module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Topology.Algebra.Order.Field | {
"line": 301,
"column": 6
} | {
"line": 301,
"column": 30
} | {
"line": 302,
"column": 4
} | [
{
"pp": "case refine_2.inl\n𝕜 : Type u_1\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nn : ℤ\nc d : 𝕜\nhc : c ≠ 0\nh : n = 0 ∧ c = d\n⊢ Tendsto (fun x ↦ d) atTop (𝓝 d)",
"ppTerm": "?refine_2.inl",
"assigned": true... | [] | exact tendsto_const_nhds | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Logic.Equiv.PartialEquiv | {
"line": 390,
"column": 38
} | {
"line": 390,
"column": 78
} | {
"line": 390,
"column": 79
} | [
{
"pp": "case inl\nα : Type u_1\nβ : Type u_2\ne : PartialEquiv α β\ns : Set α\nt : Set β\ne' : PartialEquiv α β\ninst✝¹ : (i : α) → Decidable (i ∈ s)\ninst✝ : (i : β) → Decidable (i ∈ t)\nh : e.IsImage s t\nh' : e'.IsImage s t\nx : α\nhe : x ∈ e.source\nhs : x ∈ s\n⊢ t.piecewise (↑e.symm) (↑e'.symm) (↑e x) = x... | [
"case inl\nα : Type u_1\nβ : Type u_2\ne : PartialEquiv α β\ns : Set α\nt : Set β\ne' : PartialEquiv α β\ninst✝¹ : (i : α) → Decidable (i ∈ s)\ninst✝ : (i : β) → Decidable (i ∈ t)\nh : e.IsImage s t\nh' : e'.IsImage s t\nx : α\nhe : x ∈ e.source\nhs : x ∈ s\n⊢ ↑e.symm (↑e x) = x"
] | piecewise_eq_of_mem _ _ _ ((h he).2 hs), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order.LeftRightNhds | {
"line": 370,
"column": 2
} | {
"line": 370,
"column": 13
} | {
"line": 371,
"column": 2
} | [
{
"pp": "α : Type u_4\ninst✝³ : TopologicalSpace α\ninst✝² : CommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedMonoid α\nh_nhds : ∀ (a : α), 𝓝 a = ⨅ r, ⨅ (_ : r > 1), 𝓟 {b | |a / b|ₘ < r}\na : α\n⊢ 𝓝 a = 𝓝 a",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Preorder.topology",... | [
"α : Type u_4\ninst✝³ : TopologicalSpace α\ninst✝² : CommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedMonoid α\nh_nhds : ∀ (a : α), 𝓝 a = ⨅ r, ⨅ (_ : r > 1), 𝓟 {b | |a / b|ₘ < r}\na : α\n⊢ ⨅ r, ⨅ (_ : r > 1), 𝓟 {b | |a / b|ₘ < r} = 𝓝 a"
] | rw [h_nhds] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Module.CharacterModule | {
"line": 144,
"column": 30
} | {
"line": 144,
"column": 79
} | {
"line": 144,
"column": 79
} | [
{
"pp": "R : Type uR\ninst✝⁶ : CommRing R\nA : Type uA\ninst✝⁵ : AddCommGroup A\nA' : Type u_1\ninst✝⁴ : AddCommGroup A'\nB : Type uB\ninst✝³ : AddCommGroup B\ninst✝² : Module R A\ninst✝¹ : Module R A'\ninst✝ : Module R B\n⊢ uncurry ∘ₗ curry = LinearMap.id",
"ppTerm": "?m.67",
"assigned": true,
"use... | [] | ext _ z; refine z.induction_on ?_ ?_ ?_ <;> aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.CharacterModule | {
"line": 144,
"column": 30
} | {
"line": 144,
"column": 79
} | {
"line": 144,
"column": 79
} | [
{
"pp": "R : Type uR\ninst✝⁶ : CommRing R\nA : Type uA\ninst✝⁵ : AddCommGroup A\nA' : Type u_1\ninst✝⁴ : AddCommGroup A'\nB : Type uB\ninst✝³ : AddCommGroup B\ninst✝² : Module R A\ninst✝¹ : Module R A'\ninst✝ : Module R B\n⊢ uncurry ∘ₗ curry = LinearMap.id",
"ppTerm": "?m.67",
"assigned": true,
"use... | [] | ext _ z; refine z.induction_on ?_ ?_ ?_ <;> aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Order.Basic | {
"line": 605,
"column": 32
} | {
"line": 605,
"column": 39
} | {
"line": 605,
"column": 39
} | [
{
"pp": "α : Type u\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\na✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ x ∈ s, x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen[inst✝³] a\nt : Set α := {x ... | [
"α : Type u\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\na✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ x ∈ s, x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen[inst✝³] a\nt : Set α := {x | x ∈ s ∧ x ... | mem_Ioc | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Order.Basic | {
"line": 664,
"column": 4
} | {
"line": 664,
"column": 49
} | {
"line": 665,
"column": 4
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝⁴ : TopologicalSpace α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : LinearOrder β\ninst✝ : SecondCountableTopology α\nt : Set β\nf : β → α\na✝ : Nontrivial β\nthis : Nonempty α\ns : Set β := {x | x ∈ t ∧ ∃ z, f x < z ∧ ∀ y ∈ t, x < y → z ≤ f y}\nz : β → α\nhz... | [
"α : Type u\nβ : Type v\ninst✝⁴ : TopologicalSpace α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : LinearOrder β\ninst✝ : SecondCountableTopology α\nt : Set β\nf : β → α\na✝ : Nontrivial β\nthis : Nonempty α\ns : Set β := ⋯\nz : β → α\nhz : ∀ x ∈ s, f x < z x ∧ ∀ y ∈ t, x < y → z x ≤ f y\nI : InjOn f ... | apply Set.PairwiseDisjoint.countable_of_Ioo A | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.Instances.AddCircle.Defs | {
"line": 357,
"column": 2
} | {
"line": 357,
"column": 77
} | {
"line": 358,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : AddCommGroup 𝕜\np : 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsOrderedAddMonoid 𝕜\nhp : Fact (0 < p)\na : 𝕜\ninst✝ : Archimedean 𝕜\nx y : 𝕜\nhx : x ∈ Ico a (a + p)\nhy : y ∈ Ico a (a + p)\nh : (equivIco p a) ↑x = (equivIco p a) ↑y\n⊢ ⟨x, hx⟩ = ⟨y, hy⟩",
"ppTerm": "?m.161",... | [
"𝕜 : Type u_1\ninst✝³ : AddCommGroup 𝕜\np : 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsOrderedAddMonoid 𝕜\nhp : Fact (0 < p)\na : 𝕜\ninst✝ : Archimedean 𝕜\nx y : 𝕜\nhx : x ∈ Ico a (a + p)\nhy : y ∈ Ico a (a + p)\nh : (equivIco p a) ↑x = (equivIco p a) ↑y\n⊢ (equivIco p a).toFun ((equivIco p a).invFun ⟨x, hx⟩) = ... | rw [← (equivIco p a).right_inv ⟨x, hx⟩, ← (equivIco p a).right_inv ⟨y, hy⟩] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Instances.AddCircle.Defs | {
"line": 443,
"column": 17
} | {
"line": 446,
"column": 72
} | {
"line": 447,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nB : Type u_2\ninst✝⁶ : AddCommGroup 𝕜\np : 𝕜\ninst✝⁵ : LinearOrder 𝕜\ninst✝⁴ : IsOrderedAddMonoid 𝕜\nhp : Fact (0 < p)\na : 𝕜\ninst✝³ : Archimedean 𝕜\ninst✝² : TopologicalSpace 𝕜\ninst✝¹ : OrderTopology 𝕜\nx : AddCircle p\ninst✝ : DiscreteTopology ↥(zmultiples p)\n⊢ ∀ ⦃x : AddCir... | [] | by
intro x hx
exact (eq_left_or_mem_Ioo_of_mem_Ico (equivIco p a x).2).resolve_left
(hx ∘ ((equivIco p a).symm_apply_apply x).symm.trans ∘ congrArg _) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Instances.AddCircle.Defs | {
"line": 574,
"column": 2
} | {
"line": 574,
"column": 26
} | {
"line": 575,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : Field 𝕜\np : 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\nhp : Fact (0 < p)\nn : ℕ\nh : 0 < n\n⊢ addOrderOf ↑(p / ↑n) = n",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"instHDiv",
"congrArg",... | [
"𝕜 : Type u_1\ninst✝² : Field 𝕜\np : 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\nhp : Fact (0 < p)\nn : ℕ\nh : 0 < n\n⊢ n • ↑(p / ↑n) = 0 ∧ ∀ m < n, 0 < m → m • ↑(p / ↑n) ≠ 0"
] | rw [addOrderOf_eq_iff h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Instances.AddCircle.Defs | {
"line": 619,
"column": 6
} | {
"line": 619,
"column": 18
} | {
"line": 619,
"column": 19
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\ninst✝² : Field 𝕜\np : 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\nhp : Fact (0 < p)\nu : AddCircle p\nn : ℕ\nh : 0 < n\nk : 𝕜\nhk : n • ↑k = 0\n⊢ ∃ m < n, ↑(↑m / ↑n * p) = ↑k",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
... | [
"case refine_2\n𝕜 : Type u_1\ninst✝² : Field 𝕜\np : 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\nhp : Fact (0 < p)\nu : AddCircle p\nn : ℕ\nh : 0 < n\nk : 𝕜\nhk : ↑(n • k) = 0\n⊢ ∃ m < n, ↑(↑m / ↑n * p) = ↑k"
] | ← coe_nsmul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Instances.AddCircle.Defs | {
"line": 625,
"column": 4
} | {
"line": 625,
"column": 30
} | {
"line": 625,
"column": 30
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\ninst✝² : Field 𝕜\np : 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\nhp : Fact (0 < p)\nu : AddCircle p\nn : ℕ\nh : 0 < n\nk : 𝕜\na : ℤ\nha : ↑(a / ↑n) * (p * ↑n / ↑n) + (a % ↑n) • p / ↑n = k\nh0 : ↑n ≠ 0\n⊢ ↑(↑(a.natMod ↑n) / ↑n * p) = ↑k",
