module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Topology.Algebra.Group.Basic | {
"line": 852,
"column": 2
} | {
"line": 853,
"column": 92
} | {
"line": 854,
"column": 2
} | [
{
"pp": "G : Type w\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : Group G\ninst✝⁶ : IsTopologicalGroup G\nH : Type u_1\nM : Type u_2\ninst✝⁵ : CommMonoid M\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ContinuousMul M\ninst✝² : Group H\ninst✝¹ : TopologicalSpace H\ninst✝ : IsTopologicalGroup H\nf : G →* H →* M\nhf : Continuo... | [
"G : Type w\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : Group G\ninst✝⁶ : IsTopologicalGroup G\nH : Type u_1\nM : Type u_2\ninst✝⁵ : CommMonoid M\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ContinuousMul M\ninst✝² : Group H\ninst✝¹ : TopologicalSpace H\ninst✝ : IsTopologicalGroup H\nf : G →* H →* M\nhl : ∀ (x : G), Tendsto (... | simp only [ContinuousAt, nhds_prod_eq, ← map_mul_left_nhds_one x, ← map_mul_left_nhds_one y,
prod_map_map_eq, tendsto_map'_iff, Function.comp_def, map_mul, MonoidHom.mul_apply] at * | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Algebra.GroupWithZero | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 93
} | {
"line": 177,
"column": 0
} | [
{
"pp": "α : Type u_1\nG₀ : Type u_3\ninst✝² : GroupWithZero G₀\ninst✝¹ : TopologicalSpace G₀\ninst✝ : ContinuousInv₀ G₀\nx : G₀\nl : Filter α\nf : α → G₀\nhx : x ≠ 0\n⊢ Tendsto (fun x ↦ (f x)⁻¹) l (𝓝 x⁻¹) ↔ Tendsto f l (𝓝 x)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"GroupWi... | [] | simp only [nhds_inv₀ hx, ← Filter.comap_inv, tendsto_comap_iff, Function.comp_def, inv_inv] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Algebra.GroupWithZero | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 93
} | {
"line": 177,
"column": 0
} | [
{
"pp": "α : Type u_1\nG₀ : Type u_3\ninst✝² : GroupWithZero G₀\ninst✝¹ : TopologicalSpace G₀\ninst✝ : ContinuousInv₀ G₀\nx : G₀\nl : Filter α\nf : α → G₀\nhx : x ≠ 0\n⊢ Tendsto (fun x ↦ (f x)⁻¹) l (𝓝 x⁻¹) ↔ Tendsto f l (𝓝 x)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"GroupWi... | [] | simp only [nhds_inv₀ hx, ← Filter.comap_inv, tendsto_comap_iff, Function.comp_def, inv_inv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.GroupWithZero | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 93
} | {
"line": 177,
"column": 0
} | [
{
"pp": "α : Type u_1\nG₀ : Type u_3\ninst✝² : GroupWithZero G₀\ninst✝¹ : TopologicalSpace G₀\ninst✝ : ContinuousInv₀ G₀\nx : G₀\nl : Filter α\nf : α → G₀\nhx : x ≠ 0\n⊢ Tendsto (fun x ↦ (f x)⁻¹) l (𝓝 x⁻¹) ↔ Tendsto f l (𝓝 x)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"GroupWi... | [] | simp only [nhds_inv₀ hx, ← Filter.comap_inv, tendsto_comap_iff, Function.comp_def, inv_inv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.IsUniformGroup.Basic | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 7
} | {
"line": 144,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : UniformSpace α\ninst✝¹ : Group α\ninst✝ : IsUniformGroup α\ns : Set α\nhs : TotallyBounded s\n⊢ Inv.inv = image fun x ↦ x⁻¹",
"ppTerm": "?m.124",
"assigned": true,
"usedConstants": [
"Set.ext",
"DivInvOneMonoid.toInvOneClass",
"congrArg",
"Set.... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Algebra.IsUniformGroup.Basic | {
"line": 313,
"column": 2
} | {
"line": 314,
"column": 35
} | {
"line": 315,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\n⊢ comap (fun x ↦ (op x.1, op x.2)) (comap (fun p ↦ p.2 * p.1⁻¹) (𝓝 1)) = comap (fun p ↦ p.1⁻¹ * p.2) (𝓝 1)",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"MulOpposite.opHomeomorph",... | [
"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nthis : 𝓝 1 = comap (⇑opHomeomorph) (𝓝 1)\n⊢ comap (fun x ↦ (op x.1, op x.2)) (comap (fun p ↦ p.2 * p.1⁻¹) (𝓝 1)) = comap (fun p ↦ p.1⁻¹ * p.2) (𝓝 1)"
] | have : 𝓝 (1 : G) = comap (MulOpposite.opHomeomorph) (𝓝 (1 : Gᵐᵒᵖ)) := by
simp [Homeomorph.comap_nhds_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Algebra.IsUniformGroup.Basic | {
"line": 325,
"column": 2
} | {
"line": 326,
"column": 35
} | {
"line": 327,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\n⊢ comap (fun x ↦ (op x.1, op x.2)) (comap (fun p ↦ p.1⁻¹ * p.2) (𝓝 1)) = comap (fun p ↦ p.2 * p.1⁻¹) (𝓝 1)",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"MulOpposite.opHomeomorph",... | [
"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nthis : 𝓝 1 = comap (⇑opHomeomorph) (𝓝 1)\n⊢ comap (fun x ↦ (op x.1, op x.2)) (comap (fun p ↦ p.1⁻¹ * p.2) (𝓝 1)) = comap (fun p ↦ p.2 * p.1⁻¹) (𝓝 1)"
] | have : 𝓝 (1 : G) = comap (MulOpposite.opHomeomorph) (𝓝 (1 : Gᵐᵒᵖ)) := by
simp [Homeomorph.comap_nhds_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Algebra.IsUniformGroup.Basic | {
"line": 512,
"column": 2
} | {
"line": 512,
"column": 27
} | {
"line": 513,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nδ : Type u_4\nG : Type u_5\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : AddCommGroup α\ninst✝⁶ : IsTopologicalAddGroup α\ninst✝⁵ : TopologicalSpace β\ninst✝⁴ : AddCommGroup β\ninst✝³ : TopologicalSpace δ\ninst✝² : AddCommGroup δ\ninst✝¹ : UniformSpace G\ninst✝ : AddCommGroup G\ne ... | [
"α : Type u_1\nβ : Type u_2\nδ : Type u_4\nG : Type u_5\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : AddCommGroup α\ninst✝⁶ : IsTopologicalAddGroup α\ninst✝⁵ : TopologicalSpace β\ninst✝⁴ : AddCommGroup β\ninst✝³ : TopologicalSpace δ\ninst✝² : AddCommGroup δ\ninst✝¹ : UniformSpace G\ninst✝ : AddCommGroup G\ne : β →+ α\nde... | simp_rw [forall_mem_comm] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 1209,
"column": 4
} | {
"line": 1209,
"column": 36
} | {
"line": 1209,
"column": 36
} | [
{
"pp": "G : Type w\ninst✝² : TopologicalSpace G\ninst✝¹ : MulOneClass G\ninst✝ : ContinuousMul G\nK U : Set G\nhK : IsCompact K\nhU : IsOpen[inst✝²] U\nhKU : K ⊆ U\nV : Set Gᵐᵒᵖ\nhV : V ∈ 𝓝 (op 1)\nhV' : op ⁻¹' V * K ⊆ op ⁻¹' op '' U\n⊢ op ⁻¹' V * K ⊆ U",
"ppTerm": "?m.130",
"assigned": true,
"use... | [
"G : Type w\ninst✝² : TopologicalSpace G\ninst✝¹ : MulOneClass G\ninst✝ : ContinuousMul G\nK U : Set G\nhK : IsCompact K\nhU : IsOpen[inst✝²] U\nhKU : K ⊆ U\nV : Set Gᵐᵒᵖ\nhV : V ∈ 𝓝 (op 1)\nhV' : op ⁻¹' V * K ⊆ U\n⊢ op ⁻¹' V * K ⊆ U"
] | preimage_image_eq _ op_injective | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Ring.Basic | {
"line": 89,
