module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.RingHom.Flat | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 32
} | {
"line": 158,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹⁴ : CommRing R\ninst✝¹³ : CommRing S\ninst✝¹² : Algebra R S\nA : Type u_4\nB : Type u_5\nC : Type u_6\nD : Type u_7\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : Algebra S A\ninst✝⁸ : IsScalarTower R S A\ninst✝⁷ : CommRing B\ninst✝⁶ : Algebra R B\ninst✝⁵ : Com... | [
"case refine_1\nR : Type u_1\nS : Type u_2\ninst✝¹⁴ : CommRing R\ninst✝¹³ : CommRing S\ninst✝¹² : Algebra R S\nA : Type u_4\nB : Type u_5\nC : Type u_6\nD : Type u_7\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : Algebra S A\ninst✝⁸ : IsScalarTower R S A\ninst✝⁷ : CommRing B\ninst✝⁶ : Algebra R B\ninst✝⁵ : ... | refine RingHom.Flat.comp ?_ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Derivation.ToSquareZero | {
"line": 117,
"column": 97
} | {
"line": 122,
"column": 72
} | {
"line": 124,
"column": 0
} | [
{
"pp": "R : Type u\nA : Type v\nB : Type w\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\nI : Ideal B\ninst✝¹ : Algebra A B\nhI : I ^ 2 = ⊥\ninst✝ : IsScalarTower R A B\n⊢ Derivation R A ↥I ≃ { f // (Ideal.Quotient.mkₐ R I).comp f = IsScalarT... | [] | by
refine ⟨fun d => ⟨liftOfDerivationToSquareZero I hI d, ?_⟩, fun f =>
(derivationToSquareZeroOfLift I hI f.1 f.2 :), ?_, ?_⟩
· ext x; exact liftOfDerivationToSquareZero_mk_apply I hI d x
· intro d; ext x; exact add_sub_cancel_right (d x : B) (algebraMap A B x)
· rintro ⟨f, hf⟩; ext x; exact sub_add_cancel... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.RingHom.Flat | {
"line": 268,
"column": 33
} | {
"line": 268,
"column": 73
} | {
"line": 268,
"column": 73
} | [
{
"pp": "R S T A : Type u\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : CommRing T\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra R T\ninst✝⁴ : Algebra S T\ninst✝³ : IsScalarTower R S T\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\ninst✝ : IsScalarTower R S A\nh : CommRingCat.flat (CommRingC... | [] | simp [IsScalarTower.algebraMap_eq R S T] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.RingHom.Flat | {
"line": 268,
"column": 33
} | {
"line": 268,
"column": 73
} | {
"line": 268,
"column": 73
} | [
{
"pp": "R S T A : Type u\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : CommRing T\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra R T\ninst✝⁴ : Algebra S T\ninst✝³ : IsScalarTower R S T\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\ninst✝ : IsScalarTower R S A\nh : CommRingCat.flat (CommRingC... | [] | simp [IsScalarTower.algebraMap_eq R S T] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.RingHom.Flat | {
"line": 268,
"column": 33
} | {
"line": 268,
"column": 73
} | {
"line": 268,
"column": 73
} | [
{
"pp": "R S T A : Type u\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : CommRing T\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra R T\ninst✝⁴ : Algebra S T\ninst✝³ : IsScalarTower R S T\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\ninst✝ : IsScalarTower R S A\nh : CommRingCat.flat (CommRingC... | [] | simp [IsScalarTower.algebraMap_eq R S T] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 164,
"column": 50
} | {
"line": 164,
"column": 68
} | {
"line": 165,
"column": 6
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nt : Set (PrimeSpectrum R)\nI : Set R\nhI : closure t = zeroLocus I\n⊢ I ⊆ ↑(vanishingIdeal (zeroLocus ↑(vanishingIdeal t))) ∧ t ⊆ zeroLocus ↑(vanishingIdeal t)",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toMod... | [
"R : Type u\ninst✝ : CommSemiring R\nt : Set (PrimeSpectrum R)\nI : Set R\nhI : closure t = zeroLocus I\n⊢ I ⊆ ↑(vanishingIdeal t) ∧ t ⊆ zeroLocus ↑(vanishingIdeal t)"
] | (gc R).u_l_u_eq_u, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.Factorization.LCM | {
"line": 61,
"column": 12
} | {
"line": 61,
"column": 75
} | {
"line": 62,
"column": 2
} | [
{
"pp": "case pos\na b p : ℕ\nhp : p ∈ (a.lcm b).factorization.support\nq : ℕ\nhq : q ∈ (a.lcm b).factorization.support\nh : b.factorization p ≤ a.factorization p\nh' : b.factorization q ≤ a.factorization q\n⊢ (p ^ (a.lcm b).factorization p).Coprime 1",
"ppTerm": "?pos✝",
"assigned": true,
"usedCons... | [] | simp only [coprime_one_right_eq_true, coprime_one_left_eq_true] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Nat.Factorization.LCM | {
"line": 61,
"column": 12
} | {
"line": 61,
"column": 75
} | {
"line": 62,
"column": 2
} | [
{
"pp": "case pos\na b p : ℕ\nhp : p ∈ (a.lcm b).factorization.support\nq : ℕ\nhq : q ∈ (a.lcm b).factorization.support\nh : b.factorization p ≤ a.factorization p\nh' : b.factorization q ≤ a.factorization q\n⊢ (p ^ (a.lcm b).factorization p).Coprime 1",
"ppTerm": "?pos✝",
"assigned": true,
"usedCons... | [] | simp only [coprime_one_right_eq_true, coprime_one_left_eq_true] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Factorization.LCM | {
"line": 61,
"column": 12
} | {
"line": 61,
"column": 75
} | {
"line": 62,
"column": 2
} | [
{
"pp": "case pos\na b p : ℕ\nhp : p ∈ (a.lcm b).factorization.support\nq : ℕ\nhq : q ∈ (a.lcm b).factorization.support\nh : b.factorization p ≤ a.factorization p\nh' : b.factorization q ≤ a.factorization q\n⊢ (p ^ (a.lcm b).factorization p).Coprime 1",
"ppTerm": "?pos✝",
"assigned": true,
"usedCons... | [] | simp only [coprime_one_right_eq_true, coprime_one_left_eq_true] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.Factorization.LCM | {
"line": 61,
"column": 12
} | {
"line": 61,
"column": 75
} | {
"line": 62,
"column": 2
} | [
{
"pp": "case neg\na b p : ℕ\nhp : p ∈ (a.lcm b).factorization.support\nq : ℕ\nhq : q ∈ (a.lcm b).factorization.support\nh : b.factorization p ≤ a.factorization p\nh' : ¬b.factorization q ≤ a.factorization q\n⊢ (p ^ (a.lcm b).factorization p).Coprime (q ^ (a.lcm b).factorization q)",
"ppTerm": "?neg✝",
... | [
"case neg\na b p : ℕ\nhp : p ∈ (a.lcm b).factorization.support\nq : ℕ\nhq : q ∈ (a.lcm b).factorization.support\nh : b.factorization p ≤ a.factorization p\nh' : ¬b.factorization q ≤ a.factorization q\n⊢ (p ^ (a.lcm b).factorization p).Coprime (q ^ (a.lcm b).factorization q)"
] | simp only [coprime_one_right_eq_true, coprime_one_left_eq_true] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Nat.Factorization.LCM | {
"line": 61,
"column": 12
} | {
"line": 61,
"column": 75
} | {
"line": 62,
"column": 2
} | [
{
"pp": "case neg\na b p : ℕ\nhp : p ∈ (a.lcm b).factorization.support\nq : ℕ\nhq : q ∈ (a.lcm b).factorization.support\nh : b.factorization p ≤ a.factorization p\nh' : ¬b.factorization q ≤ a.factorization q\n⊢ (p ^ (a.lcm b).factorization p).Coprime (q ^ (a.lcm b).factorization q)",
"ppTerm": "?neg✝",
... | [
"case neg\na b p : ℕ\nhp : p ∈ (a.lcm b).factorization.support\nq : ℕ\nhq : q ∈ (a.lcm b).factorization.support\nh : b.factorization p ≤ a.factorization p\nh' : ¬b.factorization q ≤ a.factorization q\n⊢ (p ^ (a.lcm b).factorization p).Coprime (q ^ (a.lcm b).factorization q)"
] | simp only [coprime_one_right_eq_true, coprime_one_left_eq_true] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Factorization.LCM | {
"line": 61,
"column": 12
} | {
"line": 61,
"column": 75
} | {
"line": 62,
"column": 2
} | [
{
"pp": "case neg\na b p : ℕ\nhp : p ∈ (a.lcm b).factorization.support\nq : ℕ\nhq : q ∈ (a.lcm b).factorization.support\nh : b.factorization p ≤ a.factorization p\nh' : ¬b.factorization q ≤ a.factorization q\n⊢ (p ^ (a.lcm b).factorization p).Coprime (q ^ (a.lcm b).factorization q)",
"ppTerm": "?neg✝",
... | [
"case neg\na b p : ℕ\nhp : p ∈ (a.lcm b).factorization.support\nq : ℕ\nhq : q ∈ (a.lcm b).factorization.support\nh : b.factorization p ≤ a.factorization p\nh' : ¬b.factorization q ≤ a.factorization q\n⊢ (p ^ (a.lcm b).factorization p).Coprime (q ^ (a.lcm b).factorization q)"
