module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.Flat.FaithfullyFlat.Basic | {
"line": 549,
"column": 94
} | {
"line": 549,
"column": 99
} | {
"line": 549,
"column": 99
} | [
{
"pp": "R : Type u_1\ninst✝⁸ : CommRing R\nS : Type u_2\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module S M\ninst✝² : IsScalarTower R S M\ninst✝¹ : FaithfullyFlat R S\ninst✝ : FaithfullyFlat S M\nN : Type (max u_1 u_3)\nx✝³ : AddCommGroup ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Spectrum.Prime.RingHom | {
"line": 105,
"column": 8
} | {
"line": 105,
"column": 27
} | {
"line": 105,
"column": 28
} | [
{
"pp": "R : Type u\nS : Type v\nS' : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : CommSemiring S'\nf : R →+* S\ne : R ≃+* S\nx : PrimeSpectrum R\n⊢ comap e.toRingHom (comap e.symm.toRingHom x) = x",
"ppTerm": "?m.67",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"R : Type u\nS : Type v\nS' : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : CommSemiring S'\nf : R →+* S\ne : R ≃+* S\nx : PrimeSpectrum R\n⊢ comap (e.symm.toRingHom.comp e.toRingHom) x = x"
] | ← comap_comp_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Spectrum.Prime.RingHom | {
"line": 109,
"column": 8
} | {
"line": 109,
"column": 27
} | {
"line": 109,
"column": 28
} | [
{
"pp": "R : Type u\nS : Type v\nS' : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : CommSemiring S'\nf : R →+* S\ne : R ≃+* S\nx : PrimeSpectrum S\n⊢ comap e.symm.toRingHom (comap e.toRingHom x) = x",
"ppTerm": "?m.96",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"R : Type u\nS : Type v\nS' : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : CommSemiring S'\nf : R →+* S\ne : R ≃+* S\nx : PrimeSpectrum S\n⊢ comap (e.toRingHom.comp e.symm.toRingHom) x = x"
] | ← comap_comp_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Sets.Opens | {
"line": 338,
"column": 20
} | {
"line": 338,
"column": 23
} | {
"line": 339,
"column": 6
} | [
{
"pp": "case mpr.refine_2\nα : Type u_2\ninst✝ : TopologicalSpace α\nB : Set (Opens α)\nh : ∀ {U : Opens α} {x : α}, x ∈ U → ∃ U' ∈ B, x ∈ U' ∧ U' ≤ U\nx : α\nsU : Set α\nhx : x ∈ sU\n⊢ IsOpen[inst✝] sU → ∃ v ∈ SetLike.coe '' B, x ∈ v ∧ v ⊆ sU",
"ppTerm": "?mpr.refine_2",
"assigned": true,
"usedCon... | [
"case mpr.refine_2\nα : Type u_2\ninst✝ : TopologicalSpace α\nB : Set (Opens α)\nh : ∀ {U : Opens α} {x : α}, x ∈ U → ∃ U' ∈ B, x ∈ U' ∧ U' ≤ U\nx : α\nsU : Set α\nhx : x ∈ sU\nhsU : IsOpen[inst✝] sU\n⊢ ∃ v ∈ SetLike.coe '' B, x ∈ v ∧ v ⊆ sU"
] | hsU | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.Sets.Opens | {
"line": 387,
"column": 2
} | {
"line": 387,
"column": 13
} | {
"line": 389,
"column": 0
} | [
{
"pp": "case hb'\nα : Type u_2\ninst✝ : TopologicalSpace α\nι : Type u_5\nb : ι → Opens α\nhb : IsBasis (range b)\nhb' : ∀ (i : ι), IsCompact ↑(b i)\nU : Set α\n⊢ ∀ (i : ι), IsCompact (b i).carrier",
"ppTerm": "?hb'",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"hb'"
],
... | [] | · exact hb' | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.LocalAtTarget | {
"line": 192,
"column": 11
} | {
"line": 192,
"column": 40
} | {
"line": 192,
"column": 41
} | [
{
"pp": "α : 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⊢ DenseRange f ↔ ∀ (i : ι), DenseRange ((U i).carrier.restrictPreimage f)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Set.restrictPreimage",
"Eq.m... | [
"α : 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⊢ closure[inst✝] (range f) = univ ↔\n ∀ (i : ι), closure[instTopologicalSpaceSubtype] (range ((U i).carrier.restrictPreimage f)) = univ"
] | denseRange_iff_closure_range, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Sets.Opens | {
"line": 395,
"column": 2
} | {
"line": 396,
"column": 47
} | {
"line": 398,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝ : TopologicalSpace α\nB : Set (Opens α)\nhB : IsBasis B\nU : Opens α\nhU : IsCompact U.carrier\nUs' : Set (Opens α)\nhsub : Us' ⊆ B\nhsup : U = sSup Us'\nt : Finset ↑Us'\nht : U.carrier ⊆ ⋃ i ∈ t, ↑↑i\n⊢ U = sSup ↑(Finset.image Subtype.val t)",
"ppTerm": "?m.84",
"assigned":... | [] | exact le_antisymm (subset_trans (a := U.carrier) ht (by simp))
(le_trans (sSup_le_sSup (by simp)) hsup.ge) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.BooleanSubalgebra | {
"line": 227,
"column": 21
} | {
"line": 227,
"column": 26
} | {
"line": 228,
"column": 2
} | [
{
"pp": "ι : Sort u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : BooleanAlgebra α\ninst✝¹ : BooleanAlgebra β\ninst✝ : BooleanAlgebra γ\nL M : BooleanSubalgebra α\nf : BoundedLatticeHom α β\ns t : Set α\na b : α\n_S : BooleanSubalgebra α\n_a : α\n⊢ _a ∈ ⊥ → _a ∈ _S",
"ppTerm": "?m.83",
"assigned... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Order.BooleanSubalgebra | {
"line": 227,
"column": 21
} | {
"line": 227,
"column": 26
} | {
"line": 228,
"column": 2
} | [
{
"pp": "ι : Sort u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : BooleanAlgebra α\ninst✝¹ : BooleanAlgebra β\ninst✝ : BooleanAlgebra γ\nL M : BooleanSubalgebra α\nf : BoundedLatticeHom α β\ns t : Set α\na b : α\n_S : BooleanSubalgebra α\n_a : α\n⊢ _a ∈ ⊥ → _a ∈ _S",
"ppTerm": "?m.83",
"assigned... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.BooleanSubalgebra | {
"line": 227,
"column": 21
} | {
"line": 227,
"column": 26
} | {
"line": 228,
"column": 2
} | [
{
"pp": "ι : Sort u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : BooleanAlgebra α\ninst✝¹ : BooleanAlgebra β\ninst✝ : BooleanAlgebra γ\nL M : BooleanSubalgebra α\nf : BoundedLatticeHom α β\ns t : Set α\na b : α\n_S : BooleanSubalgebra α\n_a : α\n⊢ _a ∈ ⊥ → _a ∈ _S",
"ppTerm": "?m.83",
"assigned... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.LocalAtTarget | {
"line": 262,
"column": 6
} | {
"line": 262,
"column": 15
} | {
"line": 262,
"column": 15
} | [
{
"pp": "case inr.refine_2\nX : Type u_6\nY : Type u_7\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nf : X → Y\nh : Continuous[inst✝², inst✝¹] f\nι : Type u_4\nU : ι → Opens Y\nhU : range f ⊆ ↑(iSup U)\nV : ι → Type u_5\ninst✝ : (i : ι) → TopologicalSpace (V i)\niV : (i : ι) → V i → X\nhiV : ∀ (i :... | [
"case inr.refine_2\nX : Type u_6\nY : Type u_7\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nf : X → Y\nh : Continuous[inst✝², inst✝¹] f\nι : Type u_4\nU : ι → Opens Y\nhU : range f ⊆ ↑(iSup U)\nV : ι → Type u_5\ninst✝ : (i : ι) → TopologicalSpace (V i)\niV : (i : ι) → V i → X\nhiV : ∀ (i : ι), Continu... | rw [hU''] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Spectral.Prespectral | {
"line": 114,
"column": 49
} | {
"line": 114,
"column": 54
} | {
"line": 114,
"column": 54
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : PrespectralSpace X\nU : Opens X\nU₁ : CompactOpens X\nh₁ : U₁ ∈ { carrier := {V | ↑V ⊆ ↑U}, lower' := ⋯ }.carrier\nU₂ : CompactOpens X\nh₂ : U₂ ∈ { carrier := {V | ↑V ⊆ ↑U}, lower' := ⋯ }.carrier\n⊢ U₁ ⊔ U₂ ∈ ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Spectral.Prespectral | {
"line": 114,
"column": 49