"column": 20
} | {
"line": 89,
"column": 67
} | {
"line": 91,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : TopologicalSpace R\ninst✝¹ : NonAssocRing R\ninst✝ : SeparatelyContinuousMul R\n⊢ Continuous[inst✝², inst✝²] fun a ↦ -a",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"Continuous",
"HMul.hMul",
"congrArg",
... | [] | by simpa using continuous_id.const_mul (-1 : R) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Order.LocalExtr | {
"line": 548,
"column": 81
} | {
"line": 548,
"column": 86
} | {
"line": 548,
"column": 86
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\nβ : Type u_2\ninst✝ : Preorder β\nb : α\nf : α → β\na : Set α\nha : a ∈ 𝓝[≤] b\nc : Set α\nhc : c ∈ 𝓝[≥] b\nh₀ : MonotoneOn f a\nh₁ : AntitoneOn f c\nthis✝ : b ∈ a\nthis : b ∈ c\nx : α\nx✝ : x ∈ a ∩ Iic b ∪ c ∩ Ici b\nh✝ : x... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Order.LocalExtr | {
"line": 548,
"column": 81
} | {
"line": 548,
"column": 86
} | {
"line": 548,
"column": 86
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\nβ : Type u_2\ninst✝ : Preorder β\nb : α\nf : α → β\na : Set α\nha : a ∈ 𝓝[≤] b\nc : Set α\nhc : c ∈ 𝓝[≥] b\nh₀ : MonotoneOn f a\nh₁ : AntitoneOn f c\nthis✝ : b ∈ a\nthis : b ∈ c\nx : α\nx✝ : x ∈ a ∩ Iic b ∪ c ∩ Ici b\nh✝ : b... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Order.LocalExtr | {
"line": 557,
"column": 81
} | {
"line": 557,
"column": 86
} | {
"line": 557,
"column": 86
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\nβ : Type u_2\ninst✝ : Preorder β\nb : α\nf : α → β\na : Set α\nha : a ∈ 𝓝[≤] b\nc : Set α\nhc : c ∈ 𝓝[≥] b\nh₀ : AntitoneOn f a\nh₁ : MonotoneOn f c\nthis✝ : b ∈ a\nthis : b ∈ c\nx : α\nx✝ : x ∈ a ∩ Iic b ∪ c ∩ Ici b\nh✝ : x... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Order.LocalExtr | {
"line": 557,
"column": 81
} | {
"line": 557,
"column": 86
} | {
"line": 557,
"column": 86
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\nβ : Type u_2\ninst✝ : Preorder β\nb : α\nf : α → β\na : Set α\nha : a ∈ 𝓝[≤] b\nc : Set α\nhc : c ∈ 𝓝[≥] b\nh₀ : AntitoneOn f a\nh₁ : MonotoneOn f c\nthis✝ : b ∈ a\nthis : b ∈ c\nx : α\nx✝ : x ∈ a ∩ Iic b ∪ c ∩ Ici b\nh✝ : b... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Algebra.Field | {
"line": 43,
"column": 2
} | {
"line": 46,
"column": 43
} | {
"line": 48,
"column": 0
} | [
{
"pp": "K : Type u_2\ninst✝⁴ : DivisionRing K\ninst✝³ : TopologicalSpace K\ninst✝² : IsTopologicalRing K\ninst✝¹ : CompactSpace K\ninst✝ : T2Space K\n⊢ Finite K",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"GroupWithZero.isOpen... | [] | suffices DiscreteTopology K by
exact finite_of_compact_of_discrete
rw [discreteTopology_iff_isOpen_singleton_zero]
exact GroupWithZero.isOpen_singleton_zero | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Field | {
"line": 43,
"column": 2
} | {
"line": 46,
"column": 43
} | {
"line": 48,
"column": 0
} | [
{
"pp": "K : Type u_2\ninst✝⁴ : DivisionRing K\ninst✝³ : TopologicalSpace K\ninst✝² : IsTopologicalRing K\ninst✝¹ : CompactSpace K\ninst✝ : T2Space K\n⊢ Finite K",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"GroupWithZero.isOpen... | [] | suffices DiscreteTopology K by
exact finite_of_compact_of_discrete
rw [discreteTopology_iff_isOpen_singleton_zero]
exact GroupWithZero.isOpen_singleton_zero | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Order.LeftRightNhds | {
"line": 192,
"column": 2
} | {
"line": 192,
"column": 52
} | {
"line": 194,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na b : α\nh : a < b\ns : Set α\n⊢ [s ∈ 𝓝[<] b, s ∈ 𝓝[Ico a b] b, s ∈ 𝓝[Ioo a b] b, ∃ l ∈ Ico a b, Ioo l b ⊆ s, ∃ l ∈ Iio b, Ioo l b ⊆ s].TFAE",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants":... | [] | simpa using! TFAE_mem_nhdsGT h.dual (ofDual ⁻¹' s) | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.Topology.Order.LeftRightNhds | {
"line": 192,
"column": 2
} | {
"line": 192,
"column": 52
} | {
"line": 194,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na b : α\nh : a < b\ns : Set α\n⊢ [s ∈ 𝓝[<] b, s ∈ 𝓝[Ico a b] b, s ∈ 𝓝[Ioo a b] b, ∃ l ∈ Ico a b, Ioo l b ⊆ s, ∃ l ∈ Iio b, Ioo l b ⊆ s].TFAE",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants":... | [] | simpa using! TFAE_mem_nhdsGT h.dual (ofDual ⁻¹' s) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order.LeftRightNhds | {
"line": 192,
"column": 2
} | {
"line": 192,
"column": 52
} | {
"line": 194,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na b : α\nh : a < b\ns : Set α\n⊢ [s ∈ 𝓝[<] b, s ∈ 𝓝[Ico a b] b, s ∈ 𝓝[Ioo a b] b, ∃ l ∈ Ico a b, Ioo l b ⊆ s, ∃ l ∈ Iio b, Ioo l b ⊆ s].TFAE",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants":... | [] | simpa using! TFAE_mem_nhdsGT h.dual (ofDual ⁻¹' s) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Order.LeftRightNhds | {
"line": 238,
"column": 2
} | {
"line": 240,
"column": 31
} | {
"line": 242,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\n⊢ 𝓝[<] a = ⊥ ↔ IsBot a ∨ ∃ b, b ⋖ a",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"OrderDual.toDual",
"Eq.mpr",
"Set.Ioi",
"Preorder.toLT",
"Equiv.in... | [] | convert! (config := { preTransparency := .default })
nhdsGT_eq_bot_iff (a := OrderDual.toDual a) using 4
exact ofDual_covBy_ofDual_iff | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order.LeftRightNhds | {
"line": 238,
"column": 2
} | {
"line": 240,
"column": 31
} | {
"line": 242,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\n⊢ 𝓝[<] a = ⊥ ↔ IsBot a ∨ ∃ b, b ⋖ a",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"OrderDual.toDual",
"Eq.mpr",
"Set.Ioi",
"Preorder.toLT",
"Equiv.in... | [] | convert! (config := { preTransparency := .default })
nhdsGT_eq_bot_iff (a := OrderDual.toDual a) using 4
exact ofDual_covBy_ofDual_iff | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Logic.Equiv.PartialEquiv | {
"line": 668,
"column": 19
} | {
"line": 668,
"column": 76
} | {
"line": 668,
"column": 76
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ne : PartialEquiv α β\ne' : PartialEquiv α γ\nx : α\nhe : x ∈ e.source\nhe' : x ∈ e'.source\n⊢ ↑e x ∈ ↑e.symm.symm.symm ⁻¹' e'.source",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PartialEquiv.symm_symm",
"con... | [] | rwa [mem_preimage, PartialEquiv.symm_symm, e.left_inv he] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Logic.Equiv.PartialEquiv | {
