module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.CategoryTheory.Limits.Shapes.FiniteMultiequalizer | {
"line": 33,
"column": 10
} | {
"line": 34,
"column": 47
} | {
"line": 35,
"column": 10
} | [
{
"pp": "case pos\nJ : MulticospanShape\ninst✝³ : Fintype J.L\ninst✝² : Fintype J.R\ninst✝¹ : DecidableEq J.L\ninst✝ : DecidableEq J.R\na : J.L\nb : J.R\nh₁ : J.fst b = a\nh₂ : J.snd b = a\n⊢ ({eqToHom ⋯ ≫ Hom.fst b} + {eqToHom ⋯ ≫ Hom.snd b}).Nodup",
"ppTerm": "?pos✝",
"assigned": true,
"usedConsta... | [
"case pos\nJ : MulticospanShape\ninst✝³ : Fintype J.L\ninst✝² : Fintype J.R\ninst✝¹ : DecidableEq J.L\ninst✝ : DecidableEq J.R\na : J.L\nb : J.R\nh₁ : J.fst b = a\nh₂ : J.snd b = a\n⊢ ¬eqToHom ⋯ ≫ Hom.fst b = eqToHom ⋯ ≫ Hom.snd b"
] | simp only [Multiset.singleton_add, Multiset.nodup_cons, Multiset.mem_singleton,
Multiset.nodup_singleton, and_true] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Limits.Shapes.FiniteMultiequalizer | {
"line": 60,
"column": 10
} | {
"line": 61,
"column": 47
} | {
"line": 62,
"column": 10
} | [
{
"pp": "case pos\nJ : MultispanShape\ninst✝³ : Fintype J.L\ninst✝² : Fintype J.R\ninst✝¹ : DecidableEq J.L\ninst✝ : DecidableEq J.R\na : J.L\nb : J.R\nh₁ : J.fst a = b\nh₂ : J.snd a = b\n⊢ ({Hom.fst a ≫ eqToHom ⋯} + {Hom.snd a ≫ eqToHom ⋯}).Nodup",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstant... | [
"case pos\nJ : MultispanShape\ninst✝³ : Fintype J.L\ninst✝² : Fintype J.R\ninst✝¹ : DecidableEq J.L\ninst✝ : DecidableEq J.R\na : J.L\nb : J.R\nh₁ : J.fst a = b\nh₂ : J.snd a = b\n⊢ ¬Hom.fst a ≫ eqToHom ⋯ = Hom.snd a ≫ eqToHom ⋯"
] | simp only [Multiset.singleton_add, Multiset.nodup_cons, Multiset.mem_singleton,
Multiset.nodup_singleton, and_true] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Category.Ring.Topology | {
"line": 120,
"column": 4
} | {
"line": 120,
"column": 40
} | {
"line": 121,
"column": 4
} | [
{
"pp": "R✝ A✝ B C : CommRingCat\ninst✝² : TopologicalSpace ↑R✝\nσ : Type v\nR A : CommRingCat\ninst✝¹ : TopologicalSpace ↑R\ninst✝ : IsTopologicalRing ↑R\n⊢ Continuous[instTopologicalSpaceProd, instTopologicalSpaceHom] fun fx ↦\n ofHom (MvPolynomial.eval₂Hom (Hom.hom fx.1) fx.2)",
"ppTerm": "?m.141",
... | [
"R✝ A✝ B C : CommRingCat\ninst✝² : TopologicalSpace ↑R✝\nσ : Type v\nR A : CommRingCat\ninst✝¹ : TopologicalSpace ↑R\ninst✝ : IsTopologicalRing ↑R\n⊢ Continuous[instTopologicalSpaceProd, inferInstance]\n ((fun f ↦ ⇑(Hom.hom f)) ∘ fun fx ↦ ofHom (MvPolynomial.eval₂Hom (Hom.hom fx.1) fx.2))"
] | refine continuous_induced_rng.mpr ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Category.Ring.Under.Property | {
"line": 151,
"column": 2
} | {
"line": 151,
"column": 34
} | {
"line": 152,
"column": 2
} | [
{
"pp": "R S : CommRingCat\ninst✝ : Algebra ↑R ↑S\nA B : Under R\nf g : A ⟶ B\nc : Fork f g := Under.equalizerFork f g\nhc : IsLimit c := Under.equalizerForkIsLimit f g\nι : R.mkUnder ↥((toAlgHom f).equalizer (toAlgHom g)) ⟶ A := ((toAlgHom f).equalizer (toAlgHom g)).val.toUnder\n⊢ PreservesLimit (parallelPair ... | [
"R S : CommRingCat\ninst✝ : Algebra ↑R ↑S\nA B : Under R\nf g : A ⟶ B\nc : Fork f g := Under.equalizerFork f g\nhc : IsLimit c := Under.equalizerForkIsLimit f g\nι : R.mkUnder ↥((toAlgHom f).equalizer (toAlgHom g)) ⟶ A := ((toAlgHom f).equalizer (toAlgHom g)).val.toUnder\nh' : (R.tensorProd S).obj (R.mkUnder ↥((toA... | let h' := (R.tensorProd S).map ι | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.Flat.Equalizer | {
"line": 61,
"column": 4
} | {
"line": 62,
"column": 21
} | {
"line": 64,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\nM : Type u_3\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : Module S M\ninst✝⁴ : IsScalarTower R S M\nN : Type u_4\nP : Type u_5\ninst✝³ : AddCommGroup N\ninst✝² : AddCommGroup P\ninst✝¹ : Module R N\n... | [] | ext
simp [smul_tmul'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Flat.Equalizer | {
"line": 61,
"column": 4
} | {
"line": 62,
"column": 21
} | {
"line": 64,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\nM : Type u_3\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : Module S M\ninst✝⁴ : IsScalarTower R S M\nN : Type u_4\nP : Type u_5\ninst✝³ : AddCommGroup N\ninst✝² : AddCommGroup P\ninst✝¹ : Module R N\n... | [] | ext
simp [smul_tmul'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Flat.Equalizer | {
"line": 116,
"column": 2
} | {
"line": 116,
"column": 49
} | {
"line": 117,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra R S\nM : Type u_3\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : Module S M\ninst✝⁵ : IsScalarTower R S M\nN : Type u_4\nP : Type u_5\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R N\... | [
"R : Type u_1\nS : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra R S\nM : Type u_3\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : Module S M\ninst✝⁵ : IsScalarTower R S M\nN : Type u_4\nP : Type u_5\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R N\ninst✝¹ : Mo... | rw [← AlgebraTensorModule.coe_lTensor (A := S)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Flat.Equalizer | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 49
} | {
"line": 135,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra R S\nM : Type u_3\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : Module S M\ninst✝⁵ : IsScalarTower R S M\nN : Type u_4\nP : Type u_5\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R N\... | [
"R : Type u_1\nS : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra R S\nM : Type u_3\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : Module S M\ninst✝⁵ : IsScalarTower R S M\nN : Type u_4\nP : Type u_5\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R N\ninst✝¹ : Mo... | rw [← AlgebraTensorModule.coe_lTensor (A := S)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.CharP.MixedCharZero | {
"line": 256,
"column": 6
} | {
"line": 256,
"column": 69
} | {
"line": 257,
"column": 6
} | [
{
"pp": "case intro.succ\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharZero R\nh : ∀ p > 0, ¬MixedCharZero R p\nI : Ideal R\nhI_ne_top : I ≠ ⊤\np : ℕ\nhp : CharP (R ⧸ I) (p + 1)\n⊢ CharP (R ⧸ I) 0",
"ppTerm": "?intro.succ",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"Mix... | [
"case intro.succ\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharZero R\nh : ∀ p > 0, ¬MixedCharZero R p\nI : Ideal R\nhI_ne_top : I ≠ ⊤\np : ℕ\nhp : CharP (R ⧸ I) (p + 1)\nh_mixed : MixedCharZero R p.succ\n⊢ CharP (R ⧸ I) 0"
] | have h_mixed : MixedCharZero R p.succ := ⟨⟨I, ⟨hI_ne_top, hp⟩⟩⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.CharP.MixedCharZero | {
"line": 255,
"column": 4
} | {
"line": 255,
"column": 15
} | {
"line": 256,
"column": 6
} | [
{
"pp": "case intro.succ\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharZero R\nh : ∀ p > 0, ¬MixedCharZero R p\nI : Ideal R\nhI_ne_top : I ≠ ⊤\np : ℕ\nhp : CharP (R ⧸ I) (p + 1)\n⊢ CharP (R ⧸ I) 0",
"ppTerm": "?intro.succ",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"Mix... | [] | | succ p => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.RingTheory.FreeCommRing | {
"line": 152,
"column": 6
} | {
"line": 152,
"column": 64
} | {
"line": 154,
"column": 0
} | [
{
"pp": "α : Type u\nR : Type v\ninst✝ : CommRing R\nf : α → R\nF : Multiplicative (Multiset α) →* R\nx : Multiplicative (Multiset α)\nF' : Multiset α →+ Additive R := MonoidHom.toAdditiveRight F\nx' : Multiset α := Multiplicative.toAdd x\n⊢ F' (Multiset.map (fun x ↦ {x}) x').sum = F' x'",
"ppTerm": "?m.138... | [] | exact DFunLike.congr_arg F (Multiset.sum_map_singleton x') | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Stream.Init | {
"line": 348,
"column": 97
} | {
"line": 351,
"column": 67
} | {
"line": 353,
"column": 0
} | [
{
"pp": "α : Type u\nn : ℕ\ns : Stream' α\n⊢ (unfolds head tail s).get n = s.get n",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Stream'.get_succ",
"Nat.recAux",
"congrArg",
"id",
"instOfNatNat",
"Stream'",
"instHAdd",
"Stre... | [] | by
induction n generalizing s with
| zero => rfl
| succ n ih => rw [get_succ, get_succ, unfolds_eq, tail_cons, ih] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Seq.Defs | {
"line": 154,
"column": 21
} | {
"line": 154,
"column": 75
} | {
"line": 154,
"column": 75
} | [
{
"pp": "α : Type u\nx y : α\ns t : Seq α\nh : cons x s = cons y t\nn : ℕ\n⊢ s.get? n = t.get? n",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Stream'.Seq",
"congrArg",
"id",
"instOfNatNat",
"instHAdd",
"HAdd.hAdd",
"Nat",
... | [] | by simp_rw [← get?_cons_succ x s n, h, get?_cons_succ] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Seq.Defs | {