} | {
"line": 114,
"column": 54
} | {
"line": 114,
"column": 54
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : PrespectralSpace X\nU : Opens X\nU₁ : CompactOpens X\nh₁ : U₁ ∈ { carrier := {V | ↑V ⊆ ↑U}, lower' := ⋯ }.carrier\nU₂ : CompactOpens X\nh₂ : U₂ ∈ { carrier := {V | ↑V ⊆ ↑U}, lower' := ⋯ }.carrier\n⊢ U₁ ⊔ U₂ ∈ ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Spectral.Prespectral | {
"line": 114,
"column": 49
} | {
"line": 114,
"column": 54
} | {
"line": 114,
"column": 54
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : PrespectralSpace X\nU : Opens X\nU₁ : CompactOpens X\nh₁ : U₁ ∈ { carrier := {V | ↑V ⊆ ↑U}, lower' := ⋯ }.carrier\nU₂ : CompactOpens X\nh₂ : U₂ ∈ { carrier := {V | ↑V ⊆ ↑U}, lower' := ⋯ }.carrier\n⊢ U₁ ⊔ U₂ ∈ ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Sober | {
"line": 100,
"column": 64
} | {
"line": 100,
"column": 72
} | {
"line": 100,
"column": 73
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\nx : α\nS : Set α\nhS : IsClosed S\nhxS : x ∈ S\nthis : closure {x} ⊆ S\n⊢ S ⊆ closure {x} ↔ ∀ (Z : Set α), IsClosed Z → x ∈ Z → S ⊆ Z",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"PartialOrder.toPreorder",
"Preorder.toLE",
... | [
"α : Type u_1\ninst✝ : TopologicalSpace α\nx : α\nS : Set α\nhS : IsClosed S\nhxS : x ∈ S\nthis : closure {x} ⊆ S\n⊢ S ⊆ ⋂₀ {t | IsClosed t ∧ {x} ⊆ t} ↔ ∀ (Z : Set α), IsClosed Z → x ∈ Z → S ⊆ Z"
] | closure, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Constructible | {
"line": 268,
"column": 4
} | {
"line": 269,
"column": 31
} | {
"line": 270,
"column": 2
} | [
{
"pp": "case open_retrocompact\nX : Type u_2\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\ns : Set X\nhfopen : IsOpenEmbedding f\nhfcomp : IsRetrocompact (range f)\nU : Set X\nhUopen : IsOpen[inst✝¹] U\nhUcomp : IsRetrocompact U\n⊢ IsConstructible (f '' U)",
"ppTerm": "... | [] | exact (hUcomp.image_of_isEmbedding hfopen.isEmbedding hfcomp).isConstructible <|
hfopen.isOpenMap _ hUopen | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Constructible | {
"line": 268,
"column": 4
} | {
"line": 269,
"column": 31
} | {
"line": 270,
"column": 2
} | [
{
"pp": "case open_retrocompact\nX : Type u_2\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\ns : Set X\nhfopen : IsOpenEmbedding f\nhfcomp : IsRetrocompact (range f)\nU : Set X\nhUopen : IsOpen[inst✝¹] U\nhUcomp : IsRetrocompact U\n⊢ IsConstructible (f '' U)",
"ppTerm": "... | [] | exact (hUcomp.image_of_isEmbedding hfopen.isEmbedding hfcomp).isConstructible <|
hfopen.isOpenMap _ hUopen | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Constructible | {
"line": 268,
"column": 4
} | {
"line": 269,
"column": 31
} | {
"line": 270,
"column": 2
} | [
{
"pp": "case open_retrocompact\nX : Type u_2\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\ns : Set X\nhfopen : IsOpenEmbedding f\nhfcomp : IsRetrocompact (range f)\nU : Set X\nhUopen : IsOpen[inst✝¹] U\nhUcomp : IsRetrocompact U\n⊢ IsConstructible (f '' U)",
"ppTerm": "... | [] | exact (hUcomp.image_of_isEmbedding hfopen.isEmbedding hfcomp).isConstructible <|
hfopen.isOpenMap _ hUopen | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.RingHom.Flat | {
"line": 238,
"column": 45
} | {
"line": 240,
"column": 44
} | {
"line": 242,
"column": 0
} | [
{
"pp": "⊢ flat.IsStableUnderCobaseChange",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"RingHom.isStableUnderCobaseChange_toMorphismProperty_iff",
"Eq.mpr",
"CategoryTheory.MorphismProperty",
"RingHom.Flat",
"CommRing",
"RingHom.Flat.isStableUnderBaseC... | [] | by
rw [flat, RingHom.isStableUnderCobaseChange_toMorphismProperty_iff]
exact RingHom.Flat.isStableUnderBaseChange | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 173,
"column": 6
} | {
"line": 173,
"column": 44
} | {
"line": 173,
"column": 45
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nx : PrimeSpectrum R\n⊢ closure {x} = zeroLocus ↑x.asIdeal",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"PrimeSpectrum.zeroLocus",
"PrimeSpectrum.vanishingIdeal",
"congrArg",
"... | [
"R : Type u\ninst✝ : CommSemiring R\nx : PrimeSpectrum R\n⊢ zeroLocus ↑(vanishingIdeal {x}) = zeroLocus ↑x.asIdeal"
] | ← zeroLocus_vanishingIdeal_eq_closure, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 177,
"column": 37
} | {
"line": 177,
"column": 75
} | {
"line": 178,
"column": 6
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nx : PrimeSpectrum R\n⊢ closure {x} ⊆ {x} ↔ x.asIdeal.IsMaximal",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"PrimeSpectrum.zeroLocus",
"PrimeSpectrum.vanishingIdeal",
"congrArg",
... | [
"R : Type u\ninst✝ : CommSemiring R\nx : PrimeSpectrum R\n⊢ zeroLocus ↑(vanishingIdeal {x}) ⊆ {x} ↔ x.asIdeal.IsMaximal"
] | ← zeroLocus_vanishingIdeal_eq_closure, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 185,
"column": 32
} | {
"line": 185,
"column": 70
} | {
"line": 186,
"column": 4
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\ns : Set (PrimeSpectrum R)\n⊢ (vanishingIdeal (closure s)).IsRadical",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"PrimeSpectrum.zeroLocus",
"PrimeSpectrum.vanishingIdeal",
"congrArg... | [
"R : Type u\ninst✝ : CommSemiring R\ns : Set (PrimeSpectrum R)\n⊢ (vanishingIdeal (zeroLocus ↑(vanishingIdeal s))).IsRadical"
] | ← zeroLocus_vanishingIdeal_eq_closure, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 215,
"column": 4
} | {
"line": 220,
"column": 20
} | {
"line": 221,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u\ninst✝¹ : CommSemiring R\ninst✝ : IsDomain R\nh : T1Space (PrimeSpectrum R)\n⊢ IsField R",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"PrimeSpectrum.mk",
"False",
"IsDomain.to_noZeroDivisors",
"Semiring.toModule",
"Cla... | [] | exact
Classical.not_not.1
(mt
(Ring.ne_bot_of_isMaximal_of_not_isField <|
(isClosed_singleton_iff_isMaximal _).1 (T1Space.t1 ⟨⊥, inferInstance⟩))
(by simp)) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 259,
"column": 35
} | {
"line": 259,
"column": 73
} | {
"line": 260,
"column": 4
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\ns : Set (PrimeSpectrum R)\n⊢ IsIrreducible (closure s) ↔ (vanishingIdeal s).IsPrime",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"PrimeSpectrum.zeroLocus",
"PrimeSpectrum.vanishingIdeal",... | [
"R : Type u\ninst✝ : CommSemiring R\ns : Set (PrimeSpectrum R)\n⊢ IsIrreducible (zeroLocus ↑(vanishingIdeal s)) ↔ (vanishingIdeal s).IsPrime"
] | ← zeroLocus_vanishingIdeal_eq_closure, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 472,
"column": 44
} | {
"line": 472,
"column": 76
} | {
"line": 472,
"column": 76
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\np✝ : PrimeSpectrum (R × S)\nh : p✝ ∈ zeroLocus ↑(RingHom.ker (RingHom.fst R S))\np : Ideal R\nhp : p.IsPrime\neq : p✝.asIdeal = p.prod ⊤\n⊢ (comap (RingHom.fst R S) { asIdeal := p, isPrime := hp }).asIdeal = p✝.asIdeal",
"ppTe... | [] | simpa [Ideal.prod] using eq.symm | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 472,