"line": 668,
"column": 19
} | {
"line": 668,
"column": 76
} | {
"line": 668,
"column": 76
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ne : PartialEquiv α β\ne' : PartialEquiv α γ\nx : α\nhe : x ∈ e.source\nhe' : x ∈ e'.source\n⊢ ↑e x ∈ ↑e.symm.symm.symm ⁻¹' e'.source",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PartialEquiv.symm_symm",
"con... | [] | rwa [mem_preimage, PartialEquiv.symm_symm, e.left_inv he] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Logic.Equiv.PartialEquiv | {
"line": 668,
"column": 19
} | {
"line": 668,
"column": 76
} | {
"line": 668,
"column": 76
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ne : PartialEquiv α β\ne' : PartialEquiv α γ\nx : α\nhe : x ∈ e.source\nhe' : x ∈ e'.source\n⊢ ↑e x ∈ ↑e.symm.symm.symm ⁻¹' e'.source",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PartialEquiv.symm_symm",
"con... | [] | rwa [mem_preimage, PartialEquiv.symm_symm, e.left_inv he] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Logic.Equiv.PartialEquiv | {
"line": 728,
"column": 83
} | {
"line": 728,
"column": 95
} | {
"line": 728,
"column": 95
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ne e' : PartialEquiv α β\nhe : e ≈ e'\ns : Set β\n⊢ e.source ∩ ↑e' ⁻¹' s = e'.source ∩ ↑e' ⁻¹' s",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"PartialEquiv.EqOnSource.source_eq",
"Set.instInter... | [
"α : Type u_1\nβ : Type u_2\ne e' : PartialEquiv α β\nhe : e ≈ e'\ns : Set β\n⊢ e'.source ∩ ↑e' ⁻¹' s = e'.source ∩ ↑e' ⁻¹' s"
] | source_eq he | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order.Basic | {
"line": 105,
"column": 22
} | {
"line": 105,
"column": 67
} | {
"line": 105,
"column": 68
} | [
{
"pp": "α : Type u\nβ : Type v\nts : TopologicalSpace α\ninst✝² : Preorder α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace β\nf : β → α\n⊢ Continuous[inst✝, ts] f ↔ ∀ (a : α), IsOpen[inst✝] (f ⁻¹' Ioi a) ∧ IsOpen[inst✝] (f ⁻¹' Iio a)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
... | [
"α : Type u\nβ : Type v\nts : TopologicalSpace α\ninst✝² : Preorder α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace β\nf : β → α\n⊢ Continuous[inst✝, Preorder.topology α] f ↔ ∀ (a : α), IsOpen[inst✝] (f ⁻¹' Ioi a) ∧ IsOpen[inst✝] (f ⁻¹' Iio a)"
] | OrderTopology.topology_eq_generate_intervals, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Order.Basic | {
"line": 106,
"column": 2
} | {
"line": 106,
"column": 7
} | {
"line": 108,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\nts : TopologicalSpace α\ninst✝² : Preorder α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace β\nf : β → α\n⊢ (∀ s ∈ {s | ∃ a, s = Ioi a ∨ s = Iio a}, IsOpen[inst✝] (f ⁻¹' s)) ↔\n ∀ (a : α), IsOpen[inst✝] (f ⁻¹' Ioi a) ∧ IsOpen[inst✝] (f ⁻¹' Iio a)",
"ppTerm": "?m.22",... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Category.Grp.EnoughInjectives | {
"line": 46,
"column": 48
} | {
"line": 46,
"column": 53
} | {
"line": 46,
"column": 53
} | [
{
"pp": "A_ : AddCommGrpCat\n⊢ (fun i ↦ { down := i 0 }) = 0",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"Eq.mpr",
"ULift.addCommGroup",
"SemilinearMapClass.distribMulActionSemiHomClass",
"CharacterModule.instLinearMapClassIntAddCircleRatO... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Category.Grp.EnoughInjectives | {
"line": 46,
"column": 48
} | {
"line": 46,
"column": 53
} | {
"line": 46,
"column": 53
} | [
{
"pp": "A_ : AddCommGrpCat\n⊢ (fun i ↦ { down := i 0 }) = 0",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"Eq.mpr",
"ULift.addCommGroup",
"SemilinearMapClass.distribMulActionSemiHomClass",
"CharacterModule.instLinearMapClassIntAddCircleRatO... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.Grp.EnoughInjectives | {
"line": 46,
"column": 48
} | {
"line": 46,
"column": 53
} | {
"line": 46,
"column": 53
} | [
{
"pp": "A_ : AddCommGrpCat\n⊢ (fun i ↦ { down := i 0 }) = 0",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"Eq.mpr",
"ULift.addCommGroup",
"SemilinearMapClass.distribMulActionSemiHomClass",
"CharacterModule.instLinearMapClassIntAddCircleRatO... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.Grp.EnoughInjectives | {
"line": 46,
"column": 59
} | {
"line": 46,
"column": 64
} | {
"line": 46,
"column": 64
} | [
{
"pp": "A_ : AddCommGrpCat\n⊢ ∀ (x y : ↑A_),\n { toFun := fun a i ↦ { down := i a }, map_zero' := ⋯ }.toFun (x + y) =\n { toFun := fun a i ↦ { down := i a }, map_zero' := ⋯ }.toFun x +\n { toFun := fun a i ↦ { down := i a }, map_zero' := ⋯ }.toFun y",
"ppTerm": "?m.35",
"assigned": true,... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Category.Grp.EnoughInjectives | {
"line": 46,
"column": 59
} | {
"line": 46,
"column": 64
} | {
"line": 46,
"column": 64
} | [
{
"pp": "A_ : AddCommGrpCat\n⊢ ∀ (x y : ↑A_),\n { toFun := fun a i ↦ { down := i a }, map_zero' := ⋯ }.toFun (x + y) =\n { toFun := fun a i ↦ { down := i a }, map_zero' := ⋯ }.toFun x +\n { toFun := fun a i ↦ { down := i a }, map_zero' := ⋯ }.toFun y",
"ppTerm": "?m.35",
"assigned": true,... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.Grp.EnoughInjectives | {
"line": 46,
"column": 59
} | {
"line": 46,
"column": 64
} | {
"line": 46,
"column": 64
} | [
{
"pp": "A_ : AddCommGrpCat\n⊢ ∀ (x y : ↑A_),\n { toFun := fun a i ↦ { down := i a }, map_zero' := ⋯ }.toFun (x + y) =\n { toFun := fun a i ↦ { down := i a }, map_zero' := ⋯ }.toFun x +\n { toFun := fun a i ↦ { down := i a }, map_zero' := ⋯ }.toFun y",
"ppTerm": "?m.35",
"assigned": true,... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.Grp.IsFinite | {
"line": 41,
"column": 4
} | {
"line": 41,
"column": 45
} | {
"line": 41,
"column": 45
} | [
{
"pp": "⊢ (of PUnit.{u + 1}).isFinite",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AddCommGrpCat.prop_isFinite_iff",
"congrArg",
"Finite",
"PUnit.addCommGroup",
"inferInstance",
"id",
"Finite.of_fintype",
"AddCommGrpCat.c... | [] | by rw [prop_isFinite_iff]; infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.CharacterModule | {
"line": 81,
"column": 17
} | {
"line": 81,
"column": 22
} | {
"line": 82,
"column": 2
} | [
{
"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\nf : A →ₗ[R] B\n⊢ ∀ (x y : CharacterModule B),\n AddMonoidHom.comp (x + y) f.toAddMonoi... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Module.CharacterModule | {
