ using h₁ _ h₃ _ <| h₂ h₄ | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 590,
"column": 2
} | {
"line": 594,
"column": 58
} | [
{
"pp": "α : Type u_4\nβ : Type u_5\ninst✝¹ : TopologicalSpace α\ninst✝ : LinearOrder β\nf g : α → β\ns : Set α\na : α\nhf : LowerSemicontinuousWithinAt f s a\nhg : LowerSemicontinuousWithinAt g s a\n⊢ LowerSemicontinuousWithinAt (fun x ↦ max (f x) (g x)) s a",
"usedConstants": [
"Eq.mpr",
"Preo... | intro b hb
simp only [lt_sup_iff] at hb ⊢
rcases hb with hb | hb
· filter_upwards [hf b hb] with x using Or.intro_left _
· filter_upwards [hg b hb] with x using Or.intro_right _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 590,
"column": 2
} | {
"line": 594,
"column": 58
} | [
{
"pp": "α : Type u_4\nβ : Type u_5\ninst✝¹ : TopologicalSpace α\ninst✝ : LinearOrder β\nf g : α → β\ns : Set α\na : α\nhf : LowerSemicontinuousWithinAt f s a\nhg : LowerSemicontinuousWithinAt g s a\n⊢ LowerSemicontinuousWithinAt (fun x ↦ max (f x) (g x)) s a",
"usedConstants": [
"Eq.mpr",
"Preo... | intro b hb
simp only [lt_sup_iff] at hb ⊢
rcases hb with hb | hb
· filter_upwards [hf b hb] with x using Or.intro_left _
· filter_upwards [hg b hb] with x using Or.intro_right _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.OuterMeasure.OfFunction | {
"line": 409,
"column": 47
} | {
"line": 415,
"column": 65
} | [
{
"pp": "α : Type u_1\nι : Sort u_2\nβ : Type u_3\nf : α → β\nm : ι → OuterMeasure β\n⊢ (comap f) (⨅ i, m i) = ⨅ i, (comap f) (m i)",
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.MeasureTheory.OuterMeasure.OfFunction.0.MeasureTheory.OuterMeasure.comap_iInf._simp_1_2",
"iInf",
"ENNR... | by
refine ext_nonempty fun s hs => ?_
refine ((comap_mono f).map_iInf_le s).antisymm ?_
simp only [comap_apply, iInf_apply' _ hs, iInf_apply' _ (hs.image _), le_iInf_iff,
Set.image_subset_iff, preimage_iUnion]
refine fun t ht => iInf_le_of_le _ (iInf_le_of_le ht <| ENNReal.tsum_le_tsum fun k => ?_)
exact ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 521,
"column": 4
} | {
"line": 522,
"column": 48
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝ : PseudoMetricSpace α\nr C : ℝ\nf : ℕ → α\nhr : r < 1\nhu : ∀ (n : ℕ), dist (f n) (f (n + 1)) ≤ C * r ^ n\nleft✝ : 0 < C\nr₀ : 0 ≤ r\n⊢ HasSum (fun n ↦ C * r ^ n) (C / (1 - r))",
"usedConstants": [
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | refine HasSum.mul_left C ?_
simpa using hasSum_geometric_of_lt_one r₀ hr | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 521,
"column": 4
} | {
"line": 522,
"column": 48
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝ : PseudoMetricSpace α\nr C : ℝ\nf : ℕ → α\nhr : r < 1\nhu : ∀ (n : ℕ), dist (f n) (f (n + 1)) ≤ C * r ^ n\nleft✝ : 0 < C\nr₀ : 0 ≤ r\n⊢ HasSum (fun n ↦ C * r ^ n) (C / (1 - r))",
"usedConstants": [
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | refine HasSum.mul_left C ?_
simpa using hasSum_geometric_of_lt_one r₀ hr | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 629,
"column": 2
} | {
"line": 635,
"column": 46
} | [
{
"pp": "ε : ℝ≥0\nhε : ε ≠ 0\nι : Type u_4\ninst✝ : Countable ι\n⊢ ∃ ε', (∀ (i : ι), 0 < ε' i) ∧ ∃ c, HasSum ε' c ∧ c < ε",
"usedConstants": [
"NNReal.instTopologicalSpace",
"Iff.mpr",
"LinearOrderedCommGroupWithZero.toLinearOrderedCommMonoidWithZero",
"Real.instLE",
"Real",
... | cases nonempty_encodable ι
obtain ⟨a, a0, aε⟩ := exists_between (pos_iff_ne_zero.2 hε)
obtain ⟨ε', hε', c, hc, hcε⟩ := posSumOfEncodable a0 ι
exact
⟨fun i ↦ ⟨ε' i, (hε' i).le⟩, fun i ↦ NNReal.coe_lt_coe.1 <| hε' i,
⟨c, hasSum_le (fun i ↦ (hε' i).le) hasSum_zero hc⟩, NNReal.hasSum_coe.1 hc,
aε.tran... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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