] | simp only [coprime_one_right_eq_true, coprime_one_left_eq_true] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.Factorization.LCM | {
"line": 61,
"column": 12
} | {
"line": 61,
"column": 75
} | {
"line": 62,
"column": 2
} | [
{
"pp": "case pos\na b p : ℕ\nhp : p ∈ (a.lcm b).factorization.support\nq : ℕ\nhq : q ∈ (a.lcm b).factorization.support\nh : ¬b.factorization p ≤ a.factorization p\nh✝ : b.factorization q ≤ a.factorization q\n⊢ Coprime 1 1",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Nat.Coprime"... | [] | simp only [coprime_one_right_eq_true, coprime_one_left_eq_true] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Nat.Factorization.LCM | {
"line": 61,
"column": 12
} | {
"line": 61,
"column": 75
} | {
"line": 62,
"column": 2
} | [
{
"pp": "case pos\na b p : ℕ\nhp : p ∈ (a.lcm b).factorization.support\nq : ℕ\nhq : q ∈ (a.lcm b).factorization.support\nh : ¬b.factorization p ≤ a.factorization p\nh✝ : b.factorization q ≤ a.factorization q\n⊢ Coprime 1 1",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Nat.Coprime"... | [] | simp only [coprime_one_right_eq_true, coprime_one_left_eq_true] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Factorization.LCM | {
"line": 61,
"column": 12
} | {
"line": 61,
"column": 75
} | {
"line": 62,
"column": 2
} | [
{
"pp": "case pos\na b p : ℕ\nhp : p ∈ (a.lcm b).factorization.support\nq : ℕ\nhq : q ∈ (a.lcm b).factorization.support\nh : ¬b.factorization p ≤ a.factorization p\nh✝ : b.factorization q ≤ a.factorization q\n⊢ Coprime 1 1",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Nat.Coprime"... | [] | simp only [coprime_one_right_eq_true, coprime_one_left_eq_true] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.Factorization.LCM | {
"line": 61,
"column": 12
} | {
"line": 61,
"column": 75
} | {
"line": 62,
"column": 2
} | [
{
"pp": "case neg\na b p : ℕ\nhp : p ∈ (a.lcm b).factorization.support\nq : ℕ\nhq : q ∈ (a.lcm b).factorization.support\nh : ¬b.factorization p ≤ a.factorization p\nh✝ : ¬b.factorization q ≤ a.factorization q\n⊢ Coprime 1 (q ^ (a.lcm b).factorization q)",
"ppTerm": "?neg✝",
"assigned": true,
"usedCo... | [] | simp only [coprime_one_right_eq_true, coprime_one_left_eq_true] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Nat.Factorization.LCM | {
"line": 61,
"column": 12
} | {
"line": 61,
"column": 75
} | {
"line": 62,
"column": 2
} | [
{
"pp": "case neg\na b p : ℕ\nhp : p ∈ (a.lcm b).factorization.support\nq : ℕ\nhq : q ∈ (a.lcm b).factorization.support\nh : ¬b.factorization p ≤ a.factorization p\nh✝ : ¬b.factorization q ≤ a.factorization q\n⊢ Coprime 1 (q ^ (a.lcm b).factorization q)",
"ppTerm": "?neg✝",
"assigned": true,
"usedCo... | [] | simp only [coprime_one_right_eq_true, coprime_one_left_eq_true] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Factorization.LCM | {
"line": 61,
"column": 12
} | {
"line": 61,
"column": 75
} | {
"line": 62,
"column": 2
} | [
{
"pp": "case neg\na b p : ℕ\nhp : p ∈ (a.lcm b).factorization.support\nq : ℕ\nhq : q ∈ (a.lcm b).factorization.support\nh : ¬b.factorization p ≤ a.factorization p\nh✝ : ¬b.factorization q ≤ a.factorization q\n⊢ Coprime 1 (q ^ (a.lcm b).factorization q)",
"ppTerm": "?neg✝",
"assigned": true,
"usedCo... | [] | simp only [coprime_one_right_eq_true, coprime_one_left_eq_true] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 192,
"column": 2
} | {
"line": 192,
"column": 63
} | {
"line": 193,
"column": 2
} | [
{
"pp": "case mp\nR : Type u\ninst✝ : CommSemiring R\nI J : Ideal R\n⊢ zeroLocus ↑I = zeroLocus ↑J → I.radical = J.radical",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"PrimeSpectrum.zeroLocus",
"PrimeSpectrum.vanishingIdeal",
"... | [
"case mpr\nR : Type u\ninst✝ : CommSemiring R\nI J : Ideal R\n⊢ I.radical = J.radical → zeroLocus ↑I = zeroLocus ↑J"
] | · intro h; simp_rw [← vanishingIdeal_zeroLocus_eq_radical, h] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 352,
"column": 2
} | {
"line": 352,
"column": 70
} | {
"line": 353,
"column": 2
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝³ : CommSemiring R\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nM : Submonoid R\ninst✝ : IsLocalization M S\np q : PrimeSpectrum S\nh :\n Ideal.map (algebraMap R S) (Ideal.comap (algebraMap R S) p.asIdeal) =\n Ideal.map (algebraMap R S) (Ideal.comap (algebraMap R S) ... | [
"R : Type u\nS : Type v\ninst✝³ : CommSemiring R\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nM : Submonoid R\ninst✝ : IsLocalization M S\np q : PrimeSpectrum S\nh : p.asIdeal = q.asIdeal\n⊢ p = q"
] | rw [IsLocalization.map_under M S, IsLocalization.map_under M S] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 957,
"column": 2
} | {
"line": 969,
"column": 31
} | {
"line": 971,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : R →+* S\n⊢ DenseRange (comap f) ↔ ∀ (I : Ideal R) (h : I ∈ minimalPrimes R), { asIdeal := I, isPrime := ⋯ } ∈ Set.range (comap f)",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Iff.mpr",
... | [] | constructor
· intro H I hI
have : I ∈ (RingHom.ker f).minimalPrimes := by
rw [denseRange_comap_iff_ker_le_nilRadical] at H
simp only [Set.mem_setOf, Ideal.IsMinimalPrime] at hI ⊢
convert! hI using 2 with p
exact ⟨fun h ↦ ⟨h.1, bot_le⟩, fun h ↦ ⟨h.1, H.trans (h.1.radical_le_iff.mpr bot_le)⟩... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 957,
"column": 2
} | {
"line": 969,
"column": 31
} | {
"line": 971,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : R →+* S\n⊢ DenseRange (comap f) ↔ ∀ (I : Ideal R) (h : I ∈ minimalPrimes R), { asIdeal := I, isPrime := ⋯ } ∈ Set.range (comap f)",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Iff.mpr",
... | [] | constructor
· intro H I hI
have : I ∈ (RingHom.ker f).minimalPrimes := by
rw [denseRange_comap_iff_ker_le_nilRadical] at H
simp only [Set.mem_setOf, Ideal.IsMinimalPrime] at hI ⊢
convert! hI using 2 with p
exact ⟨fun h ↦ ⟨h.1, bot_le⟩, fun h ↦ ⟨h.1, H.trans (h.1.radical_le_iff.mpr bot_le)⟩... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.SpecificGroups.Cyclic.Basic | {
"line": 318,
"column": 91
} | {
"line": 329,
"column": 89
} | {
"line": 330,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝³ : Group α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : IsCyclic α\nn : ℕ\nhn0 : 0 < n\ng : α\nhg : ∀ (x : α), x ∈ zpowers g\n⊢ #{a | a ^ n = 1} ≤ #(↑(zpowers (g ^ (Fintype.card α / n.gcd (Fintype.card α))))).toFinset",
"ppTerm": "?m.56",
"assigned": true,
"usedC... | [] | by
gcongr
intro x hx
let ⟨m, hm⟩ := show x ∈ Submonoid.powers g from mem_powers_iff_mem_zpowers.2 <| hg x
refine Set.mem_toFinset.2 ⟨(m / (Fintype.card α / Nat.gcd n (Fintype.card α)) : ℕ), ?_⟩
dsimp only at ⊢ hm
rw [zpow_natCast, ← pow_mul, Nat.mul_div_cancel_left', hm]
refine... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 1030,
"column": 35
} | {
"line": 1031,
"column": 89
} | {
"line": 1033,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\ne f : R\nmul : e * f = 0\nadd : e + f = 1\n⊢ zeroLocus {e} = ↑(basicOpen f)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"PrimeSpectrum.basicOpen_eq_zer... | [] | by
rw [basicOpen_eq_zeroLocus_of_mul_add f e] <;> simp only [mul, add, mul_comm, add_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Exponent | {
"line": 414,
"column": 38
} | {
"line": 414,
"column": 61
} | {
"line": 414,
"column": 61
} | [
{
"pp": "case refine_2\nG : Type u\ninst✝¹ : LeftCancelMonoid G\ninst✝ : Finite G\n_inst : Fintype G := Fintype.ofFinite G\ng : G\n⊢ orderOf g ∣ Finset.univ.lcm orderOf",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"Finset.univ",
"congrArg... | [
"case refine_2\nG : Type u\ninst✝¹ : LeftCancelMonoid G\ninst✝ : Finite G\n_inst : Fintype G := Fintype.ofFinite G\ng : G\n⊢ orderOf g ∣ exponent G"
] | lcm_orderOf_eq_exponent | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 1066,