"line": 81,
"column": 17
} | {
"line": 81,
"column": 22
} | {
"line": 82,
"column": 2
} | [
{
"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\nf : A →ₗ[R] B\n⊢ ∀ (x y : CharacterModule B),\n AddMonoidHom.comp (x + y) f.toAddMonoi... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.CharacterModule | {
"line": 81,
"column": 17
} | {
"line": 81,
"column": 22
} | {
"line": 82,
"column": 2
} | [
{
"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\nf : A →ₗ[R] B\n⊢ ∀ (x y : CharacterModule B),\n AddMonoidHom.comp (x + y) f.toAddMonoi... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.CharacterModule | {
"line": 122,
"column": 82
} | {
"line": 122,
"column": 87
} | {
"line": 123,
"column": 2
} | [
{
"pp": "case refine_1\nR : 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\nc c' : A →ₗ[R] CharacterModule B\nx : A ⊗[R] B\n⊢ (liftAddHom (c + c').toA... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Module.CharacterModule | {
"line": 122,
"column": 82
} | {
"line": 122,
"column": 87
} | {
"line": 123,
"column": 2
} | [
{
"pp": "case refine_2\nR : 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\nc c' : A →ₗ[R] CharacterModule B\nx : A ⊗[R] B\n⊢ ∀ (x : A) (y : B),\n ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Module.CharacterModule | {
"line": 122,
"column": 82
} | {
"line": 122,
"column": 87
} | {
"line": 123,
"column": 2
} | [
{
"pp": "case refine_3\nR : 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\nc c' : A →ₗ[R] CharacterModule B\nx : A ⊗[R] B\n⊢ ∀ (x y : A ⊗[R] B),\n ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Module.CharacterModule | {
"line": 124,
"column": 76
} | {
"line": 124,
"column": 81
} | {
"line": 124,
"column": 81
} | [
{
"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\nr : R\nc : A →ₗ[R] CharacterModule B\nx : A ⊗[R] B\n⊢ ∀ (x y : A ⊗[R] B),\n (liftAddHo... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Module.CharacterModule | {
"line": 124,
"column": 76
} | {
"line": 124,
"column": 81
} | {
"line": 124,
"column": 81
} | [
{
"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\nr : R\nc : A →ₗ[R] CharacterModule B\nx : A ⊗[R] B\n⊢ ∀ (x y : A ⊗[R] B),\n (liftAddHo... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.CharacterModule | {
"line": 124,
"column": 76
} | {
"line": 124,
"column": 81
} | {
"line": 124,
"column": 81
} | [
{
"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\nr : R\nc : A →ₗ[R] CharacterModule B\nx : A ⊗[R] B\n⊢ ∀ (x y : A ⊗[R] B),\n (liftAddHo... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.CharacterModule | {
"line": 144,
"column": 74
} | {
"line": 144,
"column": 79
} | {
"line": 144,
"column": 79
} | [
{
"pp": "case refine_1\nR : 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\nx✝ : CharacterModule (A ⊗[R] B)\nz : A ⊗[R] B\n⊢ ((uncurry ∘ₗ curry) x✝) 0... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Module.CharacterModule | {
"line": 144,
"column": 74
} | {
"line": 144,
"column": 79
} | {
"line": 144,
"column": 79
} | [
{
"pp": "case refine_2\nR : 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\nx✝ : CharacterModule (A ⊗[R] B)\nz : A ⊗[R] B\n⊢ ∀ (x : A) (y : B), ((uncu... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Module.CharacterModule | {
"line": 144,
"column": 74
} | {
"line": 144,
"column": 79
} | {
"line": 144,
"column": 79
} | [
{
"pp": "case refine_3\nR : 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\nx✝ : CharacterModule (A ⊗[R] B)\nz : A ⊗[R] B\n⊢ ∀ (x y : A ⊗[R] B),\n ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Module.CharacterModule | {
"line": 144,
"column": 85
} | {
"line": 144,
"column": 90
} | {
"line": 144,
"column": 90
} | [
{
"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⊢ curry ∘ₗ uncurry = LinearMap.id",
"ppTerm": "?m.106",
"assigned": true,
"us... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Module.CharacterModule | {
"line": 144,
"column": 85
} | {
"line": 144,
"column": 90
} | {
"line": 144,
"column": 90
} | [
{
"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⊢ curry ∘ₗ uncurry = LinearMap.id",
"ppTerm": "?m.106",
"assigned": true,
"us... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.CharacterModule | {
"line": 144,
"column": 85
} | {
"line": 144,
"column": 90
} | {
"line": 144,
"column": 90
} | [
{
"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⊢ curry ∘ₗ uncurry = LinearMap.id",
"ppTerm": "?m.106",
"assigned": true,
"us... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.CharacterModule | {
"line": 162,
"column": 2
} | {
"line": 165,
"column": 69
} | {
"line": 167,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ (int.divByNat n) ↑n = 0",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"Iff.mpr",
"Rat.instOfNat",
"LinearMap.id",
"Int.cast",
"AddCircle.coe_eq_zero_iff",
"Int.instAddCommMonoid",
"GroupWithZero.toMo... | [] | obtain rfl | h0 := eq_or_ne n 0
· apply map_zero
exact (AddCircle.coe_eq_zero_iff _).mpr
⟨1, by simp [mul_inv_cancel₀ (Nat.cast_ne_zero (R := ℚ).mpr h0)]⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.CharacterModule | {
"line": 162,
"column": 2
} | {
"line": 165,
"column": 69
} | {
"line": 167,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ (int.divByNat n) ↑n = 0",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"Iff.mpr",
"Rat.instOfNat",
"LinearMap.id",
"Int.cast",
"AddCircle.coe_eq_zero_iff",
"Int.instAddCommMonoid",
"GroupWithZero.toMo... | [] | obtain rfl | h0 := eq_or_ne n 0
· apply map_zero
exact (AddCircle.coe_eq_zero_iff _).mpr
⟨1, by simp [mul_inv_cancel₀ (Nat.cast_ne_zero (R := ℚ).mpr h0)]⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.CharacterModule | {
"line": 208,
"column": 34
} | {
"line": 208,
"column": 60
} | {
"line": 208,
"column": 60
} | [
{
"pp": "A : Type uA\ninst✝ : AddCommGroup A\na : A\nn : ℤ\nhn : 1 = (↑(if addOrderOf a = 0 then 2 else addOrderOf a))⁻¹.den\n⊢ a = 0",
"ppTerm": "?m.200",
"assigned": true,
"usedConstants": [
"congrArg",
"Rat",
"addOrderOf",
"AddCommGroup.toAddGroup",
"Rat.den",
... | [
"A : Type uA\ninst✝ : AddCommGroup A\na : A\nn : ℤ\nhn : 1 = if addOrderOf a = 0 then 2 else addOrderOf a\n⊢ a = 0",
"A : Type uA\ninst✝ : AddCommGroup A\na : A\nn : ℤ\nhn : 1 = (↑(if addOrderOf a = 0 then 2 else addOrderOf a))⁻¹.den\n⊢ 0 < if addOrderOf a = 0 then 2 else addOrderOf a"