"ppTerm": "... | [
"case refine_2\n𝕜 : Type u_1\ninst✝² : Field 𝕜\np : 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\nhp : Fact (0 < p)\nu : AddCircle p\nn : ℕ\nh : 0 < n\nk : 𝕜\na : ℤ\nha : ↑(a / ↑n) * p + (a % ↑n) • p / ↑n = k\nh0 : ↑n ≠ 0\n⊢ ↑(↑(a.natMod ↑n) / ↑n * p) = ↑k"
] | mul_div_cancel_right₀ p h0 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Instances.AddCircle.Defs | {
"line": 755,
"column": 2
} | {
"line": 755,
"column": 29
} | {
"line": 756,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : AddCommGroup 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsOrderedAddMonoid 𝕜\np a : 𝕜\nhp : Fact (0 < p)\ninst✝ : Archimedean 𝕜\nx : 𝕜\n⊢ Quot.mk (EndpointIdent p a) ⟨toIcoMod ⋯ a x, ⋯⟩ = Quot.mk (EndpointIdent p a) ⟨toIocMod ⋯ a x, ⋯⟩",
"ppTerm": "?m.66",
"assigned": tru... | [
"case pos\n𝕜 : Type u_1\ninst✝³ : AddCommGroup 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsOrderedAddMonoid 𝕜\np a : 𝕜\nhp : Fact (0 < p)\ninst✝ : Archimedean 𝕜\nx : 𝕜\nh : a ≡ x [PMOD p]\n⊢ Quot.mk (EndpointIdent p a) ⟨toIcoMod ⋯ a x, ⋯⟩ = Quot.mk (EndpointIdent p a) ⟨toIocMod ⋯ a x, ⋯⟩",
"case neg\n𝕜 : Type u... | by_cases h : a ≡ x [PMOD p] | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Algebra.Homology.ShortComplex.SnakeLemma | {
"line": 206,
"column": 8
} | {
"line": 206,
"column": 20
} | {
"line": 206,
"column": 20
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\nS : SnakeInput C\ninst✝ : Epi S.L₂.g\n⊢ Epi (S.v₂₃.τ₂ ≫ S.L₃.g)",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"Eq.mpr",
"CategoryTheory.ShortComplex.SnakeI... | [
"C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\nS : SnakeInput C\ninst✝ : Epi S.L₂.g\n⊢ Epi (S.L₂.g ≫ S.v₂₃.τ₃)"
] | S.v₂₃.comm₂₃ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory | {
"line": 274,
"column": 80
} | {
"line": 280,
"column": 12
} | {
"line": 282,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type s\ninst✝³ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝² : ConcreteCategory C FC\nJ : MulticospanShape\nI : MulticospanIndex J C\ninst✝¹ : HasMultiequalizer I\ninst✝ : PreservesLimit I.multicospan (forget C)\nx y : ToTyp... | [] | by
apply Concrete.limit_ext
rintro (a | b)
· apply h
· rw [← limit.w I.multicospan (WalkingMulticospan.Hom.fst b), ConcreteCategory.comp_apply,
ConcreteCategory.comp_apply]
simp [h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.ShortComplex.ConcreteCategory | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 41
} | {
"line": 69,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝⁷ : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type w\ninst✝⁶ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝⁵ : ConcreteCategory C FC\ninst✝⁴ : HasForget₂ C Ab\ninst✝³ : Preadditive C\ninst✝² : (forget₂ C Ab).Additive\ninst✝¹ : (forget₂ C Ab).PreservesHomology\ninst✝ : Has... | [
"C : Type u\ninst✝⁷ : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type w\ninst✝⁶ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝⁵ : ConcreteCategory C FC\ninst✝⁴ : HasForget₂ C Ab\ninst✝³ : Preadditive C\ninst✝² : (forget₂ C Ab).Additive\ninst✝¹ : (forget₂ C Ab).PreservesHomology\ninst✝ : HasZeroObject C... | rw [← AddCommGrpCat.epi_iff_surjective] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Monad.Equalizer | {
"line": 107,
"column": 6
} | {
"line": 107,
"column": 16
} | {
"line": 108,
"column": 6
} | [
{
"pp": "case refine_3\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nT : Comonad C\nX : T.Coalgebra\ns : Fork (CofreeEqualizer.topMap X) (CofreeEqualizer.bottomMap X)\nh₁ : s.ι.f ≫ T.map X.a = s.ι.f ≫ T.δ.app X.A\nh₂ : s.pt.a ≫ T.map s.ι.f = s.ι.f ≫ T.δ.app X.A\n⊢ ∀ {m : s.pt ⟶ (beckCoalgebraFork X).pt}, m ≫ (beck... | [
"case refine_3\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nT : Comonad C\nX : T.Coalgebra\ns : Fork (CofreeEqualizer.topMap X) (CofreeEqualizer.bottomMap X)\nh₁ : s.ι.f ≫ T.map X.a = s.ι.f ≫ T.δ.app X.A\nh₂ : s.pt.a ≫ T.map s.ι.f = s.ι.f ≫ T.δ.app X.A\nm : s.pt ⟶ (beckCoalgebraFork X).pt\nhm : m ≫ (beckCoalgebraFork... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.Monad.Adjunction | {
"line": 243,
"column": 21
} | {
"line": 243,
"column": 41
} | {
"line": 244,
"column": 8
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\nG : Comonad C\nX : G.adj.toComonad.Coalgebra\n⊢ X.a ≫ G.ε.app X.A = 𝟙 X.A",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functor",
"CategoryTheo... | [] | simpa using X.counit | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.CategoryTheory.Monad.Adjunction | {
"line": 243,
"column": 21
} | {
"line": 243,
"column": 41
} | {
"line": 244,
"column": 8
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\nG : Comonad C\nX : G.adj.toComonad.Coalgebra\n⊢ X.a ≫ G.ε.app X.A = 𝟙 X.A",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functor",
"CategoryTheo... | [] | simpa using X.counit | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monad.Adjunction | {
"line": 243,
"column": 21
} | {
"line": 243,
"column": 41
} | {
"line": 244,
"column": 8
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\nG : Comonad C\nX : G.adj.toComonad.Coalgebra\n⊢ X.a ≫ G.ε.app X.A = 𝟙 X.A",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functor",
"CategoryTheo... | [] | simpa using X.counit | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer | {
"line": 858,
"column": 8
} | {
"line": 858,
"column": 12
} | {
"line": 859,
"column": 8
} | [
{
"pp": "case left.right.snd\nC : Type u\ninst✝ : Category.{v, u} C\nJ✝ : MultispanShape\nI✝ : MultispanIndex J✝ C\nK : Multicofork I✝\nJ : MultispanShape\nI : MultispanIndex J C\nP : C\nπ : (b : J.R) → I.right b ⟶ P\nw : ∀ (a : J.L), I.fst a ≫ π (J.fst a) = I.snd a ≫ π (J.snd a)\na✝ : J.L\n⊢ I.snd a✝ ≫ π (J.sn... | [
"case left.right.snd\nC : Type u\ninst✝ : Category.{v, u} C\nJ✝ : MultispanShape\nI✝ : MultispanIndex J✝ C\nK : Multicofork I✝\nJ : MultispanShape\nI : MultispanIndex J C\nP : C\nπ : (b : J.R) → I.right b ⟶ P\nw : ∀ (a : J.L), I.fst a ≫ π (J.fst a) = I.snd a ≫ π (J.snd a)\na✝ : J.L\n⊢ I.fst a✝ ≫ π (J.fst a✝) = I.sn... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Limits.Shapes.Reflexive | {
"line": 237,
"column": 35
} | {
"line": 237,
"column": 66
} | {
"line": 237,
"column": 67
} | [
{
"pp": "W✝ X✝ Y✝ Z✝ : WalkingReflexivePair\nf : W✝.Hom X✝\ng : X✝.Hom Y✝\nh : Y✝.Hom Z✝\n⊢ (f.comp g).comp h = f.comp (g.comp h)",
"ppTerm": "?m.457",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.WalkingReflexivePair.Hom.reflexion",
"CategoryTheory.Limits.WalkingReflexive... | [
"case left.reflexion.left\n⊢ (Hom.left.comp Hom.reflexion).comp Hom.left = Hom.left.comp (Hom.reflexion.comp Hom.left)",
"case left.reflexion.right\n⊢ (Hom.left.comp Hom.reflexion).comp Hom.right = Hom.left.comp (Hom.reflexion.comp Hom.right)",
"case left.reflexion.leftCompReflexion\n⊢ (Hom.left.comp Hom.reflex... | cases f <;> cases g <;> cases h | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.Monad.Adjunction | {
"line": 346,
"column": 2
} | {
"line": 346,
"column": 6
} | {
"line": 347,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\ninst✝ : Reflective R\nX : (reflectorAdjunction R).toMonad.Algebra\n⊢ (Monad.comparison (reflectorAdjunction R)).obj ((reflector R).obj X.A) ≅ X",
"ppTerm": "?m.46",
"assigned": true,
... | [
"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\ninst✝ : Reflective R\nX : (reflectorAdjunction R).toMonad.Algebra\n⊢ X ≅ (Monad.comparison (reflectorAdjunction R)).obj ((reflector R).obj X.A)"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Monad.Limits | {
"line": 70,
"column": 23
} | {
"line": 73,
"column": 10
} | {
"line": 74,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nT : Monad C\nJ : Type u\ninst✝ : Category.{v, u} J\nD : J ⥤ T.Algebra\nc : Cone (D ⋙ T.forget)\nt : IsLimit c\nj : J\n⊢ (T.η.app c.pt ≫ t.lift (newCone D c)) ≫ c.π.app j = 𝟙 c.pt ≫ c.π.app j",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": ... | [] | by