"column": 8
} | {
"line": 1066,
"column": 19
} | {
"line": 1066,
"column": 20
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\ns : Set (PrimeSpectrum R)\nhs : IsClopen s\nh✝ : Nontrivial R\nI : Ideal R\nhI : I.FG\nJ : Ideal R\nhJ : J.FG\nhI' : zeroLocus ↑I = sᶜ\nhJ' : zeroLocus ↑J = s\nthis : I * J ≤ nilradical R\nn : ℕ\nhn : I ^ n * J ^ n ≤ ⊥\nhnz : n ≠ 0\n⊢ I ^ n ⊔ J ^ n = ⊤",
"ppTerm"... | [
"R : Type u\ninst✝ : CommSemiring R\ns : Set (PrimeSpectrum R)\nhs : IsClopen s\nh✝ : Nontrivial R\nI : Ideal R\nhI : I.FG\nJ : Ideal R\nhJ : J.FG\nhI' : zeroLocus ↑I = sᶜ\nhJ' : zeroLocus ↑J = s\nthis : I * J ≤ nilradical R\nn : ℕ\nhn : I ^ n * J ^ n ≤ ⊥\nhnz : n ≠ 0\n⊢ ⊤ ≤ I ^ n ⊔ J ^ n"
] | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.SpecificGroups.Cyclic.Basic | {
"line": 417,
"column": 2
} | {
"line": 422,
"column": 49
} | {
"line": 424,
"column": 0
} | [
{
"pp": "G : Type u_2\ninst✝² : Group G\ninst✝¹ : Finite G\ninst✝ : IsCyclic G\nd : ℕ\na b : ZMod d\nhGcard : Nat.card G = d\nh : ∀ (t : G), t ^ a.val = t ^ b.val\n⊢ a = b",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"orderOf_eq_card_of_forall_mem_zpowers",
"InvOneClass.toOn... | [] | obtain ⟨g, hg⟩ := IsCyclic.exists_generator (α := G)
specialize h g
subst hGcard
rw [pow_eq_pow_iff_modEq, orderOf_eq_card_of_forall_mem_zpowers hg,
← ZMod.natCast_eq_natCast_iff] at h
simpa [ZMod.natCast_val, ZMod.cast_id'] using h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.SpecificGroups.Cyclic.Basic | {
"line": 417,
"column": 2
} | {
"line": 422,
"column": 49
} | {
"line": 424,
"column": 0
} | [
{
"pp": "G : Type u_2\ninst✝² : Group G\ninst✝¹ : Finite G\ninst✝ : IsCyclic G\nd : ℕ\na b : ZMod d\nhGcard : Nat.card G = d\nh : ∀ (t : G), t ^ a.val = t ^ b.val\n⊢ a = b",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"orderOf_eq_card_of_forall_mem_zpowers",
"InvOneClass.toOn... | [] | obtain ⟨g, hg⟩ := IsCyclic.exists_generator (α := G)
specialize h g
subst hGcard
rw [pow_eq_pow_iff_modEq, orderOf_eq_card_of_forall_mem_zpowers hg,
← ZMod.natCast_eq_natCast_iff] at h
simpa [ZMod.natCast_val, ZMod.cast_id'] using h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Exponent | {
"line": 624,
"column": 3
} | {
"line": 624,
"column": 35
} | {
"line": 624,
"column": 35
} | [
{
"pp": "G : Type u\ninst✝ : Monoid G\nhG : Monoid.exponent G = 2\nx : G\n⊢ orderOf x = 2 → x ≠ 1",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"False",
"eq_false",
"Monoid.toMulOneClass",
"congrArg",
"False.elim",
"Nat.instAtLeas... | [] | by rintro hx rfl; norm_num at hx | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 100,
"column": 2
} | {
"line": 104,
"column": 48
} | {
"line": 105,
"column": 2
} | [
{
"pp": "case h.inr\nα : Type u_1\ninst✝² : Group α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhn : ∀ (n : ℕ), 0 < n → #{a | a ^ n = 1} ≤ n\nd : ℕ\nIH : ∀ m < d, m ∣ Fintype.card α → 0 < #{a | orderOf a = m} → #{a | orderOf a = m} = φ m\nhd : d ∣ Fintype.card α\nhpos : 0 < #{a | orderOf a = d}\nhd0 : d ≠ 0\na ... | [
"case h.inr\nα : Type u_1\ninst✝² : Group α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhn : ∀ (n : ℕ), 0 < n → #{a | a ^ n = 1} ≤ n\nd : ℕ\nIH : ∀ m < d, m ∣ Fintype.card α → 0 < #{a | orderOf a = m} → #{a | orderOf a = m} = φ m\nhd : d ∣ Fintype.card α\nhpos : 0 < #{a | orderOf a = d}\nhd0 : d ≠ 0\na : α\nha' : a... | have h2 :
(∑ m ∈ d.divisors, #{a : α | orderOf a = m}) =
∑ m ∈ d.divisors, φ m := by
rw [sum_card_orderOf_eq_card_pow_eq_one hd0, sum_totient,
← ha, card_pow_eq_one_eq_orderOf_aux hn a] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 190,
"column": 37
} | {
"line": 190,
"column": 48
} | {
"line": 190,
"column": 49
} | [
{
"pp": "G : Type u_2\nG' : Type u_3\ninst✝² : Group G\ninst✝¹ : Group G'\ninst✝ : IsCyclic G'\nf : G →* G'\nhf : f.ker ≤ center G\na b : G\nx : G'\ny : G\nhxy : f y = x\nhx : ∀ (a : ↥f.range), a ∈ zpowers ⟨x, ⋯⟩\nm : ℤ\nhm✝ : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) m = ⟨f a, ⋯⟩\nn : ℤ\nhn✝ : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) n = ⟨f b... | [
"G : Type u_2\nG' : Type u_3\ninst✝² : Group G\ninst✝¹ : Group G'\ninst✝ : IsCyclic G'\nf : G →* G'\nhf : f.ker ≤ center G\na b : G\nx : G'\ny : G\nhxy : f y = x\nhx : ∀ (a : ↥f.range), a ∈ zpowers ⟨x, ⋯⟩\nm : ℤ\nhm✝ : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) m = ⟨f a, ⋯⟩\nn : ℤ\nhn✝ : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) n = ⟨f b, ⋯⟩\nhm : x... | f.map_zpow, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 192,
"column": 37
} | {
"line": 192,
"column": 48
} | {
"line": 192,
"column": 49
} | [
{
"pp": "G : Type u_2\nG' : Type u_3\ninst✝² : Group G\ninst✝¹ : Group G'\ninst✝ : IsCyclic G'\nf : G →* G'\nhf : f.ker ≤ center G\na b : G\nx : G'\ny : G\nhxy : f y = x\nhx : ∀ (a : ↥f.range), a ∈ zpowers ⟨x, ⋯⟩\nm : ℤ\nhm✝ : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) m = ⟨f a, ⋯⟩\nn : ℤ\nhn✝ : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) n = ⟨f b... | [
"G : Type u_2\nG' : Type u_3\ninst✝² : Group G\ninst✝¹ : Group G'\ninst✝ : IsCyclic G'\nf : G →* G'\nhf : f.ker ≤ center G\na b : G\nx : G'\ny : G\nhxy : f y = x\nhx : ∀ (a : ↥f.range), a ∈ zpowers ⟨x, ⋯⟩\nm : ℤ\nhm✝ : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) m = ⟨f a, ⋯⟩\nn : ℤ\nhn✝ : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) n = ⟨f b, ⋯⟩\nhm : x... | f.map_zpow, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.CompactlyGenerated.Intervals | {
"line": 70,
"column": 6
} | {
"line": 70,
"column": 17
} | {
"line": 70,
"column": 18
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ninst✝² : CompleteLattice α\ninst✝¹ : IsModularLattice α\ninst✝ : IsCompactlyGenerated α\ns : Set ι\nf : ι → α\nh : ∀ i ∈ s, ComplementedLattice ↑(Iic (f i))\nh' : ⨆ i ∈ s, f i = ⊤\nt : (i : ι) → i ∈ s → Set α\nht : ∀ (i : ι) (a : i ∈ s), f i = sSup (t i a)\nht' : ∀ (i : ι) (... | [
"ι : Type u_1\nα : Type u_2\ninst✝² : CompleteLattice α\ninst✝¹ : IsModularLattice α\ninst✝ : IsCompactlyGenerated α\ns : Set ι\nf : ι → α\nh : ∀ i ∈ s, ComplementedLattice ↑(Iic (f i))\nh' : ⨆ i ∈ s, f i = ⊤\nt : (i : ι) → i ∈ s → Set α\nht : ∀ (i : ι) (a : i ∈ s), f i = sSup (t i a)\nht' : ∀ (i : ι) (a : i ∈ s), ... | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Sylow | {
"line": 780,
"column": 31
} | {
"line": 780,
"column": 72
} | {
"line": 780,
"column": 73
} | [
{
"pp": "G : Type u\ninst✝² : Group G\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nhn : (normalizer ↑P).Normal\n⊢ normalizer ↑↑P = ⊤",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Sylow.toSubgroup",
"Eq.mpr",
"Sylow.instSetLike",
... | [
"G : Type u\ninst✝² : Group G\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nhn : (normalizer ↑P).Normal\n⊢ normalizer ↑↑P = normalizer ↑P ⊔ normalizer ↑↑P"
] | ← normalizer_sup_eq_top' P le_normalizer, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.JordanHolder | {
"line": 105,
"column": 2
} | {
"line": 107,
"column": 51
} | {
"line": 109,
"column": 0
} | [
{
"pp": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\nx y : X\nhxz : IsMaximal x (x ⊔ y)\nhyz : IsMaximal y (x ⊔ y)\n⊢ IsMaximal (x ⊓ y) y",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"congrArg",
"Semilattic... | [] | rw [inf_comm]
rw [sup_comm] at hxz hyz
exact isMaximal_inf_left_of_isMaximal_sup hyz hxz | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.JordanHolder | {
"line": 105,
"column": 2
} | {
"line": 107,
"column": 51
} | {
"line": 109,
"column": 0
} | [
{
"pp": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\nx y : X\nhxz : IsMaximal x (x ⊔ y)\nhyz : IsMaximal y (x ⊔ y)\n⊢ IsMaximal (x ⊓ y) y",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"congrArg",
"Semilattic... | [] | rw [inf_comm]
rw [sup_comm] at hxz hyz
exact isMaximal_inf_left_of_isMaximal_sup hyz hxz | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.SimpleModule.Basic | {