"line": 363,
"column": 2
} | {
"line": 363,
"column": 82
} | {
"line": 364,
"column": 2
} | [
{
"pp": "α : Type u\nR : Seq α → Seq α → Prop\nbisim : IsBisimulation R\ns₁ s₂ : Seq α\nr : R s₁ s₂\n⊢ ↑s₁ = ↑s₂",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Stream'.Seq",
"Exists",
"Stream'",
"And",
"Stream'.IsSeq",
"Stream'.eq_of_bisim",
"Sub... | [
"case bisim\nα : Type u\nR : Seq α → Seq α → Prop\nbisim : IsBisimulation R\ns₁ s₂ : Seq α\nr : R s₁ s₂\n⊢ Stream'.IsBisimulation fun x y ↦ ∃ s s', ↑s = x ∧ ↑s' = y ∧ R s s'",
"case a\nα : Type u\nR : Seq α → Seq α → Prop\nbisim : IsBisimulation R\ns₁ s₂ : Seq α\nr : R s₁ s₂\n⊢ ∃ s s', ↑s = ↑s₁ ∧ ↑s' = ↑s₂ ∧ R s ... | apply Stream'.eq_of_bisim fun x y => ∃ s s' : Seq α, s.1 = x ∧ s'.1 = y ∧ R s s' | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Data.Seq.Defs | {
"line": 541,
"column": 2
} | {
"line": 541,
"column": 20
} | {
"line": 541,
"column": 20
} | [
{
"pp": "α : Type u\nC : Seq α → Prop\na : α\ns : Seq α\nM : a ∈ s\nh1 : ∀ (b : α) (s' : Seq α), a = b ∨ C s' → C (cons b s')\n⊢ C s",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Stream'.Seq",
"Option.some",
"Membership.mem",
"Stream'",
"Stream'.Seq.instMem... | [
"α : Type u\nC : Seq α → Prop\na : α\ns : Seq α\nh1 : ∀ (b : α) (s' : Seq α), a = b ∨ C s' → C (cons b s')\nk : ℕ\ne : some a = (↑s).get k\n⊢ C s"
] | obtain ⟨k, e⟩ := M | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Data.Seq.Defs | {
"line": 555,
"column": 6
} | {
"line": 555,
"column": 97
} | {
"line": 556,
"column": 6
} | [
{
"pp": "case succ.cons\nα : Type u\nC : Seq α → Prop\na : α\nh1 : ∀ (b : α) (s' : Seq α), a = b ∨ C s' → C (cons b s')\nk : ℕ\nIH : ∀ {s : Seq α}, some a = ↑s k → C s\nb : α\ns' : Seq α\ne : some a = ↑(cons b s') (k + 1)\n⊢ C (cons b s')",
"ppTerm": "?succ.cons",
"assigned": true,
"usedConstants": ... | [
"case succ.cons\nα : Type u\nC : Seq α → Prop\na : α\nh1 : ∀ (b : α) (s' : Seq α), a = b ∨ C s' → C (cons b s')\nk : ℕ\nIH : ∀ {s : Seq α}, some a = ↑s k → C s\nb : α\ns' : Seq α\ne : some a = ↑(cons b s') (k + 1)\nh_eq : ↑(cons b s') k.succ = ↑s' k\n⊢ C (cons b s')"
] | have h_eq : (cons b s').val (Nat.succ k) = s'.val k := by cases s' using Subtype.recOn; rfl | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.ContinuedFractions.ContinuantsRecurrence | {
"line": 52,
"column": 2
} | {
"line": 53,
"column": 36
} | {
"line": 54,
"column": 2
} | [
{
"pp": "K : Type u_1\ng : GenContFract K\nn : ℕ\ninst✝ : DivisionRing K\ngp : Pair K\nppredA predA : K\nsucc_nth_s_eq : g.s.get? (n + 1) = some gp\nnth_num_eq : g.nums n = ppredA\nsucc_nth_num_eq : g.nums (n + 1) = predA\n⊢ g.nums (n + 2) = gp.b * predA + gp.a * ppredA",
"ppTerm": "?m.46",
"assigned": ... | [
"K : Type u_1\ng : GenContFract K\nn : ℕ\ninst✝ : DivisionRing K\ngp : Pair K\npredA : K\nsucc_nth_s_eq : g.s.get? (n + 1) = some gp\nsucc_nth_num_eq : g.nums (n + 1) = predA\nppredConts : Pair K\nnth_conts_eq : g.conts n = ppredConts\nnth_num_eq : g.nums n = ppredConts.a\n⊢ g.nums (n + 2) = gp.b * predA + gp.a * p... | obtain ⟨ppredConts, nth_conts_eq, ⟨rfl⟩⟩ : ∃ conts, g.conts n = conts ∧ conts.a = ppredA :=
exists_conts_a_of_num nth_num_eq | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Data.Seq.Basic | {
"line": 310,
"column": 2
} | {
"line": 310,
"column": 11
} | {
"line": 310,
"column": 11
} | [
{
"pp": "α : Type u\ns₁ s₂ : Seq α\na : α\nh : a ∈ s₁.append s₂\n⊢ a ∈ s₁ ∨ a ∈ s₂",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Stream'.Seq",
"Membership.mem",
"Stream'.Seq.instMembership",
"Stream'.Seq.append"
],
"usedFVars": [
"α",
"s₁",
... | [
"α : Type u\ns₁ s₂ : Seq α\na : α\nh this : a ∈ s₁.append s₂\n⊢ a ∈ s₁ ∨ a ∈ s₂"
] | have := h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.ContinuedFractions.Determinant | {
"line": 89,
"column": 33
} | {
"line": 89,
"column": 78
} | {
"line": 89,
"column": 79
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\ns : SimpContFract K\nn : ℕ\nnot_terminatedAt_n : ¬(↑s).TerminatedAt n\ni : ℕ\nhi : i < n + 1\ngp : Pair K\ns_ith_eq : (↑s).s.get? i = some gp\n⊢ -(some gp.a).getD 0 = -1",
"ppTerm": "?m.169",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroC... | [
"K : Type u_1\ninst✝ : Field K\ns : SimpContFract K\nn : ℕ\nnot_terminatedAt_n : ¬(↑s).TerminatedAt n\ni : ℕ\nhi : i < n + 1\ngp : Pair K\ns_ith_eq : (↑s).s.get? i = some gp\n⊢ -(some 1).getD 0 = -1"
] | s.property i gp.a <| partNum_eq_s_a s_ith_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Colimit.Ring | {
"line": 110,
"column": 30
} | {
"line": 110,
"column": 64
} | {
"line": 110,
"column": 64
} | [
{
"pp": "case e_a.e_f\nι : Type u_1\ninst✝³ : Preorder ι\nG : ι → Type u_2\ninst✝² : (i : ι) → CommRing (G i)\nf' : (i j : ι) → i ≤ j → G i →+* G j\ninst✝¹ : Nonempty ι\ninst✝ : IsDirectedOrder ι\nq : (DirectLimit G fun i j h ↦ ⇑(f' i j h))[X]\nq₁ q₂ : (DirectLimit G fun i j h ↦ ⇑(f' i j h))[X]\nx✝¹ : ∃ i p, Po... | [] | simp_rw [RingHom.comp_apply, of_f] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Algebra.Colimit.Ring | {
"line": 110,
"column": 30
} | {
"line": 110,
"column": 64
} | {
"line": 110,
"column": 64
} | [
{
"pp": "case e_a.e_f\nι : Type u_1\ninst✝³ : Preorder ι\nG : ι → Type u_2\ninst✝² : (i : ι) → CommRing (G i)\nf' : (i j : ι) → i ≤ j → G i →+* G j\ninst✝¹ : Nonempty ι\ninst✝ : IsDirectedOrder ι\nq : (DirectLimit G fun i j h ↦ ⇑(f' i j h))[X]\nq₁ q₂ : (DirectLimit G fun i j h ↦ ⇑(f' i j h))[X]\nx✝¹ : ∃ i p, Po... | [] | simp_rw [RingHom.comp_apply, of_f] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Data.Seq.Computation | {
"line": 1064,
"column": 2
} | {
"line": 1064,
"column": 47
} | {
"line": 1064,
"column": 48
} | [
{
"pp": "α : Type u\nβ : Type v\nR : α → β → Prop\nC : Computation α → Computation β → Prop\na : α ⊕ Computation α\nb : β ⊕ Computation β\n⊢ LiftRelAux (Function.swap R) (Function.swap C) b a = LiftRelAux R C a b",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Computation.LiftRelAux... | [
"case inl.inl\nα : Type u\nβ : Type v\nR : α → β → Prop\nC : Computation α → Computation β → Prop\na : α\nb : β\n⊢ LiftRelAux (Function.swap R) (Function.swap C) (Sum.inl b) (Sum.inl a) = LiftRelAux R C (Sum.inl a) (Sum.inl b)",
"case inl.inr\nα : Type u\nβ : Type v\nR : α → β → Prop\nC : Computation α → Computat... | rcases a with a | ca <;> rcases b with b | cb | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating | {
"line": 167,
"column": 6
} | {
"line": 167,
"column": 24
} | {
"line": 168,
"column": 6
} | [
{
"pp": "case succ.inr\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : LinearOrder K\nv : K\nn✝ : ℕ\ninst✝ : FloorRing K\ng : GenContFract K := of v\nn : ℕ\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr ≠ 0\nconts : Pair K := g.contsAux (n + 2)\npconts : Pa... | [
"case succ.inr\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : LinearOrder K\nv : K\nn✝ : ℕ\ninst✝ : FloorRing K\ng : GenContFract K := of v\nn : ℕ\nifp_n : IntFractPair K\nnth_stream_eq : IntFractPair.stream v n = some ifp_n\nnth_fract_ne_zero : ifp_n.fr ≠ 0\nconts : Pair K := g.contsAux (n + 2)\npconts : Pair K := g.co... | let pB := pconts.b | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Algebra.ContinuedFractions.Computation.TerminatesIffRat | {
"line": 285,
"column": 2
} | {
"line": 285,
"column": 63
} | {
"line": 286,
"column": 2
} | [
{
"pp": "q : ℚ\nfract_q_num : ℤ := (Int.fract q).num\nn : ℕ := fract_q_num.natAbs + 1\n⊢ ∃ n, IntFractPair.stream q n = none",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"GenContFract.IntFractPair.stream",
"Rat",
"Rat.instFloorRing",
"Rat.linearOrder",
"Opt... | [
"case none\nq : ℚ\nfract_q_num : ℤ := (Int.fract q).num\nn : ℕ := fract_q_num.natAbs + 1\nstream_nth_eq : IntFractPair.stream q n = none\n⊢ ∃ n, IntFractPair.stream q n = none",
"case some\nq : ℚ\nfract_q_num : ℤ := (Int.fract q).num\nn : ℕ := fract_q_num.natAbs + 1\nifp : IntFractPair ℚ\nstream_nth_eq : IntFract... | rcases stream_nth_eq : IntFractPair.stream q n with ifp | ifp | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Algebra.ContinuedFractions.ConvergentsEquiv | {
"line": 287,
"column": 10
} | {
"line": 287,
"column": 85
} | {
"line": 287,
"column": 85
} | [
{
"pp": "K : Type u_1\ng : GenContFract K\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_partDen_ne_zero : ∀ {b : K}, g.partDens.get? (n' + 1) = some b → b ≠ 0\nnot_terminatedAt_n : ¬g.TerminatedAt (n' + 1)\ns_nth_eq : g.s.get? (n' + 1) = some { a := a, b := b }\npa pb : K\ns_n'th_eq : g.s.get? n' = ... | [