"column": 44
} | {
"line": 472,
"column": 76
} | {
"line": 472,
"column": 76
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\np✝ : PrimeSpectrum (R × S)\nh : p✝ ∈ zeroLocus ↑(RingHom.ker (RingHom.fst R S))\np : Ideal R\nhp : p.IsPrime\neq : p✝.asIdeal = p.prod ⊤\n⊢ (comap (RingHom.fst R S) { asIdeal := p, isPrime := hp }).asIdeal = p✝.asIdeal",
"ppTe... | [] | simpa [Ideal.prod] using eq.symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 472,
"column": 44
} | {
"line": 472,
"column": 76
} | {
"line": 472,
"column": 76
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\np✝ : PrimeSpectrum (R × S)\nh : p✝ ∈ zeroLocus ↑(RingHom.ker (RingHom.fst R S))\np : Ideal R\nhp : p.IsPrime\neq : p✝.asIdeal = p.prod ⊤\n⊢ (comap (RingHom.fst R S) { asIdeal := p, isPrime := hp }).asIdeal = p✝.asIdeal",
"ppTe... | [] | simpa [Ideal.prod] using eq.symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 468,
"column": 87
} | {
"line": 474,
"column": 76
} | {
"line": 476,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\n⊢ Set.range (comap (RingHom.fst R S)) = zeroLocus ↑(RingHom.ker (RingHom.fst R S))",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Set.instSProd",
"Set.ext",
"PrimeSpectrum... | [] | by
refine Set.ext fun p ↦ ⟨?_, fun h ↦ ?_⟩
· rintro ⟨I, hI, rfl⟩; exact Ideal.comap_mono bot_le
obtain ⟨p, hp, eq⟩ | ⟨p, hp, eq⟩ := p.1.ideal_prod_prime.mp p.2
· exact ⟨⟨p, hp⟩, PrimeSpectrum.ext <| by simpa [Ideal.prod] using eq.symm⟩
· refine (hp.ne_top <| (Ideal.eq_top_iff_one _).mpr ?_).elim
simpa [eq... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 483,
"column": 44
} | {
"line": 483,
"column": 76
} | {
"line": 483,
"column": 76
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\np✝ : PrimeSpectrum (R × S)\nh : p✝ ∈ zeroLocus ↑(RingHom.ker (RingHom.snd R S))\np : Ideal S\nhp : p.IsPrime\neq : p✝.asIdeal = ⊤.prod p\n⊢ (comap (RingHom.snd R S) { asIdeal := p, isPrime := hp }).asIdeal = p✝.asIdeal",
"ppTe... | [] | simpa [Ideal.prod] using eq.symm | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 483,
"column": 44
} | {
"line": 483,
"column": 76
} | {
"line": 483,
"column": 76
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\np✝ : PrimeSpectrum (R × S)\nh : p✝ ∈ zeroLocus ↑(RingHom.ker (RingHom.snd R S))\np : Ideal S\nhp : p.IsPrime\neq : p✝.asIdeal = ⊤.prod p\n⊢ (comap (RingHom.snd R S) { asIdeal := p, isPrime := hp }).asIdeal = p✝.asIdeal",
"ppTe... | [] | simpa [Ideal.prod] using eq.symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 483,
"column": 44
} | {
"line": 483,
"column": 76
} | {
"line": 483,
"column": 76
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\np✝ : PrimeSpectrum (R × S)\nh : p✝ ∈ zeroLocus ↑(RingHom.ker (RingHom.snd R S))\np : Ideal S\nhp : p.IsPrime\neq : p✝.asIdeal = ⊤.prod p\n⊢ (comap (RingHom.snd R S) { asIdeal := p, isPrime := hp }).asIdeal = p✝.asIdeal",
"ppTe... | [] | simpa [Ideal.prod] using eq.symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 585,
"column": 66
} | {
"line": 585,
"column": 71
} | {
"line": 587,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\ns : Set (PrimeSpectrum R)\nhs : IsOpen s\n⊢ ⋃₀ {s_1 | (s_1 ∈ Set.range fun r ↦ ↑(basicOpen r)) ∧ s_1 ⊆ s} = ⋃ r, ⋃ (_ : ↑(basicOpen r) ⊆ s), ↑(basicOpen r)",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"S... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 585,
"column": 66
} | {
"line": 585,
"column": 71
} | {
"line": 587,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\ns : Set (PrimeSpectrum R)\nhs : IsOpen s\n⊢ ⋃₀ {s_1 | (s_1 ∈ Set.range fun r ↦ ↑(basicOpen r)) ∧ s_1 ⊆ s} = ⋃ r, ⋃ (_ : ↑(basicOpen r) ⊆ s), ↑(basicOpen r)",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"S... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 585,
"column": 66
} | {
"line": 585,
"column": 71
} | {
"line": 587,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\ns : Set (PrimeSpectrum R)\nhs : IsOpen s\n⊢ ⋃₀ {s_1 | (s_1 ∈ Set.range fun r ↦ ↑(basicOpen r)) ∧ s_1 ⊆ s} = ⋃ r, ⋃ (_ : ↑(basicOpen r) ⊆ s), ↑(basicOpen r)",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"S... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Derivation.Basic | {
"line": 501,
"column": 11
} | {
"line": 501,
"column": 20
} | {
"line": 501,
"column": 21
} | [
{
"pp": "case neg.inr\nR : Type u_1\ninst✝⁵ : CommRing R\nM : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nK : Type u_4\ninst✝² : Field K\ninst✝¹ : Module K M\ninst✝ : Algebra R K\nD : Derivation R K M\na : K\nn : ℤ\nhn : ¬n = 0\nha : ¬a = 0\nh : n = -↑n.natAbs\n⊢ D (a ^ (-↑n.natAbs)) = -↑n.natAbs • ... | [
"case neg.inr\nR : Type u_1\ninst✝⁵ : CommRing R\nM : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nK : Type u_4\ninst✝² : Field K\ninst✝¹ : Module K M\ninst✝ : Algebra R K\nD : Derivation R K M\na : K\nn : ℤ\nhn : ¬n = 0\nha : ¬a = 0\nh : n = -↑n.natAbs\n⊢ D (a ^ ↑n.natAbs)⁻¹ = -↑n.natAbs • a ^ (-↑n.natA... | zpow_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Exponent | {
"line": 104,
"column": 2
} | {
"line": 104,
"column": 15
} | {
"line": 105,
"column": 2
} | [
{
"pp": "G : Type u\ninst✝ : Monoid G\n⊢ exponent G ≠ 0 ↔ ExponentExists G",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"Monoid.ExponentExists",
"Monoid.exponent.eq_1",
"Classical.prop... | [
"G : Type u\ninst✝ : Monoid G\n⊢ (if h : ExponentExists G then Nat.find h else 0) ≠ 0 ↔ ExponentExists G"
] | rw [exponent] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.Exponent | {
"line": 150,
"column": 2
} | {
"line": 153,
"column": 43
} | {
"line": 155,
"column": 0
} | [
{
"pp": "G : Type u\ninst✝ : Monoid G\ng : G\n⊢ g ^ exponent G = 1",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"Monoid.ExponentExists",
"Decidable",
"Classical.propDecidable",
"... | [] | by_cases h : ExponentExists G
· simp_rw [exponent, dif_pos h]
exact (Nat.find_spec h).2 g
· simp_rw [exponent, dif_neg h, pow_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Exponent | {
"line": 150,
"column": 2
} | {
"line": 153,
"column": 43
} | {
"line": 155,
"column": 0
} | [
{
"pp": "G : Type u\ninst✝ : Monoid G\ng : G\n⊢ g ^ exponent G = 1",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"Monoid.ExponentExists",
"Decidable",
"Classical.propDecidable",
"... | [] | by_cases h : ExponentExists G
· simp_rw [exponent, dif_pos h]
exact (Nat.find_spec h).2 g
· simp_rw [exponent, dif_neg h, pow_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Exponent | {
"line": 231,
"column": 57
} | {
"line": 234,
"column": 28
} | {
"line": 236,
"column": 0
} | [
{
"pp": "G : Type u\ninst✝¹ : Monoid G\ninst✝ : Fintype G\n⊢ Finset.univ.lcm orderOf ∣ exponent G",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Finset.univ",
"Finset",
"instNormalizedGCDMonoidOfStrongNormalizedGCDMonoid",
"Membership.mem",
"Monoid.order_dvd... | [] | by
apply Finset.lcm_dvd
intro g _
exact order_dvd_exponent g | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 834,
"column": 28
} | {
"line": 834,
"column": 66
} | {
