module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Topology.LocalAtTarget | {
"line": 256,
"column": 2
} | {
"line": 256,
"column": 23
} | {
"line": 257,
"column": 2
} | [
{
"pp": "X : 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), Continuous[ins... | [
"case inr\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), Continuous[inst✝... | wlog hU' : iSup U = ⊤ | Mathlib.Tactic._aux_Mathlib_Tactic_WLOG___elabRules_Mathlib_Tactic_wlog_1 | Mathlib.Tactic.wlog |
Mathlib.Order.Ideal | {
"line": 355,
"column": 2
} | {
"line": 355,
"column": 52
} | {
"line": 357,
"column": 0
} | [
{
"pp": "P : Type u_2\ninst✝¹ : SemilatticeSup P\ninst✝ : OrderBot P\nt : Ideal P\nι : Type u_3\nf : ι → P\ns : Finset ι\n⊢ s.sup f ∈ t ↔ ∀ i ∈ s, f i ∈ t",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"False",
"Finset.sup_insert",
"congrArg",
"Finset",
"Orde... | [] | induction s using Finset.induction_on <;> simp [*] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Order.Ideal | {
"line": 355,
"column": 2
} | {
"line": 355,
"column": 52
} | {
"line": 357,
"column": 0
} | [
{
"pp": "P : Type u_2\ninst✝¹ : SemilatticeSup P\ninst✝ : OrderBot P\nt : Ideal P\nι : Type u_3\nf : ι → P\ns : Finset ι\n⊢ s.sup f ∈ t ↔ ∀ i ∈ s, f i ∈ t",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"False",
"Finset.sup_insert",
"congrArg",
"Finset",
"Orde... | [] | induction s using Finset.induction_on <;> simp [*] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Ideal | {
"line": 355,
"column": 2
} | {
"line": 355,
"column": 52
} | {
"line": 357,
"column": 0
} | [
{
"pp": "P : Type u_2\ninst✝¹ : SemilatticeSup P\ninst✝ : OrderBot P\nt : Ideal P\nι : Type u_3\nf : ι → P\ns : Finset ι\n⊢ s.sup f ∈ t ↔ ∀ i ∈ s, f i ∈ t",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"False",
"Finset.sup_insert",
"congrArg",
"Finset",
"Orde... | [] | induction s using Finset.induction_on <;> simp [*] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.QuasiSeparated | {
"line": 157,
"column": 6
} | {
"line": 157,
"column": 40
} | {
"line": 158,
"column": 2
} | [
{
"pp": "case ht\nα : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : QuasiSeparatedSpace α\ns : Set (Set α)\nhf : s.Finite\nhne : s.Nonempty\nho : ∀ t ∈ s, IsOpen[inst✝¹] t ∨ IsClosed[inst✝¹] t\nhc : ∀ t ∈ s, IsCompact t\nthis :\n ∀ {α : Type u_1} [inst : TopologicalSpace α] [QuasiSeparatedSpace α] {s : Set (S... | [] | · exact isClosed_sInter (by grind) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Sets.Compacts | {
"line": 708,
"column": 2
} | {
"line": 708,
"column": 52
} | {
"line": 710,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\nι : Type u_4\nf : ι → CompactOpens α\ns : Finset ι\n⊢ ↑(s.sup f) = ⋃ i ∈ s, ↑(f i)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"False",
"Finset.sup_insert",
"Iff.of_eq",
"congrArg",
"Finset",
"OrderB... | [] | induction s using Finset.induction_on <;> simp [*] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Topology.Sets.Compacts | {
"line": 708,
"column": 2
} | {
"line": 708,
"column": 52
} | {
"line": 710,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\nι : Type u_4\nf : ι → CompactOpens α\ns : Finset ι\n⊢ ↑(s.sup f) = ⋃ i ∈ s, ↑(f i)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"False",
