module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Algebra.Category.ModuleCat.Stalk | {
"line": 79,
"column": 4
} | {
"line": 79,
"column": 17
} | [
{
"pp": "C : Type u_1\ninst✝² : SmallCategory C\ninst✝¹ : IsFiltered C\nR : C ⥤ RingCat\nM : C ⥤ Ab\ninst✝ : (i : C) → Module ↑(R.obj i) ↑(M.obj i)\nH :\n ∀ {i j : C} (f : i ⟶ j) (r : ↑(R.obj i)) (m : ↑(M.obj i)),\n (ConcreteCategory.hom (M.map f)) (r • m) = (ConcreteCategory.hom (R.map f)) r • (ConcreteCat... | rintro ⟨V, b⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.Category.ModuleCat.Stalk | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 17
} | [
{
"pp": "C : Type u_1\ninst✝² : SmallCategory C\ninst✝¹ : IsFiltered C\nR : C ⥤ RingCat\nM : C ⥤ Ab\ninst✝ : (i : C) → Module ↑(R.obj i) ↑(M.obj i)\nH :\n ∀ {i j : C} (f : i ⟶ j) (r : ↑(R.obj i)) (m : ↑(M.obj i)),\n (ConcreteCategory.hom (M.map f)) (r • m) = (ConcreteCategory.hom (R.map f)) r • (ConcreteCat... | rintro ⟨V, b⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.Category.ModuleCat.Stalk | {
"line": 113,
"column": 40
} | {
"line": 117,
"column": 38
} | [
{
"pp": "C : Type u_1\ninst✝² : SmallCategory C\ninst✝¹ : IsFiltered C\nR : C ⥤ RingCat\nM : C ⥤ Ab\ninst✝ : (i : C) → Module ↑(R.obj i) ↑(M.obj i)\nH :\n ∀ {i j : C} (f : i ⟶ j) (r : ↑(R.obj i)) (m : ↑(M.obj i)),\n (ConcreteCategory.hom (M.map f)) (r • m) = (ConcreteCategory.hom (R.map f)) r • (ConcreteCat... | by
rintro ⟨V, b⟩
refine Functor.ιColimitType_eq_of_map_eq_map _ _ _ (𝟙 _) (leftToMax _ _) ?_
dsimp
simp only [map_zero, zero_smul, *] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.CharP.MixedCharZero | {
"line": 94,
"column": 22
} | {
"line": 94,
"column": 33
} | [
{
"pp": "case mp\nR : Type u_1\ninst✝ : CommRing R\nP : Prop\nh : ∀ p > 0, MixedCharZero R p → P\nq : ℕ\nq_prime : Nat.Prime q\n⊢ MixedCharZero R q → P",
"usedConstants": [
"MixedCharZero"
]
}
] | q_mixedChar | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.CharP.MixedCharZero | {
"line": 96,
"column": 20
} | {
"line": 96,
"column": 31
} | [
{
"pp": "case mpr\nR : Type u_1\ninst✝ : CommRing R\nP : Prop\nh : ∀ (p : ℕ), Nat.Prime p → MixedCharZero R p → P\nq : ℕ\nq_pos : q > 0\n⊢ MixedCharZero R q → P",
"usedConstants": [
"MixedCharZero"
]
}
] | q_mixedChar | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.CharP.Quotient | {
"line": 49,
"column": 4
} | {
"line": 49,
"column": 56
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\np : ℕ\ninst✝ : CharP R p\nI : Ideal R\nh : ∀ (x : ℕ), ↑x ∈ I → ↑x = 0\nx : ℕ\n⊢ (Ideal.Quotient.mk I) ↑x = 0 ↔ ↑x = 0",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semiring.toModule",
"AddGroupWithOne.toAddGroup",
"Ideal.... | refine Ideal.Quotient.eq.trans (?_ : ↑x - 0 ∈ I ↔ _) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.FreeCommRing | {
"line": 242,
"column": 2
} | {
"line": 242,
"column": 36
} | [
{
"pp": "α : Type u\np : α\ns : Set α\nhps : (of p).IsSupported s\n⊢ p ∈ s",
"usedConstants": [
"Membership.mem",
"Classical.decPred",
"Set.instMembership",
"Set"
]
}
] | haveI := Classical.decPred (· ∈ s) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHaveI___1 | Lean.Parser.Tactic.tacticHaveI__ |
Mathlib.RingTheory.Flat.Equalizer | {
"line": 143,
"column": 6
} | {
"line": 143,
"column": 13
} | [
{
"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\... | ext m x | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.RingTheory.Flat.Equalizer | {
"line": 168,
"column": 6
} | {
"line": 168,
"column": 13
} | [
{
"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\... | ext m x | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.Data.Stream.Init | {
"line": 357,
"column": 2
} | {
"line": 357,
"column": 28
} | [
{
"pp": "α : Type u\ns₁ s₂ : Stream' α\n⊢ s₁ ⋈ s₂ = s₁.head :: s₂.head :: (s₁.tail ⋈ s₂.tail)",
"usedConstants": [
"Stream'.interleave",
"Stream'",
"Stream'.tail"
]
}
] | let t := tail s₁ ⋈ tail s₂ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Data.Stream.Init | {
"line": 362,
"column": 2
} | {
"line": 362,
"column": 47
} | [
{
"pp": "α : Type u\ns₁ s₂ : Stream' α\n⊢ (s₁ ⋈ s₂).tail = s₂ ⋈ s₁.tail",
"usedConstants": [
"Eq.mpr",
"Stream'.interleave",
"congrArg",
"id",
"Prod.mk",
"Stream'.corec_eq",
"Stream'.interleave.match_1",
"Stream'.corecOn",
"Stream'",
"Stream'.corec... | unfold interleave corecOn; rw [corec_eq]; rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Stream.Init | {
"line": 362,
"column": 2
} | {
"line": 362,
"column": 47
} | [
{
"pp": "α : Type u\ns₁ s₂ : Stream' α\n⊢ (s₁ ⋈ s₂).tail = s₂ ⋈ s₁.tail",
"usedConstants": [
"Eq.mpr",
"Stream'.interleave",
"congrArg",
"id",
"Prod.mk",
"Stream'.corec_eq",
"Stream'.interleave.match_1",
"Stream'.corecOn",
"Stream'",
"Stream'.corec... | unfold interleave corecOn; rw [corec_eq]; rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Stream.Init | {
"line": 564,
"column": 2
} | {
"line": 564,
"column": 38
} | [
{
"pp": "α : Type u\nn : ℕ\nx : List α\na : Stream' α\nh : n ≤ x.length\n⊢ take n (x ++ₛ a) = List.take n x",
"usedConstants": [
"Stream'.take",
"List.ext_getElem",
"congrArg",
"List.length_take",
"Stream'.appendStream'",
"Nat",
"congr",
"True",
"Nat.min... | apply List.ext_getElem (by simp [h]) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Data.Seq.Computation | {
"line": 838,
"column": 20
} | {
"line": 838,
"column": 36
} | [
{
"pp": "α : Type u\ns t : Computation α\na : α\nh1 : a ∈ s\nh2 : a ∈ t\na' : α\nma : a' ∈ s\n⊢ a' ∈ t",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Computation.mem_unique",
"Membership.mem",
"id",
"Computation",
"Computation.instMembership",
"Eq"
]
}
] | mem_unique ma h1 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Seq.Basic | {
"line": 182,
"column": 14
} | {
"line": 182,
"column": 19
} | [
{
"pp": "case zero\nα : Type u\nx : α\ns : Seq α\nn : ℕ\nh_mn : 0 < n\nh_get : s.get? 0 = some x\nl : ℕ\nhl : l + 1 = n\n⊢ x ∈ take (l + 1) s",
"usedConstants": [
"Eq.mpr",