"line": 834,
"column": 67
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nx y : PrimeSpectrum R\n⊢ x.asIdeal ≤ y.asIdeal ↔ y ∈ closure {x}",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"PrimeSpectrum.zeroLocus",
"PrimeSpectrum.vanishingIdeal",
"congrArg",
... | [
"R : Type u\ninst✝ : CommSemiring R\nx y : PrimeSpectrum R\n⊢ x.asIdeal ≤ y.asIdeal ↔ y ∈ zeroLocus ↑(vanishingIdeal {x})"
] | ← zeroLocus_vanishingIdeal_eq_closure, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.SpecificGroups.Cyclic.Basic | {
"line": 302,
"column": 16
} | {
"line": 302,
"column": 36
} | {
"line": 302,
"column": 36
} | [
{
"pp": "G : Type u_2\ninst✝ : Group G\ng : G\nn : ℕ\nh : zpowers (g ^ ↑n) ≤ zpowers g\n⊢ zpowers (g ^ ↑n) = zpowers (g ^ n)",
"ppTerm": "?m.114",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"congrArg",
"DivInvMonoid.toZPow",
"id",
"DivInvMonoid... | [] | by rw [zpow_natCast] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Exponent | {
"line": 305,
"column": 4
} | {
"line": 305,
"column": 39
} | {
"line": 306,
"column": 4
} | [
{
"pp": "case refine_1\nG : Type u_1\ninst✝¹ : Monoid G\ninst✝ : Nontrivial G\np : ℕ\nhp : Nat.Prime p\nhG : exponent G = p\ng : G\nhg : g ≠ 1\n⊢ orderOf g = p",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"E... | [
"case refine_1\nG : Type u_1\ninst✝¹ : Monoid G\ninst✝ : Nontrivial G\np : ℕ\nhp : Nat.Prime p\nhG : exponent G = p\ng : G\nhg : ¬orderOf g = 1\n⊢ orderOf g = p"
] | rw [Ne, ← orderOf_eq_one_iff] at hg | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 947,
"column": 6
} | {
"line": 947,
"column": 44
} | {
"line": 947,
"column": 45
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : R →+* S\n⊢ closure (Set.range (comap f)) = zeroLocus ↑(RingHom.ker f)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"RingHom.instRingHomClass",
"Semiring.toModule",
... | [
"R : Type u_1\nS : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : R →+* S\n⊢ zeroLocus ↑(vanishingIdeal (Set.range (comap f))) = zeroLocus ↑(RingHom.ker f)"
] | ← zeroLocus_vanishingIdeal_eq_closure, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 951,
"column": 6
} | {
"line": 951,
"column": 35
} | {
"line": 951,
"column": 36
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : R →+* S\n⊢ DenseRange (comap f) ↔ RingHom.ker f ≤ nilradical R",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"RingHom.instRingHomClass",
"Semiring.toModule",
"congr... | [
"R : Type u_1\nS : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : R →+* S\n⊢ closure (Set.range (comap f)) = Set.univ ↔ RingHom.ker f ≤ nilradical R"
] | denseRange_iff_closure_range, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Exponent | {
"line": 327,
"column": 10
} | {
"line": 327,
"column": 27
} | {
"line": 327,
"column": 28
} | [
{
"pp": "G : Type u\ninst✝ : Monoid G\nh : ∀ (g : G), 0 < orderOf g\nt : Finset ℕ\nht : ↑t = Set.range orderOf\nexponent_ne_zero_iff_range_orderOf_finite : (∀ (g : G), 0 < orderOf g) → (exponent G ≠ 0 ↔ (↑t).Finite)\na : ℕ\nha : a ∈ t\n⊢ 0 < id a",
"ppTerm": "?m.119",
"assigned": true,
"usedConstant... | [
"G : Type u\ninst✝ : Monoid G\nh : ∀ (g : G), 0 < orderOf g\nt : Finset ℕ\nht : ↑t = Set.range orderOf\nexponent_ne_zero_iff_range_orderOf_finite : (∀ (g : G), 0 < orderOf g) → (exponent G ≠ 0 ↔ (↑t).Finite)\na : ℕ\nha : a ∈ ↑t\n⊢ 0 < id a"
] | ← Finset.mem_coe, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Exponent | {
"line": 337,
"column": 8
} | {
"line": 337,
"column": 25
} | {
"line": 337,
"column": 26
} | [
{
"pp": "G : Type u\ninst✝ : Monoid G\nh : ∀ (g : G), 0 < orderOf g\nt : Finset ℕ\nht : ↑t = Set.range orderOf\nexponent_ne_zero_iff_range_orderOf_finite : (∀ (g : G), 0 < orderOf g) → (exponent G ≠ 0 ↔ (↑t).Finite)\nhtpos : 0 < t.prod id\ng : G\n⊢ orderOf g ∈ t",
"ppTerm": "?m.197",
"assigned": true,
... | [
"G : Type u\ninst✝ : Monoid G\nh : ∀ (g : G), 0 < orderOf g\nt : Finset ℕ\nht : ↑t = Set.range orderOf\nexponent_ne_zero_iff_range_orderOf_finite : (∀ (g : G), 0 < orderOf g) → (exponent G ≠ 0 ↔ (↑t).Finite)\nhtpos : 0 < t.prod id\ng : G\n⊢ orderOf g ∈ ↑t"
] | ← Finset.mem_coe, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.PGroup | {
"line": 213,
"column": 2
} | {
"line": 214,
"column": 96
} | {
"line": 215,
"column": 2
} | [
{
"pp": "p : ℕ\nG : Type u_1\ninst✝² : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\nα : Type u_2\ninst✝¹ : MulAction G α\ninst✝ : Finite α\nhpα : p ∣ Nat.card α\na : α\nha : a ∈ fixedPoints G α\n⊢ ∃ b ∈ fixedPoints G α, a ≠ b",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"D... | [
"p : ℕ\nG : Type u_1\ninst✝² : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\nα : Type u_2\ninst✝¹ : MulAction G α\ninst✝ : Finite α\nhpα : p ∣ Nat.card α\na : α\nha : a ∈ fixedPoints G α\nhpf : p ∣ Nat.card ↑(fixedPoints G α)\n⊢ ∃ b ∈ fixedPoints G α, a ≠ b"
] | have hpf : p ∣ Nat.card (fixedPoints G α) :=
Nat.modEq_zero_iff_dvd.mp ((hG.card_modEq_card_fixedPoints α).symm.trans hpα.modEq_zero_nat) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.GroupTheory.Exponent | {
"line": 624,
"column": 6
} | {
"line": 624,
"column": 19
} | {
"line": 624,
"column": 19
} | [
{
"pp": "G : Type u\ninst✝ : Monoid G\nhG : Monoid.exponent G = 2\nx : G\n⊢ orderOf x = 2 → x ≠ 1",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"False",
"Monoid.toMulOneClass",
"instOfNatNat",
"orderOf",
"MulOneClass.toMulOne",
"N... | [
"G : Type u\ninst✝ : Monoid G\nhG : Monoid.exponent G = 2\nhx : orderOf 1 = 2\n⊢ False"
] | rintro hx rfl | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.GroupTheory.Exponent | {
"line": 683,
"column": 2
} | {
"line": 683,
"column": 7
} | {
"line": 685,
"column": 0
} | [
{
"pp": "G : Type u\ninst✝ : Group G\nx y : G\nhx : orderOf x = 2\nhy : orderOf y = 2\nhxy : x ≠ y\n⊢ ¬y = 1 ∧ ¬x = 1 ∧ ¬x = y",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"False",
"eq_false",
"and_true",
"Monoid.toMulOneClas... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.GroupTheory.Rank | {
"line": 92,
"column": 2
} | {
"line": 101,
"column": 35
} | {
"line": 103,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\ns : Finset G\n⊢ rank ↥(closure ↑s) ≤ s.card",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subgroup.closure",
"congrArg",
"Function.Injective.injOn",
"Finset",
"Group.rank_le",
"Classical.propDecid... | [] | classical
let t : Finset (closure (s : Set G)) := s.preimage Subtype.val Subtype.coe_injective.injOn
have ht : closure (t : Set (closure (s : Set G))) = ⊤ := by
rw [Finset.coe_preimage]
exact closure_preimage_eq_top (s : Set G)
apply (rank_le ht).trans
suffices H : Set.InjOn Subtype.val (t : Set (closur... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.GroupTheory.Rank | {
"line": 92,
"column": 2
} | {
"line": 101,
"column": 35
} | {
"line": 103,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\ns : Finset G\n⊢ rank ↥(closure ↑s) ≤ s.card",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subgroup.closure",