"Finset.sup_insert",
"Iff.of_eq",
"congrArg",
"Finset",
"OrderB... | [] | induction s using Finset.induction_on <;> simp [*] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Sets.Compacts | {
"line": 708,
"column": 2
} | {
"line": 708,
"column": 52
} | {
"line": 710,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\nι : Type u_4\nf : ι → CompactOpens α\ns : Finset ι\n⊢ ↑(s.sup f) = ⋃ i ∈ s, ↑(f i)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"False",
"Finset.sup_insert",
"Iff.of_eq",
"congrArg",
"Finset",
"OrderB... | [] | induction s using Finset.induction_on <;> simp [*] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Sober | {
"line": 189,
"column": 8
} | {
"line": 189,
"column": 17
} | {
"line": 189,
"column": 18
} | [
{
"pp": "case h\nα : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\nf : α → β\nhf : IsClosedEmbedding f\ninst✝ : QuasiSober β\nS✝ : Set α\nhS : IsIrreducible S✝\nhS' : IsClosed[inst✝²] S✝\nhS'' : IsIrreducible (f '' S✝)\ny : α\nhx : IsGenericPoint (f y) (f '' S✝)\n⊢ f '' closu... | [
"case h\nα : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\nf : α → β\nhf : IsClosedEmbedding f\ninst✝ : QuasiSober β\nS✝ : Set α\nhS : IsIrreducible S✝\nhS' : IsClosed[inst✝²] S✝\nhS'' : IsIrreducible (f '' S✝)\ny : α\nhx : IsGenericPoint (f y) (f '' S✝)\n⊢ f '' closure[inst✝²] {... | ← hx.def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Constructible | {
"line": 138,
"column": 6
} | {
"line": 138,
"column": 32
} | {
"line": 138,
"column": 33
} | [
{
"pp": "case refine_1\nX : Type u_2\ninst✝ : TopologicalSpace X\ns : Set X\nhs : IsRetrocompact s\nt : Set X\nhtopen : IsOpen[inst✝] t\nhtcomp : IsCompact t\n⊢ IsCompact (t ∩ range Subtype.val)",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"se... | [
"case refine_1\nX : Type u_2\ninst✝ : TopologicalSpace X\ns : Set X\nhs : IsRetrocompact s\nt : Set X\nhtopen : IsOpen[inst✝] t\nhtcomp : IsCompact t\n⊢ IsCompact (t ∩ {x | x ∈ s})"
] | Subtype.range_coe_subtype, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.RingHom.FaithfullyFlat | {
"line": 55,
"column": 2
} | {
"line": 55,
"column": 29
} | {
"line": 56,
"column": 2
} | [
{
"pp": "R S T : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing T\nf : R →+* S\ng : S →+* T\nhf : f.FaithfullyFlat\nhg : g.FaithfullyFlat\n⊢ (g.comp f).FaithfullyFlat",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Algebra.algebraMap",
"CommSemiring.toSem... | [
"R S T : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing T\nf : R →+* S\ng : S →+* T\nhf : f.FaithfullyFlat\nhg : g.FaithfullyFlat\nalgInst✝² : Algebra R S := f.toAlgebra\nalgInst✝¹ : Algebra S T := g.toAlgebra\nalgInst✝ : Algebra R T := (g.comp f).toAlgebra\nscalarTowerInst✝ : IsScalarTower R ... | algebraize [f, g, g.comp f] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 963,
"column": 6
} | {
"line": 963,
"column": 91
} | {
"line": 964,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : R →+* S\nH : RingHom.ker f ≤ nilradical R\nI : Ideal R\nhI : Minimal (fun q ↦ q.IsPrime ∧ ⊥ ≤ q) I\np : Ideal R\n⊢ p.IsPrime ∧ RingHom.ker f ≤ p ↔ p.IsPrime ∧ ⊥ ≤ p",
"ppTerm": "?m.140",
"assigned": true,