"Stream'.Seq",
"congrArg",
"Membership.mem",
"Stream'.Seq.take",
"Stream'.Seq.take.eq_2",
"id... | take, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Seq.Basic | {
"line": 190,
"column": 8
} | {
"line": 190,
"column": 13
} | [
{
"pp": "case succ\nα : Type u\nx : α\nk : ℕ\nih : ∀ {s : Seq α} {n : ℕ}, k < n → s.get? k = some x → x ∈ take n s\ns : Seq α\nh_get : s.get? (k + 1) = some x\nl : ℕ\nh_mn : k + 1 < k + 1 + l + 1\ny : α\nhy : s.get? 0 = some y\n⊢ x ∈ take (k + 1 + l + 1) s",
"usedConstants": [
"Eq.mpr",
"Stream'... | take, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Seq.Basic | {
"line": 197,
"column": 2
} | {
"line": 205,
"column": 20
} | [
{
"pp": "α : Type u\ns : Seq α\nn : ℕ\n⊢ (take n s).length ≤ n",
"usedConstants": [
"Eq.mpr",
"Stream'.Seq",
"Nat.recAux",
"Preorder.toLT",
"Nat.instIsOrderedAddMonoid",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
"AddLeftCancelSemigroup.toIsLeftCancelAdd",
... | induction n generalizing s with
| zero => simp
| succ m ih =>
rw [take]
cases s.destruct with
| none => simp
| some v =>
obtain ⟨x, r⟩ := v
simpa using ih | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Data.Seq.Basic | {
"line": 197,
"column": 2
} | {
"line": 205,
"column": 20
} | [
{
"pp": "α : Type u\ns : Seq α\nn : ℕ\n⊢ (take n s).length ≤ n",
"usedConstants": [
"Eq.mpr",
"Stream'.Seq",
"Nat.recAux",
"Preorder.toLT",
"Nat.instIsOrderedAddMonoid",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
"AddLeftCancelSemigroup.toIsLeftCancelAdd",
... | induction n generalizing s with
| zero => simp
| succ m ih =>
rw [take]
cases s.destruct with
| none => simp
| some v =>
obtain ⟨x, r⟩ := v
simpa using ih | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Seq.Basic | {
"line": 197,
"column": 2
} | {
"line": 205,
"column": 20
} | [
{
"pp": "α : Type u\ns : Seq α\nn : ℕ\n⊢ (take n s).length ≤ n",
"usedConstants": [
"Eq.mpr",
"Stream'.Seq",
"Nat.recAux",
"Preorder.toLT",
"Nat.instIsOrderedAddMonoid",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
"AddLeftCancelSemigroup.toIsLeftCancelAdd",
... | induction n generalizing s with
| zero => simp
| succ m ih =>
rw [take]
cases s.destruct with
| none => simp
| some v =>
obtain ⟨x, r⟩ := v
simpa using ih | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Seq.Basic | {
"line": 212,
"column": 10
} | {
"line": 212,
"column": 15
} | [
{
"pp": "case succ\nα : Type u\nn : ℕ\nih : ∀ {s : Seq α}, (∀ (h : s.Terminates), n ≤ s.length h) → (take n s).length = n\ns : Seq α\nhle : ∀ (h : s.Terminates), n + 1 ≤ s.length h\n⊢ (take (n + 1) s).length = n + 1",
"usedConstants": [
"Eq.mpr",
"Stream'.Seq",
"congrArg",
"Stream'.S... | take, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Seq.Basic | {
"line": 302,
"column": 12
} | {
"line": 302,
"column": 28
} | [
{
"pp": "α : Type u\ns t u✝ s1 s2 : Seq α\nh : ∃ s t u, s1 = (s.append t).append u ∧ s2 = s.append (t.append u)\nu : Seq α\n⊢ match u.destruct, u.destruct with\n | none, none => True\n | some (a, s), some (a', s') => a = a' ∧ ∃ s_1 t u, s = (s_1.append t).append u ∧ s' = s_1.append (t.append u)\n | x, x_1 =>... | cases u <;> simp | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.ContinuedFractions.Computation.Translations | {
"line": 251,
"column": 21
} | {
"line": 255,
"column": 83
} | [
{
"pp": "K : Type u_1\ninst✝² : DivisionRing K\ninst✝¹ : LinearOrder K\ninst✝ : FloorRing K\nv : K\n⊢ (of v).s.get? 0 = (IntFractPair.stream v 1).bind (some ∘ fun p ↦ { a := 1, b := ↑p.b })",
"usedConstants": [
"GenContFract.s",
"Int.cast",
"Eq.mpr",
"GenContFract.IntFractPair.stream... | by
rw [of, IntFractPair.seq1]
simp only [Stream'.Seq.map, Stream'.Seq.tail, Stream'.Seq.get?, Stream'.map]
rw [← Stream'.get_succ, Stream'.get, Option.map.eq_def]
split <;> simp_all only [Option.bind_some, Option.bind_none, Function.comp_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Nat.Fib.Basic | {
"line": 227,
"column": 8
} | {
"line": 227,
"column": 69
} | [
{
"pp": "m n : ℕ\nh : n.pred.succ = n\n⊢ (fib m).gcd (fib n.pred * fib m + fib (n.pred + 1) * fib (m + 1)) = (fib m).gcd (fib (n.pred + 1) * fib (m + 1))",
"usedConstants": [
"Nat.gcd",
"Eq.mpr",
"HMul.hMul",
"Nat.gcd_add_mul_right_right",
"congrArg",
"id",
"instMul... | rw [add_comm, gcd_add_mul_right_right (fib m) _ (fib n.pred)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Nat.Fib.Basic | {
"line": 227,
"column": 8
} | {
"line": 227,
"column": 69
} | [
{
"pp": "m n : ℕ\nh : n.pred.succ = n\n⊢ (fib m).gcd (fib n.pred * fib m + fib (n.pred + 1) * fib (m + 1)) = (fib m).gcd (fib (n.pred + 1) * fib (m + 1))",
"usedConstants": [
"Nat.gcd",
"Eq.mpr",
"HMul.hMul",
"Nat.gcd_add_mul_right_right",
"congrArg",
"id",
"instMul... | rw [add_comm, gcd_add_mul_right_right (fib m) _ (fib n.pred)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Fib.Basic | {
"line": 227,
"column": 8
} | {
"line": 227,
"column": 69
} | [
{
"pp": "m n : ℕ\nh : n.pred.succ = n\n⊢ (fib m).gcd (fib n.pred * fib m + fib (n.pred + 1) * fib (m + 1)) = (fib m).gcd (fib (n.pred + 1) * fib (m + 1))",
"usedConstants": [
"Nat.gcd",
"Eq.mpr",
"HMul.hMul",
"Nat.gcd_add_mul_right_right",
"congrArg",
"id",
"instMul... | rw [add_comm, gcd_add_mul_right_right (fib m) _ (fib n.pred)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.ContinuedFractions.ConvergentsEquiv | {
"line": 170,
"column": 11
} | {
"line": 170,
"column": 44
} | [
{
"pp": "K : Type u_1\ninst✝ : DivisionRing K\nm : ℕ\nIH :\n ∀ {s : Stream'.Seq (Pair K)} (gp_succ_n : Pair K),\n s.get? (m + 1) = some gp_succ_n → convs'Aux s (m + 2) = convs'Aux (squashSeq s m) (m + 1)\ns : Stream'.Seq (Pair K)\ngp_succ_n : Pair K\ns_succ_nth_eq : s.get? (m + 1 + 1) = some gp_succ_n\ngp_h... | simpa only [convs'Aux, s_head_eq] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.ContinuedFractions.ConvergentsEquiv | {
"line": 170,
"column": 11
} | {
"line": 170,
"column": 44
} | [
{