"congrArg",
"Function.Injective.injOn",
"Finset",
"Group.rank_le",
"Classical.propDecid... | [] | classical
let t : Finset (closure (s : Set G)) := s.preimage Subtype.val Subtype.coe_injective.injOn
have ht : closure (t : Set (closure (s : Set G))) = ⊤ := by
rw [Finset.coe_preimage]
exact closure_preimage_eq_top (s : Set G)
apply (rank_le ht).trans
suffices H : Set.InjOn Subtype.val (t : Set (closur... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Rank | {
"line": 92,
"column": 2
} | {
"line": 101,
"column": 35
} | {
"line": 103,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\ns : Finset G\n⊢ rank ↥(closure ↑s) ≤ s.card",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subgroup.closure",
"congrArg",
"Function.Injective.injOn",
"Finset",
"Group.rank_le",
"Classical.propDecid... | [] | classical
let t : Finset (closure (s : Set G)) := s.preimage Subtype.val Subtype.coe_injective.injOn
have ht : closure (t : Set (closure (s : Set G))) = ⊤ := by
rw [Finset.coe_preimage]
exact closure_preimage_eq_top (s : Set G)
apply (rank_le ht).trans
suffices H : Set.InjOn Subtype.val (t : Set (closur... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 302,
"column": 4
} | {
"line": 302,
"column": 9
} | {
"line": 303,
"column": 2
} | [
{
"pp": "A : Type u_4\ninst✝³ : AddCommGroup A\ninst✝² : LinearOrder A\ninst✝¹ : IsOrderedAddMonoid A\ninst✝ : Nontrivial A\ng : A\nhs : Function.Surjective fun x ↦ x • g\nm : ℤ\nha : (fun x ↦ x • g) m ≠ 0\n⊢ g ≠ 0",
"ppTerm": "?m.84",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.add... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.GroupTheory.Sylow | {
"line": 482,
"column": 77
} | {
"line": 494,
"column": 83
} | {
"line": 496,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝³ : Group G\ninst✝² : Finite G\nG' : Type u_2\ninst✝¹ : Group G'\nf : G →* G'\nhf : Function.Surjective ⇑f\np : ℕ\ninst✝ : Fact (Nat.Prime p)\n⊢ Function.Surjective (mapSurjective hf)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Sylow.isPGroup'",
... | [] | by
have : Finite G' := Finite.of_surjective f hf
intro P
let Q₀ : Sylow p (P.comap f) := Sylow.nonempty.some
let Q : Subgroup G := Q₀.map (P.comap f).subtype
have hPQ : Q.map f ≤ P := Subgroup.map_le_iff_le_comap.mpr (Subgroup.map_subtype_le Q₀.1)
have hpQ : IsPGroup p Q := Q₀.2.map (P.comap f).subtype
ha... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 372,
"column": 85
} | {
"line": 372,
"column": 90
} | {
"line": 372,
"column": 90
} | [
{
"pp": "α : Type u_1\ninst✝ : Group α\np : ℕ\nhp : Nat.Prime p\nhα : Nat.card α = p ^ 2\nthis✝ : Finite α\nthis : Nontrivial α\nh_cyc : ¬IsCyclic α\ng : α\nhg : g ≠ 1\n⊢ p ^ 2 ≠ 0",
"ppTerm": "?m.117",
"assigned": true,
"usedConstants": [
"instPowNat",
"Eq.mpr",
"False",
"Na... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 372,
"column": 85
} | {
"line": 372,
"column": 90
} | {
"line": 372,
"column": 90
} | [
{
"pp": "α : Type u_1\ninst✝ : Group α\np : ℕ\nhp : Nat.Prime p\nhα : Nat.card α = p ^ 2\nthis✝ : Finite α\nthis : Nontrivial α\nh_cyc : ¬IsCyclic α\ng : α\nhg : g ≠ 1\n⊢ p ^ 2 ≠ 0",
"ppTerm": "?m.117",
"assigned": true,
"usedConstants": [
"instPowNat",
"Eq.mpr",
"False",
"Na... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 372,
"column": 85
} | {
"line": 372,
"column": 90
} | {
"line": 372,
"column": 90
} | [
{
"pp": "α : Type u_1\ninst✝ : Group α\np : ℕ\nhp : Nat.Prime p\nhα : Nat.card α = p ^ 2\nthis✝ : Finite α\nthis : Nontrivial α\nh_cyc : ¬IsCyclic α\ng : α\nhg : g ≠ 1\n⊢ p ^ 2 ≠ 0",
"ppTerm": "?m.117",
"assigned": true,
"usedConstants": [
"instPowNat",
"Eq.mpr",
"False",
"Na... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 1273,
"column": 4
} | {
"line": 1273,
"column": 91
} | {
"line": 1274,
"column": 4
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\n⊢ vanishingIdeal '' {s | Maximal (fun x ↦ IsClosed x ∧ IsIrreducible x) s} = {x | Minimal Ideal.IsPrime x}",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"ChainCompletePartialOrder.instOfComplete... | [
"R : Type u\ninst✝ : CommSemiring R\n⊢ {x | Minimal (fun x ↦ ∃ x₀, (IsClosed x₀ ∧ IsIrreducible x₀) ∧ vanishingIdeal x₀ = x) x} =\n {x | Minimal Ideal.IsPrime x}"
] | image_antitone_setOf_maximal (fun s t hs _ ↦ (vanishingIdeal_anti_mono_iff hs.1).symm), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Torsion | {
"line": 460,
"column": 29
} | {
"line": 460,
"column": 55
} | {
"line": 460,
"column": 56
} | [
{
"pp": "G : Type u_1\nH : Type u_2\nR : Type u_3\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nm : M\nx✝ : m ∈ (AddCommGroup.torsion M).carrier\nn : ℕ\nhn : n > 0\nhn' : n • m = 0\n⊢ r • n • m = 0",
"ppTerm": "?m.125",
"assigned": true,
"usedConstants": [
... | [] | simp only [hn', smul_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.Sylow | {
"line": 819,
"column": 2
} | {
"line": 819,
"column": 61
} | {
"line": 820,
"column": 2
} | [
{
"pp": "G : Type u\ninst✝¹ : Group G\ninst✝ : Finite G\nhn : ∀ {p : ℕ} [Fact (Nat.Prime p)] (P : Sylow p G), (↑P).Normal\nthis : Fintype G\nps : Finset ℕ := (Nat.card G).primeFactors\nP : (p : ℕ) → Sylow p G := default\n⊢ ((p : ↥ps) → (P : Sylow (↑p) G) → ↥P) ≃* G",
"ppTerm": "?m.27",
"assigned": true,... | [
"G : Type u\ninst✝¹ : Group G\ninst✝ : Finite G\nhn : ∀ {p : ℕ} [Fact (Nat.Prime p)] (P : Sylow p G), (↑P).Normal\nthis✝ : Fintype G\nps : Finset ℕ := (Nat.card G).primeFactors\nP : (p : ℕ) → Sylow p G := default\nthis : (p : ℕ) → Fintype ↥↑(P p)\n⊢ ((p : ↥ps) → (P : Sylow (↑p) G) → ↥P) ≃* G"
] | have : ∀ p, Fintype (P p) := fun p ↦ Fintype.ofFinite (P p) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Order.JordanHolder | {
"line": 203,
"column": 2
} | {
"line": 206,
"column": 95
} | {
"line": 207,
"column": 2
} | [
{
"pp": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\ni : Fin (s.length + 1)\nhx : s.toFun i ≠ last s\n⊢ s.toFun i ∈ eraseLast s",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.add_one_sub_one",
"RelSeries.last_mem... | [
"X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\ni : Fin (s.length + 1)\nhx : s.toFun i ≠ last s\nhi : ↑i < (s.length - 1).succ\n⊢ s.toFun i ∈ eraseLast s"
] | have hi : (i : ℕ) < (s.length - 1).succ := by
conv_rhs => rw [← Nat.succ_sub (length_pos_of_nontrivial ⟨_, ⟨i, rfl⟩, _, s.last_mem, hx⟩),
Nat.add_one_sub_one]
exact lt_of_le_of_ne (Nat.le_of_lt_succ i.2) (by simpa [last, s.inj, Fin.ext_iff] using hx) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.SimpleModule.Basic | {
"line": 462,
"column": 2
} | {
"line": 462,
"column": 86
} | {
"line": 463,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nS : Type u_3\ninst✝⁵ : Ring R\ninst✝⁴ : Ring S\nM : Type u_4\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nm : Submodule R M\nN : Type u_5\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nhR : IsSemisimpleRing R\nhS : IsSemisimpleRing S\nthis✝ : Module (R × S) R := Module.comp... | [
