module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Data.ZMod.ValMinAbs | {
"line": 62,
"column": 37
} | {
"line": 62,
"column": 50
} | {
"line": 62,
"column": 51
} | [
{
"pp": "case pos\nn : ℕ\na : ZMod (n + 1)\nh : a.val ≤ n.succ / 2\n⊢ ↑a.val * 2 = ↑n + 1 ↔ ↑(2 * a.val) = ↑(n + 1)",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"congrArg",
"id",
"AddMo... | [
"case pos\nn : ℕ\na : ZMod (n + 1)\nh : a.val ≤ n.succ / 2\n⊢ ↑a.val * 2 = ↑n + 1 ↔ ↑2 * ↑a.val = ↑(n + 1)"
] | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.FreeModule.Finite.Quotient | {
"line": 70,
"column": 6
} | {
"line": 70,
"column": 16
} | {
"line": 71,
"column": 4
} | [
{
"pp": "case mp\nι : Type u_1\nR : Type u_2\nM : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : IsDomain R\ninst✝¹ : IsPrincipalIdealRing R\ninst✝ : Finite ι\nN : Submodule R M\nb : Basis ι R M\nh : finrank R ↥N = finrank R M\nthis : Fintype ι\na : ι → R := smithNormalFor... | [] | exact hy i | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Invariant.Basic | {
"line": 304,
"column": 27
} | {
"line": 304,
"column": 62
} | {
"line": 305,
"column": 6
} | [
{
"pp": "case intro\nA : Type u_1\nB : Type u_2\ninst✝¹⁸ : CommRing A\ninst✝¹⁷ : CommRing B\ninst✝¹⁶ : Algebra A B\nG : Type u_3\ninst✝¹⁵ : Group G\ninst✝¹⁴ : Finite G\ninst✝¹³ : MulSemiringAction G B\ninst✝¹² : SMulCommClass G A B\nP : Ideal A\nQ : Ideal B\ninst✝¹¹ : Q.IsPrime\ninst✝¹⁰ : Q.LiesOver P\nK : Type... | [
"case intro\nA : Type u_1\nB : Type u_2\ninst✝¹⁸ : CommRing A\ninst✝¹⁷ : CommRing B\ninst✝¹⁶ : Algebra A B\nG : Type u_3\ninst✝¹⁵ : Group G\ninst✝¹⁴ : Finite G\ninst✝¹³ : MulSemiringAction G B\ninst✝¹² : SMulCommClass G A B\nP : Ideal A\nQ : Ideal B\ninst✝¹¹ : Q.IsPrime\ninst✝¹⁰ : Q.LiesOver P\nK : Type u_4\nL : Ty... | algebraMap_apply (A ⧸ P) (B ⧸ Q) L, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Galois.IsGaloisGroup | {
"line": 335,
"column": 2
} | {
"line": 335,
"column": 71
} | {
"line": 336,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝¹⁰ : Group G\ninst✝⁹ : Finite G\nA : Type u_5\nB : Type u_6\nC : Type u_7\nH : Subgroup G\ninst✝⁸ : CommRing A\ninst✝⁷ : CommRing B\ninst✝⁶ : CommRing C\ninst✝⁵ : IsDomain C\ninst✝⁴ : Algebra A C\ninst✝³ : FaithfulSMul A C\ninst✝² : MulSemiringAction G C\nhGAC : IsGaloisGroup G A C\n... | [
"G : Type u_1\ninst✝¹⁰ : Group G\ninst✝⁹ : Finite G\nA : Type u_5\nB : Type u_6\nC : Type u_7\nH : Subgroup G\ninst✝⁸ : CommRing A\ninst✝⁷ : CommRing B\ninst✝⁶ : CommRing C\ninst✝⁵ : IsDomain C\ninst✝⁴ : Algebra A C\ninst✝³ : FaithfulSMul A C\ninst✝² : MulSemiringAction G C\nhGAC : IsGaloisGroup G A C\ninst✝¹ : Alg... | let : MulSemiringAction G L := IsFractionRing.mulSemiringAction G C L | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.Invariant.Basic | {
"line": 355,
"column": 2
} | {
"line": 356,
"column": 79
} | {
"line": 357,
"column": 2
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹⁹ : CommRing A\ninst✝¹⁸ : CommRing B\ninst✝¹⁷ : Algebra A B\nG : Type u_3\ninst✝¹⁶ : Group G\ninst✝¹⁵ : Finite G\ninst✝¹⁴ : MulSemiringAction G B\ninst✝¹³ : SMulCommClass G A B\nP : Ideal A\nQ : Ideal B\ninst✝¹² : Q.IsPrime\ninst✝¹¹ : Q.LiesOver P\nK : Type u_4\nL : Ty... | [
"A : Type u_1\nB : Type u_2\ninst✝¹⁹ : CommRing A\ninst✝¹⁸ : CommRing B\ninst✝¹⁷ : Algebra A B\nG : Type u_3\ninst✝¹⁶ : Group G\ninst✝¹⁵ : Finite G\ninst✝¹⁴ : MulSemiringAction G B\ninst✝¹³ : SMulCommClass G A B\nP : Ideal A\nQ : Ideal B\ninst✝¹² : Q.IsPrime\ninst✝¹¹ : Q.LiesOver P\nK : Type u_4\nL : Type u_5\ninst... | obtain ⟨g, hg⟩ := FixedPoints.toAlgAut_surjective (MulAction.stabilizer G Q) L
(AlgEquiv.ofRingEquiv (f := f) (fun x ↦ fixed_of_fixed2 G P Q K L f x x.2)) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Invariant.Basic | {
"line": 366,
"column": 36
} | {
"line": 366,
"column": 54
} | {
"line": 366,
"column": 54
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝⁹ : CommRing A\ninst✝⁸ : CommRing B\ninst✝⁷ : Algebra A B\nG : Type u_3\ninst✝⁶ : Group G\ninst✝⁵ : Finite G\ninst✝⁴ : MulSemiringAction G B\ninst✝³ : SMulCommClass G A B\nP : Ideal A\nQ : Ideal B\ninst✝² : Q.IsPrime\ninst✝¹ : Q.LiesOver P\ninst✝ : Algebra.IsInvariant A... | [
"A : Type u_1\nB : Type u_2\ninst✝⁹ : CommRing A\ninst✝⁸ : CommRing B\ninst✝⁷ : Algebra A B\nG : Type u_3\ninst✝⁶ : Group G\ninst✝⁵ : Finite G\ninst✝⁴ : MulSemiringAction G B\ninst✝³ : SMulCommClass G A B\nP : Ideal A\nQ : Ideal B\ninst✝² : Q.IsPrime\ninst✝¹ : Q.LiesOver P\ninst✝ : Algebra.IsInvariant A B G\nthis :... | MonoidHom.coe_comp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Invariant.Basic | {
"line": 424,
"column": 2
} | {
"line": 424,
"column": 51
} | {
"line": 425,
"column": 2
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nk : Type u_3\ninst✝¹³ : CommRing A\ninst✝¹² : CommRing B\ninst✝¹¹ : Algebra A B\nG : Type u_4\ninst✝¹⁰ : Finite G\ninst✝⁹ : Group G\ninst✝⁸ : MulSemiringAction G B\ninst✝⁷ : Algebra.IsInvariant A B G\nP : Ideal A\nQ : Ideal B\ninst✝⁶ : Q.LiesOver P\ninst✝⁵ : CommRing k\ninst... | [
"A : Type u_1\nB : Type u_2\nk : Type u_3\ninst✝¹³ : CommRing A\ninst✝¹² : CommRing B\ninst✝¹¹ : Algebra A B\nG : Type u_4\ninst✝¹⁰ : Finite G\ninst✝⁹ : Group G\ninst✝⁸ : MulSemiringAction G B\ninst✝⁷ : Algebra.IsInvariant A B G\nP : Ideal A\nQ : Ideal B\ninst✝⁶ : Q.LiesOver P\ninst✝⁵ : CommRing k\ninst✝⁴ : Algebra... | obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Invariant.Basic | {
"line": 461,
"column": 4
} | {
"line": 461,
"column": 53
} | {
"line": 462,
"column": 4
} | [
{
"pp": "case refine_1\nA : Type u_1\nB : Type u_2\nk : Type u_3\ninst✝¹³ : CommRing A\ninst✝¹² : CommRing B\ninst✝¹¹ : Algebra A B\nG : Type u_4\ninst✝¹⁰ : Finite G\ninst✝⁹ : Group G\ninst✝⁸ : MulSemiringAction G B\ninst✝⁷ : Algebra.IsInvariant A B G\nP : Ideal A\nQ : Ideal B\ninst✝⁶ : Q.LiesOver P\ninst✝⁵ : C... | [
"case refine_1\nA : Type u_1\nB : Type u_2\nk : Type u_3\ninst✝¹³ : CommRing A\ninst✝¹² : CommRing B\ninst✝¹¹ : Algebra A B\nG : Type u_4\ninst✝¹⁰ : Finite G\ninst✝⁹ : Group G\ninst✝⁸ : MulSemiringAction G B\ninst✝⁷ : Algebra.IsInvariant A B G\nP : Ideal A\nQ : Ideal B\ninst✝⁶ : Q.LiesOver P\ninst✝⁵ : CommRing k\ni... | obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Invariant.Basic | {
"line": 468,
"column": 4
} | {
"line": 468,
"column": 53
} | {
"line": 469,
"column": 4
} | [
{
"pp": "case refine_2\nA : Type u_1\nB : Type u_2\nk : Type u_3\ninst✝¹³ : CommRing A\ninst✝¹² : CommRing B\ninst✝¹¹ : Algebra A B\nG : Type u_4\ninst✝¹⁰ : Finite G\ninst✝⁹ : Group G\ninst✝⁸ : MulSemiringAction G B\ninst✝⁷ : Algebra.IsInvariant A B G\nP : Ideal A\nQ : Ideal B\ninst✝⁶ : Q.LiesOver P\ninst✝⁵ : C... | [
"case refine_2\nA : Type u_1\nB : Type u_2\nk : Type u_3\ninst✝¹³ : CommRing A\ninst✝¹² : CommRing B\ninst✝¹¹ : Algebra A B\nG : Type u_4\ninst✝¹⁰ : Finite G\ninst✝⁹ : Group G\ninst✝⁸ : MulSemiringAction G B\ninst✝⁷ : Algebra.IsInvariant A B G\nP : Ideal A\nQ : Ideal B\ninst✝⁶ : Q.LiesOver P\ninst✝⁵ : CommRing k\ni... | obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Ideal.Norm.AbsNorm | {
"line": 255,
"column": 65
} | {
"line": 255,
"column": 77
} | {
"line": 256,
"column": 4
} | [
{
"pp": "S : Type u_1\ninst✝² : CommRing S\ninst✝¹ : IsDedekindDomain S\ninst✝ : Free ℤ S\nI : Ideal S\n⊢ (Quotient.mk I) ↑(toAddSubgroup I).index = 0",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHom... | [
"S : Type u_1\ninst✝² : CommRing S\ninst✝¹ : IsDedekindDomain S\ninst✝ : Free ℤ S\nI : Ideal S\n⊢ ↑(toAddSubgroup I).index = 0"
] | map_natCast, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Norm.AbsNorm | {
"line": 390,
"column": 2
} | {
"line": 390,
"column": 9
} | {
"line": 391,
"column": 4
} | [
{
"pp": "case h₁\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : IsDedekindDomain S\ninst✝¹ : Free ℤ S\ninst✝ : Module.Finite ℤ S\np : ℤ\nhp : Prime p\nthis✝ : IsAddTorsionFree S\nthis : CharZero S\nhpMax : (span {p}).IsMaximal\nhI : p ∣ ↑(absNorm 0)\n⊢ ∃ P, P.IsMaximal ∧ under ℤ P = span {p} ∧ P ∣ 0",