"K : Type u_1\ng : GenContFract K\ninst✝ : Field K\na b : K\nb_ne_zero : b ≠ 0\nn' : ℕ\nnth_partDen_ne_zero : ∀ {b : K}, g.partDens.get? (n' + 1) = some b → b ≠ 0\nnot_terminatedAt_n : ¬g.TerminatedAt (n' + 1)\ns_nth_eq : g.s.get? (n' + 1) = some { a := a, b := b }\npa pb : K\ns_n'th_eq : g.s.get? n' = some { a := ... | conts_recurrenceAux this n'th_contsAux_eq'.symm succ_n'th_contsAux_eq'.symm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.ContinuedFractions.ConvergentsEquiv | {
"line": 335,
"column": 63
} | {
"line": 335,
"column": 71
} | {
"line": 335,
"column": 72
} | [
{
"pp": "case succ.inl\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : LinearOrder K\ninst✝ : IsStrictOrderedRing K\nn : ℕ\nIH :\n ∀ {g : GenContFract K},\n (∀ {gp : Pair K} {m : ℕ}, m < n → g.s.get? m = some gp → 0 < gp.a ∧ 0 < gp.b) → g.convs n = g.convs' n\ng : GenContFract K\ns_pos : ∀ {gp : Pair K} {m : ℕ}, ... | [
"case succ.inl\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : LinearOrder K\ninst✝ : IsStrictOrderedRing K\nn : ℕ\nIH :\n ∀ {g : GenContFract K},\n (∀ {gp : Pair K} {m : ℕ}, m < n → g.s.get? m = some gp → 0 < gp.a ∧ 0 < gp.b) → g.convs n = g.convs' n\ng : GenContFract K\ns_pos : ∀ {gp : Pair K} {m : ℕ}, m < n + 1 → ... | g'_eq_g, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.CubicDiscriminant | {
"line": 185,
"column": 92
} | {
"line": 186,
"column": 11
} | {
"line": 188,
"column": 0
} | [
{
"pp": "R : Type u_1\nb c d : R\ninst✝ : Semiring R\nhb : b ≠ 0\n⊢ { a := 0, b := b, c := c, d := d }.toPoly.leadingCoeff = b",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"False",
"eq_false",
"congrArg",
"Polynomial.leadingCoeff",
"Cubic.toPoly",
"Cu... | [] | by
simp [hb] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.CubicDiscriminant | {
"line": 269,
"column": 80
} | {
"line": 270,
"column": 11
} | {
"line": 272,
"column": 0
} | [
{
"pp": "R : Type u_1\nb c d : R\ninst✝ : Semiring R\nhb : b ≠ 0\n⊢ { a := 0, b := b, c := c, d := d }.toPoly.degree = 2",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"False",
"WithBot",
"eq_false",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"WithBot.i... | [] | by
simp [hb] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.CubicDiscriminant | {
"line": 328,
"column": 86
} | {
"line": 329,
"column": 11
} | {
"line": 331,
"column": 0
} | [
{
"pp": "R : Type u_1\nb c d : R\ninst✝ : Semiring R\nhb : b ≠ 0\n⊢ { a := 0, b := b, c := c, d := d }.toPoly.natDegree = 2",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"False",
"Cubic.natDegree_of_b_ne_zero",
"eq_false",
"congrArg",
"instOfNatNat",
"... | [] | by
simp [hb] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.CubicDiscriminant | {
"line": 437,
"column": 2
} | {
"line": 439,
"column": 45
} | {
"line": 441,
"column": 0
} | [
{
"pp": "F : Type u_3\nK : Type u_4\nP : Cubic F\ninst✝¹ : Field F\ninst✝ : Field K\nφ : F →+* K\nx y z : K\nha : P.a ≠ 0\nh3 : (map φ P).roots = {x, y, z}\n⊢ map φ P = { a := φ P.a, b := φ P.a * -(x + y + z), c := φ P.a * (x * y + x * z + y * z), d := φ P.a * -(x * y * z) }",
"ppTerm": "?m.94",
"assign... | [] | apply_fun toPoly
· rw [eq_prod_three_roots ha h3, C_mul_prod_X_sub_C_eq]
· exact fun P Q ↦ (toPoly_injective P Q).mp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.CubicDiscriminant | {
"line": 437,
"column": 2
} | {
"line": 439,
"column": 45
} | {
"line": 441,
"column": 0
} | [
{
"pp": "F : Type u_3\nK : Type u_4\nP : Cubic F\ninst✝¹ : Field F\ninst✝ : Field K\nφ : F →+* K\nx y z : K\nha : P.a ≠ 0\nh3 : (map φ P).roots = {x, y, z}\n⊢ map φ P = { a := φ P.a, b := φ P.a * -(x + y + z), c := φ P.a * (x * y + x * z + y * z), d := φ P.a * -(x * y * z) }",
"ppTerm": "?m.94",
"assign... | [] | apply_fun toPoly
· rw [eq_prod_three_roots ha h3, C_mul_prod_X_sub_C_eq]
· exact fun P Q ↦ (toPoly_injective P Q).mp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.ContinuedFractions.Computation.Approximations | {
"line": 441,
"column": 8
} | {
"line": 446,
"column": 50
} | {
"line": 447,
"column": 8
} | [
{
"pp": "K : Type u_1\nv : K\nn : ℕ\ninst✝³ : Field K\ninst✝² : LinearOrder K\ninst✝¹ : IsStrictOrderedRing K\ninst✝ : FloorRing K\nnot_terminatedAt_n : ¬(of v).TerminatedAt n\ng : GenContFract K := of v\nnextConts : Pair K := g.contsAux (n + 2)\nconts : Pair K := g.contsAux (n + 1)\nconts_eq : conts = g.contsA... | [
"K : Type u_1\nv : K\nn : ℕ\ninst✝³ : Field K\ninst✝² : LinearOrder K\ninst✝¹ : IsStrictOrderedRing K\ninst✝ : FloorRing K\nnot_terminatedAt_n : ¬(of v).TerminatedAt n\ng : GenContFract K := of v\nnextConts : Pair K := g.contsAux (n + 2)\nconts : Pair K := g.contsAux (n + 1)\nconts_eq : conts = g.contsAux (n + 1)\n... | have : 0 ≤ pred_conts.b :=
haveI : (fib n : K) ≤ pred_conts.b :=
haveI : ¬g.TerminatedAt (n - 2) :=
mt (terminated_stable (n.sub_le 2)) not_terminatedAt_n
fib_le_of_contsAux_b <| Or.inr this
le_trans (mod_cast (fib n).zero_le) this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Idempotents | {
"line": 222,
"column": 6
} | {
"line": 222,
"column": 16
} | {
"line": 223,
"column": 4
} | [
{
"pp": "case inr.refine_1.zero\nR : Type u_1\nS : Type u_2\ninst✝¹ : Ring R\ninst✝ : Ring S\nf : R →+* S\nh : ∀ x ∈ RingHom.ker f, IsNilpotent x\ne₂ : R\nhe₂ : IsIdempotentElem e₂\ne₁ : R\nhe₁ : IsIdempotentElem (f e₁)\nhe₁e₂ : f e₁ * f e₂ = 0\nh✝ : Nontrivial R\na : R := e₁ - e₁ * e₂\nha : f a = f e₁\nha' : a... | [] | simp at hn | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Idempotents | {
"line": 222,
"column": 6
} | {
"line": 222,
"column": 16
} | {
"line": 223,
"column": 4
} | [
{
"pp": "case inr.refine_1.zero\nR : Type u_1\nS : Type u_2\ninst✝¹ : Ring R\ninst✝ : Ring S\nf : R →+* S\nh : ∀ x ∈ RingHom.ker f, IsNilpotent x\ne₂ : R\nhe₂ : IsIdempotentElem e₂\ne₁ : R\nhe₁ : IsIdempotentElem (f e₁)\nhe₁e₂ : f e₁ * f e₂ = 0\nh✝ : Nontrivial R\na : R := e₁ - e₁ * e₂\nha : f a = f e₁\nha' : a... | [] | simp at hn | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Idempotents | {
"line": 222,
"column": 6
} | {
"line": 222,
"column": 16
} | {
"line": 223,
"column": 4
} | [
{
"pp": "case inr.refine_1.zero\nR : Type u_1\nS : Type u_2\ninst✝¹ : Ring R\ninst✝ : Ring S\nf : R →+* S\nh : ∀ x ∈ RingHom.ker f, IsNilpotent x\ne₂ : R\nhe₂ : IsIdempotentElem e₂\ne₁ : R\nhe₁ : IsIdempotentElem (f e₁)\nhe₁e₂ : f e₁ * f e₂ = 0\nh✝ : Nontrivial R\na : R := e₁ - e₁ * e₂\nha : f a = f e₁\nha' : a... | [] | simp at hn | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Idempotents | {
"line": 230,
"column": 6
} | {
"line": 230,
"column": 16
} | {
"line": 231,
"column": 4
} | [
{
"pp": "case inr.refine_2.zero\nR : Type u_1\nS : Type u_2\ninst✝¹ : Ring R\ninst✝ : Ring S\nf : R →+* S\nh : ∀ x ∈ RingHom.ker f, IsNilpotent x\ne₂ : R\nhe₂ : IsIdempotentElem e₂\ne₁ : R\nhe₁ : IsIdempotentElem (f e₁)\nhe₁e₂ : f e₁ * f e₂ = 0\nh✝ : Nontrivial R\na : R := e₁ - e₁ * e₂\nha : f a = f e₁\nha' : a... | [] | simp at hn | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Idempotents | {
"line": 230,
"column": 6
} | {
"line": 230,
"column": 16
} | {
"line": 231,
"column": 4
} | [
{
"pp": "case inr.refine_2.zero\nR : Type u_1\nS : Type u_2\ninst✝¹ : Ring R\ninst✝ : Ring S\nf : R →+* S\nh : ∀ x ∈ RingHom.ker f, IsNilpotent x\ne₂ : R\nhe₂ : IsIdempotentElem e₂\ne₁ : R\nhe₁ : IsIdempotentElem (f e₁)\nhe₁e₂ : f e₁ * f e₂ = 0\nh✝ : Nontrivial R\na : R := e₁ - e₁ * e₂\nha : f a = f e₁\nha' : a... | [] | simp at hn | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Idempotents | {
"line": 230,
"column": 6
} | {
"line": 230,
"column": 16
} | {
"line": 231,
"column": 4
} | [
{
"pp": "case inr.refine_2.zero\nR : Type u_1\nS : Type u_2\ninst✝¹ : Ring R\ninst✝ : Ring S\nf : R →+* S\nh : ∀ x ∈ RingHom.ker f, IsNilpotent x\ne₂ : R\nhe₂ : IsIdempotentElem e₂\ne₁ : R\nhe₁ : IsIdempotentElem (f e₁)\nhe₁e₂ : f e₁ * f e₂ = 0\nh✝ : Nontrivial R\na : R := e₁ - e₁ * e₂\nha : f a = f e₁\nha' : a... | [] | simp at hn | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.ContinuedFractions.Computation.Approximations | {
"line": 473,
"column": 70
} | {
"line": 487,
"column": 46
} | {
"line": 489,
"column": 0
} | [
{
"pp": "K : Type u_1\nv : K\nn : ℕ\ninst✝³ : Field K\ninst✝² : LinearOrder K\ninst✝¹ : IsStrictOrderedRing K\ninst✝ : FloorRing K\nb : K\nnth_partDen_eq : (of v).partDens.get? n = some b\n⊢ |v - (of v).convs n| ≤ 1 / (b * (of v).dens n * (of v).dens n)",
"ppTerm": "?m.54",
"assigned": true,
"usedCo... | [] | by
have not_terminatedAt_n : ¬(of v).TerminatedAt n := by
simp [terminatedAt_iff_partDen_none, nth_partDen_eq]
refine (abs_sub_convs_le not_terminatedAt_n).trans ?_
-- One can show that `0 < (GenContFract.of v).dens n` but it's easier
-- to consider the case `(GenContFract.of v).dens n = 0`.