"ι : Type u_1\nR : Type u_2\nS : Type u_3\ninst✝⁵ : Ring R\ninst✝⁴ : Ring S\nM : Type u_4\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nm : Submodule R M\nN : Type u_5\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nhR : IsSemisimpleRing R\nhS : IsSemisimpleRing S\nthis✝ : Module (R × S) R := Module.compHom R (RingH... | let _e₂ : S →ₛₗ[.snd R S] S := { AddMonoidHom.id S with map_smul' := fun _ _ ↦ rfl } | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Algebra.Polynomial.Module.AEval | {
"line": 148,
"column": 25
} | {
"line": 148,
"column": 30
} | {
"line": 149,
"column": 2
} | [
{
"pp": "R : Type ?u.5\nA : Type ?u.7\nM : Type ?u.13\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\na : A\ninst✝⁴ : Algebra R A\ninst✝³ : AddCommMonoid M\ninst✝² : Module A M\ninst✝¹ : Module R M\ninst✝ : IsScalarTower R A M\nq : Submodule R[X] (AEval R M a)\nx✝ : AEval R M a\n⊢ x✝ ∈\n (fun p ↦\n ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Polynomial.Module.AEval | {
"line": 149,
"column": 91
} | {
"line": 149,
"column": 96
} | {
"line": 149,
"column": 96
} | [
{
"pp": "R : Type ?u.5\nA : Type ?u.7\nM : Type ?u.13\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\na : A\ninst✝⁴ : Algebra R A\ninst✝³ : AddCommMonoid M\ninst✝² : Module A M\ninst✝¹ : Module R M\ninst✝ : IsScalarTower R A M\np p' : ↥((Algebra.lsmul R R M) a).invtSubmodule\nh : p ≤ p'\nx : AEval R M a\nhx :\n ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Polynomial.Module.AEval | {
"line": 149,
"column": 91
} | {
"line": 149,
"column": 96
} | {
"line": 149,
"column": 96
} | [
{
"pp": "R : Type ?u.5\nA : Type ?u.7\nM : Type ?u.13\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\na : A\ninst✝⁴ : Algebra R A\ninst✝³ : AddCommMonoid M\ninst✝² : Module A M\ninst✝¹ : Module R M\ninst✝ : IsScalarTower R A M\np p' : ↥((Algebra.lsmul R R M) a).invtSubmodule\nh : p ≤ p'\nx : AEval R M a\nhx :\n ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Module.AEval | {
"line": 149,
"column": 91
} | {
"line": 149,
"column": 96
} | {
"line": 149,
"column": 96
} | [
{
"pp": "R : Type ?u.5\nA : Type ?u.7\nM : Type ?u.13\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\na : A\ninst✝⁴ : Algebra R A\ninst✝³ : AddCommMonoid M\ninst✝² : Module A M\ninst✝¹ : Module R M\ninst✝ : IsScalarTower R A M\np p' : ↥((Algebra.lsmul R R M) a).invtSubmodule\nh : p ≤ p'\nx : AEval R M a\nhx :\n ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.Torsion.Basic | {
"line": 937,
"column": 6
} | {
"line": 937,
"column": 63
} | {
"line": 937,
"column": 64
} | [
{
"pp": "case mpr\nR : Type w\ninst✝ : CommRing R\na b : R\nha : a ∈ R⁰\nx c : R\nh : c • (mk (R ∙ a * b)) b = (mk (R ∙ a * b)) x\n⊢ c • Submodule.Quotient.mk (a • b) = 0",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Submodule",
"Submodule.Quo... | [
"case mpr\nR : Type w\ninst✝ : CommRing R\na b : R\nha : a ∈ R⁰\nx c : R\nh : c • (mk (R ∙ a * b)) b = (mk (R ∙ a * b)) x\n⊢ c • 0 = 0"
] | (Quotient.mk_eq_zero _).mpr <| mem_span_singleton_self _, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Filtration | {
"line": 298,
"column": 2
} | {
"line": 312,
"column": 67
} | {
"line": 313,
"column": 2
} | [
{
"pp": "case mp\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF : I.Filtration M\nn₀ : ℕ\n⊢ (∀ (i : ℕ),\n single R i '' ↑(F.N i) ⊆ ↑(Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), single R i '' ↑(F.N i)))) →\n ∀ n ≥ n₀, I • F.N n = ... | [
"case mpr\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF : I.Filtration M\nn₀ : ℕ\n⊢ (∀ n ≥ n₀, I • F.N n = F.N (n + 1)) →\n ∀ (i : ℕ),\n single R i '' ↑(F.N i) ⊆ ↑(Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), single R i '' ↑(F.N i)))... | · intro H n hn
refine (F.smul_le n).antisymm ?_
intro x hx
obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_linearCombination _ _ _).mp (H _ ⟨x, hx, rfl⟩)
replace hl := congr_arg (fun f : PolynomialModule R M => f.coeff (n + 1)) hl
rw [PolynomialModule.coeff_single, Finsupp.single_apply, if_pos rfl] at hl
... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Monoidal.End | {
"line": 236,
"column": 6
} | {
"line": 236,
"column": 25
} | {
"line": 236,
"column": 25
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nM : Type u_1\ninst✝² : Category.{v_1, u_1} M\ninst✝¹ : MonoidalCategory M\nF : M ⥤ C ⥤ C\nn : M\nX : C\ninst✝ : F.Monoidal\n⊢ (ε F).app ((F.obj n).obj X) = (F.map (ρ_ n).inv).app X ≫ (δ F n (𝟙_ M)).app X",
"ppTerm": "?m.73",
"assigned": true,
"usedCo... | [
"C : Type u\ninst✝³ : Category.{v, u} C\nM : Type u_1\ninst✝² : Category.{v_1, u_1} M\ninst✝¹ : MonoidalCategory M\nF : M ⥤ C ⥤ C\nn : M\nX : C\ninst✝ : F.Monoidal\n⊢ (ε F).app ((F.obj n).obj X) = ((ρ_ (F.obj n)).inv ≫ F.obj n ◁ ε F ≫ μ F n (𝟙_ M)).app X ≫ (δ F n (𝟙_ M)).app X"
] | map_rightUnitor_inv | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.HomologicalComplex | {
"line": 738,
"column": 92
} | {
"line": 740,
"column": 6
} | {
"line": 742,
"column": 0
} | [
{
"pp": "V : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nsucc : (S : ShortComplex V) → (X₃ : V) ×' (d₂ : X₃ ⟶ S.X₁) ×' d₂ ≫ S.f = 0\nS S' : ShortComplex V\nh : S = S'\n⊢ (succ S).snd.fst = eqToHom ⋯ ≫ (succ S').snd.fst ≫ eqToHom ⋯",
"ppTerm": "?m.82",
"assigned": true,
"usedConst... | [] | by
subst h
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.HomologicalComplex | {
"line": 746,
"column": 2
} | {
"line": 746,
"column": 18
} | {
"line": 748,
"column": 0
} | [
{
"pp": "V : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nX₀ X₁ X₂ : V\nd₀ : X₁ ⟶ X₀\nd₁ : X₂ ⟶ X₁\ns : d₁ ≫ d₀ = 0\nsucc : (S : ShortComplex V) → (X₃ : V) ×' (d₂ : X₃ ⟶ S.X₁) ×' d₂ ≫ S.f = 0\nn : ℕ\n⊢ mkAux X₀ X₁ X₂ d₀ d₁ s succ n =\n { X₁ := (mk X₀ X₁ X₂ d₀ d₁ s succ).X (n + 1 + 1), X₂ :... | [] | simp [mk, mkAux] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.HomologicalComplex | {
"line": 809,
"column": 59
} | {
"line": 811,
"column": 6
} | {
"line": 813,
"column": 0
} | [
{
"pp": "V : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nsucc' : {X₀ X₁ : V} → (f : X₁ ⟶ X₀) → (X₂ : V) ×' (d : X₂ ⟶ X₁) ×' d ≫ f = 0\nX Y : V\nf g : X ⟶ Y\nh : f = g\n⊢ (succ' f).snd.fst = eqToHom ⋯ ≫ (succ' g).snd.fst",
"ppTerm": "?m.78",
"assigned": true,
"usedConstants": [
... | [] | by
subst h
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Kaehler.Basic | {
"line": 295,
"column": 4
} | {
"line": 295,
"column": 50
} | {
"line": 295,
"column": 50
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nM : Type u_1\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Module S M\ninst✝ : IsScalarTower R S M\nf f' : Ω[S⁄R] →ₗ[S] M\nhf : f.compDer (KaehlerDifferential.D R S) = f'.compDer (KaehlerDifferential.D R S... | [
"R : Type u\nS : Type v\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nM : Type u_1\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Module S M\ninst✝ : IsScalarTower R S M\nf f' : Ω[S⁄R] →ₗ[S] M\nhf : f.compDer (KaehlerDifferential.D R S) = f'.compDer (KaehlerDifferential.D R S)\nx : Ω[S⁄R... | rw [KaehlerDifferential.span_range_derivation] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Shift.Basic | {