"ppTerm... | [] | | h₁ => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.Finiteness.NilpotentKer | {
"line": 56,
"column": 10
} | {
"line": 56,
"column": 31
} | {
"line": 56,
"column": 32
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : CommRing T\ninst✝² : Algebra R S\ninst✝¹ : Algebra R T\ninst✝ : Module.Finite R T\nf : S →ₐ[R] T\nhf₁ : Function.Surjective ⇑f\nI : Ideal S\nhI : RingHom.ker f = I\nhf₂ : I ≤ nilradical S\nhf₃ : I.FG\nthis✝ : M... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : CommRing T\ninst✝² : Algebra R S\ninst✝¹ : Algebra R T\ninst✝ : Module.Finite R T\nf : S →ₐ[R] T\nhf₁ : Function.Surjective ⇑f\nI : Ideal S\nhI : RingHom.ker f = I\nhf₂ : I ≤ nilradical S\nhf₃ : I.FG\nthis✝ : Module.Finite... | Submodule.map_smul'', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.LocalRing.ResidueField.Instances | {
"line": 46,
"column": 2
} | {
"line": 46,
"column": 51
} | {
"line": 47,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝¹² : CommRing R\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : CommRing B\ninst✝⁹ : Algebra R A\ninst✝⁸ : Algebra A B\ninst✝⁷ : Algebra R B\ninst✝⁶ : IsScalarTower R A B\np : Ideal A\nq : Ideal B\ninst✝⁵ : q.LiesOver p\ninst✝⁴ : p.IsMaximal\ninst✝³ : q.IsMaximal\ninst✝²... | [
"R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝¹² : CommRing R\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : CommRing B\ninst✝⁹ : Algebra R A\ninst✝⁸ : Algebra A B\ninst✝⁷ : Algebra R B\ninst✝⁶ : IsScalarTower R A B\np : Ideal A\nq : Ideal B\ninst✝⁵ : q.LiesOver p\ninst✝⁴ : p.IsMaximal\ninst✝³ : q.IsMaximal\ninst✝² : Algebra (... | obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.LocalRing.ResidueField.Instances | {
"line": 76,
"column": 4
} | {
"line": 76,
"column": 53
} | {
"line": 77,
"column": 4
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝¹² : CommRing R\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : CommRing B\ninst✝⁹ : Algebra R A\ninst✝⁸ : Algebra A B\ninst✝⁷ : Algebra R B\ninst✝⁶ : IsScalarTower R A B\np : Ideal A\nq : Ideal B\ninst✝⁵ : q.LiesOver p\ninst✝⁴ : p.IsPrime\ninst✝³ : q.IsPrime\ninst✝² : A... | [
"R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝¹² : CommRing R\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : CommRing B\ninst✝⁹ : Algebra R A\ninst✝⁸ : Algebra A B\ninst✝⁷ : Algebra R B\ninst✝⁶ : IsScalarTower R A B\np : Ideal A\nq : Ideal B\ninst✝⁵ : q.LiesOver p\ninst✝⁴ : p.IsPrime\ninst✝³ : q.IsPrime\ninst✝² : Algebra (Loca... | obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Unramified.Basic | {
"line": 75,
"column": 11
} | {
"line": 75,
"column": 13
} | {
"line": 75,
"column": 14
} | [
{
"pp": "R : Type v\ninst✝⁵ : CommRing R\nA : Type u\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\nB : Type w\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\nI : Ideal B\ninst✝ : FormallyUnramified R A\nhI : I ^ 2 = ⊥\nf₁ : A →ₐ[R] B\n⊢ ∀ ⦃a₂ : A →ₐ[R] B⦄, (Ideal.Quotient.mkₐ R I).comp f₁ = (Ideal.Quotient.mkₐ R I).c... | [
"R : Type v\ninst✝⁵ : CommRing R\nA : Type u\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\nB : Type w\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\nI : Ideal B\ninst✝ : FormallyUnramified R A\nhI : I ^ 2 = ⊥\nf₁ f₂ : A →ₐ[R] B\n⊢ (Ideal.Quotient.mkₐ R I).comp f₁ = (Ideal.Quotient.mkₐ R I).comp f₂ → f₁ = f₂"
] | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.RamificationInertia.Inertia | {
"line": 102,
"column": 2
} | {
"line": 102,
"column": 51
} | {
"line": 103,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹⁹ : CommRing R\ninst✝¹⁸ : CommRing S\ninst✝¹⁷ : Algebra R S\np : Ideal R\nq : Ideal S\ninst✝¹⁶ : q.LiesOver p\ninst✝¹⁵ : p.IsPrime\ninst✝¹⁴ : q.IsPrime\nK : Type u_4\nL : Type u_5\ninst✝¹³ : Field K\ninst✝¹² : Field L\ninst✝¹¹ : Algebra (R ⧸ p) K\ninst✝¹⁰ : IsFractionR... | [
"R : Type u_1\nS : Type u_2\ninst✝¹⁹ : CommRing R\ninst✝¹⁸ : CommRing S\ninst✝¹⁷ : Algebra R S\np : Ideal R\nq : Ideal S\ninst✝¹⁶ : q.LiesOver p\ninst✝¹⁵ : p.IsPrime\ninst✝¹⁴ : q.IsPrime\nK : Type u_4\nL : Type u_5\ninst✝¹³ : Field K\ninst✝¹² : Field L\ninst✝¹¹ : Algebra (R ⧸ p) K\ninst✝¹⁰ : IsFractionRing (R ⧸ p) ... | obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.NumberTheory.RamificationInertia.Ramification | {
"line": 294,
"column": 2
} | {
"line": 294,
"column": 58
} | {
"line": 296,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\nS : Type v\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\np : Ideal R\nP : Ideal S\ninst✝ : IsDedekindDomain S\nhp0 : map f p ≠ ⊥\nhP : P.IsPrime\nle : map f p ≤ P\nhP0 : P ≠ ⊥\nhPirr : Irreducible P\nP' : Ideal S\nhP' : P' ∈ normalizedFactors (map f p)\nP'_eq : Associated... | [] | rwa [Multiset.count_ne_zero, associated_iff_eq.mp P'_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.RingTheory.RamificationInertia.Inertia | {
"line": 172,
"column": 2
} | {
"line": 178,
"column": 37
} | {
"line": 180,
"column": 0
} | [
{
"pp": "case pos\nR : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nq : Ideal S\nG : Type u_4\ninst✝² : Group G\ninst✝¹ : MulSemiringAction G S\ninst✝ : SMulCommClass G R S\ng : G\nhq : q.IsPrime\n⊢ (g • q).inertiaDeg R = q.inertiaDeg R",
"ppTerm": "?pos✝",
"as... | [] | · let p := q.under R
let f₀ := MulSemiringAction.toAlgAut G R S g
let := Localization.AtPrime.algebraOfLiesOver p q
let := Localization.AtPrime.algebraOfLiesOver p (g • q)
rw [inertiaDeg_eq p q, inertiaDeg_eq p (g • q)]
let e₂ := Ideal.residueFieldAlgEquiv' p (g • q) q f₀.symm (comap_symm f₀.toRingE... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.RamificationInertia.Ramification | {
"line": 370,
"column": 2
} | {
"line": 370,
"column": 9
} | {
"line": 370,
"column": 10
} | [
{
"pp": "case h₁\nR : Type u\ninst✝⁶ : CommRing R\nS : Type v\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : IsDedekindDomain S\ninst✝² : IsDedekindDomain R\ninst✝¹ : FaithfulSMul R S\nv : Ideal R\nw : Ideal S\nhv : Irreducible v\nhw : Irreducible w\nhw_bot : w ≠ ⊥\ninst✝ : w.LiesOver v\nh : 0 ≠ ⊥\n⊢ emul... | [] | | h₁ => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.Unramified.Basic | {
"line": 191,
"column": 21
} | {
"line": 191,
"column": 23
} | {
"line": 191,
"column": 24
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\nB : Type u\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nI✝ : Ideal B\nx✝ : I✝ ^ 2 = ⊥\nf₁ : R →ₐ[R] B\n⊢ ∀ ⦃a₂ : R →ₐ[R] B⦄, (Ideal.Quotient.mkₐ R I✝).comp f₁ = (Ideal.Quotient.mkₐ R I✝).comp a₂ → f₁ = a₂",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants":... | [
"R : Type u\ninst✝² : CommRing R\nB : Type u\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nI✝ : Ideal B\nx✝ : I✝ ^ 2 = ⊥\nf₁ f₂ : R →ₐ[R] B\n⊢ (Ideal.Quotient.mkₐ R I✝).comp f₁ = (Ideal.Quotient.mkₐ R I✝).comp f₂ → f₁ = f₂"
] | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.Unramified.Basic | {
"line": 202,
"column": 22
} | {
"line": 202,
"column": 24
} | {
"line": 202,
"column": 25
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : CommRing R\nA : Type u_2\nB : Type u_3\ninst✝⁶ : CommRing A\ninst✝⁵ : Algebra R A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R B\ninst✝² : FormallyUnramified R A\ne : A ≃ₐ[R] B\nC : Type u_3\ninst✝¹ : CommRing C\ninst✝ : Algebra R C\nI : Ideal C\nhI : I ^ 2 = ⊥\nf₁ : B →ₐ[R] C\n⊢ ∀ ⦃... | [
"R : Type u_1\ninst✝⁷ : CommRing R\nA : Type u_2\nB : Type u_3\ninst✝⁶ : CommRing A\ninst✝⁵ : Algebra R A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R B\ninst✝² : FormallyUnramified R A\ne : A ≃ₐ[R] B\nC : Type u_3\ninst✝¹ : CommRing C\ninst✝ : Algebra R C\nI : Ideal C\nhI : I ^ 2 = ⊥\nf₁ f₂ : B →ₐ[R] C\n⊢ (Ideal.Quoti... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.Unramified.Basic | {
"line": 219,
"column": 22
} | {
"line": 219,
"column": 24
} | {
"line": 219,
"column": 25
} | [
{