rcases (zero_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Idempotents | {
"line": 312,
"column": 4
} | {
"line": 312,
"column": 14
} | {
"line": 313,
"column": 2
} | [
{
"pp": "case inr.inl\nR : Type u_1\nS : Type u_2\ninst✝¹ : Ring R\ninst✝ : Ring S\nf : R →+* S\nh : ∀ x ∈ RingHom.ker f, IsNilpotent x\nn✝ : ℕ\ne : Fin n✝ → S\nhe : CompleteOrthogonalIdempotents e\nhe' : ∀ (i : Fin n✝), e i ∈ f.range\nh✝¹ : Nontrivial R\nh✝ : Subsingleton S\nn : ℕ\nhn : 1 ^ n = 0\n⊢ ∃ e', Comp... | [] | simp at hn | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Eigenspace.Basic | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 30
} | {
"line": 117,
"column": 0
} | [
{
"pp": "R : Type v\nM : Type w\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\nμ : R\n⊢ ⨆ i, ⨆ (h : ↑i ≤ ⊤), (f.genEigenspace μ) ↑↑⟨i, h⟩ = ⨆ k, (f.genEigenspace μ) ↑k",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Submodule",
"ENat.instNatCas... | [] | simp only [le_top, iSup_pos] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.DualNumber | {
"line": 213,
"column": 2
} | {
"line": 213,
"column": 69
} | {
"line": 214,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : TrivSqZeroExt A A\n⊢ x ∈ (inlAlgHom R A A).range ⊔ R[ ε]",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"Eq.mpr",
"NonAssocSemiring.toAdd... | [
"R : Type u_1\nA : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : TrivSqZeroExt A A\n⊢ inl x.fst + inl x.snd * ε ∈ (inlAlgHom R A A).range ⊔ R[ ε]"
] | rw [← x.inl_fst_add_inr_snd_eq, inr_eq_smul_eps, ← inl_mul_eq_smul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.DualNumber | {
"line": 214,
"column": 2
} | {
"line": 214,
"column": 35
} | {
"line": 215,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : TrivSqZeroExt A A\n⊢ inl x.fst + inl x.snd * ε ∈ (inlAlgHom R A A).range ⊔ R[ ε]",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"NonAssocSemiri... | [
"case refine_1\nR : Type u_1\nA : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : TrivSqZeroExt A A\n⊢ inl x.fst ∈ (inlAlgHom R A A).range ⊔ R[ ε]",
"case refine_2\nR : Type u_1\nA : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : TrivSqZeroExt A A... | refine add_mem ?_ (mul_mem ?_ ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.Trace | {
"line": 197,
"column": 55
} | {
"line": 197,
"column": 96
} | {
"line": 199,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Free R M\ninst✝ : Module.Finite R M\n⊢ (trace R M) id = ↑(finrank R M)",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"LinearMap.trace",
"LinearMap.id",
"Eq.m... | [] | by rw [← Module.End.one_eq_id, trace_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Trace | {
"line": 374,
"column": 2
} | {
"line": 377,
"column": 65
} | {
"line": 378,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsReduced R\nf g : End R M\nμ : R\nh_comm : Commute f g\nn : End R M := g - (algebraMap R (End R M)) μ\nhg : IsNilpotent n\n⊢ (trace R M) (f ∘ₗ g) = μ * (trace R M) f",
"ppTerm": "?m.85",
"ass... | [
"R : Type u_1\ninst✝³ : CommRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsReduced R\nf g : End R M\nμ : R\nh_comm : Commute f g\nn : End R M := g - (algebraMap R (End R M)) μ\nhg : (trace R M) (f ∘ₗ n) = 0\n⊢ (trace R M) (f ∘ₗ g) = μ * (trace R M) f"
] | replace hg : trace R M (f ∘ₗ n) = 0 := by
rw [← isNilpotent_iff_eq_zero, ← Module.End.mul_eq_comp]
refine isNilpotent_trace_of_isNilpotent (Commute.isNilpotent_mul_left ?_ hg)
exact h_comm.sub_right (Algebra.commute_algebraMap_right μ f) | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.GroupTheory.FreeGroup.Reduce | {
"line": 58,
"column": 12
} | {
"line": 58,
"column": 25
} | {
"line": 59,
"column": 2
} | [
{
"pp": "case zero\nα : Type u_1\ninst✝ : DecidableEq α\nx : α × Bool\n⊢ reduce (List.replicate 0 x) = List.replicate 0 x",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"List",
"eq_self",
"Bool",
"of_eq_true",
"Prod",
"Eq",
"List.nil"
],
"... | [] | simp [reduce] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.FreeGroup.Reduce | {
"line": 58,
"column": 12
} | {
"line": 58,
"column": 25
} | {
"line": 59,
"column": 2
} | [
{
"pp": "case zero\nα : Type u_1\ninst✝ : DecidableEq α\nx : α × Bool\n⊢ reduce (List.replicate 0 x) = List.replicate 0 x",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"List",
"eq_self",
"Bool",
"of_eq_true",
"Prod",
"Eq",
"List.nil"
],
"... | [] | simp [reduce] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.FreeGroup.Reduce | {
"line": 58,
"column": 12
} | {
"line": 58,
"column": 25
} | {
"line": 59,
"column": 2
} | [
{
"pp": "case zero\nα : Type u_1\ninst✝ : DecidableEq α\nx : α × Bool\n⊢ reduce (List.replicate 0 x) = List.replicate 0 x",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"List",
"eq_self",
"Bool",
"of_eq_true",
"Prod",
"Eq",
"List.nil"
],
"... | [] | simp [reduce] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Eisenstein.Basic | {
"line": 100,
"column": 71
} | {
"line": 122,
"column": 73
} | {
"line": 124,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\nf : R[X]\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R\nx : S\nhx : (aeval x) f = 0\nhmo : f.Monic\nhf : f.IsWeaklyEisensteinAt (R ∙ p)\n⊢ ∃ y ∈ R[x], (algebraMap R S) p * y = x ^ (Polynomial.map (algebraMap R S) f).natDegree",
"ppTerm": "?m.59",
... | [] | by
rw [aeval_def, Polynomial.eval₂_eq_eval_map, eval_eq_sum_range, range_add_one,
sum_insert notMem_range_self, sum_range, (hmo.map (algebraMap R S)).coeff_natDegree,
one_mul] at hx
replace hx := eq_neg_of_add_eq_zero_left hx
have : ∀ n < f.natDegree, p ∣ f.coeff n := by
intro n hn
exact mem_span_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.GCDMonoid.IntegrallyClosed | {
"line": 48,
"column": 4
} | {
"line": 48,
"column": 16
} | {
"line": 49,
"column": 4
} | [
{
"pp": "case h\nR : Type u_1\nA : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nX : FractionRing R\nx✝ : IsIntegral R X\np : R[X]\nhp₁ : p.Monic\nhp₂ : Polynomial.eval₂ (algebraMap R (FractionRing R)) X p = 0\nval✝ : GCDMonoid R\nx y : R\nhg : IsUnit (gcd x y)\nhe : X * (algebraMap R... | [
"case h\nR : Type u_1\nA : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nX : FractionRing R\nx✝ : IsIntegral R X\np : R[X]\nhp₁ : p.Monic\nhp₂ : Polynomial.eval₂ (algebraMap R (FractionRing R)) X p = 0\nval✝ : GCDMonoid R\nx y : R\nhg : IsUnit (gcd x y)\nhe : X * (algebraMap R (FractionRi... | rw [map_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Group.Action.Equidecomp | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 19
} | {
"line": 169,
"column": 0
} | [
{
"pp": "X : Type u_1\nG : Type u_2\ninst✝¹ : Monoid G\ninst✝ : MulAction G X\ng f : X → X\nB A : Set X\nT S : Finset G\nhg : IsDecompOn g B T\nhf : IsDecompOn f A S\nh : MapsTo f A B\n⊢ IsDecompOn (g ∘ f) (A ∩ fun a ↦ B (f a)) (T * S)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
... | [] | exact hg.comp' hf | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Quaternion | {
"line": 1253,
"column": 6
} | {
"line": 1253,
"column": 53
} | {
"line": 1253,
"column": 53
} | [
{
"pp": "R : Type u_1\nc₁ c₂ c₃ : R\n⊢ #ℍ[R,c₁,c₂,c₃] = #R ^ 4",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardinal",
"congrArg",
"CommSemiring.toSemiring",
"Cardinal.commSemiring",
"Cardinal.mk",
"QuaternionAlgebra",
"id",
... | [
"R : Type u_1\nc₁ c₂ c₃ : R\n⊢ #(R × R × R × R) = #R ^ 4"
] | mk_congr (QuaternionAlgebra.equivProd c₁ c₂ c₃) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Eisenstein.Criterion | {
"line": 203,
"column": 4
} | {
"line": 204,
"column": 32
} | {
"line": 205,
"column": 4
} | [
{
"pp": "case hfmodP2\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf : R[X]\nP : Ideal R\nhP : P.IsPrime\nhfl : f.leadingCoeff ∉ P\nhfP : ∀ (n : ℕ), ↑n < f.degree → f.coeff n ∈ P\nhfd0 : 0 < f.degree\nh0 : f.coeff 0 ∉ P ^ 2\nhu : f.IsPrimitive\n⊢ map (Ideal.Quotient.mk (ker (algebraMap R (FractionRin... | [