"pp": "K : Type u_1\ninst✝ : DivisionRing K\nm : ℕ\nIH :\n ∀ {s : Stream'.Seq (Pair K)} (gp_succ_n : Pair K),\n s.get? (m + 1) = some gp_succ_n → convs'Aux s (m + 2) = convs'Aux (squashSeq s m) (m + 1)\ns : Stream'.Seq (Pair K)\ngp_succ_n : Pair K\ns_succ_nth_eq : s.get? (m + 1 + 1) = some gp_succ_n\ngp_h... | simpa only [convs'Aux, s_head_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.ContinuedFractions.ConvergentsEquiv | {
"line": 170,
"column": 11
} | {
"line": 170,
"column": 44
} | [
{
"pp": "K : Type u_1\ninst✝ : DivisionRing K\nm : ℕ\nIH :\n ∀ {s : Stream'.Seq (Pair K)} (gp_succ_n : Pair K),\n s.get? (m + 1) = some gp_succ_n → convs'Aux s (m + 2) = convs'Aux (squashSeq s m) (m + 1)\ns : Stream'.Seq (Pair K)\ngp_succ_n : Pair K\ns_succ_nth_eq : s.get? (m + 1 + 1) = some gp_succ_n\ngp_h... | simpa only [convs'Aux, s_head_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.ContinuedFractions.Computation.TerminatesIffRat | {
"line": 188,
"column": 12
} | {
"line": 188,
"column": 61
} | [
{
"pp": "case h\nK : Type u_1\ninst✝³ : Field K\ninst✝² : LinearOrder K\ninst✝¹ : IsStrictOrderedRing K\ninst✝ : FloorRing K\nv : K\nq : ℚ\nv_eq_q : v = ↑q\nn : ℕ\n⊢ Stream'.map (Option.map (mapFr Rat.cast)) (IntFractPair.stream q) n = IntFractPair.stream v n",
"usedConstants": [
"GenContFract.IntFrac... | exact IntFractPair.coe_stream_nth_rat_eq v_eq_q n | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.CubicDiscriminant | {
"line": 397,
"column": 13
} | {
"line": 397,
"column": 26
} | [
{
"pp": "R : Type u_1\nP : Cubic R\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nh0 : P.toPoly ≠ 0\nx : R\n⊢ x ∈ P.toPoly.roots ↔ P.a * x ^ 3 + P.b * x ^ 2 + P.c * x + P.d = 0",
"usedConstants": [
"Eq.mpr",
"Polynomial.roots",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
... | mem_roots h0, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.CubicDiscriminant | {
"line": 417,
"column": 2
} | {
"line": 417,
"column": 56
} | [
{
"pp": "F : Type u_3\nK : Type u_4\nP : Cubic F\ninst✝¹ : Field F\ninst✝ : Field K\nφ : F →+* K\nha : P.a ≠ 0\n⊢ (Polynomial.map φ P.toPoly).Splits ↔ (map φ P).roots.card = 3",
"usedConstants": [
"Iff.mpr",
"GroupWithZero.toMonoidWithZero",
"RingHom.instRingHomClass",
"Cubic.map",
... | replace ha : (map φ P).a ≠ 0 := (map_ne_zero φ).mpr ha | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.Algebra.CubicDiscriminant | {
"line": 488,
"column": 2
} | {
"line": 489,
"column": 31
} | [
{
"pp": "F : Type u_3\nK : Type u_4\nP : Cubic F\ninst✝² : Field F\ninst✝¹ : Field K\nφ : F →+* K\ninst✝ : DecidableEq K\nha : P.a ≠ 0\nh3 : (Polynomial.map φ P.toPoly).Splits\nhd : P.discr ≠ 0\n⊢ (map φ P).roots.toFinset.card = 3",
"usedConstants": [
"Multiset.toFinset",
"Eq.mpr",
"Cubic.... | rwa [toFinset_card_of_nodup <| (discr_ne_zero_iff_roots_nodup ha h3).mp hd,
← splits_iff_card_roots ha] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Algebra.CubicDiscriminant | {
"line": 488,
"column": 2
} | {
"line": 489,
"column": 31
} | [
{
"pp": "F : Type u_3\nK : Type u_4\nP : Cubic F\ninst✝² : Field F\ninst✝¹ : Field K\nφ : F →+* K\ninst✝ : DecidableEq K\nha : P.a ≠ 0\nh3 : (Polynomial.map φ P.toPoly).Splits\nhd : P.discr ≠ 0\n⊢ (map φ P).roots.toFinset.card = 3",
"usedConstants": [
"Multiset.toFinset",
"Eq.mpr",
"Cubic.... | rwa [toFinset_card_of_nodup <| (discr_ne_zero_iff_roots_nodup ha h3).mp hd,
← splits_iff_card_roots ha] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.CubicDiscriminant | {
"line": 488,
"column": 2
} | {
"line": 489,
"column": 31
} | [
{
"pp": "F : Type u_3\nK : Type u_4\nP : Cubic F\ninst✝² : Field F\ninst✝¹ : Field K\nφ : F →+* K\ninst✝ : DecidableEq K\nha : P.a ≠ 0\nh3 : (Polynomial.map φ P.toPoly).Splits\nhd : P.discr ≠ 0\n⊢ (map φ P).roots.toFinset.card = 3",
"usedConstants": [
"Multiset.toFinset",
"Eq.mpr",
"Cubic.... | rwa [toFinset_card_of_nodup <| (discr_ne_zero_iff_roots_nodup ha h3).mp hd,
← splits_iff_card_roots ha] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Idempotents | {
"line": 345,
"column": 2
} | {
"line": 345,
"column": 88
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\ne₁ e₂ : R\nhe₁ : IsIdempotentElem e₁\nhe₂ : IsIdempotentElem e₂\nH' : Commute e₁ e₂\nthis✝ : (e₁ - e₂) ^ 3 = e₁ - e₂\nn : ℕ\nhn : (e₁ - e₂) ^ n = 0\nthis : (e₁ - e₂) ^ (2 * n + 1) = e₁ - e₂\n⊢ e₁ = e₂",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"No... | rwa [pow_succ, two_mul, pow_add, hn, zero_mul, zero_mul, eq_comm, sub_eq_zero] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.LinearAlgebra.Eigenspace.Basic | {
"line": 310,
"column": 2
} | {
"line": 310,
"column": 15
} | [
{
"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\nk m : ℕ∞\nhm : k ≤ m\nhk : (f.genEigenspace μ) k ≠ ⊥\n⊢ (f.genEigenspace μ) m ≠ ⊥",
"usedConstants": [
"Submodule",
"CommSemiring.toSemiring",
"AddCommGroup.toAddCommMonoi... | contrapose hk | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose_1 | Mathlib.Tactic.Contrapose.contrapose |
Mathlib.LinearAlgebra.Eigenspace.Basic | {
"line": 803,
"column": 87
} | {
"line": 806,
"column": 89
} | [
{
"pp": "R : Type v\nM : Type w\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\np : Submodule R M\nhfp : ∀ x ∈ p, f x ∈ p\nμ : R\nhμp : Disjoint (f.eigenspace μ) p\n⊢ eigenspace (LinearMap.restrict f hfp) μ = ⊥",
"usedConstants": [
"Eq.mpr",
"Disjoint.le_bot",
... | by
rw [eq_bot_iff]
intro x hx
simpa using hμp.le_bot ⟨eigenspace_restrict_le_eigenspace f hfp μ ⟨x, hx, rfl⟩, x.prop⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Field.TransferInstance | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 69
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ne : α ≃ β\ninst✝ : Field β\nadd_group_with_one : AddGroupWithOne α := e.addGroupWithOne\nneg : Neg α := e.Neg\ninv : Inv α := e.Inv\ndiv : Div α := e.div\nmul : Mul α := e.mul\nnpow : Pow α ℕ := Equiv.pow ℕ e\nzpow : Pow α ℤ := Equiv.pow ℤ e\nnnratCast : NNRatCast α := e.nnr... | apply e.injective.field _ <;> intros <;> exact e.apply_symm_apply _ | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.GCDMonoid.IntegrallyClosed | {