"pp": "R : Type u_1\ninst✝¹⁰ : CommRing R\nA : Type u_2\ninst✝⁹ : CommRing A\ninst✝⁸ : Algebra R A\nB : Type u_3\ninst✝⁷ : CommRing B\ninst✝⁶ : Algebra R B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsScalarTower R A B\ninst✝³ : FormallyUnramified R A\ninst✝² : FormallyUnramified A B\nC : Type u_3\ninst✝¹ : CommRing C\n... | [
"R : Type u_1\ninst✝¹⁰ : CommRing R\nA : Type u_2\ninst✝⁹ : CommRing A\ninst✝⁸ : Algebra R A\nB : Type u_3\ninst✝⁷ : CommRing B\ninst✝⁶ : Algebra R B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsScalarTower R A B\ninst✝³ : FormallyUnramified R A\ninst✝² : FormallyUnramified A B\nC : Type u_3\ninst✝¹ : CommRing C\ninst✝ : Alge... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.Unramified.Basic | {
"line": 233,
"column": 21
} | {
"line": 233,
"column": 23
} | {
"line": 233,
"column": 24
} | [
{
"pp": "R : Type u_1\ninst✝⁹ : CommRing R\nA : Type u_2\ninst✝⁸ : CommRing A\ninst✝⁷ : Algebra R A\nB : Type u_3\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra R B\ninst✝⁴ : Algebra A B\ninst✝³ : IsScalarTower R A B\ninst✝² : FormallyUnramified R B\nQ : Type u_3\ninst✝¹ : CommRing Q\ninst✝ : Algebra A Q\nI : Ideal Q\n... | [
"R : Type u_1\ninst✝⁹ : CommRing R\nA : Type u_2\ninst✝⁸ : CommRing A\ninst✝⁷ : Algebra R A\nB : Type u_3\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra R B\ninst✝⁴ : Algebra A B\ninst✝³ : IsScalarTower R A B\ninst✝² : FormallyUnramified R B\nQ : Type u_3\ninst✝¹ : CommRing Q\ninst✝ : Algebra A Q\nI : Ideal Q\ne : I ^ 2 = ... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.Unramified.Basic | {
"line": 252,
"column": 22
} | {
"line": 252,
"column": 24
} | {
"line": 252,
"column": 25
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : CommRing R\nA : Type u_2\nB : Type u_3\ninst✝⁶ : CommRing A\ninst✝⁵ : Algebra R A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R B\ninst✝² : FormallyUnramified R A\nf : A →ₐ[R] B\nH : Function.Surjective ⇑f\nQ : Type u_3\ninst✝¹ : CommRing Q\ninst✝ : Algebra R Q\nI : Ideal Q\nhI : I ^ ... | [
"R : Type u_1\ninst✝⁷ : CommRing R\nA : Type u_2\nB : Type u_3\ninst✝⁶ : CommRing A\ninst✝⁵ : Algebra R A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R B\ninst✝² : FormallyUnramified R A\nf : A →ₐ[R] B\nH : Function.Surjective ⇑f\nQ : Type u_3\ninst✝¹ : CommRing Q\ninst✝ : Algebra R Q\nI : Ideal Q\nhI : I ^ 2 = ⊥\nf₁ f₂... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.Unramified.Basic | {
"line": 280,
"column": 22
} | {
"line": 280,
"column": 24
} | {
"line": 280,
"column": 25
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : CommRing R\nA : Type u_2\ninst✝⁶ : CommRing A\ninst✝⁵ : Algebra R A\nB : Type u_3\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R B\ninst✝² : FormallyUnramified R A\nC : Type (max u_2 u_3)\ninst✝¹ : CommRing C\ninst✝ : Algebra B C\nI : Ideal C\nhI : I ^ 2 = ⊥\nf₁ : B ⊗[R] A →ₐ[B] C\n⊢ ∀... | [
"R : Type u_1\ninst✝⁷ : CommRing R\nA : Type u_2\ninst✝⁶ : CommRing A\ninst✝⁵ : Algebra R A\nB : Type u_3\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R B\ninst✝² : FormallyUnramified R A\nC : Type (max u_2 u_3)\ninst✝¹ : CommRing C\ninst✝ : Algebra B C\nI : Ideal C\nhI : I ^ 2 = ⊥\nf₁ f₂ : B ⊗[R] A →ₐ[B] C\n⊢ (Ideal.Quo... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.Unramified.Basic | {
"line": 304,
"column": 21
} | {
"line": 304,
"column": 23
} | {
"line": 304,
"column": 24
} | [
{
"pp": "R : Type u_1\nRₘ : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing Rₘ\nM : Submonoid R\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization M Rₘ\nQ : Type u_3\ninst✝¹ : CommRing Q\ninst✝ : Algebra R Q\nI : Ideal Q\nx✝ : I ^ 2 = ⊥\nf₁ : Rₘ →ₐ[R] Q\n⊢ ∀ ⦃a₂ : Rₘ →ₐ[R] Q⦄, (Ideal.Quotient.mkₐ R I).comp f₁ = (... | [
"R : Type u_1\nRₘ : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing Rₘ\nM : Submonoid R\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization M Rₘ\nQ : Type u_3\ninst✝¹ : CommRing Q\ninst✝ : Algebra R Q\nI : Ideal Q\nx✝ : I ^ 2 = ⊥\nf₁ f₂ : Rₘ →ₐ[R] Q\n⊢ (Ideal.Quotient.mkₐ R I).comp f₁ = (Ideal.Quotient.mkₐ R I).comp f... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.QuasiFinite.Basic | {
"line": 239,
"column": 8
} | {
"line": 239,
"column": 47
} | {
"line": 239,
"column": 48
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R T\ninst✝² : Algebra S T\ninst✝¹ : IsScalarTower R S T\ninst✝ : QuasiFinite R T\nP : Ideal S\nhP : P.IsPrime\nf : P.ResidueField ⊗[R] T →ₐ[P.ResidueField] P.F... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R T\ninst✝² : Algebra S T\ninst✝¹ : IsScalarTower R S T\ninst✝ : QuasiFinite R T\nP : Ideal S\nhP : P.IsPrime\nf : P.ResidueField ⊗[R] T →ₐ[P.ResidueField] P.Fiber T :=\n ... | ← AlgHom.coe_restrictScalars' (R := S), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.QuasiFinite.Basic | {
"line": 234,
"column": 2
} | {
"line": 244,
"column": 39
} | {
"line": 246,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R T\ninst✝² : Algebra S T\ninst✝¹ : IsScalarTower R S T\ninst✝ : QuasiFinite R T\n⊢ QuasiFinite S T",
"ppTerm": "?m.16",
"assigned": true,
"usedCon... | [] | refine ⟨fun P hP ↦ ?_⟩
let f : P.ResidueField ⊗[R] T →ₐ[P.ResidueField] P.Fiber T :=
Algebra.TensorProduct.lift (Algebra.ofId _ _)
(Algebra.TensorProduct.includeRight.restrictScalars R) fun _ _ ↦ .all _ _
have hf : Function.Surjective f := by
rw [← AlgHom.coe_restrictScalars' (R := S), ← AlgHom.coe_to... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.QuasiFinite.Basic | {
"line": 234,
"column": 2
} | {
"line": 244,
"column": 39
} | {
"line": 246,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R T\ninst✝² : Algebra S T\ninst✝¹ : IsScalarTower R S T\ninst✝ : QuasiFinite R T\n⊢ QuasiFinite S T",
"ppTerm": "?m.16",
"assigned": true,
"usedCon... | [] | refine ⟨fun P hP ↦ ?_⟩
let f : P.ResidueField ⊗[R] T →ₐ[P.ResidueField] P.Fiber T :=
Algebra.TensorProduct.lift (Algebra.ofId _ _)
(Algebra.TensorProduct.includeRight.restrictScalars R) fun _ _ ↦ .all _ _
have hf : Function.Surjective f := by
rw [← AlgHom.coe_restrictScalars' (R := S), ← AlgHom.coe_to... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.QuasiFinite.Basic | {
"line": 384,
"column": 2
} | {
"line": 384,
"column": 93
} | {
"line": 385,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R T\nf : S →ₐ[R] T\nhf : f.SurjectiveOnStalks\nq : Ideal T\ninst✝² : q.IsPrime\ninst✝¹ : (Ideal.comap f.toRingHom q).IsPrime\ninst✝ : QuasiFiniteAt R (Ideal.co... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R T\nf : S →ₐ[R] T\nhf : f.SurjectiveOnStalks\nq : Ideal T\ninst✝² : q.IsPrime\ninst✝¹ : (Ideal.comap f.toRingHom q).IsPrime\ninst✝ : QuasiFiniteAt R (Ideal.comap f.toRing... | refine .of_surjective_algHom ⟨Localization.localRingHom _ q f.toRingHom rfl, ?_⟩ (hf q ‹_›) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Smooth.Basic | {
"line": 225,
"column": 6
} | {
"line": 225,
"column": 55
} | {
"line": 226,
"column": 6
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing A\ninst✝⁷ : Algebra R A\nB : Type u_1\nP : Type u_2\nC : Type u_3\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommRing C\ninst✝³ : Algebra R C\ninst✝² : CommRing P\ninst✝¹ : Algebra R P\nP₁ : Extension R A\nP₂ : Extension R A\ninst... | [
"R : Type u\nA : Type v\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing A\ninst✝⁷ : Algebra R A\nB : Type u_1\nP : Type u_2\nC : Type u_3\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommRing C\ninst✝³ : Algebra R C\ninst✝² : CommRing P\ninst✝¹ : Algebra R P\nP₁ : Extension R A\nP₂ : Extension R A\ninst✝ : Formally... | obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Unramified.Finite | {
"line": 158,
"column": 62
} | {
"line": 231,
"column": 18
} | {
"line": 233,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : FormallyUnramified R S\ninst✝¹ : EssFiniteType R S\ninst✝ : Free R S\n⊢ Module.Finite R S",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Algebra.FormallyUnramified.one_tmul... | [] | by
classical
let I := Module.Free.ChooseBasisIndex R S
-- Let `bᵢ` be an `R`-basis of `S`.