"case hfmodP2\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf : R[X]\nP : Ideal R\nhP : P.IsPrime\nhfl : f.leadingCoeff ∉ P\nhfP : ∀ (n : ℕ), ↑n < f.degree → f.coeff n ∈ P\nhfd0 : 0 < f.degree\nh0 : f.coeff 0 ∉ P ^ 2\nhu : f.IsPrimitive\n⊢ f.coeff 0 ∉ ker (algebraMap R (FractionRing (R ⧸ P))) ^ 2"
] | rw [modByMonic_X, map_C, ne_eq, C_eq_zero, Ideal.Quotient.eq_zero_iff_mem,
← coeff_zero_eq_eval_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Group.NatPowAssoc | {
"line": 75,
"column": 37
} | {
"line": 75,
"column": 72
} | {
"line": 77,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝² : MulOneClass M\ninst✝¹ : Pow M ℕ\ninst✝ : NatPowAssoc M\nm n : ℕ\nx : M\n⊢ x ^ m * x ^ n = x ^ n * x ^ m",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Algebra.Group.NatPowAssoc.0.npow_mul_comm._simp_1_1",
"HMul.hMul",
"co... | [] | by simp only [← npow_add, add_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.PNatPowAssoc | {
"line": 69,
"column": 40
} | {
"line": 69,
"column": 72
} | {
"line": 71,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝² : Mul M\ninst✝¹ : Pow M ℕ+\ninst✝ : PNatPowAssoc M\nm n : ℕ+\nx : M\n⊢ x ^ m * x ^ n = x ^ n * x ^ m",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"congrArg",
"add_comm",
"instAddCommSemigroupPNat",
"instHAdd",
... | [] | simp only [← ppow_add, add_comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Group.PNatPowAssoc | {
"line": 69,
"column": 40
} | {
"line": 69,
"column": 72
} | {
"line": 71,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝² : Mul M\ninst✝¹ : Pow M ℕ+\ninst✝ : PNatPowAssoc M\nm n : ℕ+\nx : M\n⊢ x ^ m * x ^ n = x ^ n * x ^ m",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"congrArg",
"add_comm",
"instAddCommSemigroupPNat",
"instHAdd",
... | [] | simp only [← ppow_add, add_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.PNatPowAssoc | {
"line": 69,
"column": 40
} | {
"line": 69,
"column": 72
} | {
"line": 71,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝² : Mul M\ninst✝¹ : Pow M ℕ+\ninst✝ : PNatPowAssoc M\nm n : ℕ+\nx : M\n⊢ x ^ m * x ^ n = x ^ n * x ^ m",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"congrArg",
"add_comm",
"instAddCommSemigroupPNat",
"instHAdd",
... | [] | simp only [← ppow_add, add_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Subgroup.Order | {
"line": 34,
"column": 8
} | {
"line": 34,
"column": 16
} | {
"line": 34,
"column": 17
} | [
{
"pp": "C : Type u_1\ninst✝ : CommGroup C\nx y z : Subgroup C\nxz : x ≤ z\na : C\nha : a ∈ (x ⊔ y) ⊓ z\n⊢ a ∈ x ⊔ y ⊓ z",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"congrArg",
"Membership.mem",
"CompleteLattice.toConditionallyComplet... | [
"C : Type u_1\ninst✝ : CommGroup C\nx y z : Subgroup C\nxz : x ≤ z\na : C\nha : a ∈ x ⊔ y ∧ a ∈ z\n⊢ a ∈ x ⊔ y ⊓ z"
] | mem_inf, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GroupWithZero.Pointwise.Finset | {
"line": 37,
"column": 2
} | {
"line": 37,
"column": 52
} | {
"line": 38,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝³ : Mul α\ninst✝² : Zero α\ninst✝¹ : DecidableEq α\ns : Finset α\ninst✝ : IsLeftCancelMulZero α\n⊢ #s ≤ #(s * s)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"Finset",
"LE.le",
"instLENat",
"Finset.instEmptyCollectio... | [
"case inl\nα : Type u_1\ninst✝³ : Mul α\ninst✝² : Zero α\ninst✝¹ : DecidableEq α\ns : Finset α\ninst✝ : IsLeftCancelMulZero α\nhs : s.erase 0 = ∅\n⊢ #s ≤ #(s * s)",
"case inr\nα : Type u_1\ninst✝³ : Mul α\ninst✝² : Zero α\ninst✝¹ : DecidableEq α\ns : Finset α\ninst✝ : IsLeftCancelMulZero α\nhs : (s.erase 0).Nonem... | obtain hs | hs := (s.erase 0).eq_empty_or_nonempty | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicTopology.SimplexCategory.Basic | {
"line": 107,
"column": 12
} | {
"line": 107,
"column": 14
} | {
"line": 107,
"column": 14
} | [
{
"pp": "f : ⦋1⦌ ⟶ ⦋1⦌\ne0 : (Hom.toOrderHom f) 0 = 1\ne1 : (Hom.toOrderHom f) 1 = 0\nthis : 1 ≤ (Hom.toOrderHom f) 1\n⊢ (∃ a, f = ⦋1⦌.const ⦋1⦌ a) ∨ f = 𝟙 ⦋1⦌",
"ppTerm": "?m.1286",
"assigned": true,
"usedConstants": [
"instNeZeroNatHAdd_1",
"congrArg",
"PartialOrder.toPreorder",... | [
"f : ⦋1⦌ ⟶ ⦋1⦌\ne0 : (Hom.toOrderHom f) 0 = 1\ne1 : (Hom.toOrderHom f) 1 = 0\nthis : 1 ≤ 0\n⊢ (∃ a, f = ⦋1⦌.const ⦋1⦌ a) ∨ f = 𝟙 ⦋1⦌"
] | e1 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplexCategory.Basic | {
"line": 301,
"column": 2
} | {
"line": 301,
"column": 13
} | {
"line": 302,
"column": 2
} | [
{
"pp": "n i : ℕ\nhi : i < n + 1\n⊢ δ ⟨i, hi⟩.castSucc ≫ σ ⟨i, hi⟩ = 𝟙 ⦋n⦌",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Fin.casesOn",
"PartialOrder.toPreorder",
"CategoryTheory.CategoryStruct.id",
"SimplexCategory.δ",
"SimplexCategory.σ",
"Fin.mk",
... | [
"n i : ℕ\nhi : i < n + 1\nj : ℕ\nhj : j < ⦋n⦌.len + 1\n⊢ ↑((Hom.toOrderHom (δ ⟨i, hi⟩.castSucc ≫ σ ⟨i, hi⟩)) ⟨j, hj⟩) = ↑((Hom.toOrderHom (𝟙 ⦋n⦌)) ⟨j, hj⟩)"
] | ext ⟨j, hj⟩ | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.AlgebraicTopology.SimplexCategory.Basic | {
"line": 509,
"column": 12
} | {
"line": 509,
"column": 14
} | {
"line": 509,
"column": 14
} | [
{
"pp": "f : ⦋1⦌ ⟶ ⦋2⦌\nthis : 1 ≤ (Hom.toOrderHom f) 1\ne0 : (Hom.toOrderHom f) 0 = 1\ne1 : (Hom.toOrderHom f) 1 = 0\n⊢ (∃ i, f = δ i) ∨ ∃ a, f = ⦋1⦌.const ⦋2⦌ a",
"ppTerm": "?m.2430",
"assigned": true,
"usedConstants": [
"instNeZeroNatHAdd_1",
"congrArg",
"PartialOrder.toPreorder... | [
"f : ⦋1⦌ ⟶ ⦋2⦌\nthis : 1 ≤ 0\ne0 : (Hom.toOrderHom f) 0 = 1\ne1 : (Hom.toOrderHom f) 1 = 0\n⊢ (∃ i, f = δ i) ∨ ∃ a, f = ⦋1⦌.const ⦋2⦌ a"
] | e1 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.AlternatingFaceMapComplex | {
"line": 330,
"column": 4
} | {
"line": 336,
"column": 22
} | {
"line": 338,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Preadditive C\nA : Type u_2\ninst✝¹ : Category.{v_2, u_2} A\ninst✝ : Abelian A\nX Y : CosimplicialObject C\nf : X ⟶ Y\nn : ℕ\n⊢ f.app ⦋n⦌ ≫ (obj Y).d n (n + 1) = (obj X).d n (n + 1) ≫ f.app ⦋n + 1⦌",
"ppTerm": "?m.33",
"assigned": true,
... | [] | simp only [obj, CochainComplex.of_d, objD, Int.reduceNeg]
rw [comp_sum, sum_comp]
refine Finset.sum_congr rfl fun x _ => ?_
rw [comp_zsmul, zsmul_comp]
congr 1
symm
apply f.naturality | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.AlternatingFaceMapComplex | {
"line": 330,
"column": 4
} | {
"line": 336,
"column": 22
} | {
"line": 338,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Preadditive C\nA : Type u_2\ninst✝¹ : Category.{v_2, u_2} A\ninst✝ : Abelian A\nX Y : CosimplicialObject C\nf : X ⟶ Y\nn : ℕ\n⊢ f.app ⦋n⦌ ≫ (obj Y).d n (n + 1) = (obj X).d n (n + 1) ≫ f.app ⦋n + 1⦌",
"ppTerm": "?m.33",
"assigned": true,
... | [] | simp only [obj, CochainComplex.of_d, objD, Int.reduceNeg]
rw [comp_sum, sum_comp]
refine Finset.sum_congr rfl fun x _ => ?_
rw [comp_zsmul, zsmul_comp]
congr 1
symm
apply f.naturality | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplexCategory.Basic | {
"line": 509,
"column": 12
} | {
"line": 509,
"column": 14
} | {
"line": 509,
"column": 14
} | [
{
"pp": "f : ⦋1⦌ ⟶ ⦋2⦌\nthis : 2 ≤ (Hom.toOrderHom f) 1\ne0 : (Hom.toOrderHom f) 0 = 2\ne1 : (Hom.toOrderHom f) 1 = 0\n⊢ (∃ i, f = δ i) ∨ ∃ a, f = ⦋1⦌.const ⦋2⦌ a",
"ppTerm": "?m.2447",
"assigned": true,
"usedConstants": [
"instNeZeroNatHAdd_1",
"congrArg",
"PartialOrder.toPreorder... | [
"f : ⦋1⦌ ⟶ ⦋2⦌\nthis : 2 ≤ 0\ne0 : (Hom.toOrderHom f) 0 = 2\ne1 : (Hom.toOrderHom f) 1 = 0\n⊢ (∃ i, f = δ i) ∨ ∃ a, f = ⦋1⦌.const ⦋2⦌ a"
] | e1 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplexCategory.Basic | {
"line": 509,
"column": 12
} | {
"line": 509,
"column": 14
} | {
"line": 509,
"column": 14
} | [
{
"pp": "f : ⦋1⦌ ⟶ ⦋2⦌\nthis : 2 ≤ (Hom.toOrderHom f) 1\ne0 : (Hom.toOrderHom f) 0 = 2\ne1 : (Hom.toOrderHom f) 1 = 1\n⊢ (∃ i, f = δ i) ∨ ∃ a, f = ⦋1⦌.const ⦋2⦌ a",