rw [Category.assoc, t.fac, newCone_π_app, ← T.η.naturality_assoc, Functor.id_map,
(D.obj j).unit]
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer | {
"line": 998,
"column": 6
} | {
"line": 999,
"column": 12
} | {
"line": 999,
"column": 13
} | [
{
"pp": "case one\nC : Type u\ninst✝ : Category.{v, u} C\nJ : MultispanShape\nI : MultispanIndex J C\nc : Cofan I.left\nhc : IsColimit c\nd : Cofan I.right\nhd : IsColimit d\nK₁ K₂ : Multicofork I\nf : K₁ ⟶ K₂\n⊢ (Multicofork.toSigmaCofork hc hd K₁).ι.app WalkingParallelPair.one ≫ f.hom =\n (Multicofork.toSi... | [] | · apply Cofan.IsColimit.hom_ext hd
simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Ideal.MinimalPrime.Basic | {
"line": 77,
"column": 63
} | {
"line": 79,
"column": 64
} | {
"line": 80,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommSemiring R\nI J : Ideal R\ninst✝ : J.IsPrime\ne : I ≤ J\nS : Set (Ideal R)ᵒᵈ := {p | IsPrime p ∧ I ≤ OrderDual.ofDual p}\nh : ∃ m, OrderDual.toDual J ≤ m ∧ Maximal (fun x ↦ x ∈ S) m\n⊢ ∃ p ∈ I.minimalPrimes, p ≤ J",
"ppTerm": "?m.42",
"assigned": true,
"usedConsta... | [] | by
obtain ⟨p, hJp, hp⟩ := h
exact ⟨p, ⟨hp.prop, fun q hq hle ↦ hp.le_of_ge hq hle⟩, hJp⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Localization.Ideal | {
"line": 79,
"column": 4
} | {
"line": 79,
"column": 42
} | {
"line": 80,
"column": 4
} | [
{
"pp": "case mp\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nz✝ : S\nh : z✝ ∈ Ideal.map (algebraMap R S) I\nz : S\ny : R\nhy : y ∈ ↑I ∧ (algebraMap R S) y = z\n⊢ z ∈ ↑(map_ideal M S I)",
"ppTer... | [
"case mp\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nz✝ : S\nh : z✝ ∈ Ideal.map (algebraMap R S) I\nz : S\ny : R\nhy : y ∈ ↑I ∧ (algebraMap R S) y = z\nZ : ↥I := ⟨y, ⋯⟩\n⊢ z ∈ ↑(map_ideal M S I)"
] | let Z : { x // x ∈ I } := ⟨y, hy.left⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.Localization.Ideal | {
"line": 206,
"column": 10
} | {
"line": 206,
"column": 21
} | {
"line": 206,
"column": 22
} | [
{
"pp": "case mp.refine_1\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nJ : Ideal S\nh : J.IsPrime\nhJ : Ideal.under R J = ⊤\n⊢ J = ⊤",
"ppTerm": "?mp.refine_1",
"assigned": true,
"usedConstants": [
... | [
"case mp.refine_1\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nJ : Ideal S\nh : J.IsPrime\nhJ : Ideal.under R J = ⊤\n⊢ ⊤ ≤ J"
] | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Localization.Ideal | {
"line": 212,
"column": 11
} | {
"line": 212,
"column": 22
} | {
"line": 212,
"column": 23
} | [
{
"pp": "case mpr.refine_1\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nJ : Ideal S\nh : (Ideal.under R J).IsPrime ∧ Disjoint ↑M ↑(Ideal.under R J)\nhJ : J = ⊤\n⊢ ⊤ ≤ Ideal.under R J",
"ppTerm": "?mpr.refine... | [
"case mpr.refine_1\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nJ : Ideal S\nh : (Ideal.under R J).IsPrime ∧ Disjoint ↑M ↑(Ideal.under R J)\nhJ : ⊤ ≤ J\n⊢ ⊤ ≤ Ideal.under R J"
] | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Localization.Algebra | {
"line": 64,
"column": 32
} | {
"line": 64,
"column": 88
} | {
"line": 64,
"column": 88
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nP : Type u_3\nQ : Type u_4\ninst✝⁷ : CommSemiring R\ninst✝⁶ : CommSemiring S\ninst✝⁵ : CommSemiring P\ninst✝⁴ : CommSemiring Q\nM : Submonoid R\nT : Submonoid P\ninst✝³ : Algebra R S\ninst✝² : Algebra P Q\ninst✝¹ : IsLocalization M S\ninst✝ : IsLocalization T Q\ng : R →+* P\... | [] | by simp [RingHom.mem_ker, RingHom.mem_ker.mp x.property] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Localization.LocalizationLocalization | {
"line": 86,
"column": 2
} | {
"line": 87,
"column": 8
} | {
"line": 89,
"column": 0
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝⁸ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Algebra R S\nN : Submonoid S\nT : Type u_3\ninst✝⁵ : CommSemiring T\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsLo... | [] | · simp only [map_mul, IsScalarTower.algebraMap_apply R S T, ← eq₃, ← eq₂, ← eq₁]
ring | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Localization.LocalizationLocalization | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 6
} | {
"line": 127,
"column": 2
} | [
{
"pp": "case e_S\nR : Type u_1\ninst✝⁸ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Algebra R S\nN : Submonoid S\nT : Type u_3\ninst✝⁵ : CommSemiring T\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsLocaliz... | [
"case e_S\nR : Type u_1\ninst✝⁸ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Algebra R S\nN : Submonoid S\nT : Type u_3\ninst✝⁵ : CommSemiring T\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsLocalization N T\nH... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.RingTheory.Localization.LocalizationLocalization | {
"line": 226,
"column": 17
} | {
"line": 226,
"column": 47
} | {
"line": 226,
"column": 47
} | [
{
"pp": "R : Type u_1\ninst✝⁸ : CommSemiring R\nS : Type u_2\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Algebra R S\nT : Type u_3\ninst✝⁵ : CommSemiring T\ninst✝⁴ : Algebra R T\nM N : Submonoid R\nh : M ≤ N\ninst✝³ : IsLocalization M S\ninst✝² : IsLocalization N T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nx₁ ... | [] | by convert! e using 1 <;> simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Localization.AtPrime.Basic | {
"line": 147,
"column": 7
} | {
"line": 147,
"column": 65
} | {
"line": 149,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommSemiring R\nS : Type u_2\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Algebra R S\nP : Type u_3\ninst✝³ : CommSemiring P\nA : Type u_4\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\nI : Ideal R\nhI : I.IsPrime\ninst✝ : IsLocalization.AtPrime S I\n⊢ {p | p.IsPrime ∧ Disjoint ↑I.primeCompl ↑... | [] | ext; simp [Ideal.primeCompl, ← le_compl_iff_disjoint_left] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Localization.AtPrime.Basic | {
"line": 147,
"column": 7
} | {
"line": 147,
"column": 65
} | {
"line": 149,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommSemiring R\nS : Type u_2\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Algebra R S\nP : Type u_3\ninst✝³ : CommSemiring P\nA : Type u_4\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\nI : Ideal R\nhI : I.IsPrime\ninst✝ : IsLocalization.AtPrime S I\n⊢ {p | p.IsPrime ∧ Disjoint ↑I.primeCompl ↑... | [] | ext; simp [Ideal.primeCompl, ← le_compl_iff_disjoint_left] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Localization.AtPrime.Basic | {
"line": 268,
"column": 8
} | {
"line": 268,
"column": 30
} | {
"line": 268,
"column": 30
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommSemiring R\nP : Type u_3\ninst✝ : CommSemiring P\nI : Ideal R\nhI : I.IsPrime\nJ : Ideal P\nhJ : J.IsPrime\nf : R →+* P\nhIJ : I = Ideal.comap f J\nr : R\ns : ↥I.primeCompl\nhx : IsUnit (mk' (Localization.AtPrime J) (f r) ⟨f ↑s, ⋯⟩)\n⊢ IsUnit (mk' (Localization.AtPrime I) r s... | [
"R : Type u_1\ninst✝¹ : CommSemiring R\nP : Type u_3\ninst✝ : CommSemiring P\nI : Ideal R\nhI : I.IsPrime\nJ : Ideal P\nhJ : J.IsPrime\nf : R →+* P\nhIJ : I = Ideal.comap f J\nr : R\ns : ↥I.primeCompl\nhx : f r ∈ J.primeCompl\n⊢ r ∈ I.primeCompl"
] | AtPrime.isUnit_mk'_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.TensorProduct.Quotient | {
"line": 82,
"column": 8
} | {
"line": 82,
"column": 60
} | {
"line": 83,
"column": 6
} | [
{
"pp": "case hr\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nm : Submodule R M\nn : Submodule R N\na : ↥m ⊗[R] N\nb : M ⊗[R] ↥n\nf : ↥m ⊗[R] N →ₗ[R] (M ⧸ m) ⊗[R] (N ⧸ n) := map m.mkQ n.mkQ ∘ₗ map m.sub... | [] | · simp [g, Submodule.Quotient.mk_eq_zero _ |>.2 y.2] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Localization.Submodule | {
"line": 154,
"column": 8
} | {
"line": 154,
"column": 29
} | {
"line": 154,
"column": 30
} | [
{