"line": 471,
"column": 2
} | {
"line": 475,
"column": 16
} | {
"line": 477,
"column": 0
} | [
{
"pp": "R : Type u_2\nS : Type u_3\ninst✝² : Ring R\ninst✝¹ : Ring S\nf : R →+* S\nhf : Function.Surjective ⇑f\ninst✝ : IsSemisimpleRing R\n⊢ IsSemisimpleRing S",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"Function.bijective_id",
"Semirin... | [] | letI : Module R S := Module.compHom _ f
haveI : RingHomSurjective f := ⟨hf⟩
let e : S →ₛₗ[f] S := { AddMonoidHom.id S with map_smul' := fun _ _ ↦ rfl }
rw [IsSemisimpleRing, ← e.isSemisimpleModule_iff_of_bijective Function.bijective_id]
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.SimpleModule.Basic | {
"line": 471,
"column": 2
} | {
"line": 475,
"column": 16
} | {
"line": 477,
"column": 0
} | [
{
"pp": "R : Type u_2\nS : Type u_3\ninst✝² : Ring R\ninst✝¹ : Ring S\nf : R →+* S\nhf : Function.Surjective ⇑f\ninst✝ : IsSemisimpleRing R\n⊢ IsSemisimpleRing S",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"Function.bijective_id",
"Semirin... | [] | letI : Module R S := Module.compHom _ f
haveI : RingHomSurjective f := ⟨hf⟩
let e : S →ₛₗ[f] S := { AddMonoidHom.id S with map_smul' := fun _ _ ↦ rfl }
rw [IsSemisimpleRing, ← e.isSemisimpleModule_iff_of_bijective Function.bijective_id]
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.Torsion.Basic | {
"line": 275,
"column": 2
} | {
"line": 277,
"column": 22
} | {
"line": 279,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\na : R\n⊢ torsionBySet R M {a} = torsionBy R M a",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Submodule",
"instHSMul",
"_private.Mathlib.Algebra.Module.Torsion.B... | [] | ext x
simp only [mem_torsionBySet_iff, SetCoe.forall, Set.mem_singleton_iff, forall_eq,
mem_torsionBy_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.Torsion.Basic | {
"line": 275,
"column": 2
} | {
"line": 277,
"column": 22
} | {
"line": 279,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\na : R\n⊢ torsionBySet R M {a} = torsionBy R M a",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Submodule",
"instHSMul",
"_private.Mathlib.Algebra.Module.Torsion.B... | [] | ext x
simp only [mem_torsionBySet_iff, SetCoe.forall, Set.mem_singleton_iff, forall_eq,
mem_torsionBy_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.Torsion.Basic | {
"line": 361,
"column": 2
} | {
"line": 362,
"column": 43
} | {
"line": 364,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\na : R\n⊢ IsTorsionBy R M a ↔ torsionBy R M a = ⊤",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Module.IsTorsionBy",
"Submodule",
"Module.IsTorsio... | [] | rw [← torsionBySet_singleton_eq, ← isTorsionBySet_singleton_iff,
isTorsionBySet_iff_torsionBySet_eq_top] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Module.Torsion.Basic | {
"line": 361,
"column": 2
} | {
"line": 362,
"column": 43
} | {
"line": 364,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\na : R\n⊢ IsTorsionBy R M a ↔ torsionBy R M a = ⊤",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Module.IsTorsionBy",
"Submodule",
"Module.IsTorsio... | [] | rw [← torsionBySet_singleton_eq, ← isTorsionBySet_singleton_iff,
isTorsionBySet_iff_torsionBySet_eq_top] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.Torsion.Basic | {
"line": 361,
"column": 2
} | {
"line": 362,
"column": 43
} | {
"line": 364,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\na : R\n⊢ IsTorsionBy R M a ↔ torsionBy R M a = ⊤",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Module.IsTorsionBy",
"Submodule",
"Module.IsTorsio... | [] | rw [← torsionBySet_singleton_eq, ← isTorsionBySet_singleton_iff,
isTorsionBySet_iff_torsionBySet_eq_top] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Module.Basic | {
"line": 380,
"column": 97
} | {
"line": 382,
"column": 5
} | {
"line": 384,
"column": 0
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np : R[X]\ni : ℕ\nm : M\n⊢ (comp p) (single R i m) = p ^ i • single R 0 m",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"congrArg",
"CommSe... | [] | by
rw [comp_apply, map_single, eval_single]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Module.Basic | {
"line": 386,
"column": 2
} | {
"line": 392,
"column": 10
} | {
"line": 394,
"column": 0
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np : R[X]\nq : PolynomialModule R M\nr : R\n⊢ (eval r) ((comp p) q) = (eval (Polynomial.eval r p)) q",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_p... | [] | rw [← LinearMap.comp_apply]
induction q using induction_linear with
| zero => simp_rw [map_zero]
| add _ _ e₁ e₂ => simp_rw [map_add, e₁, e₂]
| single i m =>
rw [LinearMap.comp_apply, comp_single, eval_single, eval_smul, eval_single, eval_pow]
module | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Module.Basic | {
"line": 386,
"column": 2
} | {
"line": 392,
"column": 10
} | {
"line": 394,
"column": 0
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np : R[X]\nq : PolynomialModule R M\nr : R\n⊢ (eval r) ((comp p) q) = (eval (Polynomial.eval r p)) q",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_p... | [] | rw [← LinearMap.comp_apply]
induction q using induction_linear with
| zero => simp_rw [map_zero]
| add _ _ e₁ e₂ => simp_rw [map_add, e₁, e₂]
| single i m =>
rw [LinearMap.comp_apply, comp_single, eval_single, eval_smul, eval_single, eval_pow]
module | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.Torsion.Basic | {
"line": 606,
"column": 6
} | {
"line": 607,
"column": 49
} | {
"line": 609,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nr : R\nh : (Ideal.span {r}).IsTwoSided\nhM : IsTorsionBy R M r\n⊢ IsTorsionBySet R M ↑(Ideal.span {r})",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Module.IsTorsionBy",... | [] | finally
rwa [← isTorsionBySet_span_singleton_iff] at hM | [anonymous] | Lean.Parser.Term.whereFinally |
Mathlib.Algebra.Module.Torsion.Basic | {
"line": 908,
"column": 40
} | {
"line": 908,
"column": 51
} | {
"line": 908,
"column": 52
} | [
{
"pp": "case h\nR : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : (x : M) → Decidable (x = 0)\np : R\nhM : IsTorsion' M ↥(Submonoid.powers p)\nd : ℕ\nhd : d ≠ 0\ns : Fin d → M\nhs : span R (Set.range s) = ⊤\noj : Option (Fin d) := List.argmax (fun i ↦ p... | [
"case h\nR : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : (x : M) → Decidable (x = 0)\np : R\nhM : IsTorsion' M ↥(Submonoid.powers p)\nd : ℕ\nhd : d ≠ 0\ns : Fin d → M\nhs : span R (Set.range s) = ⊤\noj : Option (Fin d) := List.argmax (fun i ↦ pOrder hM (s ... | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Nakayama | {
"line": 228,
"column": 4
} | {
"line": 228,
"column": 8
} | {
"line": 228,
"column": 8
} | [
{
"pp": "case h.refine_2.refine_2\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nN : Submodule R M\ns : Set (M ⧸ I • N)\nhN : N.FG\nhIjac : I ≤ ⊥.jacobson\nhsspan : span R s = map (I • N).mkQ N\n⊢ span R (Quotient.out '' s) = N",
"ppTerm": "?h.ref... | [
"case h.refine_2.refine_2\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nN : Submodule R M\ns : Set (M ⧸ I • N)\nhN : N.FG\nhIjac : I ≤ ⊥.jacobson\nhsspan : span R s = map (I • N).mkQ N\n⊢ N = span R (Quotient.out '' s)"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Algebra.Category.ModuleCat.Presheaf | {
"line": 267,
"column": 17
} | {
"line": 267,
"column": 66
} | {
"line": 268,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nR : Cᵒᵖ ⥤ RingCat\nM M₁ M₂ : PresheafOfModules R\n⊢ ∀ (a b : M₁ ⟶ M₂), a + b = b + a",
"ppTerm": "?m.110",
"assigned": true,
"usedConstants": [
"Opposite",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"ModuleCat",
... | [] | intros; ext1; simp only [add_app]; apply add_comm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.Presheaf | {
"line": 267,
"column": 17
} | {
"line": 267,
"column": 66
} | {