"line": 29,
"column": 2
} | {
"line": 29,
"column": 16
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : GCDMonoid R\nM : Submonoid R\ninst✝ : IsLocalization M A\nx y : R\nhy : y ∈ M\nx' y' : R\nhx' : x = gcd x y * x'\nhy' : y = gcd x y * y'\nhu : IsUnit (gcd x' y')\n⊢ ∃ a b, IsUnit (gcd a b) ∧ mk' A x ⟨y,... | use x', y', hu | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.RingTheory.IntegralClosure.IntegrallyClosed | {
"line": 281,
"column": 2
} | {
"line": 282,
"column": 43
} | [
{
"pp": "R : Type u_4\nK : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : FaithfulSMul R K\ninst✝ : IsIntegrallyClosedIn R K\nthis : IsDomain R\n⊢ IsIntegrallyClosed R",
"usedConstants": [
"FractionRing.field",
"IsScalarTower.right",
"OreLocalization.instAl... | let f : FractionRing R →ₐ[R] K := IsFractionRing.liftAlgHom (g := Algebra.ofId _ _)
(FaithfulSMul.algebraMap_injective R K) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Algebra.Group.Action.Sigma | {
"line": 53,
"column": 4
} | {
"line": 54,
"column": 46
} | [
{
"pp": "ι : Type u_1\nM : Type u_2\nN : Type u_3\nα : ι → Type u_4\ninst✝³ : (i : ι) → SMul M (α i)\ninst✝² : (i : ι) → SMul N (α i)\na✝ : M\ni : ι\nb✝ : α i\nx✝ : (i : ι) × α i\ninst✝¹ : SMul M N\ninst✝ : ∀ (i : ι), IsScalarTower M N (α i)\na : M\nb : N\nx : (i : ι) × α i\n⊢ (a • b) • x = a • b • x",
"use... | cases x
rw [smul_mk, smul_mk, smul_mk, smul_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Action.Sigma | {
"line": 53,
"column": 4
} | {
"line": 54,
"column": 46
} | [
{
"pp": "ι : Type u_1\nM : Type u_2\nN : Type u_3\nα : ι → Type u_4\ninst✝³ : (i : ι) → SMul M (α i)\ninst✝² : (i : ι) → SMul N (α i)\na✝ : M\ni : ι\nb✝ : α i\nx✝ : (i : ι) × α i\ninst✝¹ : SMul M N\ninst✝ : ∀ (i : ι), IsScalarTower M N (α i)\na : M\nb : N\nx : (i : ι) × α i\n⊢ (a • b) • x = a • b • x",
"use... | cases x
rw [smul_mk, smul_mk, smul_mk, smul_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.ForwardDiff | {
"line": 161,
"column": 21
} | {
"line": 161,
"column": 37
} | [
{
"pp": "case h.e'_3.e_f.h\nM : Type u_1\nG : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : AddCommGroup G\nh : M\nf : M → G\nn : ℕ\ny : M\nthis✝¹ : fwdDiffₗ M G h = shiftₗ M G h - 1\nthis✝ : Commute (shiftₗ M G h) (-1)\nk : ℕ\nthis : (-1) ^ (n - k) * ↑(n.choose k) = ↑((-1) ^ (n - k) * ↑(n.choose k))\n⊢ ((-1) ^ (... | shiftₗ_pow_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Subgroup.Order | {
"line": 65,
"column": 40
} | {
"line": 65,
"column": 49
} | [
{
"pp": "case refine_1\nG : Type u_1\ninst✝¹ : Group G\nH : Type u_2\ninst✝ : Group H\nφ : G →* H\nhφ : Function.Surjective ⇑φ\nM : Subgroup H\nhM✝ : IsCoatom M\nhM : comap φ M ≠ comap φ ⊤\n⊢ comap φ M ≠ ⊤",
"usedConstants": [
"congrArg",
"Eq.mp",
"Subgroup",
"Ne",
"Subgroup.in... | comap_top | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Subgroup.Finsupp | {
"line": 65,
"column": 18
} | {
"line": 65,
"column": 46
} | [
{
"pp": "M : Type u_1\ninst✝¹ : CommGroup M\nx : M\ns : Set M\ninst✝ : Fintype ↑s\n| x ∈ closure s",
"usedConstants": [
"Subgroup.closure",
"congrArg",
"Membership.mem",
"Subtype",
"Subgroup",
"Subtype.range_coe",
"Set.range",
"CommGroup.toGroup",
"Eq.sy... | ← Subtype.range_coe (s := s) | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.Algebra.Group.Subgroup.Order | {
"line": 69,
"column": 60
} | {
"line": 69,
"column": 69
} | [
{
"pp": "case refine_2\nG : Type u_1\ninst✝¹ : Group G\nH : Type u_2\ninst✝ : Group H\nφ : G →* H\nhφ : Function.Surjective ⇑φ\nM : Subgroup H\nhM✝ : IsCoatom M\nK : Subgroup G\nhK : comap φ M < K\nhM : comap φ M < K → K = comap φ ⊤\n⊢ K = ⊤",
"usedConstants": [
"Preorder.toLT",
"congrArg",
... | comap_top | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Pointwise.Set.Card | {
"line": 67,
"column": 56
} | {
"line": 68,
"column": 48
} | [
{
"pp": "M : Type u_2\ninst✝ : DivInvMonoid M\ns t : Set M\n⊢ #↑(s / t) ≤ #↑s * #↑t",
"usedConstants": [
"Eq.mpr",
"instHDiv",
"HMul.hMul",
"Cardinal",
"congrArg",
"Cardinal.mk",
"Set.Elem",
"Cardinal.instMul",
"id",
"HDiv.hDiv",
"Cardinal.mk... | by
rw [← image2_div]; exact Cardinal.mk_image2_le | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Submonoid.Finsupp | {
"line": 58,
"column": 18
} | {
"line": 58,
"column": 46
} | [
{
"pp": "M : Type u_1\ninst✝¹ : CommMonoid M\nx : M\ns : Set M\ninst✝ : Fintype ↑s\n| x ∈ closure s",
"usedConstants": [
"Monoid.toMulOneClass",
"congrArg",
"Membership.mem",
"Subtype",
"Subtype.range_coe",
"CommMonoid.toMonoid",
"Submonoid.closure",
"Submonoi... | ← Subtype.range_coe (s := s) | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.AlgebraicTopology.SimplexCategory.Basic | {
"line": 278,
"column": 2
} | {
"line": 278,
"column": 38
} | [
{
"pp": "case a.h.h\nn : ℕ\ni : Fin (n + 2)\nj : Fin (n + 1)\nH : i ≤ j.castSucc\nk : Fin (⦋n + 1⦌.len + 1)\n⊢ j.succ.predAbove (i.castSucc.succAbove k) = i.succAbove (j.predAbove k)",
"usedConstants": [
"instOfNatNat",
"instHAdd",
"le_or_gt",
"Fin.instLinearOrder",
"HAdd.hAdd"... | rcases le_or_gt i k with (hik | hik) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.AlgebraicTopology.SimplicialSet.Dimension | {
"line": 134,
"column": 28
} | {
"line": 136,
"column": 9
} | [
{
"pp": "X : SSet\nn k : ℕ\nhk : n ≤ k\n⊢ ⊥.toSSet.degenerate k = ⊤",
"usedConstants": [
"SSet.Subcomplex.toSSet",
"Set.ext",
"False",
"Lattice.toSemilatticeSup",
"Opposite",
"CompleteLattice.toLattice",
"False.elim",
"OrderBot.toBot",
"PartialOrder.toPr... | by
ext ⟨x, hx⟩
tauto | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialSet.Subcomplex | {
"line": 330,
"column": 2
} | {
"line": 330,
"column": 19
} | [
{
"pp": "X Y : SSet\nf : X ⟶ Y\nS T : X.Subcomplex\nh : S ≤ T\n⊢ (fun S ↦ S.image f) S ≤ (fun S ↦ S.image f) T",
"usedConstants": [
"Eq.mpr",
"Opposite",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SSet.Subcomplex.image_le_iff",