"ppTerm": "?m.2464",
"assigned": true,
"usedConstants": [
"instNeZeroNatHAdd_1",
"congrArg",
"PartialOrder.toPreorder... | [
"f : ⦋1⦌ ⟶ ⦋2⦌\nthis : 2 ≤ 1\ne0 : (Hom.toOrderHom f) 0 = 2\ne1 : (Hom.toOrderHom f) 1 = 1\n⊢ (∃ i, f = δ i) ∨ ∃ a, f = ⦋1⦌.const ⦋2⦌ a"
] | e1 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplicialObject.Basic | {
"line": 666,
"column": 34
} | {
"line": 668,
"column": 61
} | {
"line": 670,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX : CosimplicialObject C\nn : ℕ\ni : Fin (n + 3)\nj : Fin (n + 2)\nH : j.succ < i\n⊢ X.δ i ≫ X.σ j = X.σ (j.castLT ⋯) ≫ X.δ (i.pred ⋯)",
"ppTerm": "?m.67",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono"... | [] | by
dsimp [δ, σ]
simp only [← X.map_comp, SimplexCategory.δ_comp_σ_of_gt' H] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialSet.Simplices | {
"line": 141,
"column": 4
} | {
"line": 141,
"column": 49
} | {
"line": 142,
"column": 4
} | [
{
"pp": "case mp\nX : SSet\nm : ℕ\nx : X _⦋m⦌\nf : ⦋m⦌ ⟶ ⦋m⦌\ninst✝ : Mono f\nh : { dim := m, simplex := (ConcreteCategory.hom (X.map f.op)) x } = { dim := m, simplex := x }\n⊢ IsIso f",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"SSet.S",
"CategoryTheory.IsIso",
"Opposi... | [
"case mp\nX : SSet\nm : ℕ\nx : X _⦋m⦌\ninst✝ : Mono (𝟙 ⦋m⦌)\nh : { dim := m, simplex := (ConcreteCategory.hom (X.map (𝟙 ⦋m⦌).op)) x } = { dim := m, simplex := x }\n⊢ IsIso (𝟙 ⦋m⦌)"
] | obtain rfl := SimplexCategory.eq_id_of_mono f | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicTopology.SimplicialSet.Degenerate | {
"line": 182,
"column": 6
} | {
"line": 182,
"column": 29
} | {
"line": 182,
"column": 30
} | [
{
"pp": "X : SSet\nn : ℕ\nx : X _⦋n⦌\nm₁ m₂ : ℕ\nf₁ : ⦋n⦌ ⟶ ⦋m₁⦌\nhf₁ : SplitEpi f₁\ny₁ : ↑(X.nonDegenerate m₁)\nhy₁ : x = (ConcreteCategory.hom (X.map f₁.op)) ↑y₁\nf₂ : ⦋n⦌ ⟶ ⦋m₂⦌\ny₂ : X _⦋m₂⦌\nhy₂ : x = (ConcreteCategory.hom (X.map f₂.op)) y₂\nthis : (ConcreteCategory.hom (X.map (g hf₁ f₂).op)) y₂ = ↑y₁\n⊢ I... | [
"X : SSet\nn : ℕ\nx : X _⦋n⦌\nm₁ m₂ : ℕ\nf₁ : ⦋n⦌ ⟶ ⦋m₁⦌\nhf₁ : SplitEpi f₁\ny₁ : ↑(X.nonDegenerate m₁)\nhy₁ : x = (ConcreteCategory.hom (X.map f₁.op)) ↑y₁\nf₂ : ⦋n⦌ ⟶ ⦋m₂⦌\ny₂ : X _⦋m₂⦌\nhy₂ : x = (ConcreteCategory.hom (X.map f₂.op)) y₂\nthis : (ConcreteCategory.hom (X.map (factorThruImage (g hf₁ f₂) ≫ image.ι (g ... | ← image.fac (g hf₁ f₂), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplicialSet.NonDegenerateSimplices | {
"line": 102,
"column": 4
} | {
"line": 102,
"column": 49
} | {
"line": 103,
"column": 4
} | [
{
"pp": "case inr\nX : SSet\nd : ℕ\nx : X _⦋d⦌\nhx : x ∈ X.nonDegenerate d\ny : X _⦋d⦌\nhy : y ∈ X.nonDegenerate d\nh : mk ↑⟨x, hx⟩ ⋯ < mk ↑⟨y, hy⟩ ⋯\nf : ⦋(mk ↑⟨x, hx⟩ ⋯).dim⦌ ⟶ ⦋(mk ↑⟨y, hy⟩ ⋯).dim⦌\nw✝ : Mono f\nhf : (ConcreteCategory.hom (X.map f.op)) (mk ↑⟨y, hy⟩ ⋯).simplex = (mk ↑⟨x, hx⟩ ⋯).simplex\n⊢ (mk... | [
"case inr\nX : SSet\nd : ℕ\nx : X _⦋d⦌\nhx : x ∈ X.nonDegenerate d\ny : X _⦋d⦌\nhy : y ∈ X.nonDegenerate d\nh : mk ↑⟨x, hx⟩ ⋯ < mk ↑⟨y, hy⟩ ⋯\nw✝ : Mono (𝟙 ⦋(mk ↑⟨x, hx⟩ ⋯).dim⦌)\nhf : (ConcreteCategory.hom (X.map (𝟙 ⦋(mk ↑⟨x, hx⟩ ⋯).dim⦌).op)) (mk ↑⟨y, hy⟩ ⋯).simplex = (mk ↑⟨x, hx⟩ ⋯).simplex\n⊢ (mk ↑⟨x, hx⟩ ⋯).... | obtain rfl := SimplexCategory.eq_id_of_mono f | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicTopology.SimplicialSet.NonDegenerateSimplices | {
"line": 115,
"column": 4
} | {
"line": 115,
"column": 49
} | {
"line": 116,
"column": 4
} | [
{
"pp": "X : SSet\nn₁ : ℕ\nx₁ : X _⦋n₁⦌\nhx₁ : x₁ ∈ X.nonDegenerate n₁\nx₂ : X _⦋n₁⦌\nhx₂ : x₂ ∈ X.nonDegenerate n₁\nh' : mk ↑⟨x₂, hx₂⟩ ⋯ ≤ mk ↑⟨x₁, hx₁⟩ ⋯\nf : ⦋(mk ↑⟨x₁, hx₁⟩ ⋯).dim⦌ ⟶ ⦋(mk ↑⟨x₂, hx₂⟩ ⋯).dim⦌\nhf : Mono f\nh : (ConcreteCategory.hom (X.map f.op)) (mk ↑⟨x₂, hx₂⟩ ⋯).simplex = (mk ↑⟨x₁, hx₁⟩ ⋯).s... | [
"X : SSet\nn₁ : ℕ\nx₁ : X _⦋n₁⦌\nhx₁ : x₁ ∈ X.nonDegenerate n₁\nx₂ : X _⦋n₁⦌\nhx₂ : x₂ ∈ X.nonDegenerate n₁\nh' : mk ↑⟨x₂, hx₂⟩ ⋯ ≤ mk ↑⟨x₁, hx₁⟩ ⋯\nhf : Mono (𝟙 ⦋(mk ↑⟨x₁, hx₁⟩ ⋯).dim⦌)\nh : (ConcreteCategory.hom (X.map (𝟙 ⦋(mk ↑⟨x₁, hx₁⟩ ⋯).dim⦌).op)) (mk ↑⟨x₂, hx₂⟩ ⋯).simplex = (mk ↑⟨x₁, hx₁⟩ ⋯).simplex\n⊢ mk ... | obtain rfl := SimplexCategory.eq_id_of_mono f | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicTopology.SimplicialSet.NonDegenerateSimplices | {
"line": 180,
"column": 4
} | {
"line": 180,
"column": 85
} | {
"line": 181,
"column": 4
} | [
{
"pp": "X : SSet\nx y : X.op.N\n⊢ mk (opObjEquiv x.simplex) ⋯ ≤ mk (opObjEquiv y.simplex) ⋯ ↔ x ≤ y",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"SSet.N.instPreorder",
"SSet.S.subcomplex",
"SSet.S.simplex",
"Eq.mpr",
"SSet.Subcomplex.ofSimplex",
"SSe... | [
"X : SSet\nx y : X.op.N\n⊢ (∃ f,\n (ConcreteCategory.hom (X.map f.op)) (mk (opObjEquiv y.simplex) ⋯).simplex =\n (mk (opObjEquiv x.simplex) ⋯).simplex) ↔\n ∃ f, (ConcreteCategory.hom (X.op.map f.op)) y.simplex = x.simplex"
] | simp only [le_iff, Subcomplex.ofSimplex_le_iff, Subcomplex.mem_ofSimplex_obj_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicTopology.SimplicialSet.NonDegenerateSimplices | {
"line": 201,
"column": 4
} | {
"line": 201,
"column": 85
} | {
"line": 202,
"column": 4
} | [
{
"pp": "X Y : SSet\ne : X ≅ Y\nx y : X.N\n⊢ mk ((ConcreteCategory.hom (e.hom.app (Opposite.op ⦋x.dim⦌))) x.simplex) ⋯ ≤\n mk ((ConcreteCategory.hom (e.hom.app (Opposite.op ⦋y.dim⦌))) y.simplex) ⋯ ↔\n x ≤ y",
"ppTerm": "?m.78",
"assigned": true,
"usedConstants": [
"SSet.N.instPreorder"... | [
"X Y : SSet\ne : X ≅ Y\nx y : X.N\n⊢ (∃ f,\n (ConcreteCategory.hom (Y.map f.op))\n (mk ((ConcreteCategory.hom (e.hom.app (Opposite.op ⦋y.dim⦌))) y.simplex) ⋯).simplex =\n (mk ((ConcreteCategory.hom (e.hom.app (Opposite.op ⦋x.dim⦌))) x.simplex) ⋯).simplex) ↔\n ∃ f, (ConcreteCategory.hom (X.ma... | simp only [le_iff, Subcomplex.ofSimplex_le_iff, Subcomplex.mem_ofSimplex_obj_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.Augment | {
"line": 221,
"column": 6
} | {
"line": 221,
"column": 13
} | {
"line": 222,
"column": 6
} | [
{
"pp": "V : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nC : CochainComplex V ℕ\nX : V\nf : X ⟶ C.X 0\nw : f ≫ C.d 0 1 = 0\ni j k✝ : ℕ\nhij : (ComplexShape.up ℕ).Rel i j\nhjk : (ComplexShape.up ℕ).Rel j k✝\nk : ℕ\n⊢ f ≫ C.d 0 (k + 1) = 0",
"ppTerm": "?m.277",
"assigned": true,
"u... | [
"case zero\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nC : CochainComplex V ℕ\nX : V\nf : X ⟶ C.X 0\nw : f ≫ C.d 0 1 = 0\ni j k : ℕ\nhij : (ComplexShape.up ℕ).Rel i j\nhjk : (ComplexShape.up ℕ).Rel j k\n⊢ f ≫ C.d 0 (0 + 1) = 0",
"case succ\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : ... | cases k | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.CategoryTheory.GradedObject.Trifunctor | {
"line": 215,
"column": 4
} | {
"line": 216,
"column": 40
} | {
"line": 217,
"column": 2
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\nC₄ : Type u_4\nC₁₂ : Type u_5\nC₂₃ : Type u_6\ninst✝⁶ : Category.{v_1, u_1} C₁\ninst✝⁵ : Category.{v_2, u_2} C₂\ninst✝⁴ : Category.{v_3, u_3} C₃\ninst✝³ : Category.{v_4, u_4} C₄\ninst✝² : Category.{v_5, u_5} C₁₂\ninst✝¹ : Category.{v_6, u_6} C₂₃\nF : C₁ ⥤ C₂... | [] | simp only [ι_mapTrifunctorMapMap, categoryOfGradedObjects_id, Functor.map_id,
NatTrans.id_app, id_comp, comp_id] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.GradedObject.Trifunctor | {
"line": 335,
"column": 2
} | {
"line": 335,
"column": 82
} | {
"line": 336,
"column": 2
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\nC₄ : Type u_4\nC₁₂ : Type u_5\nC₂₃ : Type u_6\ninst✝⁷ : Category.{v_1, u_1} C₁\ninst✝⁶ : Category.{v_2, u_2} C₂\ninst✝⁵ : Category.{v_3, u_3} C₃\ninst✝⁴ : Category.{v_4, u_4} C₄\ninst✝³ : Category.{v_5, u_5} C₁₂\ninst✝² : Category.{v_6, u_6} C₂₃\nF₁₂ : C₁ ⥤ ... | [