"line": 268,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nR : Cᵒᵖ ⥤ RingCat\nM M₁ M₂ : PresheafOfModules R\n⊢ ∀ (a b : M₁ ⟶ M₂), a + b = b + a",
"ppTerm": "?m.110",
"assigned": true,
"usedConstants": [
"Opposite",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"ModuleCat",
... | [] | intros; ext1; simp only [add_app]; apply add_comm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.HomologicalComplex | {
"line": 344,
"column": 21
} | {
"line": 346,
"column": 23
} | {
"line": 348,
"column": 0
} | [
{
"pp": "ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC X✝ Y✝ : HomologicalComplex V c\na₁✝ a₂✝ : X✝ ⟶ Y✝\nh : (forget V c).map a₁✝ = (forget V c).map a₂✝\n⊢ a₁✝ = a₂✝",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Homologica... | [] | by
ext i
exact congr_fun h i | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Shift.Basic | {
"line": 394,
"column": 2
} | {
"line": 394,
"column": 6
} | {
"line": 395,
"column": 2
} | [
{
"pp": "C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddMonoid A\ninst✝ : HasShift C A\nX Y : C\nf : X ⟶ Y\n⊢ (shiftFunctor C 0).map f = (shiftZero A X).hom ≫ f ≫ (shiftZero A Y).inv",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQ... | [
"C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddMonoid A\ninst✝ : HasShift C A\nX Y : C\nf : X ⟶ Y\n⊢ (shiftZero A X).hom ≫ f ≫ (shiftZero A Y).inv = (shiftFunctor C 0).map f"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Algebra.Homology.HomologicalComplex | {
"line": 803,
"column": 4
} | {
"line": 805,
"column": 16
} | {
"line": 806,
"column": 2
} | [
{
"pp": "case zero\nι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nX₀ X₁ X₂ : V\nd₀ : X₁ ⟶ X₀\nd₁ : X₂ ⟶ X₁\ns : d₁ ≫ d₀ = 0\nsucc : (S : ShortComplex V) → (X₃ : V) ×' (d₂ : X₃ ⟶ S.X₁) ×' d₂ ≫ S.f = 0\nsucc' : {X₀ X₁ : V} → (f : X₁ ⟶ X₀) → (X₂ : V) ×' (d : X₂ ⟶ X₁) ×' d ≫ f = ... | [] | apply eqToIso
dsimp [mk', mk, of, mkAux, of.d]
rw [id_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.HomologicalComplex | {
"line": 803,
"column": 4
} | {
"line": 805,
"column": 16
} | {
"line": 806,
"column": 2
} | [
{
"pp": "case zero\nι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nX₀ X₁ X₂ : V\nd₀ : X₁ ⟶ X₀\nd₁ : X₂ ⟶ X₁\ns : d₁ ≫ d₀ = 0\nsucc : (S : ShortComplex V) → (X₃ : V) ×' (d₂ : X₃ ⟶ S.X₁) ×' d₂ ≫ S.f = 0\nsucc' : {X₀ X₁ : V} → (f : X₁ ⟶ X₀) → (X₂ : V) ×' (d : X₂ ⟶ X₁) ×' d ≫ f = ... | [] | apply eqToIso
dsimp [mk', mk, of, mkAux, of.d]
rw [id_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Single | {
"line": 210,
"column": 35
} | {
"line": 213,
"column": 23
} | {
"line": 216,
"column": 0
} | [
{
"pp": "V : Type u\ninst✝² : Category.{v, u} V\ninst✝¹ : HasZeroMorphisms V\ninst✝ : HasZeroObject V\nA B : V\nf : A ⟶ B\n⊢ ((single₀ V).map f).f 0 = f",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ChainComplex",
"HomologicalComplex.instCategory",
"Nat... | [] | by
rw [HomologicalComplex.single_map_f_self]
dsimp [HomologicalComplex.singleObjXSelf, HomologicalComplex.singleObjXIsoOfEq]
rw [comp_id, id_comp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.Single | {
"line": 280,
"column": 35
} | {
"line": 283,
"column": 23
} | {
"line": 285,
"column": 0
} | [
{
"pp": "V : Type u\ninst✝² : Category.{v, u} V\ninst✝¹ : HasZeroMorphisms V\ninst✝ : HasZeroObject V\nA B : V\nf : A ⟶ B\n⊢ ((single₀ V).map f).f 0 = f",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HomologicalComplex.instCategory",
"Nat.instOne",
"Homo... | [] | by
rw [HomologicalComplex.single_map_f_self]
dsimp [HomologicalComplex.singleObjXSelf, HomologicalComplex.singleObjXIsoOfEq]
rw [comp_id, id_comp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Kaehler.Basic | {
"line": 831,
"column": 37
} | {
"line": 831,
"column": 70
} | {
"line": 831,
"column": 71
} | [
{
"pp": "case h\nR : Type u\ninst✝⁶ : CommRing R\nA : Type u_2\nB : Type u_3\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R A\ninst✝² : Algebra A B\ninst✝¹ : Algebra R B\ninst✝ : IsScalarTower R A B\nh : Function.Surjective ⇑(algebraMap A B)\nx : A →₀ A\nhx : x ∈ ↑(mapRange.linearMap (Algebra.lin... | [
"case h\nR : Type u\ninst✝⁶ : CommRing R\nA : Type u_2\nB : Type u_3\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R A\ninst✝² : Algebra A B\ninst✝¹ : Algebra R B\ninst✝ : IsScalarTower R A B\nh : Function.Surjective ⇑(algebraMap A B)\nx : A →₀ A\nhx : x ∈ ↑(mapRange.linearMap (Algebra.linearMap A B) ... | ← Algebra.algebraMap_eq_smul_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.Homotopy | {
"line": 348,
"column": 83
} | {
"line": 350,
"column": 41
} | {
"line": 353,
"column": 0
} | [
{
"pp": "ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D : HomologicalComplex V c\nk₂ k₁ k₀ : ι\nr₂₁ : c.Rel k₂ k₁\nr₁₀ : c.Rel k₁ k₀\nhom : (i j : ι) → C.X i ⟶ D.X j\n⊢ (nullHomotopicMap hom).f k₁ = C.d k₁ k₀ ≫ hom k₀ k₁ + hom k₁ k₂ ≫ D.d k₂ k₁",
"ppTerm... | [] | by
dsimp only [nullHomotopicMap]
rw [dNext_eq hom r₁₀, prevD_eq hom r₂₁] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplex | {
"line": 372,
"column": 2
} | {
"line": 372,
"column": 25
} | {
"line": 374,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nF G K : CochainComplex C ℤ\nn₁ n₂ n₁₂ : ℤ\nz₁ : Cochain F G n₁\nk : Rˣ\nz₂ : Cochain G K n₂\nh : n₁ + n₂ = n₁₂\n⊢ z₁.comp (k • z₂) h = k • z₁.comp z₂ h",
"ppTerm": "?m.60",
"assign... | [] | apply Cochain.comp_smul | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplex | {
"line": 372,
"column": 2
} | {
"line": 372,
"column": 25
} | {
"line": 374,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nF G K : CochainComplex C ℤ\nn₁ n₂ n₁₂ : ℤ\nz₁ : Cochain F G n₁\nk : Rˣ\nz₂ : Cochain G K n₂\nh : n₁ + n₂ = n₁₂\n⊢ z₁.comp (k • z₂) h = k • z₁.comp z₂ h",
"ppTerm": "?m.60",
"assign... | [] | apply Cochain.comp_smul | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplex | {
"line": 372,
"column": 2
} | {
"line": 372,
"column": 25
} | {
"line": 374,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nF G K : CochainComplex C ℤ\nn₁ n₂ n₁₂ : ℤ\nz₁ : Cochain F G n₁\nk : Rˣ\nz₂ : Cochain G K n₂\nh : n₁ + n₂ = n₁₂\n⊢ z₁.comp (k • z₂) h = k • z₁.comp z₂ h",
"ppTerm": "?m.60",
"assign... | [] | apply Cochain.comp_smul | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Homotopy | {
"line": 450,
"column": 42
} | {
"line": 450,
"column": 48
} | {
"line": 452,
"column": 0
} | [
{
"pp": "case a\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nP Q : ChainComplex V ℕ\nf : (i j : ℕ) → P.X i ⟶ Q.X j\n⊢ ¬1 = 0",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.t... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Algebra.Homology.Homotopy | {
"line": 580,
"column": 44
} | {
"line": 580,
"column": 50
} | {
"line": 582,
"column": 0
} | [
{
"pp": "case a\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nP Q : CochainComplex V ℕ\nf : (i j : ℕ) → P.X i ⟶ Q.X j\n⊢ ¬1 = 0",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Algebra.Homology.HomotopyCategory.MappingCone | {
"line": 320,
"column": 20
} | {
"line": 320,
"column": 41
} | {
"line": 320,
"column": 41
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{v, u_1} C\ninst✝³ : Category.{v', u_2} D\ninst✝² : Preadditive C\ninst✝¹ : Preadditive D\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝ : HasHomotopyCofiber φ\nK : CochainComplex C ℤ\nn m : ℤ\nα : Cochain F K m\nβ : Cochain G K n\nh : m + 1 = n\n⊢ 1 + m = n",... | [] | by rw [← h, add_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.HomotopyCofiber | {
"line": 752,
"column": 31
} | {
"line": 752,
"column": 89
} | {
"line": 753,
"column": 4
} | [
{
"pp": "C : Type u_1\ninst✝¹⁰ : Category.{v_1, u_1} C\ninst✝⁹ : Preadditive C\nι : Type u_2\nc : ComplexShape ι\nF : HomologicalComplex C c\ninst✝⁸ : DecidableRel c.Rel\nD : Type u_3\ninst✝⁷ : Category.{v_2, u_3} D\ninst✝⁶ : Preadditive D\nH : C ⥤ D\ninst✝⁵ : H.Additive\ninst✝⁴ : ∀ (i : ι), HasBinaryBiproduct ... | [