"pp": "case mp.refine_4\nR : Type u_1\ninst✝⁷ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝⁶ : CommSemiring S\ninst✝⁵ : Algebra R S\ninst✝⁴ : IsLocalization M S\nN : Type u_3\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : Module S N\ninst✝ : IsScalarTower R S N\nx : N\na : Set N\nh : x ∈ Su... | [
"case mp.refine_4\nR : Type u_1\ninst✝⁷ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝⁶ : CommSemiring S\ninst✝⁵ : Algebra R S\ninst✝⁴ : IsLocalization M S\nN : Type u_3\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : Module S N\ninst✝ : IsScalarTower R S N\nx : N\na : Set N\nh : x ∈ Submodule.span... | mul_comm (mk' S _ _), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.OreLocalization.OreSet | {
"line": 59,
"column": 4
} | {
"line": 59,
"column": 24
} | {
"line": 61,
"column": 0
} | [
{
"pp": "case mpr\nR : Type u_1\ninst✝ : Monoid R\nS : Submonoid R\nH : ∀ (r₁ r₂ : R) (s : ↥S), r₁ * ↑s = r₂ * ↑s → ∃ s', ↑s' * r₁ = ↑s' * r₂\nr' : R → ↥S → R\ns' : R → ↥S → ↥S\nh : ∀ (r : R) (s : ↥S), ↑(s' r s) * r = r' r s * ↑s\n⊢ Nonempty (OreSet S)",
"ppTerm": "?mpr",
"assigned": true,
"usedCons... | [] | exact ⟨H, r', s', h⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.Matrix.Transvection | {
"line": 230,
"column": 39
} | {
"line": 233,
"column": 70
} | {
"line": 235,
"column": 0
} | [
{
"pp": "n : Type u_1\nR : Type u₂\ninst✝² : DecidableEq n\ninst✝¹ : CommRing R\ninst✝ : Fintype n\nM : Matrix n n R\nhM : ∀ (t : TransvectionStruct n R), Commute t.toMatrix M\n⊢ M ∈ Set.range ⇑(scalar n)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"add_mul",
"Distrib.leftD... | [] | by
refine mem_range_scalar_of_commute_single ?_
intro i j hij
simpa [transvection, mul_add, add_mul] using! (hM ⟨i, j, hij, 1⟩).eq | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Laurent | {
"line": 367,
"column": 62
} | {
"line": 367,
"column": 69
} | {
"line": 367,
"column": 69
} | [
{
"pp": "case mul_T\nR : Type u_1\ninst✝ : Semiring R\nQ : R[T;T⁻¹] → Prop\nQf : ∀ (f : R[X]), Q (toLaurent f)\nQT : ∀ (f : R[T;T⁻¹]), Q (f * T 1) → Q f\nf : R[X]\nn : ℕ\n⊢ Q (toLaurent f * T (-↑n))",
"ppTerm": "?mul_T",
"assigned": true,
"usedConstants": [],
"usedFVars": [],
"usedGoals": [
... | [
"case mul_T\nR : Type u_1\ninst✝ : Semiring R\nQ : R[T;T⁻¹] → Prop\nQf : ∀ (f : R[X]), Q (toLaurent f)\nQT : ∀ (f : R[T;T⁻¹]), Q (f * T 1) → Q f\nf : R[X]\nn : ℕ\n⊢ Q (toLaurent f * T (-↑n))"
] | | _ f n
=> | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.LinearAlgebra.Matrix.Adjugate | {
"line": 247,
"column": 2
} | {
"line": 247,
"column": 36
} | {
"line": 248,
"column": 2
} | [
{
"pp": "n : Type v\nα : Type w\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA : Matrix n n α\nb : n → α\n⊢ A.cramer b = A.adjugate *ᵥ b",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
"NonUnitalCommRing.toNonUnitalNonAs... | [
"n : Type v\nα : Type w\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA : Matrix n n α\nb : n → α\n⊢ A.cramer b = Aᵀᵀ.adjugate *ᵥ b"
] | nth_rw 2 [← A.transpose_transpose] | Mathlib.Tactic._aux_Mathlib_Tactic_NthRewrite___macroRules_Mathlib_Tactic_tacticNth_rw______1 | Mathlib.Tactic.tacticNth_rw_____ |
Mathlib.LinearAlgebra.Matrix.Transvection | {
"line": 417,
"column": 6
} | {
"line": 417,
"column": 46
} | {
"line": 418,
"column": 6
} | [
{
"pp": "case neg\n𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nk n : ℕ\nhn : n < r\nIH : ((List.drop (n + 1) (listTransvecCol M)).prod * M) (inl i) (inr ()) = if n + 1 ≤ ↑i then 0 else M (inl i) (inr ())\nhn' : n < (listTransvecCo... | [
"case neg.inl\n𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nk n : ℕ\nhn : n < r\nIH : ((List.drop (n + 1) (listTransvecCol M)).prod * M) (inl i) (inr ()) = if n + 1 ≤ ↑i then 0 else M (inl i) (inr ())\nhn' : n < (listTransvecCol M).len... | rcases le_or_gt (n + 1) i with (hi | hi) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.LinearAlgebra.Matrix.Invertible | {
"line": 159,
"column": 6
} | {
"line": 159,
"column": 72
} | {
"line": 160,
"column": 4
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nα : Type u_3\ninst✝⁷ : Fintype n\ninst✝⁶ : DecidableEq n\ninst✝⁵ : Fintype m\ninst✝⁴ : DecidableEq m\ninst✝³ : Ring α\nA : Matrix n n α\nU : Matrix n m α\nC : Matrix m m α\nV : Matrix m n α\ninst✝² : Invertible A\ninst✝¹ : Invertible C\ninst✝ : Invertible (⅟C + V * ⅟A * U)\n... | [] | rw [sub_right_inj, Matrix.add_mul, Matrix.add_mul, Matrix.add_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Matrix.Invertible | {
"line": 159,
"column": 6
} | {
"line": 159,
"column": 72
} | {
"line": 160,
"column": 4
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nα : Type u_3\ninst✝⁷ : Fintype n\ninst✝⁶ : DecidableEq n\ninst✝⁵ : Fintype m\ninst✝⁴ : DecidableEq m\ninst✝³ : Ring α\nA : Matrix n n α\nU : Matrix n m α\nC : Matrix m m α\nV : Matrix m n α\ninst✝² : Invertible A\ninst✝¹ : Invertible C\ninst✝ : Invertible (⅟C + V * ⅟A * U)\n... | [] | rw [sub_right_inj, Matrix.add_mul, Matrix.add_mul, Matrix.add_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.Invertible | {
"line": 159,
"column": 6
} | {
"line": 159,
"column": 72
} | {
"line": 160,
"column": 4
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nα : Type u_3\ninst✝⁷ : Fintype n\ninst✝⁶ : DecidableEq n\ninst✝⁵ : Fintype m\ninst✝⁴ : DecidableEq m\ninst✝³ : Ring α\nA : Matrix n n α\nU : Matrix n m α\nC : Matrix m m α\nV : Matrix m n α\ninst✝² : Invertible A\ninst✝¹ : Invertible C\ninst✝ : Invertible (⅟C + V * ⅟A * U)\n... | [] | rw [sub_right_inj, Matrix.add_mul, Matrix.add_mul, Matrix.add_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.Transvection | {
"line": 421,
"column": 8
} | {
"line": 421,
"column": 38
} | {
"line": 422,
"column": 2
} | [
{
"pp": "case neg.inr.hnc\n𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nk n : ℕ\nhn : n < r\nIH : ((List.drop (n + 1) (listTransvecCol M)).prod * M) (inl i) (inr ()) = if n + 1 ≤ ↑i then 0 else M (inl i) (inr ())\nhn' : n < (listTr... | [] | · simpa only [not_le] using hi | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Matrix.Trace | {
"line": 166,
"column": 70
} | {
"line": 167,
"column": 41
} | {
"line": 169,
"column": 0
} | [
{
"pp": "m : Type u_2\nn : Type u_3\np : Type u_4\nR : Type u_6\ninst✝³ : Fintype m\ninst✝² : Fintype n\ninst✝¹ : Fintype p\ninst✝ : NonUnitalCommSemiring R\nA : Matrix m n R\nB : Matrix n p R\nC : Matrix p m R\n⊢ (A * (B * C)).trace = (C * (A * B)).trace",
"ppTerm": "?m.26",
"assigned": true,
"used... | [] | by
rw [← Matrix.mul_assoc, trace_mul_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.Transvection | {
"line": 494,
"column": 6
} | {
"line": 494,
"column": 46
} | {
"line": 495,
"column": 6
} | [
{
"pp": "case neg\n𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nn : ℕ\nIH : n ≤ r → (M * (List.take n (listTransvecRow M)).prod) (inr ()) (inl i) = if n ≤ ↑i then M (inr ()) (inl i) else 0\nhk : n + 1 ≤ r\nhnr : n < r\nn' : Fin r :... | [
"case neg.inl\n𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nn : ℕ\nIH : n ≤ r → (M * (List.take n (listTransvecRow M)).prod) (inr ()) (inl i) = if n ≤ ↑i then M (inr ()) (inl i) else 0\nhk : n + 1 ≤ r\nhnr : n < r\nn' : Fin r := ⟨n, hn... | rcases le_or_gt (n + 1) i with (hi | hi) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.LinearAlgebra.Matrix.NonsingularInverse | {
"line": 363,
"column": 2
} | {
"line": 363,
"column": 28
} | {
"line": 365,
"column": 0
} | [
{
"pp": "m : Type u\ninst✝² : DecidableEq m\nK : Type u_3\ninst✝¹ : Field K\ninst✝ : Fintype m\nA : Matrix m m K\n⊢ Function.Injective A.mulVec ↔ Function.Injective fun v ↦ v ᵥ* Aᵀ",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | [] | simp_rw [vecMul_transpose] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.RingTheory.MatrixPolynomialAlgebra | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 40
} | {
"line": 101,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\nn : Type w\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\nm : Matrix n n R[X]\nk : ℕ\ni j : n\n⊢ (matPolyEquiv m).coeff k i j = (m i j).coeff k",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"CommSemiring.toSemiring",