"C₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\nC₄ : Type u_4\nC₁₂ : Type u_5\nC₂₃ : Type u_6\ninst✝⁷ : Category.{v_1, u_1} C₁\ninst✝⁶ : Category.{v_2, u_2} C₂\ninst✝⁵ : Category.{v_3, u_3} C₃\ninst✝⁴ : Category.{v_4, u_4} C₄\ninst✝³ : Category.{v_5, u_5} C₁₂\ninst✝² : Category.{v_6, u_6} C₂₃\nF₁₂ : C₁ ⥤ C₂ ⥤ C₁₂\nG ... | let p' : I₁ × I₂ × I₃ → ρ₁₂.I₁₂ × I₃ := fun ⟨i₁, i₂, i₃⟩ => ⟨ρ₁₂.p ⟨i₁, i₂⟩, i₃⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Algebra.Homology.TotalComplexShift | {
"line": 171,
"column": 2
} | {
"line": 181,
"column": 80
} | {
"line": 182,
"column": 2
} | [
{
"pp": "case pos\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nK : HomologicalComplex₂ C (up ℤ) (up ℤ)\nx : ℤ\ninst✝ : K.HasTotal (up ℤ)\nn₀ n₁ n₀' n₁' : ℤ\nh₀ : n₀ + x = n₀'\nh₁ : n₁ + x = n₁'\nh : (up ℤ).Rel n₀ n₁\n⊢ ((shiftFunctor₁ C x).obj K).D₂ (up ℤ) n₀ n₁ ≫ (K.totalShift₁XIso x ... | [
"case neg\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nK : HomologicalComplex₂ C (up ℤ) (up ℤ)\nx : ℤ\ninst✝ : K.HasTotal (up ℤ)\nn₀ n₁ n₀' n₁' : ℤ\nh₀ : n₀ + x = n₀'\nh₁ : n₁ + x = n₁'\nh : ¬(up ℤ).Rel n₀ n₁\n⊢ ((shiftFunctor₁ C x).obj K).D₂ (up ℤ) n₀ n₁ ≫ (K.totalShift₁XIso x n₁ n₁' h₁).... | · apply total.hom_ext
intro p q hpq
dsimp at h hpq
dsimp [totalShift₁XIso]
rw [ι_D₂_assoc, Linear.comp_units_smul, ι_totalDesc_assoc, ι_D₂,
((shiftFunctor₁ C x).obj K).d₂_eq _ _ rfl _ (by dsimp; lia),
K.d₂_eq _ _ rfl _ (by dsimp; lia), smul_smul,
Linear.units_smul_comp, Category.assoc,... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Homology.CochainComplexPlus | {
"line": 99,
"column": 44
} | {
"line": 99,
"column": 57
} | {
"line": 99,
"column": 57
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasLimitsOfShape WalkingCospan C\nX Y : Plus C\nf : X ⟶ Y\nx✝ : Mono f.hom\n⊢ Mono ((ι C).map f)",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"x✝"
],
"usedGoals... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.DerivedCategory.SmallShiftedHom | {
"line": 44,
"column": 13
} | {
"line": 44,
"column": 15
} | {
"line": 44,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\nK L : CochainComplex C ℤ\nn : ℤ\ninst✝ : HasSmallLocalizedShiftedHom (HomologicalComplex.quasiIso C (ComplexShape.up ℤ)) ℤ K L\nx : CohomologyClass K L n\nthis : HasDerivedCategory C := HasDerivedCategory.standard C\ny₁ : Cocycle K L n\n⊢ ∀ (b... | [
"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\nK L : CochainComplex C ℤ\nn : ℤ\ninst✝ : HasSmallLocalizedShiftedHom (HomologicalComplex.quasiIso C (ComplexShape.up ℤ)) ℤ K L\nx : CohomologyClass K L n\nthis : HasDerivedCategory C := HasDerivedCategory.standard C\ny₁ y₂ : Cocycle K L n\n⊢ y₁ ≈ y₂ →\n ... | y₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.Homology.HomotopyCategory.KInjective | {
"line": 130,
"column": 4
} | {
"line": 130,
"column": 16
} | {
"line": 132,
"column": 0
} | [
{
"pp": "case inr\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\nK L : CochainComplex C ℤ\nf : K ⟶ L\nα : Cochain K L (-1)\nn : ℤ\nhα : (δ (-1) 0 α).EqUpTo (Cochain.ofHom f) n\nhK : (HomologicalComplex.sc' K n (n + 1) (n + 2)).Exact\ninst✝ : Injective (L.X (n + 1))\nu : K.X (n + 1) ⟶ L.X (n ... | [] | simp [hβ, u] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Abelian.EpiWithInjectiveKernel | {
"line": 63,
"column": 63
} | {
"line": 63,
"column": 76
} | {
"line": 63,
"column": 76
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsIso f\n⊢ IsIso { X₁ := 0, X₂ := X, X₃ := Y, f := 0, g := f, zero := ⋯ }.g",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"inst✝"
],
"usedGoals": []
... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle | {
"line": 85,
"column": 4
} | {
"line": 85,
"column": 77
} | {
"line": 86,
"column": 2
} | [
{
"pp": "case neg\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\ninst✝ : HasZeroObject C\nX : C\nK : CochainComplex C ℤ\np q n : ℤ\nh : p + n = q\nα : Cochain ((singleFunctor C p).obj X) K n\np' q' : ℤ\nhpq' : p' + n = q'\nhp : ¬p' = p\n⊢ ((fun f ↦ fromSingleMk f h)\n ((fun α ↦ (Homol... | [] | · exact (HomologicalComplex.isZero_single_obj_X _ _ _ _ hp).eq_of_src _ _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle | {
"line": 326,
"column": 79
} | {
"line": 334,
"column": 23
} | {
"line": 336,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\ninst✝ : HasZeroObject C\nX : C\nK : CochainComplex C ℤ\nq n : ℤ\nα : Cocycle K ((singleFunctor C q).obj X) n\np : ℤ\nh : p + n = q\np' : ℤ\nhp' : p' + 1 = p\n⊢ ∃ f, ∃ (hf : K.d p' p ≫ f = 0), toSingleMk f h p' hp' hf = α",
"ppTerm": "?... | [] | by
obtain ⟨f, hf⟩ := Cochain.toSingleMk_surjective α.1 p h
have hα := ((n + 1).negOnePow • α).δ_eq_zero (n + 1)
rw [coe_units_smul, δ_units_smul, ← hf, Cochain.δ_toSingleMk _ _ _ p' (by lia),
smul_smul, Int.units_mul_self, one_smul] at hα
refine ⟨f, ?_, ?_⟩
· simpa [← cancel_mono (HomologicalComplex.singl... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.Factorizations.CM5a | {
"line": 157,
"column": 4
} | {
"line": 157,
"column": 37
} | {
"line": 158,
"column": 4
} | [
{
"pp": "case inl\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\nK L : CochainComplex C ℤ\ninst✝ : EnoughInjectives C\nn₀ n₁ : ℤ\nhn₁ : n₀ + 1 = n₁\ni : ℤ\nhi✝ : i ≤ n₀\nhi : i < n₀\nφ : (shortComplexFunctor' C (ComplexShape.up ℤ) (i - 1) i (i + 1)).obj (mid K L n₁) ⟶\n (shortComplexFunctor... | [
"case inl\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\nK L : CochainComplex C ℤ\ninst✝ : EnoughInjectives C\nn₀ n₁ : ℤ\nhn₁ : n₀ + 1 = n₁\ni : ℤ\nhi✝ : i ≤ n₀\nhi : i < n₀\nφ : (shortComplexFunctor' C (ComplexShape.up ℤ) (i - 1) i (i + 1)).obj (mid K L n₁) ⟶\n (shortComplexFunctor' C (Complex... | have : IsIso φ.τ₁ := isIso_π_f .. | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle | {
"line": 370,
"column": 4
} | {
"line": 370,
"column": 66
} | {
"line": 371,
"column": 4
} | [
{
"pp": "case mp\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\ninst✝ : HasZeroObject C\nX : C\nK : CochainComplex C ℤ\np q : ℤ\nf : K.X p ⟶ X\nn : ℤ\nh : p + n = q\np' : ℤ\nhp' : p' + 1 = p\nhf : K.d p' p ≫ f = 0\np'' : ℤ\nhp'' : p + 1 = p''\nα : Cochain K ((singleFunctor C q).obj X) (n - 1)\... | [
"case mp\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\ninst✝ : HasZeroObject C\nX : C\nK : CochainComplex C ℤ\np q : ℤ\nf : K.X p ⟶ X\nn : ℤ\nh : p + n = q\np' : ℤ\nhp' : p' + 1 = p\nhf : K.d p' p ≫ f = 0\np'' : ℤ\nhp'' : p + 1 = p''\nα : Cochain K ((singleFunctor C q).obj X) (n - 1)\nhα : δ (n -... | obtain ⟨g, hg⟩ := Cochain.toSingleMk_surjective α p'' (by lia) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicTopology.ModelCategory.Basic | {
"line": 149,
"column": 31
} | {
"line": 149,
"column": 39
} | {
"line": 149,
"column": 39
} | [
{
"pp": "C : Type u\ninst✝⁸ : Category.{v, u} C\ninst✝⁷ : CategoryWithFibrations C\ninst✝⁶ : CategoryWithCofibrations C\ninst✝⁵ : CategoryWithWeakEquivalences C\ninst✝⁴ : HasFiniteLimits C\ninst✝³ : HasFiniteColimits C\ninst✝² : (weakEquivalences C).HasTwoOutOfThreeProperty\ninst✝¹ : (cofibrations C).IsWeakFact... | [
"C : Type u\ninst✝⁸ : Category.{v, u} C\ninst✝⁷ : CategoryWithFibrations C\ninst✝⁶ : CategoryWithCofibrations C\ninst✝⁵ : CategoryWithWeakEquivalences C\ninst✝⁴ : HasFiniteLimits C\ninst✝³ : HasFiniteColimits C\ninst✝² : (weakEquivalences C).HasTwoOutOfThreeProperty\ninst✝¹ : (cofibrations C).IsWeakFactorizationSys... | ← hf.fac | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.Factorizations.CM5a | {
"line": 433,
"column": 5
} | {
"line": 433,
"column": 18
} | {
"line": 433,
"column": 18
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Abelian C\nK L : CochainComplex C ℤ\nf : K ⟶ L\ninst✝³ : EnoughInjectives C\ninst✝² : Mono f\nn : ℤ\ninst✝¹ : K.IsStrictlyGE (n + 1)\ninst✝ : L.IsStrictlyGE (n + 1)\n⊢ Mono { mid := L, ι := f, π := 𝟙 L, ι_π := ⋯ }.ι",
"ppTerm": "?m.62",
"a... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.Factorizations.CM5a | {
"line": 439,
"column": 8
} | {
"line": 440,
"column": 41
} | {