] | Rat.inv_natCast_den_of_pos | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Instances.AddCircle.Defs | {
"line": 678,
"column": 68
} | {
"line": 690,
"column": 28
} | {
"line": 692,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : Field 𝕜\np : 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\nhp : Fact (0 < p)\nn : ℕ\n⊢ Nat.card { u // addOrderOf u = n } = n.totient",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Nat.gcd",
"Eq.mpr",
"Nat.Coprime",
... | [] | by
rcases n.eq_zero_or_pos with (rfl | hn)
· simp only [Nat.totient_zero, addOrderOf_eq_zero_iff]
rcases em (∃ u : AddCircle p, ¬IsOfFinAddOrder u) with (⟨u, hu⟩ | h)
· have : Infinite { u : AddCircle p // ¬IsOfFinAddOrder u } := by
rw [← coe_setOf, infinite_coe_iff]
exact infinite_not_isOfF... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Category.ModuleCat.AB | {
"line": 43,
"column": 2
} | {
"line": 47,
"column": 51
} | {
"line": 49,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : Ring R\ninst✝ : Small.{v, u} R\nX Y : ModuleCat R\nf g : X ⟶ Y\nh : ∀ (G : ModuleCat R), ObjectProperty.singleton (of R (Shrink.{v, u} R)) G → ∀ (h : G ⟶ X), h ≫ f = h ≫ g\n⊢ f = g",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Algebra... | [] | simp only [ObjectProperty.singleton_iff, ModuleCat.hom_ext_iff, hom_comp,
LinearMap.ext_iff, LinearMap.coe_comp, Function.comp_apply, forall_eq'] at h
ext x
simpa using h (ModuleCat.ofHom ((LinearMap.toSpanSingleton R X x).comp
(Shrink.linearEquiv R R : Shrink R →ₗ[R] R))) 1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.AB | {
"line": 43,
"column": 2
} | {
"line": 47,
"column": 51
} | {
"line": 49,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : Ring R\ninst✝ : Small.{v, u} R\nX Y : ModuleCat R\nf g : X ⟶ Y\nh : ∀ (G : ModuleCat R), ObjectProperty.singleton (of R (Shrink.{v, u} R)) G → ∀ (h : G ⟶ X), h ≫ f = h ≫ g\n⊢ f = g",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Algebra... | [] | simp only [ObjectProperty.singleton_iff, ModuleCat.hom_ext_iff, hom_comp,
LinearMap.ext_iff, LinearMap.coe_comp, Function.comp_apply, forall_eq'] at h
ext x
simpa using h (ModuleCat.ofHom ((LinearMap.toSpanSingleton R X x).comp
(Shrink.linearEquiv R R : Shrink R →ₗ[R] R))) 1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.ModuleCat.Adjunctions | {
"line": 172,
"column": 8
} | {
"line": 172,
"column": 13
} | {
"line": 173,
"column": 6
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nX Y : Type u\nf : X ⟶ Y\nX' : Type u\n⊢ (μIso R X X').inv ≫ (free R).map f ▷ (free R).obj X' ≫ (μIso R Y X').hom =\n (μIso R X X').inv ≫ (μIso R X X').hom ≫ (free R).map (f ▷ X')",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Category.ModuleCat.Adjunctions | {
"line": 175,
"column": 8
} | {
"line": 175,
"column": 13
} | {
"line": 176,
"column": 6
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nX Y X' : Type u\nf : X ⟶ Y\n⊢ (μIso R X' X).inv ≫ (free R).obj X' ◁ (free R).map f ≫ (μIso R X' Y).hom =\n (μIso R X' X).inv ≫ (μIso R X' X).hom ≫ (free R).map (X' ◁ f)",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Catego... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Category.ModuleCat.Adjunctions | {
"line": 188,
"column": 8
} | {
"line": 188,
"column": 13
} | {
"line": 189,
"column": 6
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nX : Type u\n⊢ 𝟙 ((free R).obj X) =\n (λ_ ((free R).obj X)).inv ≫ (εIso R).hom ▷ (free R).obj X ≫ (μIso R (𝟙_ (Type u)) X).hom ≫ (free R).map (λ_ X).hom",
"ppTerm": "?m.251",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ModuleCat.freeMk",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Category.ModuleCat.Adjunctions | {
"line": 191,
"column": 8
} | {
"line": 191,
"column": 13
} | {
"line": 191,
"column": 14
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nX : Type u\n⊢ 𝟙 ((free R).obj X) =\n (ρ_ ((free R).obj X)).inv ≫ (free R).obj X ◁ (εIso R).hom ≫ (μIso R X (𝟙_ (Type u))).hom ≫ (free R).map (ρ_ X).hom",
"ppTerm": "?m.275",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Monoi... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Category.ModuleCat.Adjunctions | {
"line": 337,
"column": 8
} | {
"line": 337,
"column": 45
} | {
"line": 338,
"column": 8
} | [
{
"pp": "case single.single\nR : Type u_1\ninst✝⁴ : CommRing R\nC : Type u\ninst✝³ : Category.{v, u} C\nD : Type u\ninst✝² : Category.{v, u} D\ninst✝¹ : Preadditive D\ninst✝ : Linear R D\nF : C ⥤ D\nX Y Z : Free R C\nf' : X ⟶ Y\nr : R\ng' : Y ⟶ Z\ns : R\n⊢ (sum (single f' r ≫ single g' s) fun f' r ↦ r • F.map f... | [
"case single.single\nR : Type u_1\ninst✝⁴ : CommRing R\nC : Type u\ninst✝³ : Category.{v, u} C\nD : Type u\ninst✝² : Category.{v, u} D\ninst✝¹ : Preadditive D\ninst✝ : Linear R D\nF : C ⥤ D\nX Y Z : Free R C\nf' : X ⟶ Y\nr : R\ng' : Y ⟶ Z\ns : R\n⊢ ((single (f' ≫ g') (r * s)).sum fun f' r ↦ r • F.map f') =\n ((s... | rw [single_comp_single _ _ f' g' r s] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Limits.Types.Pullbacks | {
"line": 55,
"column": 10
} | {
"line": 55,
"column": 15
} | {
"line": 55,
"column": 15
} | [
{
"pp": "X Y Z : Type u\nX' Y' Z' : Type v\nf✝ : X ⟶ Z\ng✝ : Y ⟶ Z\nf' : X' ⟶ Z'\ng' : Y' ⟶ Z'\nf : X ⟶ Z\ng : Y ⟶ Z\n⊢ ∀ (s : PullbackCone f g), (↾fun x ↦ ⟨((hom s.fst) x, (hom s.snd) x), ⋯⟩) ≫ (pullbackCone f g).fst = s.fst",
"ppTerm": "?m.143",
"assigned": true,
"usedConstants": [
"Eq.mpr",... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.Limits.Types.Pullbacks | {
"line": 55,
"column": 10
} | {
"line": 55,
"column": 15
} | {
"line": 55,
"column": 15
} | [
{
"pp": "X Y Z : Type u\nX' Y' Z' : Type v\nf✝ : X ⟶ Z\ng✝ : Y ⟶ Z\nf' : X' ⟶ Z'\ng' : Y' ⟶ Z'\nf : X ⟶ Z\ng : Y ⟶ Z\n⊢ ∀ (s : PullbackCone f g), (↾fun x ↦ ⟨((hom s.fst) x, (hom s.snd) x), ⋯⟩) ≫ (pullbackCone f g).fst = s.fst",
"ppTerm": "?m.143",
"assigned": true,
"usedConstants": [
"Eq.mpr",... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Types.Pullbacks | {
"line": 55,
"column": 10
} | {
"line": 55,
"column": 15
} | {
"line": 55,
"column": 15
} | [
{
"pp": "X Y Z : Type u\nX' Y' Z' : Type v\nf✝ : X ⟶ Z\ng✝ : Y ⟶ Z\nf' : X' ⟶ Z'\ng' : Y' ⟶ Z'\nf : X ⟶ Z\ng : Y ⟶ Z\n⊢ ∀ (s : PullbackCone f g), (↾fun x ↦ ⟨((hom s.fst) x, (hom s.snd) x), ⋯⟩) ≫ (pullbackCone f g).fst = s.fst",
"ppTerm": "?m.143",
"assigned": true,
"usedConstants": [