"C : Type u_1\ninst✝¹⁰ : Category.{v_1, u_1} C\ninst✝⁹ : Preadditive C\nι : Type u_2\nc : ComplexShape ι\nF : HomologicalComplex C c\ninst✝⁸ : DecidableRel c.Rel\nD : Type u_3\ninst✝⁷ : Category.{v_2, u_3} D\ninst✝⁶ : Preadditive D\nH : C ⥤ D\ninst✝⁵ : H.Additive\ninst✝⁴ : ∀ (i : ι), HasBinaryBiproduct (F.X i) (F.X... | homotopyCofiber.inr_mapHomologicalComplexObjIso_hom_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Quotient | {
"line": 72,
"column": 15
} | {
"line": 74,
"column": 53
} | {
"line": 76,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nr : HomRel C\n⊢ ∀ {X Y Z : C} (f : X ⟶ Y) {g g' : Y ⟶ Z}, CompClosure r g g' → CompClosure r (f ≫ g) (f ≫ g')",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"CategoryTheory.Category.assoc",
"CategoryTheory.CategoryStruct.toQ... | [] | by
rintro a b e f _ _ ⟨c, d, g, h₁, h₂, i, h⟩
simpa using CompClosure.intro _ _ (f ≫ g) _ _ i h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.HomotopyCofiber | {
"line": 763,
"column": 31
} | {
"line": 763,
"column": 89
} | {
"line": 764,
"column": 4
} | [
{
"pp": "C : Type u_1\ninst✝¹⁰ : Category.{v_1, u_1} C\ninst✝⁹ : Preadditive C\nι : Type u_2\nc : ComplexShape ι\nF : HomologicalComplex C c\ninst✝⁸ : DecidableRel c.Rel\nD : Type u_3\ninst✝⁷ : Category.{v_2, u_3} D\ninst✝⁶ : Preadditive D\nH : C ⥤ D\ninst✝⁵ : H.Additive\ninst✝⁴ : ∀ (i : ι), HasBinaryBiproduct ... | [
"C : Type u_1\ninst✝¹⁰ : Category.{v_1, u_1} C\ninst✝⁹ : Preadditive C\nι : Type u_2\nc : ComplexShape ι\nF : HomologicalComplex C c\ninst✝⁸ : DecidableRel c.Rel\nD : Type u_3\ninst✝⁷ : Category.{v_2, u_3} D\ninst✝⁶ : Preadditive D\nH : C ⥤ D\ninst✝⁵ : H.Additive\ninst✝⁴ : ∀ (i : ι), HasBinaryBiproduct (F.X i) (F.X... | homotopyCofiber.inr_mapHomologicalComplexObjIso_hom_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Quotient.Preadditive | {
"line": 71,
"column": 42
} | {
"line": 71,
"column": 91
} | {
"line": 72,
"column": 6
} | [
{
"pp": "case mk.mk.mk\nC : Type ?u.2\ninst✝² : Category.{v_1, ?u.2} C\ninst✝¹ : Preadditive C\nr : HomRel C\ninst✝ : Congruence r\nhr : ∀ ⦃X Y : C⦄ (f₁ f₂ g₁ g₂ : X ⟶ Y), r f₁ f₂ → r g₁ g₂ → r (f₁ + g₁) (f₂ + g₂)\nP Q : Quotient r\niZ : Zero (P ⟶ Q) := { zero := Quot.mk (HomRel.CompClosure r) 0 }\niA : Add (P ... | [] | exact congr_arg (functor r).map (add_assoc _ _ _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Shift.CommShift | {
"line": 114,
"column": 2
} | {
"line": 116,
"column": 38
} | {
"line": 118,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\nF : C ⥤ D\nA : Type u_4\ninst✝² : AddMonoid A\ninst✝¹ : HasShift C A\ninst✝ : HasShift D A\na : A\ne : shiftFunctor C a ⋙ F ≅ F ⋙ shiftFunctor D a\n⊢ isoAdd' ⋯ e (isoZero F A) = e",
"ppTerm": "?m.59",
"a... | [] | ext X
simp [shiftFunctorAdd'_add_zero_hom_app, ← Functor.map_comp_assoc,
shiftFunctorAdd'_add_zero_inv_app] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Shift.CommShift | {
"line": 114,
"column": 2
} | {
"line": 116,
"column": 38
} | {
"line": 118,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\nF : C ⥤ D\nA : Type u_4\ninst✝² : AddMonoid A\ninst✝¹ : HasShift C A\ninst✝ : HasShift D A\na : A\ne : shiftFunctor C a ⋙ F ≅ F ⋙ shiftFunctor D a\n⊢ isoAdd' ⋯ e (isoZero F A) = e",
"ppTerm": "?m.59",
"a... | [] | ext X
simp [shiftFunctorAdd'_add_zero_hom_app, ← Functor.map_comp_assoc,
shiftFunctorAdd'_add_zero_inv_app] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexShift | {
"line": 247,
"column": 2
} | {
"line": 247,
"column": 51
} | {
"line": 248,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nK L : CochainComplex C ℤ\nn : ℤ\nγ : Cochain K L n\na n' : ℤ\nhn' : n' + a = n\n⊢ (-γ).rightShift a n' hn' = -γ.rightShift a n' hn'",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"Hom... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nK L : CochainComplex C ℤ\nn : ℤ\nγ : Cochain K L n\na n' : ℤ\nhn' : n' + a = n\n⊢ (rightShiftAddEquiv K L n a n' hn') (-γ) = -γ.rightShift a n' hn'"
] | change rightShiftAddEquiv K L n a n' hn' (-γ) = _ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexShift | {
"line": 413,
"column": 63
} | {
"line": 413,
"column": 82
} | {
"line": 413,
"column": 83
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nK L M : CochainComplex C ℤ\nn : ℤ\nγ : Cochain K L n\na n' : ℤ\nhn' : n + a = n'\nγ' : Cochain L M 0\n⊢ Int.negOnePow 0 • (γ.leftShift a n' hn').comp γ' ⋯ = (γ.leftShift a n' hn').comp γ' ⋯",
"ppTerm": "?m.89",
"assigned": true,
... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nK L M : CochainComplex C ℤ\nn : ℤ\nγ : Cochain K L n\na n' : ℤ\nhn' : n + a = n'\nγ' : Cochain L M 0\n⊢ 1 • (γ.leftShift a n' hn').comp γ' ⋯ = (γ.leftShift a n' hn').comp γ' ⋯"
] | Int.negOnePow_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Triangulated.Triangulated | {
"line": 171,
"column": 2
} | {
"line": 171,
"column": 41
} | {
"line": 172,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nX₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C\nu₁₂ : X₁ ⟶ X₂\nu₂₃ : X₂ ⟶ X₃\nu₁₃ : X₁ ⟶ X₃\ncomm : u₁₂ ≫ u₂₃ = u₁₃\nv₁₂ : X₂ ⟶ Z₁... | [
"C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nX₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C\nu₁₂ : X₁ ⟶ X₂\nu₂₃ : X₂ ⟶ X₃\nu₁₃ : X₁ ⟶ X₃\ncomm : u₁₂ ≫ u₂₃ = u₁₃\nv₁₂ : X₂ ⟶ Z₁₂\nw₁₂ : Z₁₂... | have rel₂₃ := H.triangleMorphism₂.comm₃ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Triangulated.Triangulated | {
"line": 158,
"column": 2
} | {
"line": 194,
"column": 77
} | {
"line": 196,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nX₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C\nu₁₂ : X₁ ⟶ X₂\nu₂₃ : X₂ ⟶ X₃\nu₁₃ : X₁ ⟶ X₃\ncomm : u₁₂ ≫ u₂₃ = u₁₃\nv₁₂ : X₂ ⟶ Z₁... | [] | let iso₁₂ := isoTriangleOfIso₁₂ _ _ h₁₂ h₁₂' e₁ e₂ comm₁₂
let iso₂₃ := isoTriangleOfIso₁₂ _ _ h₂₃ h₂₃' e₂ e₃ comm₂₃
let iso₁₃ := isoTriangleOfIso₁₂ _ _ h₁₃ h₁₃' e₁ e₃ (by
dsimp; rw [← comm, assoc, ← comm', ← reassoc_of% comm₁₂, comm₂₃])
have eq₁₂ := iso₁₂.hom.comm₂
have eq₁₂' := iso₁₂.hom.comm₃
have eq₁₃ ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Triangulated.Triangulated | {
"line": 158,
"column": 2
} | {
"line": 194,
"column": 77
} | {
"line": 196,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nX₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C\nu₁₂ : X₁ ⟶ X₂\nu₂₃ : X₂ ⟶ X₃\nu₁₃ : X₁ ⟶ X₃\ncomm : u₁₂ ≫ u₂₃ = u₁₃\nv₁₂ : X₂ ⟶ Z₁... | [] | let iso₁₂ := isoTriangleOfIso₁₂ _ _ h₁₂ h₁₂' e₁ e₂ comm₁₂
let iso₂₃ := isoTriangleOfIso₁₂ _ _ h₂₃ h₂₃' e₂ e₃ comm₂₃
let iso₁₃ := isoTriangleOfIso₁₂ _ _ h₁₃ h₁₃' e₁ e₃ (by
dsimp; rw [← comm, assoc, ← comm', ← reassoc_of% comm₁₂, comm₂₃])
have eq₁₂ := iso₁₂.hom.comm₂
have eq₁₂' := iso₁₂.hom.comm₃
have eq₁₃ ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.HomotopyCategory.DegreewiseSplit | {
"line": 217,
"column": 8
} | {
"line": 217,
"column": 40
} | {
"line": 219,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v, u_1} C\ninst✝¹ : Preadditive C\nS : ShortComplex (CochainComplex C ℤ)\nσ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting\ninst✝ : HasBinaryBiproducts C\nK L : CochainComplex C ℤ\nφ : K ⟶ L\nn : ℤ\n⊢ (snd φ).v n n ⋯ ≫ ((triangleRotateShortComplex φ).map (... | [] | by simp [ext_from_iff φ _ _ rfl] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.HomologySequence | {
"line": 146,
"column": 4
} | {
"line": 146,
"column": 59
} | {
"line": 147,
"column": 4
} | [
{
"pp": "C : Type u_1\nι✝ : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nc : ComplexShape ι✝\nK : HomologicalComplex C c\ni j : ι✝\nhij : c.Rel i j\ninst✝ : CategoryWithHomology C\nS : ShortComplex C :=\n { X₁ := K.homology i, X₂ := K.opcycles i, X₃ := K.cycles j, f := K.homologyι i, g := ... | [