"used... | [] | exact ⟨fun h ↦ ⟨h.1, bot_le⟩, fun h ↦ ⟨h.1, H.trans (h.1.radical_le_iff.mpr bot_le)⟩⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 996,
"column": 11
} | {
"line": 996,
"column": 23
} | {
"line": 996,
"column": 24
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nx : PrimeSpectrum R\n⊢ StableUnderSpecialization {x} ↔ x.asIdeal.IsMaximal",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"StableUnderSpecialization",
"congrArg",
"CommSemiring.toSemiring",
"PartialOrder.... | [
"R : Type u\ninst✝ : CommSemiring R\nx : PrimeSpectrum R\n⊢ StableUnderSpecialization {x} ↔ IsMax x"
] | ← isMax_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 1040,
"column": 2
} | {
"line": 1040,
"column": 39
} | {
"line": 1041,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nx : R\nhx : x ∈ {e | IsIdempotentElem e}\ny : R\nhy : y ∈ {e | IsIdempotentElem e}\neq : basicOpen x = basicOpen y\nne : x ≠ y\n⊢ False",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"False",
"HMul.hMul",
"PrimeSpectrum.basicO... | [
"case inr\nR : Type u\ninst✝ : CommSemiring R\nx : R\nhx : x ∈ {e | IsIdempotentElem e}\ny : R\nhy : y ∈ {e | IsIdempotentElem e}\neq : basicOpen x = basicOpen y\nne : x ≠ y\nthis :\n ∀ x ∈ {e | IsIdempotentElem e}, ∀ y ∈ {e | IsIdempotentElem e}, basicOpen x = basicOpen y → x ≠ y → x * y ≠ x → False\nne' : ¬x * y... | wlog ne' : x * y ≠ x generalizing x y | Mathlib.Tactic._aux_Mathlib_Tactic_WLOG___elabRules_Mathlib_Tactic_wlog_1 | Mathlib.Tactic.wlog |
Mathlib.Data.ZMod.QuotientGroup | {
"line": 196,
"column": 59
} | {
"line": 198,
"column": 32
} | {
"line": 200,
"column": 0
} | [
{
"pp": "G : Type u_3\ninst✝ : Group G\nH : Subgroup G\ng : G\nq : orbitRel.Quotient (↥(zpowers g)) (G ⧸ H)\nk : ℤ\n⊢ (H.quotientEquivSigmaZMod g) (g ^ k • Quotient.out q) = ⟨q, ↑k⟩",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"instHSMul",
"... | [] | by
rw [apply_eq_iff_eq_symm_apply, quotientEquivSigmaZMod_symm_apply, ZMod.coe_intCast,
zpow_smul_mod_minimalPeriod] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 1041,
"column": 2
} | {
"line": 1042,
"column": 34
} | {
"line": 1043,
"column": 2
} | [
{
"pp": "case inr\nR : Type u\ninst✝ : CommSemiring R\nx : R\nhx : x ∈ {e | IsIdempotentElem e}\ny : R\nhy : y ∈ {e | IsIdempotentElem e}\neq : basicOpen x = basicOpen y\nne : x ≠ y\nthis :\n ∀ x ∈ {e | IsIdempotentElem e}, ∀ y ∈ {e | IsIdempotentElem e}, basicOpen x = basicOpen y → x ≠ y → x * y ≠ x → False\n... | [
"R : Type u\ninst✝ : CommSemiring R\nx : R\nhx : x ∈ {e | IsIdempotentElem e}\ny : R\nhy : y ∈ {e | IsIdempotentElem e}\neq : basicOpen x = basicOpen y\nne : x ≠ y\nne' : x * y ≠ x\n⊢ False"
] | · apply this y hy x hx eq.symm ne.symm
rwa [mul_comm, of_not_not ne'] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 1063,
"column": 37
} | {
"line": 1063,
"column": 47
} | {
"line": 1064,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\ns : Set (PrimeSpectrum R)\nhs : IsClopen s\nh✝ : Nontrivial R\nI : Ideal R\nhI : I.FG\nJ : Ideal R\nhJ : J.FG\nhI' : zeroLocus ↑I = sᶜ\nhJ' : zeroLocus ↑J = s\nthis : I * J ≤ nilradical R\nhn : (I * J) ^ 0 ≤ 0\n⊢ False",
"ppTerm": "?m.249",
"assigned": true,
... | [] | simp at hn | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 1168,
"column": 2
} | {