"id",
"LE.le",
... | rw [image_le_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicTopology.SimplicialSet.Subcomplex | {
"line": 342,
"column": 2
} | {
"line": 342,
"column": 19
} | [
{
"pp": "X Y : SSet\nB : X.Subcomplex\nf : Y ⟶ X\n⊢ (B.preimage f).image f ≤ B",
"usedConstants": [
"Eq.mpr",
"le_refl",
"Opposite",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SSet.Subcomplex.image_le_iff",
"id",
"LE.le",
"SSet.Subcom... | rw [image_le_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicTopology.SimplicialSet.Subcomplex | {
"line": 342,
"column": 2
} | {
"line": 342,
"column": 19
} | [
{
"pp": "X Y : SSet\nB : X.Subcomplex\nf : Y ⟶ X\n⊢ (B.preimage f).image f ≤ B",
"usedConstants": [
"Eq.mpr",
"le_refl",
"Opposite",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SSet.Subcomplex.image_le_iff",
"id",
"LE.le",
"SSet.Subcom... | rw [image_le_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialSet.Subcomplex | {
"line": 342,
"column": 2
} | {
"line": 342,
"column": 19
} | [
{
"pp": "X Y : SSet\nB : X.Subcomplex\nf : Y ⟶ X\n⊢ (B.preimage f).image f ≤ B",
"usedConstants": [
"Eq.mpr",
"le_refl",
"Opposite",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SSet.Subcomplex.image_le_iff",
"id",
"LE.le",
"SSet.Subcom... | rw [image_le_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.NonDegenerateSimplices | {
"line": 99,
"column": 4
} | {
"line": 99,
"column": 48
} | [
{
"pp": "case inr\nX : SSet\ny : X.N\nd : ℕ\nx : X _⦋d⦌\nhx : x ∈ X.nonDegenerate d\nh : mk ↑⟨x, hx⟩ ⋯ < y\nh' : (mk ↑⟨x, hx⟩ ⋯).dim = y.dim\nf : ⦋(mk ↑⟨x, hx⟩ ⋯).dim⦌ ⟶ ⦋y.dim⦌\nw✝ : Mono f\nhf : (ConcreteCategory.hom (X.map f.op)) y.simplex = (mk ↑⟨x, hx⟩ ⋯).simplex\n⊢ (mk ↑⟨x, hx⟩ ⋯).dim < y.dim",
"usedC... | obtain ⟨d', ⟨y, hy⟩, rfl⟩ := y.mk_surjective | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicTopology.SimplicialSet.Finite | {
"line": 37,
"column": 63
} | {
"line": 39,
"column": 9
} | [
{
"pp": "X : SSet\ninst✝ : X.Finite\nn : ℕ\nx y : ↑(X.nonDegenerate n)\nh : (fun x ↦ N.mk ↑x ⋯) x = (fun x ↦ N.mk ↑x ⋯) y\n⊢ x = y",
"usedConstants": [
"Iff.mpr",
"SSet.S.simplex",
"Eq.mpr",
"SSet.S",
"Opposite",
"Iff.of_eq",
"SSet.N.mk",
"congrArg",
"SS... | by
rw [N.ext_iff, S.ext_iff'] at h
aesop | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialSet.CompStructTruncated | {
"line": 123,
"column": 45
} | {
"line": 123,
"column": 65
} | [
{
"pp": "X Y : Truncated 2\nx y : X.obj (Opposite.op { obj := ⦋0⦌, property := Edge._proof_1 })\ne : Edge x y\n⊢ (ConcreteCategory.hom (X.map (δ₂ 0 Edge._proof_2 _proof_2 ≫ σ₂ 0 _proof_2 Edge._proof_2).op)) e.edge = e.edge",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.ObjectProperty.FullSubcate... | δ₂_zero_comp_σ₂_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplicialSet.NerveNondegenerate | {
"line": 67,
"column": 4
} | {
"line": 67,
"column": 45
} | [
{
"pp": "case succ.h\nX : Type u_1\ninst✝ : PartialOrder X\nn : ℕ\ns : nerve X _⦋n + 1⦌\ni : Fin (n + 1)\nthis : s.obj i.castSucc ≤ s.obj i.succ\n⊢ s.obj i.castSucc = s.obj i.succ ↔ ¬s.obj i.castSucc < s.obj i.succ",
"usedConstants": [
"_private.Mathlib.AlgebraicTopology.SimplicialSet.NerveNondegenera... | grind [lt_self_iff_false, LE.le.lt_or_eq] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.AlgebraicTopology.SimplicialSet.Nerve | {
"line": 210,
"column": 2
} | {
"line": 210,
"column": 21
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nx₀ x₁ x₂ : C\nf₀₁ : x₀ ⟶ x₁\nf₁₂ : x₁ ⟶ x₂\nf₀₂ : x₀ ⟶ x₂\nh' : (edgeMk f₀₁).CompStruct (edgeMk f₁₂) (edgeMk (f₀₁ ≫ f₁₂)) :=\n Edge.CompStruct.mk (ComposableArrows.mk₂ f₀₁ f₁₂) ⋯ ⋯ ⋯\nx✝ : Nonempty ((edgeMk f₀₁).CompStruct (edgeMk f₁₂) (edgeMk f₀₂))\nh : (edgeMk ... | rw [← Arrow.mk_inj] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.TotalComplex | {
"line": 235,
"column": 8
} | {
"line": 235,
"column": 44
} | [
{
"pp": "case pos\nC : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Preadditive C\nI₁ : Type u_2\nI₂ : Type u_3\nI₁₂ : Type u_4\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nK : HomologicalComplex₂ C c₁ c₂\nc₁₂ : ComplexShape I₁₂\ninst✝² : TotalComplexShape c₁ c₂ c₁₂\ninst✝¹ : DecidableEq I₁₂\ninst✝ : K.Ha... | by_cases h₄ : c₂.Rel i₂ (c₂.next i₂) | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Algebra.Homology.TotalComplex | {
"line": 246,
"column": 8
} | {
"line": 246,
"column": 44
} | [
{
"pp": "case neg\nC : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Preadditive C\nI₁ : Type u_2\nI₂ : Type u_3\nI₁₂ : Type u_4\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nK : HomologicalComplex₂ C c₁ c₂\nc₁₂ : ComplexShape I₁₂\ninst✝² : TotalComplexShape c₁ c₂ c₁₂\ninst✝¹ : DecidableEq I₁₂\ninst✝ : K.Ha... | by_cases h₄ : c₂.Rel i₂ (c₂.next i₂) | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Algebra.Homology.TotalComplexSymmetry | {
"line": 99,
"column": 12
} | {
"line": 99,
"column": 46
} | [
{
"pp": "C : Type u_1\nI₁ : Type u_2\nI₂ : Type u_3\nJ : Type u_4\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nK : HomologicalComplex₂ C c₁ c₂\nc : ComplexShape J\ninst✝⁴ : TotalComplexShape c₁ c₂ c\ninst✝³ : TotalComplexShape c₂ c₁ c\ninst✝² : TotalComple... | ← ComplexShape.next_π₁ c₁ c h₂ i₁, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.BifunctorHomotopy | {
"line": 160,
"column": 6
} | {
"line": 160,
"column": 42
} | [
{
"pp": "case pos\nC₁ : Type u_1\nC₂ : Type u_2\nD : Type u_3\nI₁ : Type u_4\nI₂ : Type u_5\nJ : Type u_6\ninst✝¹¹ : Category.{v_1, u_1} C₁\ninst✝¹⁰ : Category.{v_2, u_2} C₂\ninst✝⁹ : Category.{v_3, u_3} D\ninst✝⁸ : Preadditive C₁\ninst✝⁷ : Preadditive C₂\ninst✝⁶ : Preadditive D\nc₁ : ComplexShape I₁\nc₂ : Comp... | by_cases h₄ : c₂.Rel i₂ (c₂.next i₂) | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Algebra.Homology.BifunctorHomotopy | {
"line": 166,
"column": 6
} | {
"line": 166,
"column": 42
} | [
{