"Eq.mpr",... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Types.Pullbacks | {
"line": 55,
"column": 21
} | {
"line": 55,
"column": 26
} | {
"line": 55,
"column": 26
} | [
{
"pp": "X Y Z : Type u\nX' Y' Z' : Type v\nf✝ : X ⟶ Z\ng✝ : Y ⟶ Z\nf' : X' ⟶ Z'\ng' : Y' ⟶ Z'\nf : X ⟶ Z\ng : Y ⟶ Z\n⊢ ∀ (s : PullbackCone f g), (↾fun x ↦ ⟨((hom s.fst) x, (hom s.snd) x), ⋯⟩) ≫ (pullbackCone f g).snd = s.snd",
"ppTerm": "?m.144",
"assigned": true,
"usedConstants": [
"Eq.mpr",... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.Limits.Types.Pullbacks | {
"line": 55,
"column": 21
} | {
"line": 55,
"column": 26
} | {
"line": 55,
"column": 26
} | [
{
"pp": "X Y Z : Type u\nX' Y' Z' : Type v\nf✝ : X ⟶ Z\ng✝ : Y ⟶ Z\nf' : X' ⟶ Z'\ng' : Y' ⟶ Z'\nf : X ⟶ Z\ng : Y ⟶ Z\n⊢ ∀ (s : PullbackCone f g), (↾fun x ↦ ⟨((hom s.fst) x, (hom s.snd) x), ⋯⟩) ≫ (pullbackCone f g).snd = s.snd",
"ppTerm": "?m.144",
"assigned": true,
"usedConstants": [
"Eq.mpr",... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Types.Pullbacks | {
"line": 55,
"column": 21
} | {
"line": 55,
"column": 26
} | {
"line": 55,
"column": 26
} | [
{
"pp": "X Y Z : Type u\nX' Y' Z' : Type v\nf✝ : X ⟶ Z\ng✝ : Y ⟶ Z\nf' : X' ⟶ Z'\ng' : Y' ⟶ Z'\nf : X ⟶ Z\ng : Y ⟶ Z\n⊢ ∀ (s : PullbackCone f g), (↾fun x ↦ ⟨((hom s.fst) x, (hom s.snd) x), ⋯⟩) ≫ (pullbackCone f g).snd = s.snd",
"ppTerm": "?m.144",
"assigned": true,
"usedConstants": [
"Eq.mpr",... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.SnakeLemma | {
"line": 357,
"column": 6
} | {
"line": 357,
"column": 25
} | {
"line": 357,
"column": 26
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nS : SnakeInput C\n⊢ S.op.δ.unop = S.δ.op.unop",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"Eq.mpr",
"CategoryTheory.ShortComplex.SnakeInput.L₃",
"Op... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nS : SnakeInput C\n⊢ S.op.δ.unop = S.δ"
] | Quiver.Hom.unop_op, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.ShortComplex.ConcreteCategory | {
"line": 120,
"column": 61
} | {
"line": 126,
"column": 85
} | {
"line": 128,
"column": 0
} | [
{
"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\nS : ShortCo... | [] | by
dsimp [cyclesMk]
-- `abCyclesIso_inv_apply_iCycles` is not in `simp`-normal form, so we first
-- have to simplify it.
have := abCyclesIso_inv_apply_iCycles (S.map (forget₂ C Ab)) ⟨x₂, hx₂⟩
simp only [map_X₂, map_X₃, map_g] at this
rw [← ConcreteCategory.comp_apply, S.mapCyclesIso_hom_iCycles (forget₂ C A... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer | {
"line": 186,
"column": 27
} | {
"line": 186,
"column": 32
} | {
"line": 186,
"column": 32
} | [
{
"pp": "case left\nJ : MultispanShape\ninst✝¹ : Small.{t, w} J.L\ninst✝ : Small.{t, w'} J.R\na✝ : J.L\n⊢ ∃ a, Sum.elim left right a = left a✝",
"ppTerm": "?left",
"assigned": true,
"usedConstants": [
"False",
"CategoryTheory.Limits.WalkingMultispan.left.injEq",
"CategoryTheory.Lim... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer | {
"line": 186,
"column": 27
} | {
"line": 186,
"column": 32
} | {
"line": 186,
"column": 32
} | [
{
"pp": "case right\nJ : MultispanShape\ninst✝¹ : Small.{t, w} J.L\ninst✝ : Small.{t, w'} J.R\na✝ : J.R\n⊢ ∃ a, Sum.elim left right a = right a✝",
"ppTerm": "?right",
"assigned": true,
"usedConstants": [
"False",
"CategoryTheory.Limits.WalkingMultispan.right.injEq",
"CategoryTheory... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Homology.ShortComplex.ModuleCat | {
"line": 55,
"column": 2
} | {
"line": 55,
"column": 7
} | {
"line": 57,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nS : ShortComplex (ModuleCat R)\n⊢ (∀ (x₂ : ↑S.X₂), (ConcreteCategory.hom S.g) x₂ = 0 → ∃ x₁, (ConcreteCategory.hom S.f) x₁ = x₂) ↔\n (ModuleCat.Hom.hom S.g).ker ≤ (ModuleCat.Hom.hom S.f).range",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Su... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Homology.ShortComplex.ModuleCat | {
"line": 60,
"column": 2
} | {
"line": 60,
"column": 7
} | {
"line": 62,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nS : ShortComplex (ModuleCat R)\n⊢ (ModuleCat.Hom.hom S.g).ker ≤ (ModuleCat.Hom.hom S.f).range ↔\n (ModuleCat.Hom.hom S.f).range = (ModuleCat.Hom.hom S.g).ker",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"Ri... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Homology.ShortComplex.ModuleCat | {
"line": 88,
"column": 26
} | {
"line": 88,
"column": 31
} | {
"line": 88,
"column": 31
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nS : ShortComplex (ModuleCat R)\nX₁ X₂ X₃ : ModuleCat R\nf : X₁ ⟶ X₂\ng : X₂ ⟶ X₃\nhfg : (ModuleCat.Hom.hom f).range ≤ (ModuleCat.Hom.hom g).ker\n⊢ f ≫ g = 0",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Submodule",
"RingHomSurjective.ids"... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Homology.ShortComplex.ModuleCat | {
"line": 88,
"column": 26
} | {
"line": 88,
"column": 31
} | {
"line": 88,
"column": 31
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nS : ShortComplex (ModuleCat R)\nX₁ X₂ X₃ : ModuleCat R\nf : X₁ ⟶ X₂\ng : X₂ ⟶ X₃\nhfg : (ModuleCat.Hom.hom f).range ≤ (ModuleCat.Hom.hom g).ker\n⊢ f ≫ g = 0",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Submodule",
"RingHomSurjective.ids"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.ModuleCat | {
"line": 88,
"column": 26
} | {
"line": 88,
"column": 31
} | {
"line": 88,
"column": 31
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nS : ShortComplex (ModuleCat R)\nX₁ X₂ X₃ : ModuleCat R\nf : X₁ ⟶ X₂\ng : X₂ ⟶ X₃\nhfg : (ModuleCat.Hom.hom f).range ≤ (ModuleCat.Hom.hom g).ker\n⊢ f ≫ g = 0",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Submodule",
"RingHomSurjective.ids"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.ModuleCat | {
"line": 110,
"column": 11
} | {
"line": 110,
"column": 16
} | {
"line": 111,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nS : ShortComplex (ModuleCat R)\n⊢ ModuleCat.ofHom (ModuleCat.Hom.hom S.g).ker.subtype ≫ S.g = 0",
"ppTerm": "?m.148",
"assigned": true,
"usedConstants": [
"Submodule",
"LinearMap.comp.congr_simp",
"CategoryTheory.CategoryStruct.toQuiver",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Homology.ShortComplex.ModuleCat | {
"line": 110,
"column": 11
} | {
"line": 110,
"column": 16
} | {
"line": 111,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nS : ShortComplex (ModuleCat R)\n⊢ ModuleCat.ofHom (ModuleCat.Hom.hom S.g).ker.subtype ≫ S.g = 0",