"Matrix",
... | [
"case refine_1\nR : Type u_1\ninst✝² : CommSemiring R\nn : Type w\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\nm : Matrix n n R[X]\nk : ℕ\ni j : n\n⊢ (matPolyEquiv 0).coeff k i j = (0 i j).coeff k",
"case refine_2\nR : Type u_1\ninst✝² : CommSemiring R\nn : Type w\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\nm : Ma... | refine Matrix.induction_on' m ?_ ?_ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Polynomial.Identities | {
"line": 100,
"column": 4
} | {
"line": 100,
"column": 44
} | {
"line": 102,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type v\nT : Type w\nι : Type x\nk✝ : Type y\nA : Type z\na b : R\nm n : ℕ\ninst✝ : CommRing R\nx y : R\nk : ℕ\nz : R\nhz : x ^ (k + 1) - y ^ (k + 1) = z * (x - y)\n⊢ x ^ (k + 2) - y ^ (k + 2) = (z * x + y ^ (k + 1)) * (x - y)",
"ppTerm": "?m.105",
"assigned": true,
"usedCons... | [] | linear_combination (norm := ring) x * hz | Mathlib.Tactic.LinearCombination._aux_Mathlib_Tactic_LinearCombination___elabRules_Mathlib_Tactic_LinearCombination_linearCombination_1 | Mathlib.Tactic.LinearCombination.linearCombination |
Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff | {
"line": 70,
"column": 2
} | {
"line": 71,
"column": 56
} | {
"line": 72,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\nn : Type v\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\nM : Matrix n n R\n⊢ (∑ x ∈ univ.erase (Equiv.refl n), ↑↑(Equiv.Perm.sign x) * ∏ i, M.charmatrix (x i) i +\n ↑↑(Equiv.Perm.sign (Equiv.refl n)) * ∏ i, M.charmatrix ((Equiv.refl n) i) i -\n ∏ i, (X - C... | [
"R : Type u\ninst✝² : CommRing R\nn : Type v\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\nM : Matrix n n R\n⊢ (∑ x ∈ univ.erase (Equiv.refl n), ↑↑(Equiv.Perm.sign x) * ∏ i, M.charmatrix (x i) i).degree < ↑(Fintype.card n - 1)"
] | simp only [charmatrix_apply_eq, one_mul, Equiv.Perm.sign_refl, id, Int.cast_one,
Units.val_one, add_sub_cancel_right, Equiv.coe_refl] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Module.SpanRank | {
"line": 257,
"column": 2
} | {
"line": 257,
"column": 37
} | {
"line": 259,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np : Submodule R M\nhp : p.FG\n⊢ p.spanRank < ℵ₀",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Preorder.toLT",
"Cardinal",
"PartialOrder.toPreorder",
... | [] | exact spanRank_finite_iff_fg.mpr hp | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Module.SpanRank | {
"line": 423,
"column": 2
} | {
"line": 424,
"column": 72
} | {
"line": 426,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : Semiring R\nT : Type v\ninst✝ : Semiring T\nf : R ≃+* T\nI : Ideal R\n⊢ Submodule.spanFinrank (map f I) = Submodule.spanFinrank I",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroOneClass",
"Semiring.toModule",
... | [] | rw [Submodule.spanFinrank, Submodule.spanFinrank, ← Cardinal.toNat_lift.{u, v},
← Cardinal.toNat_lift.{v, u}, I.lift_spanRank_map_eq_of_ringEquiv f] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Module.SpanRank | {
"line": 423,
"column": 2
} | {
"line": 424,
"column": 72
} | {
"line": 426,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : Semiring R\nT : Type v\ninst✝ : Semiring T\nf : R ≃+* T\nI : Ideal R\n⊢ Submodule.spanFinrank (map f I) = Submodule.spanFinrank I",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroOneClass",
"Semiring.toModule",
... | [] | rw [Submodule.spanFinrank, Submodule.spanFinrank, ← Cardinal.toNat_lift.{u, v},
← Cardinal.toNat_lift.{v, u}, I.lift_spanRank_map_eq_of_ringEquiv f] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.SpanRank | {
"line": 423,
"column": 2
} | {
"line": 424,
"column": 72
} | {
"line": 426,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : Semiring R\nT : Type v\ninst✝ : Semiring T\nf : R ≃+* T\nI : Ideal R\n⊢ Submodule.spanFinrank (map f I) = Submodule.spanFinrank I",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroOneClass",
"Semiring.toModule",
... | [] | rw [Submodule.spanFinrank, Submodule.spanFinrank, ← Cardinal.toNat_lift.{u, v},
← Cardinal.toNat_lift.{v, u}, I.lift_spanRank_map_eq_of_ringEquiv f] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff | {
"line": 310,
"column": 6
} | {
"line": 310,
"column": 37
} | {
"line": 310,
"column": 38
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\nn : Type v\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\nM : Matrix n n R\na✝ : Nontrivial R\nt : R[T;T⁻¹] := T 1\nt_inv : R[T;T⁻¹] := T (-1)\np : R[T;T⁻¹] := ((scalar n) t - M.map ⇑LaurentPolynomial.C).det\nq : R[T;T⁻¹] := (1 - (scalar n) t * M.map ⇑LaurentPolynomial.C).... | [
"R : Type u\ninst✝² : CommRing R\nn : Type v\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\nM : Matrix n n R\na✝ : Nontrivial R\nt : R[T;T⁻¹] := T 1\nt_inv : R[T;T⁻¹] := T (-1)\np : R[T;T⁻¹] := ((scalar n) t - M.map ⇑LaurentPolynomial.C).det\nq : R[T;T⁻¹] := (1 - (scalar n) t * M.map ⇑LaurentPolynomial.C).det\nht : t_... | ← mul_one (Fintype.card n : ℤ), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.IntegralClosure.Algebra.Basic | {
"line": 267,
"column": 17
} | {
"line": 267,
"column": 42
} | {
"line": 268,
"column": 2
} | [
{
"pp": "G : Type u_1\nR : Type u_2\nK : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing K\ninst✝³ : Algebra R K\ninst✝² : Group G\ninst✝¹ : MulSemiringAction G K\ninst✝ : SMulCommClass G R K\ng : G\n⊢ g • 0 = 0",
"ppTerm": "?m.95",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike... | [] | by ext; exact smul_zero g | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.IntegralNormalization | {
"line": 178,
"column": 2
} | {
"line": 180,
"column": 84
} | {
"line": 182,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : Semiring R\nA : Type u_1\ninst✝ : Semiring A\nf : R →+* A\np : R[X]\nH : f p.leadingCoeff ≠ 0\n⊢ (map f p).integralNormalization = map f p.integralNormalization",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Polynomial.integralNormalization",
"If... | [] | ext i
simp [integralNormalization_coeff, degree_map_eq_of_leadingCoeff_ne_zero _ H, apply_ite f,
leadingCoeff_map_of_leadingCoeff_ne_zero _ H, natDegree_map_eq_iff.mpr (.inl H)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.IntegralNormalization | {
"line": 178,
"column": 2
} | {
"line": 180,
"column": 84
} | {
"line": 182,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : Semiring R\nA : Type u_1\ninst✝ : Semiring A\nf : R →+* A\np : R[X]\nH : f p.leadingCoeff ≠ 0\n⊢ (map f p).integralNormalization = map f p.integralNormalization",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Polynomial.integralNormalization",
"If... | [] | ext i
simp [integralNormalization_coeff, degree_map_eq_of_leadingCoeff_ne_zero _ H, apply_ite f,
leadingCoeff_map_of_leadingCoeff_ne_zero _ H, natDegree_map_eq_iff.mpr (.inl H)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Splits | {
"line": 170,
"column": 37
} | {
"line": 170,
"column": 51
} | {
"line": 170,
"column": 51
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝ : CommSemiring R\nS : Submonoid R[X] := MonoidHom.mrange C\nhS : ↑S = {x | ∃ a, C a = x}\na : R\nmg : Multiset R[X]\nhg : mg.prod ∈ Submonoid.closure {x | ∃ a, X + C a = x}\nj : R[X] → R\nhj : ∀ y ∈ mg, X + C (j y) = y\nhmg : mg = Multiset.map (fun x ↦ X + C x) (Multi... | [
"case refine_1\nR : Type u_1\ninst✝ : CommSemiring R\nS : Submonoid R[X] := MonoidHom.mrange C\nhS : ↑S = {x | ∃ a, C a = x}\na : R\nmg : Multiset R[X]\nhg : mg.prod ∈ Submonoid.closure {x | ∃ a, X + C a = x}\nj : R[X] → R\nhj : ∀ y ∈ mg, X + C (j y) = y\nhmg : mg = Multiset.map (fun x ↦ X + C x) (Multiset.map j mg... | leadingCoeff_C | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Splits | {
"line": 186,
"column": 27
} | {
"line": 186,
"column": 45
} | {
"line": 186,
"column": 45