"pp": "case neg\nC₁ : Type u_1\nC₂ : Type u_2\nD : Type u_3\nI₁ : Type u_4\nI₂ : Type u_5\nJ : Type u_6\ninst✝¹¹ : Category.{v_1, u_1} C₁\ninst✝¹⁰ : Category.{v_2, u_2} C₂\ninst✝⁹ : Category.{v_3, u_3} D\ninst✝⁸ : Preadditive C₁\ninst✝⁷ : Preadditive C₂\ninst✝⁶ : Preadditive D\nc₁ : ComplexShape I₁\nc₂ : Comp... | by_cases h₄ : c₂.Rel i₂ (c₂.next i₂) | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Algebra.Homology.CommSq | {
"line": 65,
"column": 47
} | {
"line": 67,
"column": 44
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nX₁ X₂ X₃ X₄ : C\ninst✝ : HasBinaryBiproduct X₂ X₃\nf : X₁ ⟶ X₂\ng : X₁ ⟶ X₃\ninl : X₂ ⟶ X₄\ninr : X₃ ⟶ X₄\nsq : CommSq f g inl inr\nh : IsColimit (PushoutCocone.mk inl inr ⋯)\ns : Cofork (biprod.lift f (-g)) 0\n⊢ f ≫ biprod.inl ≫ s.π... | by
rw [← sub_eq_zero, ← assoc, ← assoc, ← Preadditive.sub_comp]
convert! s.condition <;> cat_disch | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Triangulated.TStructure.Basic | {
"line": 111,
"column": 2
} | {
"line": 131,
"column": 47
} | [
{
"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\nt : TStructure C\n⊢ Monotone t.le",
"usedConstants": [
"Int.instAddSemigroup",
"Eq.mpr",
... | let H := fun (a : ℕ) => ∀ (n : ℤ), t.le n ≤ t.le (n + a)
suffices ∀ (a : ℕ), H a by
intro n₀ n₁ h
obtain ⟨a, ha⟩ := Int.nonneg_def.1 h
obtain rfl : n₁ = n₀ + a := by lia
apply this
have H_zero : H 0 := fun n => by
simp only [Nat.cast_zero, add_zero]
rfl
have H_one : H 1 := fun n X hX => by... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Triangulated.TStructure.Basic | {
"line": 111,
"column": 2
} | {
"line": 131,
"column": 47
} | [
{
"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\nt : TStructure C\n⊢ Monotone t.le",
"usedConstants": [
"Int.instAddSemigroup",
"Eq.mpr",
... | let H := fun (a : ℕ) => ∀ (n : ℤ), t.le n ≤ t.le (n + a)
suffices ∀ (a : ℕ), H a by
intro n₀ n₁ h
obtain ⟨a, ha⟩ := Int.nonneg_def.1 h
obtain rfl : n₁ = n₀ + a := by lia
apply this
have H_zero : H 0 := fun n => by
simp only [Nat.cast_zero, add_zero]
rfl
have H_one : H 1 := fun n X hX => by... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Triangulated.TStructure.Basic | {
"line": 242,
"column": 2
} | {
"line": 242,
"column": 41
} | [
{
"pp": "C : Type u_1\ninst✝⁷ : Category.{v_1, u_1} C\ninst✝⁶ : Preadditive C\ninst✝⁵ : HasZeroObject C\ninst✝⁴ : HasShift C ℤ\ninst✝³ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝² : Pretriangulated C\nt : TStructure C\nX Y : C\nf : X ⟶ Y\nn₀ n₁ : ℤ\nh : n₀ < n₁\ninst✝¹ : t.IsLE X n₀\ninst✝ : t.IsGE Y n₁\nth... | apply (shiftFunctor C n₀).map_injective | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexCohomology | {
"line": 152,
"column": 4
} | {
"line": 153,
"column": 78
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nK L : CochainComplex C ℤ\nn : ℤ\nx : Cocycle K L n\nh✝ : Nonempty (Homotopy (Cocycle.equivHomShift.symm x) 0)\nγ : Cochain K ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj L) (-1)\nh : (↑x).rightShift n 0 ⋯ = δ (-1... | exact ⟨n - 1, by simp, n.negOnePow • γ.rightUnshift _ (by lia),
by simp [Cochain.δ_rightUnshift _ _ _ _ _ (zero_add n), smul_smul, ← h]⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Homology.Embedding.ExtendHomotopy | {
"line": 101,
"column": 54
} | {
"line": 101,
"column": 71
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝³ : Category.{v_1, u_3} C\ninst✝² : HasZeroObject C\ninst✝¹ : Preadditive C\nK L : HomologicalComplex C c\nf g : K ⟶ L\nh : Homotopy f g\ne : c.Embedding c'\ninst✝ : e.IsRelIff\ni j : ι\nhij' : ¬c'.Rel (e.f j) (e.... | rwa [← e.rel_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Algebra.Homology.Embedding.ExtendHomotopy | {
"line": 101,
"column": 54
} | {
"line": 101,
"column": 71
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝³ : Category.{v_1, u_3} C\ninst✝² : HasZeroObject C\ninst✝¹ : Preadditive C\nK L : HomologicalComplex C c\nf g : K ⟶ L\nh : Homotopy f g\ne : c.Embedding c'\ninst✝ : e.IsRelIff\ni j : ι\nhij' : ¬c'.Rel (e.f j) (e.... | rwa [← e.rel_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Embedding.ExtendHomotopy | {
"line": 101,
"column": 54
} | {
"line": 101,
"column": 71
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝³ : Category.{v_1, u_3} C\ninst✝² : HasZeroObject C\ninst✝¹ : Preadditive C\nK L : HomologicalComplex C c\nf g : K ⟶ L\nh : Homotopy f g\ne : c.Embedding c'\ninst✝ : e.IsRelIff\ni j : ι\nhij' : ¬c'.Rel (e.f j) (e.... | rwa [← e.rel_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Embedding.ExtendHomotopy | {
"line": 128,
"column": 6
} | {
"line": 129,
"column": 69
} | [
{
"pp": "case pos\nι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝³ : Category.{v_1, u_3} C\ninst✝² : HasZeroObject C\ninst✝¹ : Preadditive C\nK L : HomologicalComplex C c\nf g : K ⟶ L\ne : c.Embedding c'\ninst✝ : e.IsRelIff\nh : Homotopy (extendMap f e) (extendMap g e... | · have hi' : c'.Rel (e.f i) (e.f (c.next i)) := by rwa [e.rel_iff]
simp [dNext_eq _ hi, dNext_eq _ hi', K.extend_d_eq _ rfl rfl] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Preadditive.Injective.LiftingProperties | {
"line": 50,
"column": 18
} | {
"line": 50,
"column": 30
} | [
{
"pp": "case mp\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasZeroMorphisms C\nI Z : C\nhZ : IsZero Z\na✝ : Injective I\nA B : C\ni : A ⟶ B\n⊢ MorphismProperty.monomorphisms C i → HasLiftingProperty i 0",
"usedConstants": [
"CategoryTheory.MorphismProperty.monomorphisms"
]
}
] | (_ : Mono i) | Lean.Elab.Tactic.evalIntro | Lean.Parser.Term.typeAscription |
Mathlib.Algebra.Homology.BifunctorAssociator | {
"line": 802,
"column": 8
} | {
"line": 802,
"column": 59
} | [
{
"pp": "case neg\nC₁ : Type u_1\nC₂ : Type u_2\nC₁₂ : Type u_3\nC₂₃ : Type u_4\nC₃ : Type u_5\nC₄ : Type u_6\ninst✝³³ : Category.{v_1, u_1} C₁\ninst✝³² : Category.{v_2, u_2} C₂\ninst✝³¹ : Category.{v_3, u_5} C₃\ninst✝³⁰ : Category.{v_4, u_6} C₄\ninst✝²⁹ : Category.{v_5, u_3} C₁₂\ninst✝²⁸ : Category.{v_6, u_4} ... | mapBifunctor₁₂.d₃_eq_zero _ _ _ _ _ _ _ _ _ _ _ h₁, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.ModelCategory.LeftHomotopy | {