"C : Type u_1\nι✝ : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nc : ComplexShape ι✝\nK : HomologicalComplex C c\ni j : ι✝\nhij : c.Rel i j\ninst✝ : CategoryWithHomology C\nS : ShortComplex C :=\n { X₁ := K.homology i, X₂ := K.opcycles i, X₃ := K.cycles j, f := K.homologyι i, g := K.opcyclesTo... | rw [← ShortComplex.exact_iff_of_epi_of_isIso_of_mono π] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.PathCategory.Basic | {
"line": 122,
"column": 6
} | {
"line": 125,
"column": 29
} | {
"line": 127,
"column": 0
} | [
{
"pp": "case cons\nV : Type u₁\ninst✝¹ : Quiver V\nC : Type ?u.6\ninst✝ : Category.{v_1, ?u.6} C\nφ : V ⥤q C\nX✝ Y✝ Z✝ : Paths V\nf : X✝ ⟶ Y✝\nb✝ c✝ : Paths V\ng' : Quiver.Path Y✝ b✝\np : b✝ ⟶ c✝\nih :\n Quiver.Path.rec (𝟙 (φ.obj X✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) (f ≫ g') =\n Quiver.Path.rec (𝟙 (φ... | [] | have : f ≫ Quiver.Path.cons g' p = (f ≫ g').cons p := by apply Quiver.Path.comp_cons
rw [this]
simp only at ih ⊢
rw [ih, Category.assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.PathCategory.Basic | {
"line": 122,
"column": 6
} | {
"line": 125,
"column": 29
} | {
"line": 127,
"column": 0
} | [
{
"pp": "case cons\nV : Type u₁\ninst✝¹ : Quiver V\nC : Type ?u.6\ninst✝ : Category.{v_1, ?u.6} C\nφ : V ⥤q C\nX✝ Y✝ Z✝ : Paths V\nf : X✝ ⟶ Y✝\nb✝ c✝ : Paths V\ng' : Quiver.Path Y✝ b✝\np : b✝ ⟶ c✝\nih :\n Quiver.Path.rec (𝟙 (φ.obj X✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) (f ≫ g') =\n Quiver.Path.rec (𝟙 (φ... | [] | have : f ≫ Quiver.Path.cons g' p = (f ≫ g').cons p := by apply Quiver.Path.comp_cons
rw [this]
simp only at ih ⊢
rw [ih, Category.assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Localization.Construction | {
"line": 235,
"column": 6
} | {
"line": 235,
"column": 17
} | {
"line": 236,
"column": 6
} | [
{
"pp": "case mpr.cons.inl\nC : Type uC\ninst✝¹ : Category.{uC', uC} C\nW : MorphismProperty C\nP : MorphismProperty W.Localization\ninst✝ : P.IsStableUnderComposition\nhP₁ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), P (W.Q.map f)\nhP₂ : ∀ ⦃X Y : C⦄ (w : X ⟶ Y) (hw : W w), P (wInv w hw)\nX Y : W.Localization\nf : X ⟶ Y\na✝ : ⊤ ... | [
"case mpr.cons.inr\nC : Type uC\ninst✝¹ : Category.{uC', uC} C\nW : MorphismProperty C\nP : MorphismProperty W.Localization\ninst✝ : P.IsStableUnderComposition\nhP₁ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), P (W.Q.map f)\nhP₂ : ∀ ⦃X Y : C⦄ (w : X ⟶ Y) (hw : W w), P (wInv w hw)\nX Y : W.Localization\nf : X ⟶ Y\na✝ : ⊤ f\nG : Paths... | · apply hP₁ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions | {
"line": 293,
"column": 44
} | {
"line": 293,
"column": 47
} | {
"line": 293,
"column": 47
} | [
{
"pp": "case refine_2\nC : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\ninst✝ : W.HasLeftCalculusOfFractions\nX Y : C\nf f' : L.obj X ⟶ L.obj Y\nφ : W.LeftFraction X Y\nhφ : f = φ.map L ⋯\nφ' : W.LeftFrac... | [
"case refine_2\nC : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\ninst✝ : W.HasLeftCalculusOfFractions\nX Y : C\nf f' : L.obj X ⟶ L.obj Y\nφ : W.LeftFraction X Y\nhφ : f = φ.map L ⋯\nφ' : W.LeftFraction X Y\nhφ... | hφ' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Localization.CalculusOfFractions.Preadditive | {
"line": 201,
"column": 2
} | {
"line": 201,
"column": 82
} | {
"line": 202,
"column": 2
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\ninst✝² : Preadditive C\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\ninst✝ : W.HasLeftCalculusOfFractions\nX Y Z : C\nf : L.obj X ⟶ L.obj Y\ng₁ g₂ : L.obj Y ⟶ L.obj Z\nα : W.LeftFraction X Y\n... | [
"C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\ninst✝² : Preadditive C\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\ninst✝ : W.HasLeftCalculusOfFractions\nX Y Z : C\nf : L.obj X ⟶ L.obj Y\ng₁ g₂ : L.obj Y ⟶ L.obj Z\nα : W.LeftFraction X Y\nhα : f = α.m... | obtain ⟨γ, hγ₁, hγ₂⟩ := (RightFraction₂.mk _ α.hs β.f β.f').exists_leftFraction₂ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Localization.CalculusOfFractions | {
"line": 389,
"column": 19
} | {
"line": 389,
"column": 39
} | {
"line": 389,
"column": 39
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\nW : MorphismProperty C\ninst✝ : W.HasLeftCalculusOfFractions\nX Y Z : C\nz₁✝ : Hom W X Y\nz₂✝ : Hom W Y Z\na₁ a₂ : W.LeftFraction X Y\nb : W.LeftFraction Y Z\nU : C\nt₁ : a₁.Y' ⟶ U\nt₂ : a₂.Y' ⟶ U\nhst : a₁.s ≫ ... | [] | rw [← hst]; exact ht | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Localization.CalculusOfFractions | {
"line": 389,
"column": 19
} | {
"line": 389,
"column": 39
} | {
"line": 389,
"column": 39
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\nW : MorphismProperty C\ninst✝ : W.HasLeftCalculusOfFractions\nX Y Z : C\nz₁✝ : Hom W X Y\nz₂✝ : Hom W Y Z\na₁ a₂ : W.LeftFraction X Y\nb : W.LeftFraction Y Z\nU : C\nt₁ : a₁.Y' ⟶ U\nt₂ : a₂.Y' ⟶ U\nhst : a₁.s ≫ ... | [] | rw [← hst]; exact ht | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Localization.CalculusOfFractions | {
"line": 410,
"column": 6
} | {
"line": 411,
"column": 51
} | {
"line": 412,
"column": 6
} | [
{
"pp": "case refine_2.refine_2\nC : Type u_1\nD : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\nW : MorphismProperty C\ninst✝ : W.HasLeftCalculusOfFractions\nX Y Z : C\nz₁✝ : Hom W X Y\nz₂✝ : Hom W Y Z\na₁ a₂ : W.LeftFraction X Y\nb : W.LeftFraction Y Z\nU : C\nt₁ : a₁.Y' ⟶ U\nt₂ : ... | [
"case refine_2.refine_2\nC : Type u_1\nD : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\nW : MorphismProperty C\ninst✝ : W.HasLeftCalculusOfFractions\nX Y Z : C\nz₁✝ : Hom W X Y\nz₂✝ : Hom W Y Z\na₁ a₂ : W.LeftFraction X Y\nb : W.LeftFraction Y Z\nU : C\nt₁ : a₁.Y' ⟶ U\nt₂ : a₂.Y' ⟶ U\nh... | obtain ⟨q, fac₃⟩ := exists_leftFraction (RightFraction.mk (z₁.s ≫ w₁.s)
(W.comp_mem _ _ z₁.hs w₁.hs) (z₂.s ≫ w₂.s)) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.ObjectProperty.FiniteProducts | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 93
} | {
"line": 82,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : HasTerminal C\nP Q : ObjectProperty C\ninst✝¹ : Q.IsClosedUnderBinaryProducts\ninst✝ : Q.IsClosedUnderLimitsOfShape (Discrete PEmpty.{1})\nh : P ≤ Q\n⊢ P.binaryProductsClosure ≤ Q",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": ... | [
"C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : HasTerminal C\nP Q : ObjectProperty C\ninst✝¹ : Q.IsClosedUnderBinaryProducts\ninst✝ : Q.IsClosedUnderLimitsOfShape (Discrete PEmpty.{1})\nh : P ≤ Q\nthis : Q.IsClosedUnderIsomorphisms := IsClosedUnderBinaryProducts.closedUnderIsomorphisms Q\n⊢ P.binaryProduct... | letI : Q.IsClosedUnderIsomorphisms := IsClosedUnderBinaryProducts.closedUnderIsomorphisms Q | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLetI___1 | Lean.Parser.Tactic.tacticLetI__ |
Mathlib.CategoryTheory.ObjectProperty.FiniteProducts | {
"line": 199,
"column": 4
} | {
"line": 201,
"column": 13
} | {
"line": 203,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nP : ObjectProperty C\nH : ∀ (J : Type w) [Finite J], P.IsClosedUnderColimitsOfShape (Discrete J)\nJ : Type\nx✝ : Finite J\n⊢ P.IsClosedUnderColimitsOfShape (Discrete J)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"CategoryTheo... | [] | rw [P.isClosedUnderColimitsOfShape_iff_of_equivalence
(Discrete.equivalence (equivShrink.{w} _))]
exact H _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.ObjectProperty.FiniteProducts | {
"line": 199,
"column": 4
} | {
"line": 201,
"column": 13
} | {
"line": 203,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nP : ObjectProperty C\nH : ∀ (J : Type w) [Finite J], P.IsClosedUnderColimitsOfShape (Discrete J)\nJ : Type\nx✝ : Finite J\n⊢ P.IsClosedUnderColimitsOfShape (Discrete J)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"CategoryTheo... | [] | rw [P.isClosedUnderColimitsOfShape_iff_of_equivalence
(Discrete.equivalence (equivShrink.{w} _))]
exact H _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.ObjectProperty.FiniteProducts | {
"line": 229,