"ppTerm": "?m.148",
"assigned": true,
"usedConstants": [
"Submodule",
"LinearMap.comp.congr_simp",
"CategoryTheory.CategoryStruct.toQuiver",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.ModuleCat | {
"line": 110,
"column": 11
} | {
"line": 110,
"column": 16
} | {
"line": 111,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nS : ShortComplex (ModuleCat R)\n⊢ ModuleCat.ofHom (ModuleCat.Hom.hom S.g).ker.subtype ≫ S.g = 0",
"ppTerm": "?m.148",
"assigned": true,
"usedConstants": [
"Submodule",
"LinearMap.comp.congr_simp",
"CategoryTheory.CategoryStruct.toQuiver",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.ModuleCat | {
"line": 112,
"column": 11
} | {
"line": 112,
"column": 16
} | {
"line": 113,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nS : ShortComplex (ModuleCat R)\n⊢ (ModuleCat.kernelIsLimit S.g).lift (KernelFork.ofι S.f ⋯) ≫ ModuleCat.ofHom S.moduleCatToCycles.range.mkQ = 0",
"ppTerm": "?m.149",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"RingHomSurjective.... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Homology.ShortComplex.ModuleCat | {
"line": 112,
"column": 11
} | {
"line": 112,
"column": 16
} | {
"line": 113,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nS : ShortComplex (ModuleCat R)\n⊢ (ModuleCat.kernelIsLimit S.g).lift (KernelFork.ofι S.f ⋯) ≫ ModuleCat.ofHom S.moduleCatToCycles.range.mkQ = 0",
"ppTerm": "?m.149",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"RingHomSurjective.... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.ModuleCat | {
"line": 112,
"column": 11
} | {
"line": 112,
"column": 16
} | {
"line": 113,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nS : ShortComplex (ModuleCat R)\n⊢ (ModuleCat.kernelIsLimit S.g).lift (KernelFork.ofι S.f ⋯) ≫ ModuleCat.ofHom S.moduleCatToCycles.range.mkQ = 0",
"ppTerm": "?m.149",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"RingHomSurjective.... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monad.Adjunction | {
"line": 170,
"column": 8
} | {
"line": 170,
"column": 67
} | {
"line": 170,
"column": 68
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\nh : L ⊣ R\nX✝ Y✝ : D\nf : X✝ ⟶ Y✝\n⊢ R.map (L.map (R.map f)) ≫ R.map (h.counit.app Y✝) = R.map (h.counit.app X✝) ≫ R.map f",
"ppTerm": "?m.105",
"assigned": true,
"usedConstants": [
... | [] | rw [← R.map_comp, Adjunction.counit_naturality, R.map_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.ComposableArrows.Basic | {
"line": 256,
"column": 10
} | {
"line": 256,
"column": 17
} | {
"line": 256,
"column": 17
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nn m : ℕ\nF✝ G✝ : ComposableArrows C n\nF G : ComposableArrows C 1\nleft : F.obj' 0 homMk₁._proof_4 ⟶ G.obj' 0 homMk₁._proof_4\nright : F.obj' 1 homMk₁._proof_5 ⟶ G.obj' 1 homMk₁._proof_5\nw : F.map' 0 1 homMk₁._proof_4 homMk₁._proof_5 ≫ right = left ≫ G.map'... | [] | exact w | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Monad.Comonadicity | {
"line": 112,
"column": 23
} | {
"line": 112,
"column": 28
} | {
"line": 113,
"column": 6
} | [
{
"pp": "C : Type u₁\nD : Type u₂\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Category.{v₁, u₂} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nA : adj.toComonad.Coalgebra\nB : C\ninst✝ : HasEqualizer (G.map A.a) (adj.unit.app (G.obj A.A))\nf : (comparison adj).obj B ⟶ A\n⊢ (fun f ↦ { f := (adj.homEquiv B A.A).symm (f ≫ equa... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.Monad.Comonadicity | {
"line": 112,
"column": 23
} | {
"line": 112,
"column": 28
} | {
"line": 113,
"column": 6
} | [
{
"pp": "C : Type u₁\nD : Type u₂\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Category.{v₁, u₂} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nA : adj.toComonad.Coalgebra\nB : C\ninst✝ : HasEqualizer (G.map A.a) (adj.unit.app (G.obj A.A))\nf : (comparison adj).obj B ⟶ A\n⊢ (fun f ↦ { f := (adj.homEquiv B A.A).symm (f ≫ equa... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monad.Comonadicity | {
"line": 112,
"column": 23
} | {
"line": 112,
"column": 28
} | {
"line": 113,
"column": 6
} | [
{
"pp": "C : Type u₁\nD : Type u₂\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Category.{v₁, u₂} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nA : adj.toComonad.Coalgebra\nB : C\ninst✝ : HasEqualizer (G.map A.a) (adj.unit.app (G.obj A.A))\nf : (comparison adj).obj B ⟶ A\n⊢ (fun f ↦ { f := (adj.homEquiv B A.A).symm (f ≫ equa... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.ComposableArrows.Basic | {
"line": 543,
"column": 8
} | {
"line": 543,
"column": 15
} | {
"line": 544,
"column": 6
} | [
{
"pp": "case zero\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nn m : ℕ\nF✝ G✝ : ComposableArrows C n\nF G : ComposableArrows C (n + 1)\nα : F.obj' 0 ⋯ ⟶ G.obj' 0 ⋯\nβ : F.δ₀ ⟶ G.δ₀\nw : F.map' 0 1 homMk₁._proof_4 ⋯ ≫ app' β 0 ⋯ = α ≫ G.map' 0 1 homMk₁._proof_4 ⋯\nhi : 0 < n + 1\n⊢ (F.map' 0 (0 + 1) ⋯ hi ≫\n ... | [] | exact w | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.ComposableArrows.Basic | {
"line": 543,
"column": 8
} | {
"line": 543,
"column": 15
} | {
"line": 544,
"column": 6
} | [
{
"pp": "case zero\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nn m : ℕ\nF✝ G✝ : ComposableArrows C n\nF G : ComposableArrows C (n + 1)\nα : F.obj' 0 ⋯ ⟶ G.obj' 0 ⋯\nβ : F.δ₀ ⟶ G.δ₀\nw : F.map' 0 1 homMk₁._proof_4 ⋯ ≫ app' β 0 ⋯ = α ≫ G.map' 0 1 homMk₁._proof_4 ⋯\nhi : 0 < n + 1\n⊢ (F.map' 0 (0 + 1) ⋯ hi ≫\n ... | [] | exact w | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.ComposableArrows.Basic | {
"line": 543,
"column": 8
} | {
"line": 543,
"column": 15
} | {
"line": 544,
"column": 6
} | [
{
"pp": "case zero\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nn m : ℕ\nF✝ G✝ : ComposableArrows C n\nF G : ComposableArrows C (n + 1)\nα : F.obj' 0 ⋯ ⟶ G.obj' 0 ⋯\nβ : F.δ₀ ⟶ G.δ₀\nw : F.map' 0 1 homMk₁._proof_4 ⋯ ≫ app' β 0 ⋯ = α ≫ G.map' 0 1 homMk₁._proof_4 ⋯\nhi : 0 < n + 1\n⊢ (F.map' 0 (0 + 1) ⋯ hi ≫\n ... | [] | exact w | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Pointwise | {
"line": 160,
"column": 46
} | {
"line": 163,
"column": 67
} | {
"line": 165,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝² : Group M\nR : Type u_4\ninst✝¹ : Ring R\nP : Ideal R\ninst✝ : MulSemiringAction M R\n⊢ inertia M P ≤ MulAction.stabilizer M P",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Submodule.ins... | [] | by