"line": 697,
"column": 2
} | {
"line": 697,
"column": 48
} | {
"line": 698,
"column": 2
} | [
{
"pp": "C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddCommMonoid A\ninst✝ : HasShift C A\nm₁ m₂ m₃ : A\nX : C\n⊢ (((shiftFunctorComm C m₁ (m₂ + m₃)).hom.app X ≫ (shiftFunctor C m₁).map ((shiftFunctorAdd C m₂ m₃).hom.app X)) ≫\n (shiftFunctorComm C m₁ m₃).inv.app ((shiftFunctor C m₂).... | [
"C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddCommMonoid A\ninst✝ : HasShift C A\nm₁ m₂ m₃ : A\nX : C\n⊢ (shiftFunctorComm C m₁ (m₂ + m₃)).hom.app X ≫\n (shiftFunctor C m₁).map ((shiftFunctorAdd C m₂ m₃).hom.app X) ≫\n (shiftFunctorComm C m₁ m₃).inv.app ((shiftFunctor C m₂).obj X) ≫... | simp only [Category.assoc, Iso.hom_inv_id_app] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Kaehler.Basic | {
"line": 504,
"column": 6
} | {
"line": 504,
"column": 33
} | {
"line": 504,
"column": 34
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\n⊢ (x𝖣1) = 0",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Submodule",
"Submodule.Quotient.instZeroQuotient... | [
"R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\n⊢ (x𝖣(algebraMap R S) 1) = 0"
] | ← (algebraMap R S).map_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Kaehler.Basic | {
"line": 735,
"column": 6
} | {
"line": 735,
"column": 59
} | {
"line": 736,
"column": 6
} | [
{
"pp": "case a.tmul\nR : Type u\ninst✝⁶ : CommRing R\nA : Type u_2\nB : Type u_3\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R A\ninst✝² : Algebra A B\ninst✝¹ : Algebra R B\ninst✝ : IsScalarTower R A B\nr : B\ns : Ω[A⁄R]\n⊢ (mapBaseChange R A B) (r ⊗ₜ[A] s) ∈ (map R A B B).ker",
"ppTerm": "... | [
"case a.tmul\nR : Type u\ninst✝⁶ : CommRing R\nA : Type u_2\nB : Type u_3\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R A\ninst✝² : Algebra A B\ninst✝¹ : Algebra R B\ninst✝ : IsScalarTower R A B\nr : B\nx : A →₀ A\n⊢ (mapBaseChange R A B) (r ⊗ₜ[A] (linearCombination A ⇑(D R A)) x) ∈ (map R A B B).ke... | obtain ⟨x, rfl⟩ := linearCombination_surjective _ _ s | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Kaehler.Basic | {
"line": 808,
"column": 4
} | {
"line": 808,
"column": 29
} | {
"line": 809,
"column": 4
} | [
{
"pp": "case mpr\nR : Type u\ninst✝⁶ : CommRing R\nA : Type u_2\nB : Type u_3\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R A\ninst✝² : Algebra A B\ninst✝¹ : Algebra R B\ninst✝ : IsScalarTower R A B\nh : Function.Surjective ⇑(algebraMap A B)\nx : Ω[A⁄R]\nhx :\n x ∈\n Submodule.map (linearCo... | [
"case mpr\nR : Type u\ninst✝⁶ : CommRing R\nA : Type u_2\nB : Type u_3\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R A\ninst✝² : Algebra A B\ninst✝¹ : Algebra R B\ninst✝ : IsScalarTower R A B\nh : Function.Surjective ⇑(algebraMap A B)\nx : A →₀ A\nhx : x ∈ ↑(mapRange.linearMap (Algebra.linearMap A B... | obtain ⟨x, hx, rfl⟩ := hx | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex | {
"line": 254,
"column": 2
} | {
"line": 257,
"column": 35
} | {
"line": 259,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\nι : Type u_2\nc : ComplexShape ι\nK : HomologicalComplex C c\ni j : ι\ninst✝ : K.HasHomology j\n⊢ K.d i j ≫ K.pOpcycles j = 0",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryT... | [] | by_cases hij : c.Rel i j
· obtain rfl := c.prev_eq' hij
exact (K.sc j).f_pOpcycles
· rw [K.shape _ _ hij, zero_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex | {
"line": 254,
"column": 2
} | {
"line": 257,
"column": 35
} | {
"line": 259,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\nι : Type u_2\nc : ComplexShape ι\nK : HomologicalComplex C c\ni j : ι\ninst✝ : K.HasHomology j\n⊢ K.d i j ≫ K.pOpcycles j = 0",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryT... | [] | by_cases hij : c.Rel i j
· obtain rfl := c.prev_eq' hij
exact (K.sc j).f_pOpcycles
· rw [K.shape _ _ hij, zero_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Kaehler.Basic | {
"line": 846,
"column": 2
} | {
"line": 848,
"column": 39
} | {
"line": 850,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝⁶ : CommRing R\nA : Type u_2\nB : Type u_3\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R A\ninst✝² : Algebra A B\ninst✝¹ : Algebra R B\ninst✝ : IsScalarTower R A B\nh : Function.Surjective ⇑(algebraMap A B)\n⊢ Function.Surjective ⇑(mapBaseChange R A B)",
"ppTerm": "... | [] | have := subsingleton_of_surjective A B h
rw [← LinearMap.range_eq_top, range_mapBaseChange, ← top_le_iff]
exact fun x _ ↦ Subsingleton.elim _ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Kaehler.Basic | {
"line": 846,
"column": 2
} | {
"line": 848,
"column": 39
} | {
"line": 850,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝⁶ : CommRing R\nA : Type u_2\nB : Type u_3\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R A\ninst✝² : Algebra A B\ninst✝¹ : Algebra R B\ninst✝ : IsScalarTower R A B\nh : Function.Surjective ⇑(algebraMap A B)\n⊢ Function.Surjective ⇑(mapBaseChange R A B)",
"ppTerm": "... | [] | have := subsingleton_of_surjective A B h
rw [← LinearMap.range_eq_top, range_mapBaseChange, ← top_le_iff]
exact fun x _ ↦ Subsingleton.elim _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Homotopy | {
"line": 366,
"column": 55
} | {
"line": 366,
"column": 63
} | {
"line": 366,
"column": 64
} | [
{
"pp": "ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D : HomologicalComplex V c\nk₁ k₀ : ι\nr₁₀ : c.Rel k₁ k₀\nhk₀ : ∀ (l : ι), ¬c.Rel k₀ l\nhom : (i j : ι) → C.X i ⟶ D.X j\n⊢ C.d k₀ (c.next k₀) ≫ hom (c.next k₀) k₀ + hom k₀ k₁ ≫ D.d k₁ k₀ = hom k₀ k₁ ≫ D.d... | [
"ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D : HomologicalComplex V c\nk₁ k₀ : ι\nr₁₀ : c.Rel k₁ k₀\nhk₀ : ∀ (l : ι), ¬c.Rel k₀ l\nhom : (i j : ι) → C.X i ⟶ D.X j\n⊢ 0 ≫ hom (c.next k₀) k₀ + hom k₀ k₁ ≫ D.d k₁ k₀ = hom k₀ k₁ ≫ D.d k₁ k₀",
"case a\nι : Type ... | C.shape, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.Homotopy | {
"line": 401,
"column": 6
} | {
"line": 401,
"column": 14
} | {
"line": 401,
"column": 15
} | [
{
"pp": "ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D : HomologicalComplex V c\nk₀ : ι\nhk₀ : ∀ (l : ι), ¬c.Rel k₀ l\nhk₀' : ∀ (l : ι), ¬c.Rel l k₀\nhom : (i j : ι) → C.X i ⟶ D.X j\n⊢ C.d k₀ (c.next k₀) ≫ hom (c.next k₀) k₀ + hom k₀ (c.prev k₀) ≫ D.d (c.pr... | [
"ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D : HomologicalComplex V c\nk₀ : ι\nhk₀ : ∀ (l : ι), ¬c.Rel k₀ l\nhk₀' : ∀ (l : ι), ¬c.Rel l k₀\nhom : (i j : ι) → C.X i ⟶ D.X j\n⊢ 0 ≫ hom (c.next k₀) k₀ + hom k₀ (c.prev k₀) ≫ D.d (c.prev k₀) k₀ = 0",
"case a\nι ... | C.shape, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.HomotopyCofiber | {
"line": 329,
"column": 6
} | {
"line": 330,
"column": 74
} | {
"line": 331,
"column": 6
} | [
{
"pp": "case pos\nC : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Preadditive C\nι : Type u_2\nc : ComplexShape ι\nF G K : HomologicalComplex C c\nφ : F ⟶ G\ninst✝¹ : HasHomotopyCofiber φ\ninst✝ : DecidableRel c.Rel\nα : G ⟶ K\nhα : Homotopy (φ ≫ α) 0\nj : ι\nhjk : c.Rel j (c.next j)\nH : φ.f (c.next j)... | [