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝ : CommSemiring R\nf : R[X]\nhf : f.Splits\nh : Irreducible f\na✝ : Nontrivial R\na : R\nm : Multiset R\nhm : f = C f.leadingCoeff * ((X + C a) ::ₘ Multiset.map (fun x ↦ X + C x) m).prod\nha : a ∈ a ::ₘ m\n⊢ f.natDegree ≤ 1",
"ppTerm": "?inr",
"assigned": true,
... | [
"case inr\nR : Type u_1\ninst✝ : CommSemiring R\nf : R[X]\nhf : f.Splits\nh : Irreducible f\na✝ : Nontrivial R\na : R\nm : Multiset R\nhm : f = C f.leadingCoeff * ((X + C a) * (Multiset.map (fun x ↦ X + C x) m).prod)\nha : a ∈ a ::ₘ m\n⊢ f.natDegree ≤ 1"
] | Multiset.prod_cons | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Splits | {
"line": 206,
"column": 2
} | {
"line": 206,
"column": 30
} | {
"line": 207,
"column": 2
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝ : CommSemiring R\nf g : R[X]\nhf : f.Splits\nhg✝ : g.natDegree ≤ 1\nh : Invertible g.leadingCoeff\nhg : g.natDegree = 1\nm : Multiset R\nhm : f = C f.leadingCoeff * (Multiset.map (fun x ↦ X + C x) m).prod\n⊢ ∀ f ∈ Multiset.map (fun p ↦ p.comp g) (Multiset.map (fun x ↦ X + ... | [
"case inr\nR : Type u_1\ninst✝ : CommSemiring R\nf g : R[X]\nhf : f.Splits\nhg✝ : g.natDegree ≤ 1\nh : Invertible g.leadingCoeff\nhg : g.natDegree = 1\nm : Multiset R\nhm : f = C f.leadingCoeff * (Multiset.map (fun x ↦ X + C x) m).prod\n⊢ ∀ (f : R[X]), (∃ a, (∃ a_1 ∈ m, X + C a_1 = a) ∧ a.comp g = f) → f.Splits"
] | simp only [Multiset.mem_map] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Polynomial.Splits | {
"line": 456,
"column": 4
} | {
"line": 456,
"column": 24
} | {
"line": 456,
"column": 24
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nih : ∀ {f g : R[X]}, f ≠ 0 → g ≠ 0 → ∀ (p : R[X]), f * g = p → p.Splits → p.natDegree = n → f.Splits ∧ g.Splits\ng : R[X]\nhg₀ : g ≠ 0\np : R[X]\nh : p.Splits\nhn : p.natDegree = n + 1\na : R\nha : eval a p = 0\nf : R[X]\nhf₀ : (X - C a) * f... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nih : ∀ {f g : R[X]}, f ≠ 0 → g ≠ 0 → ∀ (p : R[X]), f * g = p → p.Splits → p.natDegree = n → f.Splits ∧ g.Splits\ng : R[X]\nhg₀ : g ≠ 0\np : R[X]\nh : p.Splits\nhn : p.natDegree = n + 1\na : R\nha : eval a p = 0\nf : R[X]\nhf₀ : (X - C a) * f ≠ 0\nhp : (... | rw [mul_assoc] at hp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Basic | {
"line": 152,
"column": 2
} | {
"line": 152,
"column": 59
} | {
"line": 154,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nS T : Subalgebra R A\n⊢ Algebra.IsIntegral R ↥(S ⊔ T) ↔ Algebra.IsIntegral R ↥S ∧ Algebra.IsIntegral R ↥T",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
... | [] | simp_rw [← le_integralClosure_iff_isIntegral, sup_le_iff] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Basic | {
"line": 152,
"column": 2
} | {
"line": 152,
"column": 59
} | {
"line": 154,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nS T : Subalgebra R A\n⊢ Algebra.IsIntegral R ↥(S ⊔ T) ↔ Algebra.IsIntegral R ↥S ∧ Algebra.IsIntegral R ↥T",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
... | [] | simp_rw [← le_integralClosure_iff_isIntegral, sup_le_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Basic | {
"line": 152,
"column": 2
} | {
"line": 152,
"column": 59
} | {
"line": 154,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nS T : Subalgebra R A\n⊢ Algebra.IsIntegral R ↥(S ⊔ T) ↔ Algebra.IsIntegral R ↥S ∧ Algebra.IsIntegral R ↥T",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
... | [] | simp_rw [← le_integralClosure_iff_isIntegral, sup_le_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Algebraic.Integral | {
"line": 318,
"column": 2
} | {
"line": 322,
"column": 23
} | {
"line": 324,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : NoZeroDivisors R\na b : S\nha : IsAlgebraic R a\nhb : IsAlgebraic R b\n⊢ IsAlgebraic R (a * b)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
... | [] | have ⟨ra, a0, int_a⟩ := ha.exists_integral_multiple
have ⟨rb, b0, int_b⟩ := hb.exists_integral_multiple
refine IsAlgebraic.iff_exists_smul_integral.mpr ⟨_, mul_ne_zero a0 b0, ?_⟩
simp_rw [Algebra.smul_def, map_mul, mul_mul_mul_comm, ← Algebra.smul_def]
exact int_a.mul int_b | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Algebraic.Integral | {
"line": 318,
"column": 2
} | {
"line": 322,
"column": 23
} | {
"line": 324,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : NoZeroDivisors R\na b : S\nha : IsAlgebraic R a\nhb : IsAlgebraic R b\n⊢ IsAlgebraic R (a * b)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
... | [] | have ⟨ra, a0, int_a⟩ := ha.exists_integral_multiple
have ⟨rb, b0, int_b⟩ := hb.exists_integral_multiple
refine IsAlgebraic.iff_exists_smul_integral.mpr ⟨_, mul_ne_zero a0 b0, ?_⟩
simp_rw [Algebra.smul_def, map_mul, mul_mul_mul_comm, ← Algebra.smul_def]
exact int_a.mul int_b | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Localization.Integral | {
"line": 503,
"column": 4
} | {
"line": 503,
"column": 23
} | {
"line": 504,
"column": 4
} | [
{
"pp": "case mp\nR : Type u_1\ninst✝⁹ : CommRing R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nK : Type u_4\ninst✝⁶ : Field K\ninst✝⁵ : IsDomain R\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : Module.IsTorsionFree R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : S... | [
"case mp\nR : Type u_1\ninst✝⁹ : CommRing R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nK : Type u_4\ninst✝⁶ : Field K\ninst✝⁵ : IsDomain R\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : Module.IsTorsionFree R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : S), IsAlgebra... | rw [div_eq_mul_inv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Algebraic.Integral | {
"line": 467,
"column": 6
} | {
"line": 467,
"column": 21
} | {
"line": 467,
"column": 22
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Ring A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra S A\ninst✝² : IsScalarTower R S A\ninst✝¹ : NoZeroDivisors S\na : A\ninst✝ : Algebra.IsIntegral R S\nha : ¬Transcendental S a\n⊢ ¬Transcenden... | [
"R : Type u_1\nS : Type u_2\nA : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Ring A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra S A\ninst✝² : IsScalarTower R S A\ninst✝¹ : NoZeroDivisors S\na : A\ninst✝ : Algebra.IsIntegral R S\nha : ¬¬IsAlgebraic S a\n⊢ ¬¬IsAlgebraic R a"
] | Transcendental, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Algebraic.Integral | {
"line": 472,
"column": 6
} | {
"line": 472,
"column": 21
} | {
"line": 472,
"column": 22
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Ring A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra S A\ninst✝² : IsScalarTower R S A\ninst✝¹ : NoZeroDivisors S\na : A\ninst✝ : Algebra.IsAlgebraic R S\nha : ¬Transcendental S a\n⊢ ¬Transcende... | [
"R : Type u_1\nS : Type u_2\nA : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Ring A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra S A\ninst✝² : IsScalarTower R S A\ninst✝¹ : NoZeroDivisors S\na : A\ninst✝ : Algebra.IsAlgebraic R S\nha : ¬¬IsAlgebraic S a\n⊢ ¬¬IsAlgebraic R a"
] | Transcendental, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Algebraic.Integral | {
"line": 664,
"column": 2
} | {
"line": 665,
"column": 56
} | {
"line": 667,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nalg : Algebra.IsAlgebraic R S\ninst✝¹ : IsDomain S\ninst✝ : FaithfulSMul R S\nσ : Type u\n⊢ Module.rank (FractionRing (MvPolynomial σ R)) (FractionRing (MvPolynomial σ S)) = lift.{u, u_2} (Module.rank R S)",
... | [] | have := IsDomain.of_faithfulSMul R S
rw [rank_fractionRing, rank_mvPolynomial_mvPolynomial] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Algebraic.Integral | {
"line": 664,
"column": 2
} | {
"line": 665,
"column": 56
} | {
"line": 667,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nalg : Algebra.IsAlgebraic R S\ninst✝¹ : IsDomain S\ninst✝ : FaithfulSMul R S\nσ : Type u\n⊢ Module.rank (FractionRing (MvPolynomial σ R)) (FractionRing (MvPolynomial σ S)) = lift.{u, u_2} (Module.rank R S)",
... | [] | have := IsDomain.of_faithfulSMul R S