"column": 4
} | {
"line": 231,
"column": 74
} | {
"line": 232,
"column": 4
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\nP : ObjectProperty C\ninst✝¹ : P.ContainsZero\ninst✝ : P.IsClosedUnderIsomorphisms\nX : C\np : P.ColimitOfShape (Discrete PEmpty.{1}) X\nZ : C\nhZ : IsZero Z\nhZ₂ : P Z\n⊢ P X",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Cate... | [
"C : Type u_1\ninst✝² : Category.{v_1, u_1} C\nP : ObjectProperty C\ninst✝¹ : P.ContainsZero\ninst✝ : P.IsClosedUnderIsomorphisms\nX : C\np : P.ColimitOfShape (Discrete PEmpty.{1}) X\nZ : C\nhZ : IsZero Z\nhZ₂ : P Z\nhX : IsInitial X\n⊢ P X"
] | have hX : IsInitial X :=
(IsColimit.equivOfNatIsoOfIso p.diag.uniqueFromEmpty _ _
(by exact Cocone.ext (Iso.refl _) (by rintro ⟨⟨⟩⟩))).1 p.isColimit | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Homology.HomotopyCategory.HomologicalFunctor | {
"line": 36,
"column": 8
} | {
"line": 36,
"column": 56
} | {
"line": 36,
"column": 56
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nn : ℤ\nT : Pretriangulated.Triangle (HomotopyCategory C (ComplexShape.up ℤ))\nhT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles\n⊢ ∃ T' e, ((Pretriangulated.shortComplexOfDistTriangle T' ⋯).map (homologyFunctor C (ComplexShap... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nn : ℤ\nT : Pretriangulated.Triangle (HomotopyCategory C (ComplexShape.up ℤ))\nhT✝ : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles\nhT : ∃ S σ, Nonempty (T ≅ CochainComplex.trianglehOfDegreewiseSplit S σ)\n⊢ ∃ T' e, ((Pretriangulated.short... | distinguished_iff_iso_trianglehOfDegreewiseSplit | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Shift.InducedShiftSequence | {
"line": 112,
"column": 4
} | {
"line": 114,
"column": 30
} | {
"line": 115,
"column": 4
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nA : Type u_3\ninst✝⁹ : Category.{v_1, u_1} C\ninst✝⁸ : Category.{v_2, u_2} D\ninst✝⁷ : Category.{v_3, u_3} A\nL : C ⥤ D\nF : D ⥤ A\nG : C ⥤ A\ne : L ⋙ F ≅ G\nM : Type u_4\ninst✝⁶ : AddMonoid M\ninst✝⁵ : HasShift C M\ninst✝⁴ : G.ShiftSequence M\nF' : M → D ⥤ A\ne' : (m : M) →... | [
"C : Type u_1\nD : Type u_2\nA : Type u_3\ninst✝⁹ : Category.{v_1, u_1} C\ninst✝⁸ : Category.{v_2, u_2} D\ninst✝⁷ : Category.{v_3, u_3} A\nL : C ⥤ D\nF : D ⥤ A\nG : C ⥤ A\ne : L ⋙ F ≅ G\nM : Type u_4\ninst✝⁶ : AddMonoid M\ninst✝⁵ : HasShift C M\ninst✝⁴ : G.ShiftSequence M\nF' : M → D ⥤ A\ne' : (m : M) → L ⋙ F' m ≅ ... | simp only [← NatTrans.naturality_assoc, induced.shiftIso_hom_app_obj,
← Functor.map_comp_assoc, ← Functor.map_comp, Iso.inv_hom_id_app, comp_obj,
Functor.map_id, id_comp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Shift.ShiftedHom | {
"line": 218,
"column": 2
} | {
"line": 221,
"column": 76
} | {
"line": 223,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝⁴ : Category.{v_2, u_2} D\nM : Type u_4\ninst✝³ : AddMonoid M\ninst✝² : HasShift C M\ninst✝¹ : HasShift D M\nX Y Z : C\na b c : M\nf : ShiftedHom X Y a\ng : ShiftedHom Y Z b\nh : b + a = c\nF : C ⥤ D\ninst✝ : F.CommShift M\n⊢ (f.comp g h)... | [] | dsimp [comp, map]
simp only [Functor.map_comp, assoc, ← Functor.commShiftIso_hom_naturality_assoc]
simp only [F.commShiftIso_add' h, Functor.CommShift.isoAdd'_hom_app,
← Functor.map_comp_assoc, Iso.inv_hom_id_app, Functor.comp_obj, comp_id] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Shift.ShiftedHom | {
"line": 218,
"column": 2
} | {
"line": 221,
"column": 76
} | {
"line": 223,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝⁴ : Category.{v_2, u_2} D\nM : Type u_4\ninst✝³ : AddMonoid M\ninst✝² : HasShift C M\ninst✝¹ : HasShift D M\nX Y Z : C\na b c : M\nf : ShiftedHom X Y a\ng : ShiftedHom Y Z b\nh : b + a = c\nF : C ⥤ D\ninst✝ : F.CommShift M\n⊢ (f.comp g h)... | [] | dsimp [comp, map]
simp only [Functor.map_comp, assoc, ← Functor.commShiftIso_hom_naturality_assoc]
simp only [F.commShiftIso_add' h, Functor.CommShift.isoAdd'_hom_app,
← Functor.map_comp_assoc, Iso.inv_hom_id_app, Functor.comp_obj, comp_id] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Triangulated.Subcategory | {
"line": 473,
"column": 4
} | {
"line": 473,
"column": 28
} | {
"line": 474,
"column": 4
} | [
{
"pp": "case mp\nC : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : HasZeroObject C\ninst✝⁴ : HasShift C ℤ\ninst✝³ : Preadditive C\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nP : ObjectProperty C\ninst✝ : P.IsStableUnderShift ℤ\nY Z : C\ng : Y ⟶ Z\n⊢ (∃ Z_1 g_1 h, ∃ (_ : T... | [
"case mp\nC : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : HasZeroObject C\ninst✝⁴ : HasShift C ℤ\ninst✝³ : Preadditive C\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nP : ObjectProperty C\ninst✝ : P.IsStableUnderShift ℤ\nY Z✝ : C\ng✝ : Y ⟶ Z✝\nZ : C\ng : Z✝ ⟶ Z\nh : Z ⟶ (shift... | rintro ⟨Z, g, h, H, mem⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.CategoryTheory.Triangulated.Subcategory | {
"line": 475,
"column": 4
} | {
"line": 475,
"column": 28
} | {
"line": 476,
"column": 4
} | [
{
"pp": "case mpr\nC : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : HasZeroObject C\ninst✝⁴ : HasShift C ℤ\ninst✝³ : Preadditive C\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nP : ObjectProperty C\ninst✝ : P.IsStableUnderShift ℤ\nY Z : C\ng : Y ⟶ Z\n⊢ (∃ X f h, ∃ (_ : Tria... | [
"case mpr\nC : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : HasZeroObject C\ninst✝⁴ : HasShift C ℤ\ninst✝³ : Preadditive C\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\nP : ObjectProperty C\ninst✝ : P.IsStableUnderShift ℤ\nY Z✝ : C\ng✝ : Y ⟶ Z✝\nZ : C\ng : Z ⟶ Y\nh : Z✝ ⟶ (shif... | rintro ⟨Z, g, h, H, mem⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.Homology.HomotopyCategory.ShiftSequence | {
"line": 59,
"column": 8
} | {
"line": 59,
"column": 29
} | {
"line": 59,
"column": 29
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nn i i' : ℤ\nhi : n + i = i'\n⊢ n + (up ℤ).prev i = (up ℤ).prev i'",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne",
"AddRightCance... | [] | simp only [prev]; lia | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.HomotopyCategory.ShiftSequence | {
"line": 59,
"column": 8
} | {
"line": 59,
"column": 29
} | {
"line": 59,
"column": 29
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nn i i' : ℤ\nhi : n + i = i'\n⊢ n + (up ℤ).prev i = (up ℤ).prev i'",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne",
"AddRightCance... | [] | simp only [prev]; lia | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Triangulated.Subcategory | {
"line": 677,
"column": 4
} | {
"line": 677,
"column": 24
} | {
"line": 678,
"column": 4
} | [
{
"pp": "C : Type u_1\ninst✝¹⁴ : Category.{v_1, u_1} C\ninst✝¹³ : HasZeroObject C\ninst✝¹² : HasShift C ℤ\ninst✝¹¹ : Preadditive C\ninst✝¹⁰ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝⁹ : Pretriangulated C\nD : Type u_2\ninst✝⁸ : Category.{v_2, u_2} D\ninst✝⁷ : Preadditive D\ninst✝⁶ : HasZeroObject D\ninst✝⁵... | [
"C : Type u_1\ninst✝¹⁴ : Category.{v_1, u_1} C\ninst✝¹³ : HasZeroObject C\ninst✝¹² : HasShift C ℤ\ninst✝¹¹ : Preadditive C\ninst✝¹⁰ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝⁹ : Pretriangulated C\nD : Type u_2\ninst✝⁸ : Category.{v_2, u_2} D\ninst✝⁷ : Preadditive D\ninst✝⁶ : HasZeroObject D\ninst✝⁵ : HasShift ... | intro _ _ X₁ X₂ f hf | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Algebra.Homology.HomotopyCategory.Triangulated | {
"line": 112,
"column": 30
} | {
"line": 112,
"column": 36
} | {
"line": 112,
"column": 36
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : HasBinaryBiproducts C\nX₁ X₂ X₃ : CochainComplex C ℤ\nf : X₁ ⟶ X₂\ng : X₂ ⟶ X₃\n⊢ -1 + -1 = -2",
"ppTerm": "?m.148",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"Int.instDecidableEq",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
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