let b : Basis I R S := Module.Free.chooseBasis R S
-- Let `∑ₛ fᵢ ⊗ bᵢ : S ⊗[R] S` (summing over some finite `s`) be an element such that
-- `∑ₛ fᵢbᵢ = 1` and `∀ x : S, xfᵢ ⊗ bᵢ = aᵢ ⊗ xfᵢ` which exists since `S` is un... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Etale.Basic | {
"line": 221,
"column": 2
} | {
"line": 221,
"column": 85
} | {
"line": 222,
"column": 2
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\n⊢ Etale R A ↔ FormallyUnramified R A ∧ Smooth R A",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Algebra.smooth_iff",
"congrArg",
"CommSemiring.toSemiring",
... | [
"R : Type u\nA : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\n⊢ autoParam (FormallyUnramified R A ∧ FormallySmooth R A) formallyEtale._autoParam ∧\n autoParam (FinitePresentation R A) finitePresentation._autoParam ↔\n FormallyUnramified R A ∧\n autoParam (FormallySmooth R A) S... | rw [etale_iff, FormallyEtale.iff_formallyUnramified_and_formallySmooth, smooth_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Smooth.Basic | {
"line": 363,
"column": 2
} | {
"line": 363,
"column": 51
} | {
"line": 364,
"column": 2
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nH :\n ∀ ⦃B : Type (max u v)⦄ [inst : CommRing B] [inst_1 : Algebra R B] (I : Ideal B),\n I ^ 2 = ⊥ → Function.Surjective (Ideal.Quotient.mkₐ R I).comp\nP : Generators R A A := Generators.self R A\nf : P.Ring →ₐ[R... | [
"R : Type u\nA : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nH :\n ∀ ⦃B : Type (max u v)⦄ [inst : CommRing B] [inst_1 : Algebra R B] (I : Ideal B),\n I ^ 2 = ⊥ → Function.Surjective (Ideal.Quotient.mkₐ R I).comp\nP : Generators R A A := Generators.self R A\nf : P.Ring →ₐ[R] A := IsSca... | obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Smooth.Kaehler | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 38
} | {
"line": 96,
"column": 2
} | [
{
"pp": "R : Type u_1\nP : Type u_2\nS : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing P\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R P\ninst✝² : Algebra P S\ninst✝¹ : Algebra R S\ninst✝ : IsScalarTower R P S\ng : S →ₐ[R] P\nhf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥\nhg : (IsScalarTower.toAlgHom R P S).comp g =... | [
"R : Type u_1\nP : Type u_2\nS : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing P\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R P\ninst✝² : Algebra P S\ninst✝¹ : Algebra R S\ninst✝ : IsScalarTower R P S\ng : S →ₐ[R] P\nhf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥\nhg : (IsScalarTower.toAlgHom R P S).comp g = AlgHom.id R... | change g (p • s) * m = p * (g s * m) | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.RingTheory.Smooth.Basic | {
"line": 491,
"column": 2
} | {
"line": 492,
"column": 78
} | {
"line": 493,
"column": 2
} | [
{
"pp": "R : Type u_4\nRₘ : Type u_6\ninst✝³ : CommRing R\ninst✝² : CommRing Rₘ\nM : Submonoid R\ninst✝¹ : Algebra R Rₘ\ninst✝ : IsLocalization M Rₘ\nQ : Type (max u_4 u_6)\nx✝¹ : CommRing Q\nx✝ : Algebra R Q\nI : Ideal Q\ne : I ^ 2 = ⊥\nf : Rₘ →ₐ[R] Q ⧸ I\nthis : ∀ (x : ↥M), IsUnit ((algebraMap R Q) ↑x)\n⊢ ∃ a... | [
"R : Type u_4\nRₘ : Type u_6\ninst✝³ : CommRing R\ninst✝² : CommRing Rₘ\nM : Submonoid R\ninst✝¹ : Algebra R Rₘ\ninst✝ : IsLocalization M Rₘ\nQ : Type (max u_4 u_6)\nx✝¹ : CommRing Q\nx✝ : Algebra R Q\nI : Ideal Q\ne : I ^ 2 = ⊥\nf : Rₘ →ₐ[R] Q ⧸ I\nthis✝ : ∀ (x : ↥M), IsUnit ((algebraMap R Q) ↑x)\nthis : Rₘ →ₐ[R] ... | let this : Rₘ →ₐ[R] Q :=
{ IsLocalization.lift this with commutes' := IsLocalization.lift_eq this } | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.Smooth.Kaehler | {
"line": 118,
"column": 46
} | {
"line": 126,
"column": 46
} | {
"line": 128,
"column": 0
} | [
{
"pp": "R : Type u_1\nP : Type u_2\nS : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing P\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R P\ninst✝² : Algebra P S\ninst✝¹ : Algebra R S\ninst✝ : IsScalarTower R P S\ng : S →ₐ[R] P\nhf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥\nhg : (IsScalarTower.toAlgHom R P S).comp g =... | [] | by
letI := g.toRingHom.toAlgebra
haveI := isScalarTower_of_section_of_ker_sqZero g hf' hg
simp only [retractionOfSectionOfKerSqZero, LinearMap.coe_restrictScalars,
LinearMap.liftBaseChange_tmul, SetLike.val_smul_of_tower]
-- The issue is a mismatch between `RingHom.ker (algebraMap P S)` and
-- `RingHom.ke... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Unramified.Locus | {
"line": 113,
"column": 2
} | {
"line": 117,
"column": 79
} | {
"line": 119,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\n⊢ unramifiedLocus R A = (Module.support A Ω[A⁄R])ᶜ",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"Algebra.to_smulCommClass",
"IsLocalizedModule... | [] | ext p
simp only [Set.mem_compl_iff, Module.notMem_support_iff]
have := IsLocalizedModule.iso p.asIdeal.primeCompl
(KaehlerDifferential.map R R A (Localization.AtPrime p.asIdeal))
exact (Algebra.formallyUnramified_iff _ _).trans this.subsingleton_congr.symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Unramified.Locus | {
"line": 113,
"column": 2
} | {
"line": 117,
"column": 79
} | {
"line": 119,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\n⊢ unramifiedLocus R A = (Module.support A Ω[A⁄R])ᶜ",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"Algebra.to_smulCommClass",
"IsLocalizedModule... | [] | ext p
simp only [Set.mem_compl_iff, Module.notMem_support_iff]
have := IsLocalizedModule.iso p.asIdeal.primeCompl
(KaehlerDifferential.map R R A (Localization.AtPrime p.asIdeal))
exact (Algebra.formallyUnramified_iff _ _).trans this.subsingleton_congr.symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Unramified.Field | {
"line": 46,
"column": 22
} | {
"line": 46,
"column": 24
} | {
"line": 46,
"column": 25
} | [
{
"pp": "K : Type u_1\nL : Type u_3\ninst✝⁵ : Field K\ninst✝⁴ : Field L\ninst✝³ : Algebra K L\ninst✝² : Algebra.IsSeparable K L\nB : Type u_3\ninst✝¹ : CommRing B\ninst✝ : Algebra K B\nI : Ideal B\nhI : I ^ 2 = ⊥\nf₁ : L →ₐ[K] B\n⊢ ∀ ⦃a₂ : L →ₐ[K] B⦄, (Ideal.Quotient.mkₐ K I).comp f₁ = (Ideal.Quotient.mkₐ K I).... | [
"K : Type u_1\nL : Type u_3\ninst✝⁵ : Field K\ninst✝⁴ : Field L\ninst✝³ : Algebra K L\ninst✝² : Algebra.IsSeparable K L\nB : Type u_3\ninst✝¹ : CommRing B\ninst✝ : Algebra K B\nI : Ideal B\nhI : I ^ 2 = ⊥\nf₁ f₂ : L →ₐ[K] B\n⊢ (Ideal.Quotient.mkₐ K I).comp f₁ = (Ideal.Quotient.mkₐ K I).comp f₂ → f₁ = f₂"
] | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.Kaehler.JacobiZariski | {
"line": 278,
"column": 4
} | {
"line": 285,
"column": 7
} | {
"line": 287,
"column": 0
} | [
{
"pp": "case mul_X\nR : Type u₁\nS : Type u₂\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nT : Type u₃\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nι : Type w₁\nι' : Type w₃\nQ : Generators S T ι\nQ' : Generators S T ι'\nf : Q.Hom Q'\nthis✝ :... | [] | simp only [map_mul, Hom.toAlgHom_X, δAux_mul, algebraMap_apply, Hom.algebraMap_toAlgHom,
← @IsScalarTower.algebraMap_smul Q'.Ring T, algebraMap_self, δAux_X,
RingHom.id_apply, coe_eval₂Hom, IH, Hom.aeval_val, smul_add, map_aeval, tmul_add, tmul_smul,
← @IsScalarTower.algebraMap_smul Q.Ring T, smul_zer... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Kaehler.JacobiZariski | {
"line": 278,
"column": 4
} | {
"line": 285,
"column": 7
} | {
"line": 287,
"column": 0
} | [
{
"pp": "case mul_X\nR : Type u₁\nS : Type u₂\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nT : Type u₃\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nι : Type w₁\nι' : Type w₃\nQ : Generators S T ι\nQ' : Generators S T ι'\nf : Q.Hom Q'\nthis✝ :... | [] | simp only [map_mul, Hom.toAlgHom_X, δAux_mul, algebraMap_apply, Hom.algebraMap_toAlgHom,
← @IsScalarTower.algebraMap_smul Q'.Ring T, algebraMap_self, δAux_X,
RingHom.id_apply, coe_eval₂Hom, IH, Hom.aeval_val, smul_add, map_aeval, tmul_add, tmul_smul,
← @IsScalarTower.algebraMap_smul Q.Ring T, smul_zer... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Kaehler.JacobiZariski | {
"line": 272,
"column": 2
} | {
"line": 272,
"column": 50
} | {
"line": 273,
"column": 2
} | [
{
"pp": "R : Type u₁\nS : Type u₂\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nT : Type u₃\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nι : Type w₁\nι' : Type w₃\nQ : Generators S T ι\nQ' : Generators S T ι'\nf : Q.Hom Q'\nx : Q.Ring\nthis✝ :... | [] | induction x using MvPolynomial.induction_on with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.Smooth.Kaehler | {