refine fun σ hσ ↦ SetLike.ext fun x ↦ ?_
rw [Ideal.mem_pointwise_smul_iff_inv_smul_mem,
← P.add_mem_iff_left (a := x) ((inv_mem hσ) x), add_sub_cancel] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.LocalizedModule.Submodule | {
"line": 52,
"column": 46
} | {
"line": 52,
"column": 73
} | {
"line": 52,
"column": 74
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : AddCommMonoid N\ninst✝⁶ : Module R M\ninst✝⁵ : Module R N\ninst✝⁴ : Algebra R S\ninst✝³ : Module S N\ninst✝² : IsScalarTower R S N\np : Submonoid R\ninst✝¹ : IsL... | [
"R : Type u_1\nS : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : AddCommMonoid N\ninst✝⁶ : Module R M\ninst✝⁵ : Module R N\ninst✝⁴ : Algebra R S\ninst✝³ : Module S N\ninst✝² : IsScalarTower R S N\np : Submonoid R\ninst✝¹ : IsLocalization ... | IsLocalizedModule.mk'_smul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.LocalizedModule.Submodule | {
"line": 76,
"column": 40
} | {
"line": 76,
"column": 91
} | {
"line": 76,
"column": 91
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : AddCommMonoid N\ninst✝⁶ : Module R M\ninst✝⁵ : Module R N\ninst✝⁴ : Algebra R S\ninst✝³ : Module S N\ninst✝² : IsScalarTower R S N\np : Submonoid R\ninst✝¹ : IsL... | [] | by rintro _ ⟨m, hm, rfl⟩; exact ⟨m, hm, 1, by simp⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Ideal.Colon | {
"line": 147,
"column": 11
} | {
"line": 147,
"column": 16
} | {
"line": 148,
"column": 2
} | [
{
"pp": "case add\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\nS : Set M\nr : R\nh : r ∈ N.colon S\ns x✝ y✝ : M\nhx✝ : x✝ ∈ span R S\nhy✝ : y✝ ∈ span R S\na✝¹ : r • x✝ ∈ N\na✝ : r • y✝ ∈ N\n⊢ r • (x✝ + y✝) ∈ N",
"ppTerm": "?add",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Ideal.Colon | {
"line": 147,
"column": 11
} | {
"line": 147,
"column": 16
} | {
"line": 148,
"column": 2
} | [
{
"pp": "case add\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\nS : Set M\nr : R\nh : r ∈ N.colon S\ns x✝ y✝ : M\nhx✝ : x✝ ∈ span R S\nhy✝ : y✝ ∈ span R S\na✝¹ : r • x✝ ∈ N\na✝ : r • y✝ ∈ N\n⊢ r • (x✝ + y✝) ∈ N",
"ppTerm": "?add",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Colon | {
"line": 147,
"column": 11
} | {
"line": 147,
"column": 16
} | {
"line": 148,
"column": 2
} | [
{
"pp": "case add\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\nS : Set M\nr : R\nh : r ∈ N.colon S\ns x✝ y✝ : M\nhx✝ : x✝ ∈ span R S\nhy✝ : y✝ ∈ span R S\na✝¹ : r • x✝ ∈ N\na✝ : r • y✝ ∈ N\n⊢ r • (x✝ + y✝) ∈ N",
"ppTerm": "?add",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.LocalizedModule.Submodule | {
"line": 198,
"column": 2
} | {
"line": 201,
"column": 89
} | {
"line": 202,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u_1\nS : Type u_2\nN : Type u_4\ninst✝⁷ : CommSemiring R\ninst✝⁶ : CommSemiring S\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : Module R N\ninst✝³ : Algebra R S\ninst✝² : Module S N\ninst✝¹ : IsScalarTower R S N\np : Submonoid R\ninst✝ : IsLocalization p S\nI : Submodule R R\nN' : Submodu... | [
"case refine_2\nR : Type u_1\nS : Type u_2\nN : Type u_4\ninst✝⁷ : CommSemiring R\ninst✝⁶ : CommSemiring S\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : Module R N\ninst✝³ : Algebra R S\ninst✝² : Module S N\ninst✝¹ : IsScalarTower R S N\np : Submonoid R\ninst✝ : IsLocalization p S\nI : Submodule R R\nN' : Submodule S N\nr : ... | · refine smul_induction_on ((Submodule.restrictScalars_mem _ _ _).mp hx) ?_ fun _ _ ↦ add_mem
rintro _ ⟨r, hr, s, rfl⟩ n hn
rw [← IsLocalization.mk'_eq_mk', IsLocalization.mk'_eq_mul_mk'_one, mul_smul, algebraMap_smul]
exact smul_mem_smul hr ((Submodule.restrictScalars_mem _ _ _).mpr <| smul_mem _ _ hn) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Localization.LocalizationLocalization | {
"line": 97,
"column": 2
} | {
"line": 103,
"column": 73
} | {
"line": 104,
"column": 2
} | [
{
"pp": "R : 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\... | [
"R : 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\nx y : R\nz ... | suffices (algebraMap R S) (x * z' : R) = (algebraMap R S) (y * z') by
obtain ⟨c, eq₃ : ↑c * (x * z') = ↑c * (y * z')⟩ := (IsLocalization.eq_iff_exists M S).mp this
refine ⟨⟨c * z', ?_⟩, ?_⟩
· rw [mem_localizationLocalizationSubmodule]
refine ⟨z, c * s, ?_⟩
rw [map_mul, ← eq₂, Submonoid.coe_mul, ... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.RingTheory.Localization.LocalizationLocalization | {
"line": 233,
"column": 4
} | {
"line": 236,
"column": 36
} | {
"line": 237,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommSemiring R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nM N : Submonoid R\ninst✝ : IsLocalization M S\nh : M ≤ N\nh' : ∀ (x : ↥N), ∃ m, m * ↑x ∈ M\ny : ↥N\n⊢ IsUnit ((algebraMap R S) ↑y)",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
... | [] | obtain ⟨m, hm⟩ := h' y
have := IsLocalization.map_units S ⟨_, hm⟩
rw [map_mul] at this
exact (IsUnit.mul_iff.mp this).2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Localization.LocalizationLocalization | {
"line": 233,
"column": 4
} | {
"line": 236,
"column": 36
} | {
"line": 237,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommSemiring R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nM N : Submonoid R\ninst✝ : IsLocalization M S\nh : M ≤ N\nh' : ∀ (x : ↥N), ∃ m, m * ↑x ∈ M\ny : ↥N\n⊢ IsUnit ((algebraMap R S) ↑y)",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
... | [] | obtain ⟨m, hm⟩ := h' y
have := IsLocalization.map_units S ⟨_, hm⟩
rw [map_mul] at this
exact (IsUnit.mul_iff.mp this).2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Localization.LocalizationLocalization | {
"line": 287,
"column": 6
} | {
"line": 287,
"column": 33
} | {
"line": 287,
"column": 34
} | [
{
"pp": "R : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\ninst✝⁸ : IsDomain R\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsFractionRing R T\nthis✝ :... | [
"R : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\ninst✝⁸ : IsDomain R\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsFractionRing R T\nthis✝ : Nontrivial ... | ← (algebraMap R S).map_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
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