"line": 209,
"column": 6
} | {
"line": 209,
"column": 38
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : CategoryWithWeakEquivalences C\nX Y : C\nf g : X ⟶ Y\nP : Cylinder X\nh : P.LeftHomotopy f g\nL : C ⥤ (weakEquivalences C).Localization := (weakEquivalences C).Q\n⊢ AreEqualizedByLocalization (weakEquivalences C) f g",
"usedConstants": [
"Eq.mpr... | areEqualizedByLocalization_iff L | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.ModelCategory.RightHomotopy | {
"line": 212,
"column": 6
} | {
"line": 212,
"column": 38
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : CategoryWithWeakEquivalences C\nX Y : C\nf g : X ⟶ Y\nP : PathObject Y\nh : P.RightHomotopy f g\nL : C ⥤ (weakEquivalences C).Localization := (weakEquivalences C).Q\n⊢ AreEqualizedByLocalization (weakEquivalences C) f g",
"usedConstants": [
"Eq.... | areEqualizedByLocalization_iff L | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.Projective.Resolution | {
"line": 220,
"column": 8
} | {
"line": 220,
"column": 48
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasProjectiveResolutions C\nX : C\n⊢ (HomotopyCategory.quotient C (ComplexShape.down ℕ)).map\n (ProjectiveResolution.lift (𝟙 X) (projectiveResolution X) (projectiveResolution X)) =\n 𝟙 ((HomotopyCategory.quotient C (ComplexSh... | ← (HomotopyCategory.quotient _ _).map_id | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.Projective.Resolution | {
"line": 303,
"column": 2
} | {
"line": 304,
"column": 57
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : EnoughProjectives C\nZ : C\nn : ℕ\n⊢ (HomologicalComplex.sc' (ofComplex Z) (n + 1 + 1) (n + 1) n).Exact",
"usedConstants": [
"PSigma.snd",
"CategoryTheory.Abelian.toPreadditive",
"Eq.mpr",
"CategoryTheory.Pr... | simp only [HomologicalComplex.sc', HomologicalComplex.shortComplexFunctor', ofComplex,
ChainComplex.mk', ChainComplex.mk, ChainComplex.of_d] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Abelian.Projective.Resolution | {
"line": 320,
"column": 14
} | {
"line": 320,
"column": 30
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : EnoughProjectives C\nZ : C\n⊢ (ofComplex Z).d 1 0 ≫ Projective.π Z = 0",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"Eq.mpr",
"CategoryTheory.ProjectiveResolution.ofComplex",
"CategoryTheory.... | ofComplex_d_1_0, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.Projective.Resolution | {
"line": 323,
"column": 6
} | {
"line": 329,
"column": 19
} | [
{
"pp": "case zero\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : EnoughProjectives C\nZ : C\n⊢ QuasiIsoAt (((ofComplex Z).toSingle₀Equiv Z).symm ⟨Projective.π Z, ⋯⟩) 0",
"usedConstants": [
"CategoryTheory.ShortComplex.QuasiIso",
"Iff.mpr",
"CategoryTheory.Abelian.toP... | rw [ChainComplex.quasiIsoAt₀_iff, ShortComplex.quasiIso_iff_of_zeros']
· dsimp
refine (ShortComplex.exact_and_epi_g_iff_of_iso ?_).2
⟨exact_d_f (Projective.π Z), by dsimp; infer_instance⟩
exact ShortComplex.isoMk (Iso.refl _) (Iso.refl _) (Iso.refl _)
(by simp [ofComplex]) (by ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Abelian.Projective.Resolution | {
"line": 323,
"column": 6
} | {
"line": 329,
"column": 19
} | [
{
"pp": "case zero\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : EnoughProjectives C\nZ : C\n⊢ QuasiIsoAt (((ofComplex Z).toSingle₀Equiv Z).symm ⟨Projective.π Z, ⋯⟩) 0",
"usedConstants": [
"CategoryTheory.ShortComplex.QuasiIso",
"Iff.mpr",
"CategoryTheory.Abelian.toP... | rw [ChainComplex.quasiIsoAt₀_iff, ShortComplex.quasiIso_iff_of_zeros']
· dsimp
refine (ShortComplex.exact_and_epi_g_iff_of_iso ?_).2
⟨exact_d_f (Projective.π Z), by dsimp; infer_instance⟩
exact ShortComplex.isoMk (Iso.refl _) (Iso.refl _) (Iso.refl _)
(by simp [ofComplex]) (by ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.LocalCohomology | {
"line": 226,
"column": 4
} | {
"line": 227,
"column": 30
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nI J K : Ideal R\nhJK : J.radical = K.radical\nL : Ideal R\nhL : J ≤ L.radical\n⊢ K ≤ L.radical",
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"congrArg",
"CommSemiring.toSemiring",
"PartialOrder.toPreorder",
"Preorder.toLE",
... | rw [← Ideal.radical_le_radical_iff] at hL ⊢
exact hJK.symm.trans_le hL | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.LocalCohomology | {
"line": 226,
"column": 4
} | {
"line": 227,
"column": 30
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nI J K : Ideal R\nhJK : J.radical = K.radical\nL : Ideal R\nhL : J ≤ L.radical\n⊢ K ≤ L.radical",
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"congrArg",
"CommSemiring.toSemiring",
"PartialOrder.toPreorder",
"Preorder.toLE",
... | rw [← Ideal.radical_le_radical_iff] at hL ⊢
exact hJK.symm.trans_le hL | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Abelian.LeftDerived | {
"line": 190,
"column": 2
} | {
"line": 200,
"column": 5
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nD : Type u_1\ninst✝⁵ : Category.{v_1, u_1} D\ninst✝⁴ : Abelian C\ninst✝³ : HasProjectiveResolutions C\ninst✝² : Abelian D\nF G : C ⥤ D\ninst✝¹ : F.Additive\ninst✝ : G.Additive\nα : F ⟶ G\nX : C\nP : ProjectiveResolution X\n⊢ (NatTrans.leftDerivedToHomotopyCategor... | rw [← cancel_mono (P.isoLeftDerivedToHomotopyCategoryObj G).hom, assoc, assoc,
Iso.inv_hom_id, comp_id]
dsimp [isoLeftDerivedToHomotopyCategoryObj, Functor.mapHomotopyCategoryFactors,
NatTrans.leftDerivedToHomotopyCategory]
rw [assoc]
erw [id_comp, comp_id]
obtain ⟨β, hβ⟩ := (HomotopyCategory.quotient... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Abelian.LeftDerived | {
"line": 190,
"column": 2
} | {
"line": 200,
"column": 5
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nD : Type u_1\ninst✝⁵ : Category.{v_1, u_1} D\ninst✝⁴ : Abelian C\ninst✝³ : HasProjectiveResolutions C\ninst✝² : Abelian D\nF G : C ⥤ D\ninst✝¹ : F.Additive\ninst✝ : G.Additive\nα : F ⟶ G\nX : C\nP : ProjectiveResolution X\n⊢ (NatTrans.leftDerivedToHomotopyCategor... | rw [← cancel_mono (P.isoLeftDerivedToHomotopyCategoryObj G).hom, assoc, assoc,