"line": 344,
"column": 4
} | {
"line": 344,
"column": 53
} | {
"line": 345,
"column": 4
} | [
{
"pp": "case refine_1\nR : Type u_1\nP : Type u_2\nS : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing P\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R P\ninst✝² : Algebra P S\ninst✝¹ : Algebra R S\ninst✝ : IsScalarTower R P S\nhf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥\nhf : Surjective ⇑(algebraMap P S)\nP' : Type... | [
"case refine_1\nR : Type u_1\nP : Type u_2\nS : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing P\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R P\ninst✝² : Algebra P S\ninst✝¹ : Algebra R S\ninst✝ : IsScalarTower R P S\nhf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥\nhf : Surjective ⇑(algebraMap P S)\nP' : Type u_2 := P ⧸ ... | obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Kaehler.JacobiZariski | {
"line": 294,
"column": 2
} | {
"line": 294,
"column": 50
} | {
"line": 295,
"column": 2
} | [
{
"pp": "R : Type u₁\nS : Type u₂\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nT : Type u₃\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nι : Type w₁\nσ : Type w₂\nQ : Generators S T ι\nP : Generators R S σ\nx : (Q.comp P).Ring\nthis✝ : AddComm... | [] | induction x using MvPolynomial.induction_on with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.Unramified.Field | {
"line": 143,
"column": 2
} | {
"line": 143,
"column": 65
} | {
"line": 144,
"column": 2
} | [
{
"pp": "K : Type u_1\nA : Type u_2\ninst✝⁴ : Field K\ninst✝³ : CommRing A\ninst✝² : Algebra K A\ninst✝¹ : FormallyUnramified K A\ninst✝ : EssFiniteType K A\nx : A\nhx : IsNilpotent x\nf : A →ₐ[K] AlgebraicClosure K ⊗[K] A := TensorProduct.includeRight\nthis : Function.Injective ⇑f\nM : Ideal (AlgebraicClosure ... | [
"K : Type u_1\nA : Type u_2\ninst✝⁴ : Field K\ninst✝³ : CommRing A\ninst✝² : Algebra K A\ninst✝¹ : FormallyUnramified K A\ninst✝ : EssFiniteType K A\nx : A\nhx : IsNilpotent x\nf : A →ₐ[K] AlgebraicClosure K ⊗[K] A := TensorProduct.includeRight\nthis : Function.Injective ⇑f\nM : Ideal (AlgebraicClosure K ⊗[K] A)\nh... | generalize algebraMap _ (Localization.AtPrime M) (f x) = y at * | Lean.Elab.Tactic.evalGeneralize | Lean.Parser.Tactic.generalize |
Mathlib.RingTheory.Kaehler.JacobiZariski | {
"line": 407,
"column": 4
} | {
"line": 407,
"column": 44
} | {
"line": 408,
"column": 4
} | [
{
"pp": "case hz\nR : Type u₁\nS : Type u₂\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nT : Type u₃\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nι : Type w₁\nσ : Type w₂\nQ : Generators S T ι\nP : Generators R S σ\nx : ↥Q.ker\nhx : Extension.... | [
"case hz\nR : Type u₁\nS : Type u₂\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nT : Type u₃\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nι : Type w₁\nσ : Type w₂\nQ : Generators S T ι\nP : Generators R S σ\nx : ↥Q.ker\nhx : Extension.Cotangent.mk... | rw [← CotangentSpace.compEquiv_symm_inr] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Smooth.Kaehler | {
"line": 396,
"column": 6
} | {
"line": 397,
"column": 42
} | {
"line": 398,
"column": 6
} | [
{
"pp": "R : Type u_1\nS : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : Extension R S\nx : ↥P.ker\nhx : ((map P.toInfinitesimal) (mk x)).val = val 0\n⊢ ↑x ∈ P.ker ^ 2",
"ppTerm": "?m.199",
"assigned": true,
"usedConstants": [
"Ideal.cotangentIdeal",
"Algeb... | [
"R : Type u_1\nS : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : Extension R S\nx : ↥P.ker\nhx : P.toInfinitesimal.toRingHom ↑x ∈ P.ker.cotangentIdeal ^ 2\n⊢ ↑x ∈ P.ker ^ 2"
] | simp only [map_mk, Hom.toAlgHom_apply, val_mk, val_zero, Ideal.toCotangent_eq_zero,
Extension.ker_infinitesimal] at hx | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Smooth.Kaehler | {
"line": 405,
"column": 4
} | {
"line": 405,
"column": 53
} | {
"line": 406,
"column": 4
} | [
{
"pp": "case right\nR : Type u_1\nS : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : Extension R S\nx : P.infinitesimal.Ring\nhx : x ∈ P.infinitesimal.ker\n⊢ ∃ a, (map P.toInfinitesimal) a = mk ⟨x, hx⟩",
"ppTerm": "?right",
"assigned": true,
"usedConstants": [
"A... | [
"case right\nR : Type u_1\nS : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : Extension R S\nx : P.Ring\nhx : (Ideal.Quotient.mk (P.ker ^ 2)) x ∈ P.infinitesimal.ker\n⊢ ∃ a, (map P.toInfinitesimal) a = mk ⟨(Ideal.Quotient.mk (P.ker ^ 2)) x, hx⟩"
] | obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Etale.Kaehler | {
"line": 246,
"column": 4
} | {
"line": 246,
"column": 57
} | {
"line": 247,
"column": 4
} | [
{
"pp": "case left\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R T\ninst✝² : Algebra S T\ninst✝¹ : IsScalarTower R S T\nP : Extension R S\nQ : Extension R T\nf : P.Hom Q\nalg : Algebra P.Ring Q.Ring\nhalg : alge... | [
"case left\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R T\ninst✝² : Algebra S T\ninst✝¹ : IsScalarTower R S T\nP : Extension R S\nQ : Extension R T\nf : P.Hom Q\nalg : Algebra P.Ring Q.Ring\nhalg : algebraMap P.Rin... | apply (Extension.tensorCotangent f halg H₂).injective | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Unramified.LocalRing | {
"line": 299,
"column": 60
} | {
"line": 299,
"column": 89
} | {
"line": 299,
"column": 89
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\np : Ideal R\ninst✝⁷ : p.IsPrime\nq : Ideal S\ninst✝⁶ : q.IsPrime\nhq : p.primesOver S = {q}\ninst✝⁵ : Module.Finite R S\ninst✝⁴ : FaithfulSMul R S\ninst✝³ : q.LiesOver p\ninst✝² : Algebra.IsUnramifiedAt R q\nin... | [] | simpa using! Submodule.fg_bot | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.RingTheory.Unramified.LocalRing | {
"line": 299,
"column": 60
} | {
"line": 299,
"column": 89
} | {
"line": 299,
"column": 89
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\np : Ideal R\ninst✝⁷ : p.IsPrime\nq : Ideal S\ninst✝⁶ : q.IsPrime\nhq : p.primesOver S = {q}\ninst✝⁵ : Module.Finite R S\ninst✝⁴ : FaithfulSMul R S\ninst✝³ : q.LiesOver p\ninst✝² : Algebra.IsUnramifiedAt R q\nin... | [] | simpa using! Submodule.fg_bot | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Unramified.LocalRing | {
"line": 299,
"column": 60
} | {
"line": 299,
"column": 89
} | {
"line": 299,
"column": 89
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\np : Ideal R\ninst✝⁷ : p.IsPrime\nq : Ideal S\ninst✝⁶ : q.IsPrime\nhq : p.primesOver S = {q}\ninst✝⁵ : Module.Finite R S\ninst✝⁴ : FaithfulSMul R S\ninst✝³ : q.LiesOver p\ninst✝² : Algebra.IsUnramifiedAt R q\nin... | [] | simpa using! Submodule.fg_bot | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Countable | {
"line": 34,
"column": 2
} | {
"line": 34,
"column": 64
} | {
"line": 35,
"column": 2
} | [
{
"pp": "M : Type u_1\nR : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Countable R\ninst✝ : Module.Finite R M\n⊢ Countable M",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Submodule",
"Exists",
"Submodule.instTop",
"Exists... | [
"M : Type u_1\nR : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Countable R\ninst✝ : Module.Finite R M\nn : ℕ\ns : Fin n → M\nh : Submodule.span R (Set.range s) = ⊤\n⊢ Countable M"
] | obtain ⟨n, s, h⟩ := Module.Finite.exists_fin (R := R) (M := M) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.Enumerative.InclusionExclusion | {
"line": 131,
"column": 6
} | {
"line": 131,
"column": 91
} | {
"line": 132,
"column": 6
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nG : Type u_3\ninst✝¹ : AddCommGroup G\ninst✝ : DecidableEq α\ns : Finset ι\nS : ι → Finset α\nf : α → G\n⊢ (∑ t ∈ s.powerset, (-1) ^ #t • if ht : t.Nonempty then ∑ a ∈ t.inf' ht S, f a else ∑ a ∈ s.biUnion S, f a) =\n ∑ a ∈ s.biUnion S, (∏ i ∈ s, (1 - (↑(S i)).indicator 1... | [
"ι : Type u_1\nα : Type u_2\nG : Type u_3\ninst✝¹ : AddCommGroup G\ninst✝ : DecidableEq α\ns : Finset ι\nS : ι → Finset α\nf : α → G\n⊢ (∑ t ∈ s.powerset, (-1) ^ #t • if ht : t.Nonempty then ∑ a ∈ t.inf' ht S, f a else ∑ a ∈ s.biUnion S, f a) =\n ∑ y ∈ s.powerset, ∑ x ∈ s.biUnion S, ((-1) ^ #y * ((∏ i ∈ s \\ y, ... | simp only [Int.reduceNeg, prod_sub, sum_comm (s := s.biUnion S), sum_smul, mul_assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Module.Torsion.PrimaryComponent | {
"line": 97,
"column": 2
} | {
"line": 99,
"column": 77
} | {
"line": 100,
"column": 2
} | [
{
"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)\nx : M\n⊢ x ∈ Submodule.map (torsionBySet A M ↑J).subtype (primaryComponent (↥(torsio... | [