rw [rank_fractionRing, rank_mvPolynomial_mvPolynomial] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.SurjectiveOnStalks | {
"line": 184,
"column": 4
} | {
"line": 186,
"column": 38
} | {
"line": 187,
"column": 4
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝⁴ : CommRing R\nS : Type u_2\ninst✝³ : CommRing S\nT : Type u_3\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\ninst✝ : Algebra R S\nhf : (algebraMap R T).SurjectiveOnStalks\ng : T →+* S ⊗[R] T := Algebra.TensorProduct.includeRight.toRingHom\nJ : Ideal (S ⊗[R] T)\nhJ : J.I... | [
"case refine_2\nR : Type u_1\ninst✝⁴ : CommRing R\nS : Type u_2\ninst✝³ : CommRing S\nT : Type u_3\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\ninst✝ : Algebra R S\nhf : (algebraMap R T).SurjectiveOnStalks\ng : T →+* S ⊗[R] T := Algebra.TensorProduct.includeRight.toRingHom\nJ : Ideal (S ⊗[R] T)\nhJ : J.IsPrime\nx : ... | simp only [tmul_smul, Algebra.TensorProduct.algebraMap_apply, Algebra.algebraMap_self,
RingHomCompTriple.comp_apply, Algebra.smul_mul_assoc, Algebra.TensorProduct.tmul_mul_tmul,
one_mul, mul_one, id_apply, ← e] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Ideal.MinimalPrime.Localization | {
"line": 65,
"column": 4
} | {
"line": 65,
"column": 22
} | {
"line": 66,
"column": 4
} | [
{
"pp": "case mpr\nR : Type u_1\ninst✝ : CommSemiring R\nI : Ideal R\nx : R\n⊢ (∃ y ∉ I.radical, x * y ∈ I.radical) → ∃ i, I.IsMinimalPrime i ∧ x ∈ i",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"HMul.hMul",
"CommSemiring.toSemiring",
"Members... | [
"case mpr\nR : Type u_1\ninst✝ : CommSemiring R\nI : Ideal R\nx y : R\nhy : y ∉ I.radical\nhx : x * y ∈ I.radical\n⊢ ∃ i, I.IsMinimalPrime i ∧ x ∈ i"
] | rintro ⟨y, hy, hx⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.RingTheory.Localization.AsSubring | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 8
} | {
"line": 88,
"column": 4
} | [
{
"pp": "A : Type u_1\nK : Type u_2\ninst✝³ : CommRing A\nS : Submonoid A\nhS✝ : S ≤ A⁰\ninst✝² : CommRing K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nhS : S ≤ A⁰\nx✝ : K\n⊢ x✝ ∈ {x | ∃ a s, ∃ (hs : s ∈ S), x = IsLocalization.mk' K a ⟨s, ⋯⟩} ↔\n x✝ ∈ ↑(mapToFractionRing K S (Localization S) hS).rang... | [
"A : Type u_1\nK : Type u_2\ninst✝³ : CommRing A\nS : Submonoid A\nhS✝ : S ≤ A⁰\ninst✝² : CommRing K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nhS : S ≤ A⁰\nx✝ : K\n⊢ x✝ ∈ ↑(mapToFractionRing K S (Localization S) hS).range ↔\n x✝ ∈ {x | ∃ a s, ∃ (hs : s ∈ S), x = IsLocalization.mk' K a ⟨s, ⋯⟩}"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.RingTheory.Localization.AsSubring | {
"line": 131,
"column": 4
} | {
"line": 131,
"column": 8
} | {
"line": 132,
"column": 4
} | [
{
"pp": "A : Type u_1\nK : Type u_2\ninst✝³ : CommRing A\nS : Submonoid A\nhS : S ≤ A⁰\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx✝ : K\n⊢ x✝ ∈ {x | ∃ a s, ∃ (_ : s ∈ S), x = (algebraMap A K) a * ((algebraMap A K) s)⁻¹} ↔\n x✝ ∈ ↑(mapToFractionRing K S (Localization S) hS).range",
... | [
"A : Type u_1\nK : Type u_2\ninst✝³ : CommRing A\nS : Submonoid A\nhS : S ≤ A⁰\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx✝ : K\n⊢ x✝ ∈ ↑(mapToFractionRing K S (Localization S) hS).range ↔\n x✝ ∈ {x | ∃ a s, ∃ (_ : s ∈ S), x = (algebraMap A K) a * ((algebraMap A K) s)⁻¹}"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.RingTheory.Ideal.MinimalPrime.Localization | {
"line": 197,
"column": 6
} | {
"line": 200,
"column": 33
} | {
"line": 201,
"column": 4
} | [
{
"pp": "case mpr.refine_1\nR : Type u_1\ninst✝³ : CommSemiring R\nS : Submonoid R\nA : Type u_2\ninst✝² : CommSemiring A\ninst✝¹ : Algebra R A\ninst✝ : IsLocalization S A\nJ : Ideal R\np : Ideal A\nhp : p ∈ Ideal.under R ⁻¹' J.minimalPrimes\n⊢ p.IsPrime",
"ppTerm": "?mpr.refine_1",
"assigned": true,
... | [] | rw [IsLocalization.isPrime_iff_isPrime_disjoint S A, IsLocalization.disjoint_under_iff S]
refine ⟨hp.isPrime, ?_⟩
rintro rfl
exact hp.isPrime.ne_top rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.MinimalPrime.Localization | {
"line": 197,
"column": 6
} | {
"line": 200,
"column": 33
} | {
"line": 201,
"column": 4
} | [
{
"pp": "case mpr.refine_1\nR : Type u_1\ninst✝³ : CommSemiring R\nS : Submonoid R\nA : Type u_2\ninst✝² : CommSemiring A\ninst✝¹ : Algebra R A\ninst✝ : IsLocalization S A\nJ : Ideal R\np : Ideal A\nhp : p ∈ Ideal.under R ⁻¹' J.minimalPrimes\n⊢ p.IsPrime",
"ppTerm": "?mpr.refine_1",
"assigned": true,
... | [] | rw [IsLocalization.isPrime_iff_isPrime_disjoint S A, IsLocalization.disjoint_under_iff S]
refine ⟨hp.isPrime, ?_⟩
rintro rfl
exact hp.isPrime.ne_top rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.MinimalPrime.Localization | {
"line": 209,
"column": 2
} | {
"line": 209,
"column": 48
} | {
"line": 210,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommSemiring R\nS : Submonoid R\nA : Type u_2\ninst✝² : CommSemiring A\ninst✝¹ : Algebra R A\ninst✝ : IsLocalization S A\nJ : Ideal A\n⊢ (Ideal.comap (algebraMap R A) J).minimalPrimes =\n Ideal.comap (algebraMap R A) '' Ideal.under R ⁻¹' (Ideal.under R J).minimalPrimes",
"... | [
"R : Type u_1\ninst✝³ : CommSemiring R\nS : Submonoid R\nA : Type u_2\ninst✝² : CommSemiring A\ninst✝¹ : Algebra R A\ninst✝ : IsLocalization S A\nJ : Ideal A\n⊢ (Ideal.comap (algebraMap R A) J).minimalPrimes ⊆ Set.range (Ideal.comap (algebraMap R A))"
] | refine (Set.image_preimage_eq_iff.mpr ?_).symm | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Compactness.Bases | {
"line": 35,
"column": 4
} | {
"line": 35,
"column": 62
} | {
"line": 36,
"column": 2
} | [
{
"pp": "case refine_1\nX : Type u_1\nι : Type u_2\ninst✝ : TopologicalSpace X\nb : ι → Set X\nhb : IsTopologicalBasis (range b)\nU : Set X\nhUc : IsCompact U\nhUo : IsOpen U\nY : Type u_1\nf' : Y → ι\ne : U = ⋃ i, (b ∘ f') i\nhf' : ∀ (i : Y), b (f' i) = (b ∘ f') i\nt : Finset Y\nht : U ⊆ ⋃ i ∈ t, (b ∘ f') i\ni... | [] | simpa using subset_iUnion₂ (s := fun i _ => b (f' i)) i hi | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.Sets.Opens | {
"line": 241,
"column": 19
} | {
"line": 241,
"column": 73
} | {
"line": 243,
"column": 0
} | [
{
"pp": "case h_option\nα : Type u_2\ninst✝² : TopologicalSpace α\nι : Type u_5\ninst✝¹ : Finite ι\nα✝ : Type u_5\ninst✝ : Fintype α✝\nih : ∀ (U : α✝ → Opens α), ↑(⨅ i, U i) = ⋂ i, ↑(U i)\nU : Option α✝ → Opens α\n⊢ ↑(⨅ i, U i) = ⋂ i, ↑(U i)",
"ppTerm": "?h_option",
"assigned": true,
"usedConstants"... | [] | rw [iInf_option, Set.iInter_option, Opens.coe_inf, ih] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Sets.Opens | {
"line": 241,
"column": 19
} | {
"line": 241,
"column": 73
} | {
"line": 243,
"column": 0
} | [
{
"pp": "case h_option\nα : Type u_2\ninst✝² : TopologicalSpace α\nι : Type u_5\ninst✝¹ : Finite ι\nα✝ : Type u_5\ninst✝ : Fintype α✝\nih : ∀ (U : α✝ → Opens α), ↑(⨅ i, U i) = ⋂ i, ↑(U i)\nU : Option α✝ → Opens α\n⊢ ↑(⨅ i, U i) = ⋂ i, ↑(U i)",
"ppTerm": "?h_option",
"assigned": true,
"usedConstants"... | [] | rw [iInf_option, Set.iInter_option, Opens.coe_inf, ih] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Sets.Opens | {
"line": 241,
"column": 19
} | {
"line": 241,
"column": 73
} | {
"line": 243,
"column": 0
} | [
{
"pp": "case h_option\nα : Type u_2\ninst✝² : TopologicalSpace α\nι : Type u_5\ninst✝¹ : Finite ι\nα✝ : Type u_5\ninst✝ : Fintype α✝\nih : ∀ (U : α✝ → Opens α), ↑(⨅ i, U i) = ⋂ i, ↑(U i)\nU : Option α✝ → Opens α\n⊢ ↑(⨅ i, U i) = ⋂ i, ↑(U i)",
"ppTerm": "?h_option",
"assigned": true,
"usedConstants"... | [] | rw [iInf_option, Set.iInter_option, Opens.coe_inf, ih] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Sets.Opens | {
"line": 274,
"column": 57
} | {
"line": 274,
"column": 65
} | {
"line": 274,
"column": 66
} | [
{
"pp": "α : Type u_2\ninst✝ : TopologicalSpace α\nU V : Opens α\nx : α\n⊢ (∃ W, ↑(W ⊓ U) ⊆ ↑V ∧ x ∈ W) ↔ ∃ t ⊆ ↑V ∪ (↑U)ᶜ, IsOpen[inst✝] t ∧ x ∈ t",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"TopologicalSpace.Opens.instCompleteLattice",
"Compl.compl",