Iso.inv_hom_id, comp_id]
dsimp [isoLeftDerivedToHomotopyCategoryObj, Functor.mapHomotopyCategoryFactors,
NatTrans.leftDerivedToHomotopyCategory]
rw [assoc]
erw [id_comp, comp_id]
obtain ⟨β, hβ⟩ := (HomotopyCategory.quotient... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ModelCategory.Lifting | {
"line": 108,
"column": 6
} | {
"line": 108,
"column": 40
} | [
{
"pp": "case h\nC : 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 : CochainC... | Int.negOnePow_even 2 ⟨1, by simp⟩, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.ModelCategory.Lifting | {
"line": 112,
"column": 48
} | {
"line": 112,
"column": 57
} | [
{
"pp": "case h\nC : 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 : CochainC... | ← ι.comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.ModelCategory.Lifting | {
"line": 153,
"column": 12
} | {
"line": 153,
"column": 21
} | [
{
"pp": "case h\nC : 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 : Cochai... | simp [hα] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.GradedObject.Monoidal | {
"line": 87,
"column": 66
} | {
"line": 88,
"column": 32
} | [
{
"pp": "I : Type u\ninst✝³ : AddMonoid I\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : MonoidalCategory C\nX₁ X₂ : GradedObject I C\ninst✝ : X₁.HasTensor X₂\nA : C\nk : I\nf : (i₁ i₂ : I) → i₁ + i₂ = k → (X₁ i₁ ⊗ X₂ i₂ ⟶ A)\ni₁ i₂ : I\nhi : i₁ + i₂ = k\n⊢ ιTensorObj X₁ X₂ i₁ i₂ k hi ≫ tensorObjDesc f... | by
apply ι_mapBifunctorMapObjDesc | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.SpectralObject.Homology | {
"line": 129,
"column": 75
} | {
"line": 131,
"column": 11
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\ninst✝ : Category.{v_2, u_2} ι\nX : SpectralObject C ι\ni₁ i₂ i₃ i₄ i₅ i₆ : ι\nf₂ : i₁ ⟶ i₂\nf₃ : i₂ ⟶ i₃\nf₄ : i₃ ⟶ i₄\nf₅ : i₄ ⟶ i₅\nf₆ : i₅ ⟶ i₆\nf₂₃ : i₁ ⟶ i₃\nh₂₃ : f₂ ≫ f₃ = f₂₃\nf₅₆ : i₄ ⟶ i₆\nh₅₆ : f₅ ≫ f₆ = f₅₆\nn₁ ... | by
simp only [← map_comp]
cat_disch | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.SpectralObject.FirstPage | {
"line": 123,
"column": 9
} | {
"line": 123,
"column": 60
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝⁴ : Category.{?u.9037, u_1} C\ninst✝³ : Abelian C\ninst✝² : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\ninst✝¹ : data.HasFirstPageComputation\ninst✝ : X.HasSpectralSequence data\npq pq' : κ\nh... | by rw [hj, ← data.hc₀₂ r₀ pq pq' hpq, data.hi₀₁ pq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.SpectralObject.Page | {
"line": 157,
"column": 29
} | {
"line": 158,
"column": 81
} | [
{
"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 : ι\nf : i ⟶ j\ng : j ⟶ k\nhg : IsIso g\nn₀ n₁ : ℤ\nhn₁ : n₀ + 1 = n₁\n⊢ X.δ f g n₀ n₁ hn₁ = 0",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditiv... | by
simpa only [Preadditive.IsIso.comp_right_eq_zero] using X.zero₁ f g _ rfl n₀ n₁ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Artinian.Module | {
"line": 137,
"column": 6
} | {
"line": 137,
"column": 21
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\n⊢ (∀ (a : Set (Submodule R M)), a.Nonempty → ∃ M' ∈ a, ∀ I ∈ a, ¬I < M') ↔ IsArtinian R M",
"usedConstants": [
"isArtinian_iff",
"Eq.mpr",
"Submodule",
"Preorder.toLT",
"cong... | isArtinian_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Artinian.Module | {
"line": 401,
"column": 2
} | {
"line": 401,
"column": 51
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nN : Submodule R M\ninst✝ : IsArtinianRing R\nhN : N.FG\n⊢ IsArtinian R ↥N",
"usedConstants": [
"Submodule",
"congrArg",
"AddCommGroup.toAddCommMonoid",
"isArtinian_of_fg_of_artinian'",... | rw [← Module.Finite.iff_fg] at hN; infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Artinian.Module | {
"line": 401,
"column": 2
} | {
"line": 401,
"column": 51
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nN : Submodule R M\ninst✝ : IsArtinianRing R\nhN : N.FG\n⊢ IsArtinian R ↥N",
"usedConstants": [
"Submodule",
"congrArg",
"AddCommGroup.toAddCommMonoid",
"isArtinian_of_fg_of_artinian'",... | rw [← Module.Finite.iff_fg] at hN; infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Artinian.Module | {
"line": 462,
"column": 40
} | {
"line": 462,
"column": 65
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Semiring R\ninst✝ : IsArtinianRing Rᵐᵒᵖ\nx : R\n⊢ IsUnit (MulOpposite.op x) ↔ IsRightRegular (MulOpposite.op x)",
"usedConstants": [
"Eq.mpr",
"congrArg",
"MulOpposite",
"IsUnit",
"id",
"instDistribOfSemiring",
"IsRightRegular",
... | isUnit_iff_isRightRegular | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Artinian.Module | {
"line": 582,
"column": 6
} | {
"line": 582,
"column": 18
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsArtinianRing R\nn : ℕ\nthis : Fintype (MaximalSpectrum R)\n⊢ nilradical R ^ n = ⨅ I, I.asIdeal ^ n",
"usedConstants": [
"MaximalSpectrum.asIdeal",
"Eq.mpr",
"iInf",
"Semiring.toModule",
"IsScalarTower.right",
"congrArg... | ← iInf_univ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Basic | {
"line": 274,
"column": 20
} | {
"line": 276,
"column": 74
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\nN : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : LieRingModule L N\ninst✝ : LieMo... | by
ext n
simp only [smul_sub, smul_lie, smul_apply, LieHom.lie_apply, map_smul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.IdealOperations | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 66
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\nN : LieSubmodule R L M\ninst✝ : LieAlgebra R L\n⊢ ⁅⊥, N⁆ = ⊥",
"usedConstants": [
"le_bot_iff",
"LieAlgebra.toModule",
"LieSubmodu... | suffices ⁅(⊥ : LieIdeal R L), N⁆ ≤ ⊥ by exact le_bot_iff.mp this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
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