"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)\nx : M\n⊢ ((∃ n, ∀ a ∈ I ^ n, a • x = 0) ∧ ∀ a ∈ J, a • x = 0) ↔ x = 0"
] | simp only [mem_map, primaryComponent_mem, mem_torsionBySet_iff, SetLike.coe_sort_coe,
Subtype.forall, subtype_apply, Subtype.exists, SetLike.mk_smul_mk, mk_eq_zero, exists_and_left,
exists_prop, exists_eq_right_right, Submodule.map_bot, Submodule.mem_bot] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Order.Group.Pointwise.CompleteLattice | {
"line": 101,
"column": 26
} | {
"line": 101,
"column": 36
} | {
"line": 101,
"column": 36
} | [
{
"pp": "M : Type u_1\ninst✝³ : CompleteLattice M\ninst✝² : Group M\ninst✝¹ : MulLeftMono M\ninst✝ : MulRightMono M\ns : Set M\n⊢ sInf ((fun x ↦ x⁻¹) '' s) = (sSup s)⁻¹",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"iInf",
"DivInvOneMonoid.toInvOneClass",
... | [
"M : Type u_1\ninst✝³ : CompleteLattice M\ninst✝² : Group M\ninst✝¹ : MulLeftMono M\ninst✝ : MulRightMono M\ns : Set M\n⊢ ⨅ a ∈ s, a⁻¹ = (sSup s)⁻¹"
] | sInf_image | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.NNReal.Defs | {
"line": 903,
"column": 15
} | {
"line": 903,
"column": 27
} | {
"line": 904,
"column": 2
} | [
{
"pp": "⊢ ⟨|0|, ⋯⟩ = 0",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Subtype.mk.congr_simp",
"Real.instLE",
"Real",
"Real.lattice",
"Real.instZero",
"abs",
"congrArg",
"covariant_swap_... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.NNReal.Defs | {
"line": 904,
"column": 14
} | {
"line": 904,
"column": 26
} | {
"line": 905,
"column": 2
} | [
{
"pp": "⊢ ⟨|1|, ⋯⟩ = 1",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"MulOne.toOne",
"Subtype.mk.congr_simp",
"Real.instLE",
"Real",
"Real.lattice",
"Real.instZero",
"abs",
"congrArg",
... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.ENNReal.Real | {
"line": 70,
"column": 79
} | {
"line": 73,
"column": 34
} | {
"line": 75,
"column": 0
} | [
{
"pp": "a b : ℝ≥0∞\nh : a ≤ b\nht : b = ∞ → a = ∞\n⊢ a.toReal ≤ b.toReal",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Real.instLE",
"Real",
"mt",
"ENNReal.toReal_nonneg",
"Ne",
"LE.le",
"Or.casesOn",
"ENNReal.toReal",
"ENNReal.inst... | [] | by
rcases eq_or_ne a ∞ with rfl | ha
· exact toReal_nonneg
· exact toReal_mono (mt ht ha) h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.ENNReal.Operations | {
"line": 409,
"column": 40
} | {
"line": 409,
"column": 89
} | {
"line": 410,
"column": 2
} | [
{
"pp": "case inl\na b c : ℝ≥0∞\nh : 0 < b → b < a → c ≠ ∞\nhab : a ≤ b\n⊢ (a - b) * c = a * c - b * c",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"HMul.hMul",
"ENNReal.instOrderedSub",
"tsub_eq_zero_of_le",
"ENNReal.ins... | [] | simp [hab, mul_left_mono hab, tsub_eq_zero_of_le] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.ENNReal.Operations | {
"line": 409,
"column": 40
} | {
"line": 409,
"column": 89
} | {
"line": 410,
"column": 2
} | [
{
"pp": "case inl\na b c : ℝ≥0∞\nh : 0 < b → b < a → c ≠ ∞\nhab : a ≤ b\n⊢ (a - b) * c = a * c - b * c",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"HMul.hMul",
"ENNReal.instOrderedSub",
"tsub_eq_zero_of_le",
"ENNReal.ins... | [] | simp [hab, mul_left_mono hab, tsub_eq_zero_of_le] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.ENNReal.Operations | {
"line": 409,
"column": 40
} | {
"line": 409,
"column": 89
} | {
"line": 410,
"column": 2
} | [
{
"pp": "case inl\na b c : ℝ≥0∞\nh : 0 < b → b < a → c ≠ ∞\nhab : a ≤ b\n⊢ (a - b) * c = a * c - b * c",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"HMul.hMul",
"ENNReal.instOrderedSub",
"tsub_eq_zero_of_le",
"ENNReal.ins... | [] | simp [hab, mul_left_mono hab, tsub_eq_zero_of_le] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.EReal.Basic | {
"line": 466,
"column": 2
} | {
"line": 466,
"column": 51
} | {
"line": 467,
"column": 2
} | [
{
"pp": "x y : ℝ\n⊢ WithBot.some '' WithTop.some '' Ioo x y = Ioo ↑x ↑y",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"Real",
"WithBot.some",
"WithBot",
"WithBot.image_coe_Ioo",
"WithTop.instPreorder",
"con... | [
"x y : ℝ\n⊢ Ioo ↑↑x ↑↑y = Ioo ↑x ↑y"
] | rw [WithTop.image_coe_Ioo, WithBot.image_coe_Ioo] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.EReal.Basic | {
"line": 807,
"column": 72
} | {
"line": 807,
"column": 85
} | {
"line": 807,
"column": 86
} | [
{
"pp": "m n : ℕ\n⊢ ↑↑(m * n) = ↑↑m * ↑↑n",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"HMul.hMul",
"congrArg",
"EReal",
"Real.semiring",
"id",
"AddMonoidWithOne.toNatCast"... | [
"m n : ℕ\n⊢ ↑(↑m * ↑n) = ↑↑m * ↑↑n"
] | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.ENNReal.Inv | {
"line": 836,
"column": 2
} | {
"line": 844,
"column": 66
} | {
"line": 846,
"column": 0
} | [
{
"pp": "ι : Sort u_1\nf : ι → ℝ≥0∞\na : ℝ≥0∞\nhinfty : a = ∞ → ⨅ i, f i = 0 → ∃ i, f i = 0\nh₀ : a = 0 → Nonempty ι\n⊢ a * ⨅ i, f i = ⨅ i, a * f i",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Preorder.toLT",
"iInf",
"iInf_eq_top",
... | [] | obtain rfl | ha₀ := eq_or_ne a 0
· simp [h₀ rfl]
obtain rfl | ha := eq_or_ne a ∞
· obtain ⟨i, hi⟩ | hf := em (∃ i, f i = 0)
· rw [iInf_eq_bot.2, iInf_eq_bot.2, bot_eq_zero, mul_zero] <;>
exact fun _ _ ↦ ⟨i, by simpa [hi]⟩
· rw [top_mul (mt (hinfty rfl) hf), eq_comm, iInf_eq_top]
exact fun i ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.ENNReal.Inv | {
"line": 836,
"column": 2
} | {
"line": 844,
"column": 66
} | {
"line": 846,
"column": 0
} | [
{
"pp": "ι : Sort u_1\nf : ι → ℝ≥0∞\na : ℝ≥0∞\nhinfty : a = ∞ → ⨅ i, f i = 0 → ∃ i, f i = 0\nh₀ : a = 0 → Nonempty ι\n⊢ a * ⨅ i, f i = ⨅ i, a * f i",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Preorder.toLT",
"iInf",
"iInf_eq_top",
... | [] | obtain rfl | ha₀ := eq_or_ne a 0
· simp [h₀ rfl]
obtain rfl | ha := eq_or_ne a ∞
· obtain ⟨i, hi⟩ | hf := em (∃ i, f i = 0)
· rw [iInf_eq_bot.2, iInf_eq_bot.2, bot_eq_zero, mul_zero] <;>
exact fun _ _ ↦ ⟨i, by simpa [hi]⟩
· rw [top_mul (mt (hinfty rfl) hf), eq_comm, iInf_eq_top]
exact fun i ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.ENNReal.Inv | {
"line": 961,
"column": 22
} | {
"line": 961,
"column": 35
} | {
"line": 961,
"column": 36
} | [
{
"pp": "case coe.inr\na : ℝ≥0\nha : a ≠ 0\n⊢ (↑a⁻¹).toNNReal = (↑a).toNNReal⁻¹",
"ppTerm": "?coe.inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.ofNNReal",
"congrArg",
"ENNReal.toNNReal_coe",
"NNReal.instInv",
"id",
"NNReal",
"ENNReal.to... | [
"case coe.inr\na : ℝ≥0\nha : a ≠ 0\n⊢ a⁻¹ = (↑a).toNNReal⁻¹"
] | toNNReal_coe, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Metrizable.Basic | {
"line": 200,
"column": 2
} | {
"line": 203,
"column": 55
} | {
"line": 205,
"column": 0
} | [
{
"pp": "X : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : PseudoMetrizableSpace X\ns : Set X\nhs : IsSeparable s\n⊢ SeparableSpace ↑s",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Function.Injective.injOn",
"Membership.mem",
"Exists",... | [] | rcases hs.exists_countable_dense_subset with ⟨t, hts, htc, hst⟩
lift t to Set s using hts
refine ⟨⟨t, countable_of_injective_of_countable_image Subtype.coe_injective.injOn htc, ?_⟩⟩
rwa [IsInducing.subtypeVal.dense_iff, Subtype.forall] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Metrizable.Basic | {
"line": 200,
"column": 2
} | {
"line": 203,
"column": 55
} | {
"line": 205,
"column": 0
} | [
{
"pp": "X : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : PseudoMetrizableSpace X\ns : Set X\nhs : IsSeparable s\n⊢ SeparableSpace ↑s",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Function.Injective.injOn",
"Membership.mem",
"Exists",... | [] | rcases hs.exists_countable_dense_subset with ⟨t, hts, htc, hst⟩
lift t to Set s using hts
refine ⟨⟨t, countable_of_injective_of_countable_image Subtype.coe_injective.injOn htc, ?_⟩⟩
rwa [IsInducing.subtypeVal.dense_iff, Subtype.forall] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Pseudo.Lemmas | {
"line": 24,
"column": 37
} | {
"line": 25,
"column": 74
} | {
"line": 27,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ninst✝ : PseudoMetricSpace α\nx : ℝ\n⊢ 𝓝 x = ⨅ r, ⨅ (_ : r > 0), 𝓟 {b | |x - b| < r}",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Real",
"Preorder.toLT",
"iInf",
"Real.lattice",
"Real.instZero",
"Iff.of_eq",
... | [] | by
simp only [nhds_basis_ball.eq_biInf, ball, Real.dist_eq, abs_sub_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Order.DenselyOrdered | {
"line": 154,
"column": 2
} | {
"line": 160,