"TopologicalSpa... | [
"α : Type u_2\ninst✝ : TopologicalSpace α\nU V : Opens α\nx : α\n⊢ (∃ W, ↑W ∩ ↑U ⊆ ↑V ∧ x ∈ W) ↔ ∃ t ⊆ ↑V ∪ (↑U)ᶜ, IsOpen[inst✝] t ∧ x ∈ t"
] | coe_inf, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Sets.OpenCover | {
"line": 66,
"column": 19
} | {
"line": 66,
"column": 30
} | {
"line": 66,
"column": 31
} | [
{
"pp": "ι : Type u_1\nX : Type u_3\ninst✝¹ : TopologicalSpace X\nu : ι → Opens X\nhu : iSup u = ⊤\ninst✝ : CompactSpace X\n⊢ ∃ s, IsOpenCover fun i ↦ u ↑i",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"eq_top_iff",
"TopologicalSpace.Opens.instCompleteLattice",
"Topolog... | [
"ι : Type u_1\nX : Type u_3\ninst✝¹ : TopologicalSpace X\nu : ι → Opens X\nhu : ⊤ ≤ iSup u\ninst✝ : CompactSpace X\n⊢ ∃ s, IsOpenCover fun i ↦ u ↑i"
] | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Sets.Opens | {
"line": 347,
"column": 8
} | {
"line": 347,
"column": 17
} | {
"line": 347,
"column": 18
} | [
{
"pp": "case mp\nα : Type u_2\ninst✝ : TopologicalSpace α\nB : Set (Opens α)\nhB : IsBasis B\nU : Opens α\n⊢ ↑U = ↑(sSup {V | V ∈ B ∧ V ≤ U})",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"TopologicalSpace.Opens.instCompleteLattice",
"TopologicalSpace.Opens.ins... | [
"case mp\nα : Type u_2\ninst✝ : TopologicalSpace α\nB : Set (Opens α)\nhB : IsBasis B\nU : Opens α\n⊢ ↑U = ⋃ i ∈ {V | V ∈ B ∧ V ≤ U}, ↑i"
] | coe_sSup, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.LocallyClosed | {
"line": 103,
"column": 6
} | {
"line": 103,
"column": 28
} | {
"line": 103,
"column": 29
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\n⊢ IsClosed[instTopologicalSpaceSubtype] (coborder s ↓∩ s)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Membership.mem",
"Set.Elem",
"id",
"LE.le",
"Set.instInte... | [
"X : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\n⊢ coborder s ∩ closure[inst✝] (coborder s ∩ s) ⊆ s"
] | isClosed_preimage_val, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.LocalAtTarget | {
"line": 199,
"column": 2
} | {
"line": 199,
"column": 32
} | {
"line": 201,
"column": 0
} | [
{
"pp": "case e_a\nα : Type u_1\nβ : Type u_2\ninst✝ : TopologicalSpace β\nf : α → β\nι : Type u_3\nU : ι → Opens β\nhU : IsOpenCover U\n⊢ univ = ⋃ i, {x | x ∈ U i}",
"ppTerm": "?e_a✝",
"assigned": true,
"usedConstants": [
"TopologicalSpace.IsOpenCover.iSup_set_eq_univ",
"Set.univ",
... | [] | exact hU.iSup_set_eq_univ.symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Sets.Opens | {
"line": 425,
"column": 24
} | {
"line": 425,
"column": 42
} | {
"line": 425,
"column": 42
} | [
{
"pp": "case refine_1\nα : Type u_2\ninst✝ : TopologicalSpace α\ns : Opens α\nH : ∀ (ι : Type u_2) (s_1 : ι → Opens α), s ≤ iSup s_1 → ∃ t, s ≤ t.sup s_1\nι : Type u_2\nU : ι → Set α\nhU : ∀ (i : ι), IsOpen[inst✝] (U i)\nhU' : ↑s ⊆ ⋃ i, U i\nt : Finset ι\nht : s ≤ t.sup fun i ↦ { carrier := U i, is_open' := ⋯ ... | [
"case refine_1\nα : Type u_2\ninst✝ : TopologicalSpace α\ns : Opens α\nH : ∀ (ι : Type u_2) (s_1 : ι → Opens α), s ≤ iSup s_1 → ∃ t, s ≤ t.sup s_1\nι : Type u_2\nU : ι → Set α\nhU : ∀ (i : ι), IsOpen[inst✝] (U i)\nhU' : ↑s ⊆ ⋃ i, U i\nt : Finset ι\nht : s ≤ t.sup fun i ↦ { carrier := U i, is_open' := ⋯ }\n⊢ ⨆ a ∈ t... | Finset.sup_eq_iSup | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.BooleanSubalgebra | {
"line": 400,
"column": 2
} | {
"line": 400,
"column": 63
} | {
"line": 401,
"column": 2
} | [
{
"pp": "case refine_3\nα : Type u_2\ninst✝ : BooleanAlgebra α\ns : Set α\nisSublattice : IsSublattice s\nbot_mem : ⊥ ∈ s\ntop_mem : ⊤ ∈ s\na : α\nt✝ : Finset (↑s × ↑s)\nx✝² : ↑s × ↑s\nt : Finset (↑s × ↑s)\nx✝¹ : ∃ t_1, (t.sup fun x ↦ ↑x.1 \\ ↑x.2)ᶜ = t_1.sup fun x ↦ ↑x.1 \\ ↑x.2\nx y : ↑s\nx✝ : (x, y) ∉ t\ntc ... | [
"case refine_3\nα : Type u_2\ninst✝ : BooleanAlgebra α\ns : Set α\nisSublattice : IsSublattice s\nbot_mem : ⊥ ∈ s\ntop_mem : ⊤ ∈ s\na : α\nt✝¹ : Finset (↑s × ↑s)\nx✝⁵ : ↑s × ↑s\nt✝ : Finset (↑s × ↑s)\nx✝⁴ : ∃ t, (t✝.sup fun x ↦ ↑x.1 \\ ↑x.2)ᶜ = t.sup fun x ↦ ↑x.1 \\ ↑x.2\nx y : ↑s\nx✝³ : (x, y) ∉ t✝\ntc✝ : Finset (... | refine tc.induction ⟨∅, by simp⟩ fun ⟨z, w⟩ tc _ ⟨t, eq⟩ ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.QuasiSeparated | {
"line": 67,
"column": 4
} | {
"line": 67,
"column": 8
} | {
"line": 68,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : IsEmbedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen[inst✝] U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen[inst✝] V\nhV'' : IsCompact V\n⊢ U ∩ V = f '' (f ⁻¹... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nH : IsQuasiSeparated s\nh : IsEmbedding f\nU V : Set β\nhU : U ⊆ f '' s\nhU' : IsOpen[inst✝] U\nhU'' : IsCompact U\nhV : V ⊆ f '' s\nhV' : IsOpen[inst✝] V\nhV'' : IsCompact V\n⊢ f '' (f ⁻¹' U ∩ f ⁻¹' V) = U ∩... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Order.Ideal | {
"line": 335,
"column": 2
} | {
"line": 335,
"column": 57
} | {
"line": 337,
"column": 0
} | [
{
"pp": "P : Type u_1\ninst✝¹ : PartialOrder P\ninst✝ : OrderTop P\na : P\n⊢ (principal a).IsProper ↔ a ≠ ⊤",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Order.Ideal.isProper_iff_top_notMem",
"Eq.mpr",
"congrArg",
"Iff.rfl",
"PartialOrder.toPreorder",
... | [] | rw [isProper_iff_top_notMem, mem_principal, top_le_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.Ideal | {
"line": 335,
"column": 2
} | {
"line": 335,
"column": 57
} | {
"line": 337,
"column": 0
} | [
{
"pp": "P : Type u_1\ninst✝¹ : PartialOrder P\ninst✝ : OrderTop P\na : P\n⊢ (principal a).IsProper ↔ a ≠ ⊤",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Order.Ideal.isProper_iff_top_notMem",
"Eq.mpr",
"congrArg",
"Iff.rfl",
"PartialOrder.toPreorder",
... | [] | rw [isProper_iff_top_notMem, mem_principal, top_le_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Ideal | {
"line": 335,
"column": 2
} | {
"line": 335,
"column": 57
} | {
"line": 337,
"column": 0
} | [
{
"pp": "P : Type u_1\ninst✝¹ : PartialOrder P\ninst✝ : OrderTop P\na : P\n⊢ (principal a).IsProper ↔ a ≠ ⊤",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Order.Ideal.isProper_iff_top_notMem",
"Eq.mpr",
"congrArg",
"Iff.rfl",
"PartialOrder.toPreorder",
... | [] | rw [isProper_iff_top_notMem, mem_principal, top_le_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Ideal | {
"line": 460,
"column": 6
} | {
"line": 460,
"column": 36
} | {
"line": 462,
"column": 0
} | [
{
"pp": "P : Type u_1\ninst✝¹ : SemilatticeSup P\ninst✝ : OrderBot P\nx : P\nS✝ S : Set (Ideal P)\ns : Ideal P\nhs : s ∈ S\n⊢ ⋂ s ∈ S, ↑s ⊆ ↑s",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"PartialOrder.toPreorder",
"Preorder.toLE",
"Set.biInter_subset_of_mem",
"S... | [] | exact biInter_subset_of_mem hs | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Sober | {
"line": 193,
"column": 18
} | {
"line": 212,
"column": 38
} | {
"line": 214,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\nf : α → β\nhf : IsOpenEmbedding f\ninst✝ : QuasiSober β\nS✝ : Set α\nhS : IsIrreducible S✝\nhS' : IsClosed[inst✝²] S✝\n⊢ ∃ x, IsGenericPoint x S✝",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
... | [] | by
have hS'' := hS.image f hf.continuous.continuousOn
obtain ⟨x, hx⟩ := QuasiSober.sober hS''.closure isClosed_closure
obtain ⟨T, hT, rfl⟩ := hf.isInducing.isClosed_iff.mp hS'
rw [image_preimage_eq_inter_range] at hx hS''
have hxT : x ∈ T := by
rw [← hT.closure_eq]
exact closure_mono int... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Constructible | {
"line": 314,
"column": 40
} | {
"line": 314,
"column": 52
} | {
"line": 314,
"column": 52
} | [
{
"pp": "X : Type u_2\ninst✝ : TopologicalSpace X\nι : Type u_4\nU : ι → Opens X\nhU : IsOpenCover U\nh₁ : ∀ (i : ι), IsRetrocompact ↑(U i)\nh₂ : ∀ (i : ι), IsQuasiSeparated ↑(U i)\nV₁ V₂ : Set X\nho₁ : IsOpen[inst✝] V₁\nhc₁ : IsCompact V₁\nho₂ : IsOpen[inst✝] V₂\nhc₂ : IsCompact V₂\nt : Finset ι\nht : V₁ ⊆ ⋃ i... | [] | by simpa [*] | [anonymous] | Lean.Parser.Term.byTactic |
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