"line": 440,
"column": 41
} | [
{
"pp": "case hX\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Abelian C\nK L : CochainComplex C ℤ\nf : K ⟶ L\ninst✝³ : EnoughInjectives C\ninst✝² : Mono f\nn : ℤ\ninst✝¹ : K.IsStrictlyGE (n + 1)\ninst✝ : L.IsStrictlyGE (n + 1)\ni : ℤ\nhi : i ≤ n + ↑0\n⊢ IsZero (homology K i)",
"ppTerm": "?hX",
... | [] | rw [← exactAt_iff_isZero_homology]
exact exactAt_of_isGE _ (n + 1) i | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Factorizations.CM5a | {
"line": 439,
"column": 8
} | {
"line": 440,
"column": 41
} | {
"line": 440,
"column": 41
} | [
{
"pp": "case hX\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Abelian C\nK L : CochainComplex C ℤ\nf : K ⟶ L\ninst✝³ : EnoughInjectives C\ninst✝² : Mono f\nn : ℤ\ninst✝¹ : K.IsStrictlyGE (n + 1)\ninst✝ : L.IsStrictlyGE (n + 1)\ni : ℤ\nhi : i ≤ n + ↑0\n⊢ IsZero (homology K i)",
"ppTerm": "?hX",
... | [] | rw [← exactAt_iff_isZero_homology]
exact exactAt_of_isGE _ (n + 1) i | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Factorizations.CM5a | {
"line": 439,
"column": 8
} | {
"line": 440,
"column": 41
} | {
"line": 440,
"column": 41
} | [
{
"pp": "case hY\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Abelian C\nK L : CochainComplex C ℤ\nf : K ⟶ L\ninst✝³ : EnoughInjectives C\ninst✝² : Mono f\nn : ℤ\ninst✝¹ : K.IsStrictlyGE (n + 1)\ninst✝ : L.IsStrictlyGE (n + 1)\ni : ℤ\nhi : i ≤ n + ↑0\n⊢ IsZero (homology L i)",
"ppTerm": "?hY",
... | [] | rw [← exactAt_iff_isZero_homology]
exact exactAt_of_isGE _ (n + 1) i | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Factorizations.CM5a | {
"line": 439,
"column": 8
} | {
"line": 440,
"column": 41
} | {
"line": 440,
"column": 41
} | [
{
"pp": "case hY\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Abelian C\nK L : CochainComplex C ℤ\nf : K ⟶ L\ninst✝³ : EnoughInjectives C\ninst✝² : Mono f\nn : ℤ\ninst✝¹ : K.IsStrictlyGE (n + 1)\ninst✝ : L.IsStrictlyGE (n + 1)\ni : ℤ\nhi : i ≤ n + ↑0\n⊢ IsZero (homology L i)",
"ppTerm": "?hY",
... | [] | rw [← exactAt_iff_isZero_homology]
exact exactAt_of_isGE _ (n + 1) i | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.LocalCohomology | {
"line": 205,
"column": 15
} | {
"line": 205,
"column": 17
} | {
"line": 206,
"column": 6
} | [
{
"pp": "case h\nR : Type u\ninst✝ : CommRing R\nI J K : Ideal R\nhR : IsNoetherian R R\nJ' : SelfLERadical J\nj1 : CostructuredArrow (idealPowersToSelfLERadical J) J'\n⊢ ∀ (j₂ : CostructuredArrow (idealPowersToSelfLERadical J) J'), Zigzag j1 j₂",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [... | [
"case h\nR : Type u\ninst✝ : CommRing R\nI J K : Ideal R\nhR : IsNoetherian R R\nJ' : SelfLERadical J\nj1 j2 : CostructuredArrow (idealPowersToSelfLERadical J) J'\n⊢ Zigzag j1 j2"
] | j2 | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.Homology.ModelCategory.Lifting | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 26
} | {
"line": 92,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nA B X Y : CochainComplex C ℤ\nt : A ⟶ X\ni : A ⟶ B\np : X ⟶ Y\nb : B ⟶ Y\nsq : CommSq t i p b\nhsq : (n : ℤ) → ⋯.LiftStruct\nQ : CochainComplex C ℤ\nπ : B ⟶ Q\nhπ : i ≫ π = 0\nhQ : IsColimit (CokernelCofork.ofπ π hπ)\nK : CochainComplex C... | [] | exact ⟨l', by cat_disch⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Homology.ModelCategory.Injective | {
"line": 145,
"column": 6
} | {
"line": 145,
"column": 41
} | {
"line": 147,
"column": 4
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{u_2, u_1} C\ninst✝² : Abelian C\nA : CochainComplex C ℤ\nhA : CochainComplex.plus C A\nB : CochainComplex C ℤ\nhB : CochainComplex.plus C B\nX Y : CochainComplex C ℤ\nhY : CochainComplex.plus C Y\ni : A ⟶ B\ninst✝¹ : Mono (ObjectProperty.homMk i)\nhi : Mono i\np : X ⟶ Y... | [] | exact isKInjective_of_injective _ d | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Homology.ModelCategory.Injective | {
"line": 185,
"column": 10
} | {
"line": 185,
"column": 23
} | {
"line": 185,
"column": 23
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{u_2, u_1} C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nK : CochainComplex C ℤ\nn : ℤ\nhn : K.IsStrictlyGE n\nL : CochainComplex C ℤ\nm : ℤ\nhm : L.IsStrictlyGE m\nf : { obj := K, property := ⋯ }.obj ⟶ { obj := L, property := ⋯ }.obj\nd : ℤ\nleft✝² : K.IsStrictlyGE... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.ModelCategory.Injective | {
"line": 200,
"column": 17
} | {
"line": 200,
"column": 30
} | {
"line": 200,
"column": 30
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{u_2, u_1} C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nK : CochainComplex C ℤ\nn : ℤ\nhn : K.IsStrictlyGE n\nL : CochainComplex C ℤ\nm : ℤ\nhm : L.IsStrictlyGE m\nf : { obj := K, property := ⋯ }.obj ⟶ { obj := L, property := ⋯ }.obj\nd : ℤ\nleft✝ : K.IsStrictlyGE ... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.GradedObject.Unitor | {
"line": 116,
"column": 7
} | {
"line": 116,
"column": 37
} | {
"line": 116,
"column": 37
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nI : Type u_3\nJ : Type u_4\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Category.{v_2, u_2} D\ninst✝⁴ : Zero I\ninst✝³ : DecidableEq I\ninst✝² : HasInitial C\nF : C ⥤ D ⥤ D\nX : C\ne : F.obj X ≅ 𝟭 D\ninst✝¹ : ∀ (Y : D), PreservesColimit (Functor.empty C) (F.flip.obj Y)\np : I ... | [
"C : Type u_1\nD : Type u_2\nI : Type u_3\nJ : Type u_4\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Category.{v_2, u_2} D\ninst✝⁴ : Zero I\ninst✝³ : DecidableEq I\ninst✝² : HasInitial C\nF : C ⥤ D ⥤ D\nX : C\ne : F.obj X ≅ 𝟭 D\ninst✝¹ : ∀ (Y : D), PreservesColimit (Functor.empty C) (F.flip.obj Y)\np : I × J → J\nhp ... | CofanMapObjFun.ιMapObj_iso_inv | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.GradedObject.Unitor | {
"line": 236,
"column": 7
} | {
"line": 236,
"column": 37
} | {
"line": 236,
"column": 37
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nI : Type u_3\nJ : Type u_4\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Category.{v_2, u_2} D\ninst✝⁴ : Zero I\ninst✝³ : DecidableEq I\ninst✝² : HasInitial C\nF : D ⥤ C ⥤ D\nY : C\ne : F.flip.obj Y ≅ 𝟭 D\ninst✝¹ : ∀ (X : D), PreservesColimit (Functor.empty C) (F.obj X)\np : J ... | [
"C : Type u_1\nD : Type u_2\nI : Type u_3\nJ : Type u_4\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Category.{v_2, u_2} D\ninst✝⁴ : Zero I\ninst✝³ : DecidableEq I\ninst✝² : HasInitial C\nF : D ⥤ C ⥤ D\nY : C\ne : F.flip.obj Y ≅ 𝟭 D\ninst✝¹ : ∀ (X : D), PreservesColimit (Functor.empty C) (F.obj X)\np : J × I → J\nhp ... | CofanMapObjFun.ιMapObj_iso_inv | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.SpectralObject.Page | {
"line": 201,
"column": 10
} | {
"line": 205,
"column": 37
} | {
"line": 205,
"column": 38
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} ι\ninst✝ : Abelian C\nX : SpectralObject C ι\ni j k l : ι\nf₁ : i ⟶ j\nf₂ : j ⟶ k\nf₃ : k ⟶ l\nf₁₂ : i ⟶ k\nh₁₂ : f₁ ≫ f₂ = f₁₂\nf₂₃ : j ⟶ l\nh₂₃ : f₂ ≫ f₃ = f₂₃\nn₀ n₁ n₂ : ℤ\nhn₁ : n₀ + 1 = n₁\nhn₂ : n₁ + 1 = n₂\... | [] | by
refine (IsColimit.equivOfNatIsoOfIso ?_ _ _ ?_).2
(cokernelIsCokernel (X.δToCycles f₁ f₂ f₃ n₀ n₁))
· exact parallelPair.ext (Iso.refl _) (Iso.refl _) (by simpa) (by simp)
· exact Cofork.ext (Iso.refl _) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence | {
"line": 128,
"column": 15
} | {
"line": 129,
"column": 45
} | {
"line": 131,
"column": 0
} | [
{
"pp": "ι : Type u_2\nκ : Type u_3\ninst✝ : Preorder ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\nr : ℤ\nhr : r₀ ≤ r\npq' : κ\ni₂ i₃ : ι\nhi₂ : i₂ = data.i₂ pq'\nhi₃ : i₃ = data.i₃ r pq' ⋯\n⊢ i₂ ≤ i₃",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Cate... | [] | by
simpa only [hi₂, hi₃] using data.le₂₃ r pq' | [anonymous] | Lean.Parser.Term.byTactic |
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