"case pos\nC : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Preadditive C\nι : Type u_2\nc : ComplexShape ι\nF G K : HomologicalComplex C c\nφ : F ⟶ G\ninst✝¹ : HasHomotopyCofiber φ\ninst✝ : DecidableRel c.Rel\nα : G ⟶ K\nhα : Homotopy (φ ≫ α) 0\nj : ι\nhjk : c.Rel j (c.next j)\nH : φ.f (c.next j) ≫ α.f (c.ne... | simp only [comp_add, d_sndX_assoc _ _ _ hjk, add_comp, assoc, H,
d_fstX_assoc _ _ _ _ hjk, neg_comp, dNext, AddMonoidHom.mk'_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.Homotopy | {
"line": 528,
"column": 33
} | {
"line": 528,
"column": 40
} | {
"line": 529,
"column": 2
} | [
{
"pp": "case hnc\nι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf g : C ⟶ D\nh k : D ⟶ E\ni✝ : ι\nP Q : ChainComplex V ℕ\ne : P ⟶ Q\nzero : P.X 0 ⟶ Q.X 1\ncomm_zero : e.f 0 = zero ≫ Q.d 1 0\none : P.X 1 ⟶ Q.X 2\ncomm_one : e.f 1... | [] | exact w | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Homology.Homotopy | {
"line": 580,
"column": 6
} | {
"line": 580,
"column": 34
} | {
"line": 580,
"column": 34
} | [
{
"pp": "case a\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nP Q : CochainComplex V ℕ\nf : (i j : ℕ) → P.X i ⟶ Q.X j\n⊢ ¬(ComplexShape.up ℕ).Rel ((ComplexShape.up ℕ).prev 0) 0",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instOne",
"congr... | [
"case a\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nP Q : CochainComplex V ℕ\nf : (i j : ℕ) → P.X i ⟶ Q.X j\n⊢ ¬(ComplexShape.up ℕ).Rel 0 0"
] | CochainComplex.prev_nat_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplex | {
"line": 636,
"column": 19
} | {
"line": 636,
"column": 24
} | {
"line": 637,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nF G K L : CochainComplex C ℤ\nn m : ℤ\nx✝ : Cocycle F G n\n⊢ 1 • x✝ = x✝",
"ppTerm": "?m.82",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"CochainComplex.H... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplex | {
"line": 636,
"column": 19
} | {
"line": 636,
"column": 24
} | {
"line": 637,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nF G K L : CochainComplex C ℤ\nn m : ℤ\nx✝ : Cocycle F G n\n⊢ 1 • x✝ = x✝",
"ppTerm": "?m.82",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"CochainComplex.H... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplex | {
"line": 636,
"column": 19
} | {
"line": 636,
"column": 24
} | {
"line": 637,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nF G K L : CochainComplex C ℤ\nn m : ℤ\nx✝ : Cocycle F G n\n⊢ 1 • x✝ = x✝",
"ppTerm": "?m.82",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"CochainComplex.H... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplex | {
"line": 638,
"column": 20
} | {
"line": 638,
"column": 25
} | {
"line": 639,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nF G K L : CochainComplex C ℤ\nn m : ℤ\nx✝ : R\n⊢ x✝ • 0 = 0",
"ppTerm": "?m.83",
"assigned": true,
"usedConstants": [
"CochainComplex.HomComplex.instModuleCochain",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplex | {
"line": 638,
"column": 20
} | {
"line": 638,
"column": 25
} | {
"line": 639,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nF G K L : CochainComplex C ℤ\nn m : ℤ\nx✝ : R\n⊢ x✝ • 0 = 0",
"ppTerm": "?m.83",
"assigned": true,
"usedConstants": [
"CochainComplex.HomComplex.instModuleCochain",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplex | {
"line": 638,
"column": 20
} | {
"line": 638,
"column": 25
} | {
"line": 639,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nF G K L : CochainComplex C ℤ\nn m : ℤ\nx✝ : R\n⊢ x✝ • 0 = 0",
"ppTerm": "?m.83",
"assigned": true,
"usedConstants": [
"CochainComplex.HomComplex.instModuleCochain",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplex | {
"line": 639,
"column": 23
} | {
"line": 639,
"column": 28
} | {
"line": 640,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nF G K L : CochainComplex C ℤ\nn m : ℤ\nx✝² : R\nx✝¹ x✝ : Cocycle F G n\n⊢ x✝² • (x✝¹ + x✝) = x✝² • x✝¹ + x✝² • x✝",
"ppTerm": "?m.84",
"assigned": true,
"usedConstants": [
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplex | {
"line": 639,
"column": 23
} | {
"line": 639,
"column": 28
} | {
"line": 640,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nF G K L : CochainComplex C ℤ\nn m : ℤ\nx✝² : R\nx✝¹ x✝ : Cocycle F G n\n⊢ x✝² • (x✝¹ + x✝) = x✝² • x✝¹ + x✝² • x✝",
"ppTerm": "?m.84",
"assigned": true,
"usedConstants": [
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplex | {
"line": 639,
"column": 23
} | {
"line": 639,
"column": 28
} | {
"line": 640,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nF G K L : CochainComplex C ℤ\nn m : ℤ\nx✝² : R\nx✝¹ x✝ : Cocycle F G n\n⊢ x✝² • (x✝¹ + x✝) = x✝² • x✝¹ + x✝² • x✝",
"ppTerm": "?m.84",
"assigned": true,
"usedConstants": [
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplex | {
"line": 641,
"column": 18
} | {
"line": 641,
"column": 23
} | {
"line": 643,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nF G K L : CochainComplex C ℤ\nn m : ℤ\n⊢ ∀ (x : Cocycle F G n), 0 • x = 0",
"ppTerm": "?m.116",
"assigned": true,
"usedConstants": [
"CochainComplex.HomComplex.instModule... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplex | {
"line": 641,
"column": 18
} | {
"line": 641,
"column": 23
} | {
"line": 643,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nF G K L : CochainComplex C ℤ\nn m : ℤ\n⊢ ∀ (x : Cocycle F G n), 0 • x = 0",
"ppTerm": "?m.116",
"assigned": true,
"usedConstants": [
"CochainComplex.HomComplex.instModule... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplex | {
"line": 641,
"column": 18
} | {
"line": 641,
"column": 23
} | {
"line": 643,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nF G K L : CochainComplex C ℤ\nn m : ℤ\n⊢ ∀ (x : Cocycle F G n), 0 • x = 0",
"ppTerm": "?m.116",
"assigned": true,
"usedConstants": [
"CochainComplex.HomComplex.instModule... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Homotopy | {
"line": 657,
"column": 33
} | {
"line": 657,
"column": 40
} | {
"line": 658,
"column": 2
} | [
{
"pp": "case hnc\nι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf g : C ⟶ D\nh k : D ⟶ E\ni✝ : ι\nP Q : CochainComplex V ℕ\ne : P ⟶ Q\nzero : P.X 1 ⟶ Q.X 0\ncomm_zero : e.f 0 = P.d 0 1 ≫ zero\none : P.X 2 ⟶ Q.X 1\ncomm_one : e.f... | [] | exact w | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Homology.HomotopyCategory.MappingCone | {
"line": 517,
"column": 61
} | {
"line": 519,
"column": 13
} | {
"line": 521,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v, u_1} C\ninst✝¹ : Preadditive C\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝ : HasHomotopyCofiber φ\nK : CochainComplex C ℤ\nα : Cocycle K F 1\nβ : Cochain K G 0\neq : δ 0 1 β + (↑α).comp (Cochain.ofHom φ) ⋯ = 0\np q : ℤ\nhpq : p + 0 = q\n⊢ (lift φ α β eq).f p ≫ (snd φ... | [] | by
obtain rfl : q = p := by lia
simp [lift] | [anonymous] | Lean.Parser.Term.byTactic |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.