"column": 70
} | {
"line": 162,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b : α\n⊢ Ioc a b ⊆ closure[inst✝³] (interior (Ioc a b))",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Set.Ioc",
"False",
"Trans.trans",
... | [] | rcases eq_or_ne a b with (rfl | h)
· simp
· calc
Ioc a b ⊆ Icc a b := Ioc_subset_Icc_self
_ = closure (Ioo a b) := (closure_Ioo h).symm
_ ⊆ closure (interior (Ioc a b)) :=
closure_mono (interior_maximal Ioo_subset_Ioc_self isOpen_Ioo) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order.DenselyOrdered | {
"line": 154,
"column": 2
} | {
"line": 160,
"column": 70
} | {
"line": 162,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b : α\n⊢ Ioc a b ⊆ closure[inst✝³] (interior (Ioc a b))",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Set.Ioc",
"False",
"Trans.trans",
... | [] | rcases eq_or_ne a b with (rfl | h)
· simp
· calc
Ioc a b ⊆ Icc a b := Ioc_subset_Icc_self
_ = closure (Ioo a b) := (closure_Ioo h).symm
_ ⊆ closure (interior (Ioc a b)) :=
closure_mono (interior_maximal Ioo_subset_Ioc_self isOpen_Ioo) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Order.IsLUB | {
"line": 73,
"column": 17
} | {
"line": 75,
"column": 52
} | {
"line": 77,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\ns : Set α\na : α\nhsa : a ∈ upperBounds s\nhsf : a ∈ closure[inst✝²] s\n⊢ IsLUB s a",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Filter.mem_principal_self",
"congrArg",
... | [] | by
rw [mem_closure_iff_clusterPt, ClusterPt, inf_comm] at hsf
exact isLUB_of_mem_nhds hsa (mem_principal_self s) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.Pseudo.Pi | {
"line": 35,
"column": 2
} | {
"line": 38,
"column": 39
} | {
"line": 39,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : PseudoMetricSpace α\nX : β → Type u_3\ninst✝¹ : Fintype β\ninst✝ : (b : β) → PseudoMetricSpace (X b)\n⊢ PseudoMetricSpace ((b : β) → X b)",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"NNDist.nndist",
"Finset.univ",
"PseudoEM... | [
"α : Type u_1\nβ : Type u_2\ninst✝² : PseudoMetricSpace α\nX : β → Type u_3\ninst✝¹ : Fintype β\ninst✝ : (b : β) → PseudoMetricSpace (X b)\ni : PseudoMetricSpace ((b : β) → X b) :=\n PseudoEMetricSpace.toPseudoMetricSpaceOfDist (fun f g ↦ ↑(Finset.univ.sup fun b ↦ nndist (f b) (g b))) ⋯ ⋯\n⊢ PseudoMetricSpace ((b ... | let i := PseudoEMetricSpace.toPseudoMetricSpaceOfDist
(fun f g : ∀ b, X b => ((sup univ fun b => nndist (f b) (g b) : ℝ≥0) : ℝ))
(fun f g => NNReal.zero_le_coe)
(fun f g => by simp [edist_pi_def]) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Topology.MetricSpace.Pseudo.Pi | {
"line": 69,
"column": 2
} | {
"line": 70,
"column": 27
} | {
"line": 72,
"column": 0
} | [
{
"pp": "β : Type u_2\nX : β → Type u_3\ninst✝¹ : Fintype β\ninst✝ : (b : β) → PseudoMetricSpace (X b)\nf g : (b : β) → X b\nr : ℝ\nhr : 0 < r\n⊢ dist f g < r ↔ ∀ (b : β), dist (f b) (g b) < r",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Real.instLE",
"Real",
"pseudoM... | [] | lift r to ℝ≥0 using hr.le
exact nndist_pi_lt_iff hr | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Pseudo.Pi | {
"line": 69,
"column": 2
} | {
"line": 70,
"column": 27
} | {
"line": 72,
"column": 0
} | [
{
"pp": "β : Type u_2\nX : β → Type u_3\ninst✝¹ : Fintype β\ninst✝ : (b : β) → PseudoMetricSpace (X b)\nf g : (b : β) → X b\nr : ℝ\nhr : 0 < r\n⊢ dist f g < r ↔ ∀ (b : β), dist (f b) (g b) < r",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Real.instLE",
"Real",
"pseudoM... | [] | lift r to ℝ≥0 using hr.le
exact nndist_pi_lt_iff hr | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Order.Monotone | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 33
} | {
"line": 85,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : TopologicalSpace α\ninst✝² : OrderTopology α\ninst✝¹ : LinearOrder β\ns : Set α\nf : α → β\ninst✝ : SecondCountableTopology α\nhf : MonotoneOn f s\na✝ : Nontrivial α\nt : Set β := {c | ∃ x y, x ∈ s ∧ y ∈ s ∧ x < y ∧ f x = c ∧ f y = c}\nx y : ... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : TopologicalSpace α\ninst✝² : OrderTopology α\ninst✝¹ : LinearOrder β\ns : Set α\nf : α → β\ninst✝ : SecondCountableTopology α\nhf : MonotoneOn f s\na✝ : Nontrivial α\nt : Set β := {c | ∃ x y, x ∈ s ∧ y ∈ s ∧ x < y ∧ f x = c ∧ f y = c}\nx y : β → α\nhxs :... | rw [hfx _ hd, hfy _ hc] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Order.IntermediateValue | {
"line": 73,
"column": 2
} | {
"line": 76,
"column": 32
} | {
"line": 78,
"column": 0
} | [
{
"pp": "X : Type u\nα : Type v\ninst✝⁴ : TopologicalSpace X\ninst✝³ : LinearOrder α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderClosedTopology α\ninst✝ : PreconnectedSpace X\na b : X\nf g : X → α\nhf : Continuous[inst✝⁴, inst✝²] f\nhg : Continuous[inst✝⁴, inst✝²] g\nha : f a ≤ g a\nhb : g b ≤ f b\n⊢ ∃ x, f x =... | [] | obtain ⟨x, _, hfg, hgf⟩ : (univ ∩ { x | f x ≤ g x ∧ g x ≤ f x }).Nonempty :=
isPreconnected_closed_iff.1 PreconnectedSpace.isPreconnected_univ _ _ (isClosed_le hf hg)
(isClosed_le hg hf) (fun _ _ => le_total _ _) ⟨a, trivial, ha⟩ ⟨b, trivial, hb⟩
exact ⟨x, le_antisymm hfg hgf⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order.IntermediateValue | {
"line": 73,
"column": 2
} | {
"line": 76,
"column": 32
} | {
"line": 78,
"column": 0
} | [
{
"pp": "X : Type u\nα : Type v\ninst✝⁴ : TopologicalSpace X\ninst✝³ : LinearOrder α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderClosedTopology α\ninst✝ : PreconnectedSpace X\na b : X\nf g : X → α\nhf : Continuous[inst✝⁴, inst✝²] f\nhg : Continuous[inst✝⁴, inst✝²] g\nha : f a ≤ g a\nhb : g b ≤ f b\n⊢ ∃ x, f x =... | [] | obtain ⟨x, _, hfg, hgf⟩ : (univ ∩ { x | f x ≤ g x ∧ g x ≤ f x }).Nonempty :=
isPreconnected_closed_iff.1 PreconnectedSpace.isPreconnected_univ _ _ (isClosed_le hf hg)
(isClosed_le hg hf) (fun _ _ => le_total _ _) ⟨a, trivial, ha⟩ ⟨b, trivial, hb⟩
exact ⟨x, le_antisymm hfg hgf⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Order.IntermediateValue | {
"line": 272,
"column": 4
} | {
"line": 273,
"column": 37
} | {
"line": 274,
"column": 2
} | [
{
"pp": "case pos\nα : Type u\ninst✝² : TopologicalSpace α\ninst✝¹ : ConditionallyCompleteLinearOrder α\ninst✝ : OrderTopology α\ns : Set α\nhs : IsPreconnected s\nhne : s.Nonempty\nhs' : IsConnected s\nhb : BddBelow s\nha : BddAbove s\n⊢ {Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo... | [] | simp only [insert_subset_iff, mem_insert_iff, mem_singleton_iff, true_or, or_true,
singleton_subset_iff, and_self] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 404,
"column": 2
} | {
"line": 406,
"column": 99
} | {
"line": 408,
"column": 0
} | [
{
"pp": "E : Type u_5\nF : Type u_6\ninst✝¹ : SeminormedGroup E\ninst✝ : SeminormedGroup F\nx : E\ny : F\n⊢ ‖x‖ₑ = ‖y‖ₑ ↔ ‖x‖ = ‖y‖",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"ENNReal.ofNNReal",
"norm_nonneg'",
"ENNRea... | [] | simp only [← ofReal_norm']
refine ⟨fun h ↦ ?_, fun h ↦ by congr⟩
exact (Real.toNNReal_eq_toNNReal_iff (norm_nonneg' _) (norm_nonneg' _)).mp (ENNReal.coe_inj.mp h) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 404,
"column": 2
} | {
"line": 406,
"column": 99
} | {
"line": 408,
"column": 0
} | [
{
"pp": "E : Type u_5\nF : Type u_6\ninst✝¹ : SeminormedGroup E\ninst✝ : SeminormedGroup F\nx : E\ny : F\n⊢ ‖x‖ₑ = ‖y‖ₑ ↔ ‖x‖ = ‖y‖",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"ENNReal.ofNNReal",
"norm_nonneg'",
"ENNRea... | [] | simp only [← ofReal_norm']
refine ⟨fun h ↦ ?_, fun h ↦ by congr⟩
exact (Real.toNNReal_eq_toNNReal_iff (norm_nonneg' _) (norm_nonneg' _)).mp (ENNReal.coe_inj.mp h) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 449,
"column": 62
} | {
"line": 450,
"column": 98
} | {
"line": 452,
"column": 0
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\na : E\nn : ℕ\n⊢ ‖a ^ n‖₊ ≤ ↑n * ‖a‖₊",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.instLE",
"Real",
"HMul.hMul",
"norm_pow_le_mul_norm",
... | [] | by
simpa only [← NNReal.coe_le_coe, NNReal.coe_mul, NNReal.coe_natCast] using! norm_pow_le_mul_norm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.ProperSpace | {
"line": 124,
"column": 26
} | {
"line": 127,
"column": 68
} | {
"line": 129,
"column": 0
} | [
{
"pp": "α✝ : Type u\nβ✝ : Type v\nX : Type u_1\nι : Type u_2\ninst✝⁴ : PseudoMetricSpace α✝\nα : Type u_3\nβ : Type u_4\ninst✝³ : PseudoMetricSpace α\ninst✝² : PseudoMetricSpace β\ninst✝¹ : ProperSpace α\ninst✝ : ProperSpace β\n⊢ ∀ (x : α × β) (r : ℝ), IsCompact (closedBall x r)",
"ppTerm": "?m.6",
"as... | [] | by
rintro ⟨x, y⟩ r
rw [← closedBall_prod_same x y]
exact (isCompact_closedBall x r).prod (isCompact_closedBall y r) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 1003,
"column": 76
} | {
"line": 1003,
"column": 86
} | {
"line": 1003,
"column": 86
} | [
{
"pp": "E : Type u_5\ninst✝ : NormedGroup E\na b : E\n⊢ a / b = 1 ↔ a = b",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHDiv",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"congrArg",
"Group.toDivisionMonoid",
"DivisionMo... | [
"E : Type u_5\ninst✝ : NormedGroup E\na b : E\n⊢ a = b ↔ a = b"
] | div_eq_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 770,
"column": 8
} | {
"line": 770,
"column": 81
} | {
"line": 770,
"column": 82
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI J I' J' : FractionalIdeal R⁰ K\nH : I * J' = I' * J\nh : J ≤ I\nh' : J' ≤ I'\nhJ' : J' ≠ 0\nhI : I ≠ 0\nthis : J' ⊓ span... | [
"case neg\nR : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI J I' J' : FractionalIdeal R⁰ K\nH : I * J' = I' * J\nh : J ≤ I\nh' : J' ≤ I'\nhJ' : J' ≠ 0\nhI : I ≠ 0\nthis : (spanSingleton R⁰ (I'... | ← inv_mul_eq_iff_eq_mul₀ (by simpa [spanSingleton_eq_zero_iff] using H'), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order.IntermediateValue | {
"line": 695,
"column": 2
} | {
"line": 697,
"column": 7
} | {
"line": 699,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : ConditionallyCompleteLinearOrder α\ninst✝⁴ : OrderTopology α\ninst✝³ : DenselyOrdered α\nδ : Type u_1\ninst✝² : LinearOrder δ\ninst✝¹ : TopologicalSpace δ\ninst✝ : OrderClosedTopology δ\na : α\nf : α → δ\nhf : ContinuousOn f (Iic a)\nhbot : Tendsto f at... | [] | intro y hy
have := intermediate_value_Iic' hf hbot (mem_Ici_of_Ioi hy)
grind | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order.IntermediateValue | {
"line": 695,
"column": 2
} | {
"line": 697,
"column": 7
} | {
"line": 699,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : ConditionallyCompleteLinearOrder α\ninst✝⁴ : OrderTopology α\ninst✝³ : DenselyOrdered α\nδ : Type u_1\ninst✝² : LinearOrder δ\ninst✝¹ : TopologicalSpace δ\ninst✝ : OrderClosedTopology δ\na : α\nf : α → δ\nhf : ContinuousOn f (Iic a)\nhbot : Tendsto f at... | [] | intro y hy
have := intermediate_value_Iic' hf hbot (mem_Ici_of_Ioi hy)
grind | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Bounded | {
"line": 240,
"column": 2
} | {
"line": 240,
"column": 38
} | {
"line": 242,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝ : PseudoMetricSpace α\nf : β → α\nhf : Tendsto (Prod.map f f) (Filter.cofinite ×ˢ Filter.cofinite) (𝓤 α)\ns : Set β\nhsf : s.Finite\nhs1 : ∀ x ∈ sᶜ ×ˢ sᶜ, Prod.map f f x ∈ {p | dist p.1 p.2 < 1}\nx : β\nhx : x ∈ sᶜ\ny : β\nhy : y ∈ sᶜ\n⊢ dist (f x) (f y) ≤ 1",
"ppTerm... | [] | exact le_of_lt (hs1 (x, y) ⟨hx, hy⟩) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.MetricSpace.Bounded | {
"line": 416,
"column": 2
} | {
"line": 416,
"column": 36
} | {
"line": 418,
"column": 0
} | [
{
"pp": "α : Type u\ns : Set α\ninst✝ : PseudoMetricSpace α\nhs : s.Subsingleton\n⊢ diam s = 0",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Real",
"Real.instZero",
"congrArg",
"Metric.ediam_subsingleton",
"Metric.ediam",
"ENNReal.toReal",
"True"... | [] | simp [diam, ediam_subsingleton hs] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.MetricSpace.Bounded | {
"line": 416,
"column": 2
} | {
"line": 416,
"column": 36
} | {
"line": 418,
"column": 0
} | [
{
"pp": "α : Type u\ns : Set α\ninst✝ : PseudoMetricSpace α\nhs : s.Subsingleton\n⊢ diam s = 0",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Real",
"Real.instZero",
"congrArg",
"Metric.ediam_subsingleton",
"Metric.ediam",
"ENNReal.toReal",
"True"... | [] | simp [diam, ediam_subsingleton hs] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Bounded | {
"line": 416,
"column": 2
} | {
"line": 416,
"column": 36
} | {
"line": 418,
"column": 0
} | [
{
"pp": "α : Type u\ns : Set α\ninst✝ : PseudoMetricSpace α\nhs : s.Subsingleton\n⊢ diam s = 0",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Real",
"Real.instZero",
"congrArg",
"Metric.ediam_subsingleton",
"Metric.ediam",
"ENNReal.toReal",
"True"... | [] | simp [diam, ediam_subsingleton hs] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.AtTopBot.Archimedean | {
"line": 198,
"column": 2
} | {
"line": 208,
"column": 28
} | {
"line": 210,
"column": 0
} | [
{
"pp": "α : Type u_1\nR : Type u_2\nl : Filter α\nf : α → R\nr : R\ninst✝³ : Semiring R\ninst✝² : LinearOrder R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : Archimedean R\nhr : 0 < r\nhf : Tendsto f l atTop\n⊢ Tendsto (fun x ↦ f x * r) l atTop",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
... | [] | refine tendsto_atTop.2 fun b => ?_
obtain ⟨n : ℕ, hn : 1 ≤ n • r⟩ := Archimedean.arch 1 hr
have hn' : 1 ≤ (n : R) * r := by rwa [nsmul_eq_mul] at hn
filter_upwards [tendsto_atTop.1 hf (max b 0 * n)] with x hx
calc
b ≤ max b 0 * 1 := by
{ rw [mul_one]
exact le_max_left _ _ }
_ ≤ max b 0 * (n * ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.AtTopBot.Archimedean | {
"line": 198,
"column": 2
} | {
"line": 208,
"column": 28
} | {
"line": 210,
"column": 0
} | [
{
"pp": "α : Type u_1\nR : Type u_2\nl : Filter α\nf : α → R\nr : R\ninst✝³ : Semiring R\ninst✝² : LinearOrder R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : Archimedean R\nhr : 0 < r\nhf : Tendsto f l atTop\n⊢ Tendsto (fun x ↦ f x * r) l atTop",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
... | [] | refine tendsto_atTop.2 fun b => ?_
obtain ⟨n : ℕ, hn : 1 ≤ n • r⟩ := Archimedean.arch 1 hr
have hn' : 1 ≤ (n : R) * r := by rwa [nsmul_eq_mul] at hn
filter_upwards [tendsto_atTop.1 hf (max b 0 * n)] with x hx
calc
b ≤ max b 0 * 1 := by
{ rw [mul_one]
exact le_max_left _ _ }
_ ≤ max b 0 * (n * ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Instances.Nat | {
"line": 68,
"column": 30
} | {
"line": 68,
"column": 96
} | {
"line": 70,
"column": 0
} | [
{
"pp": "⊢ (cocompact ℕ).NeBot",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"congrArg",
"_private.Mathlib.Topology.Instances.Nat.0.Nat.instNoncompactSpace._simp_1",
"Filter.NeBot",
"instTopologicalSpaceNat",
"Filter.cocompact",
"instDiscreteTopologyNat... | [] | by simp only [Filter.cocompact_eq_cofinite, Filter.cofinite_neBot] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.LiminfLimsup | {
"line": 432,
"column": 72
} | {
"line": 433,
"column": 61
} | {
"line": 435,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : CompleteLattice α\nu : β → α\n⊢ liminf u ⊤ = ⨅ i, u i",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Filter.limsInf",
"iInf",
"Filter.liminf",
"congrArg",
"Filter.map",
"Filter.map_top",
... | [] | by
rw [liminf, map_top, limsInf_principal_eq_sInf, sInf_range] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.EReal.Operations | {
"line": 415,
"column": 2
} | {
"line": 415,
"column": 22
} | {
"line": 416,
"column": 2
} | [
{
"pp": "case coe.coe\nx y : ℝ\nhy : 0 ≤ ↑y\n⊢ (↑x - ↑y).toENNReal = (↑x).toENNReal - (↑y).toENNReal",
"ppTerm": "?coe.coe",
"assigned": true,
"usedConstants": [
"Real.instLE",
"Real",
"EReal.toENNReal",
"HSub.hSub",
"EReal",
"LE.le",
"SubNegZeroMonoid.toSub... | [
"case pos\nx y : ℝ\nhy : 0 ≤ ↑y\nhxy : x ≤ y\n⊢ (↑x - ↑y).toENNReal = (↑x).toENNReal - (↑y).toENNReal",
"case neg\nx y : ℝ\nhy : 0 ≤ ↑y\nhxy : ¬x ≤ y\n⊢ (↑x - ↑y).toENNReal = (↑x).toENNReal - (↑y).toENNReal"
] | by_cases hxy : x ≤ y | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Data.EReal.Operations | {
"line": 592,
"column": 2
} | {
"line": 592,
"column": 21
} | {
"line": 593,
"column": 2
} | [
{
"pp": "x : EReal\nh : 0 < x\n⊢ ⊤ * x = ⊤",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"EReal",
"instTopEReal",
"id",
"EReal.mul_comm",
"Top.top",
"Eq",
"EReal.instMul",
"instHMul"
... | [
"x : EReal\nh : 0 < x\n⊢ x * ⊤ = ⊤"
] | rw [EReal.mul_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.EReal.Operations | {
"line": 596,
"column": 2
} | {
"line": 596,
"column": 21
} | {
"line": 597,
"column": 2
} | [
{
"pp": "x : EReal\nh : x < 0\n⊢ ⊤ * x = ⊥",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"EReal",
"instTopEReal",
"id",
"Bot.bot",
"EReal.mul_comm",
"Top.top",
"Eq",
"instBotEReal",
... | [
"x : EReal\nh : x < 0\n⊢ x * ⊤ = ⊥"
] | rw [EReal.mul_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
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