"line": 1169,
"column": 36
} | {
"line": 1170,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nhf : f.IsIntegral\ns : Set (PrimeSpectrum S)\nhs : IsClosed s\ny : PrimeSpectrum R\nx : PrimeSpectrum S\nhx : x ∈ s\ne : comap f x ⤳ y\nalgInst✝ : Algebra R S := f.toAlgebra\nalgebraizeInst✝ : Algebra.IsIntegral R S\n⊢ y ... | [
"R : Type u_1\nS : Type u_2\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nhf : f.IsIntegral\ns : Set (PrimeSpectrum S)\nhs : IsClosed s\ny : PrimeSpectrum R\nx : PrimeSpectrum S\nhx : x ∈ s\ne : comap f x ⤳ y\nalgInst✝ : Algebra R S := f.toAlgebra\nalgebraizeInst✝ : Algebra.IsIntegral R S\nq : Ideal S\nhq₁... | obtain ⟨q, hq₁, hq₂, hq₃⟩ := Ideal.exists_ideal_over_prime_of_isIntegral y.asIdeal x.asIdeal
((le_iff_specializes _ _).mpr e) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.GroupTheory.PGroup | {
"line": 368,
"column": 56
} | {
"line": 368,
"column": 83
} | {
"line": 368,
"column": 83
} | [
{
"pp": "p : ℕ\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : Fact (Nat.Prime p)\nhG : Nat.card G = p ^ 2\n⊢ p * Nat.card (G ⧸ center G) ∣ Nat.card G",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"HMul.hMul",
"MulZeroClass.toMul",
"congrArg",
... | [
"p : ℕ\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : Fact (Nat.Prime p)\nhG : Nat.card G = p ^ 2\n⊢ p * Nat.card (G ⧸ center G) ∣ Nat.card ↥(center G) * (center G).index"
] | ← (center G).card_mul_index | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.PGroup | {
"line": 366,
"column": 31
} | {
"line": 372,
"column": 30
} | {
"line": 374,
"column": 0
} | [
{
"pp": "p : ℕ\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : Fact (Nat.Prime p)\nhG : Nat.card G = p ^ 2\n⊢ IsCyclic (G ⧸ center G)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"IsPGroup.card_center_eq_prime_pow",
"Eq.mpr",
"Nat.instMulZeroClass",
"Dvd.dvd",
"Na... | [] | by
apply isCyclic_of_card_dvd_prime (p := p)
rw [← mul_dvd_mul_iff_left (NeZero.ne p), ← sq, ← hG, ← (center G).card_mul_index]
apply mul_dvd_mul_right
rcases card_center_eq_prime_pow hG zero_lt_two with ⟨k, hk0, hk⟩
rw [hk]
exact dvd_pow_self p hk0.ne' | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 198,
"column": 20
} | {
"line": 198,
"column": 25
} | {
"line": 200,
"column": 0
} | [
{
"pp": "G : Type u_2\nG' : Type u_3\ninst✝² : Group G\ninst✝¹ : Group G'\ninst✝ : IsCyclic G'\nf : G →* G'\nhf : f.ker ≤ center G\na b : G\nx : G'\ny : G\nhxy : f y = x\nhx : ∀ (a : ↥f.range), a ∈ zpowers ⟨x, ⋯⟩\nm : ℤ\nhm✝ : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) m = ⟨f a, ⋯⟩\nn : ℤ\nhn✝ : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) n = ⟨f b... | [] | group | Mathlib.Tactic.Group._aux_Mathlib_Tactic_Group___macroRules_Mathlib_Tactic_Group_group_1 | Mathlib.Tactic.Group.group |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 198,
"column": 20
} | {
"line": 198,
"column": 25
} | {
"line": 200,
"column": 0
} | [
{
"pp": "G : Type u_2\nG' : Type u_3\ninst✝² : Group G\ninst✝¹ : Group G'\ninst✝ : IsCyclic G'\nf : G →* G'\nhf : f.ker ≤ center G\na b : G\nx : G'\ny : G\nhxy : f y = x\nhx : ∀ (a : ↥f.range), a ∈ zpowers ⟨x, ⋯⟩\nm : ℤ\nhm✝ : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) m = ⟨f a, ⋯⟩\nn : ℤ\nhn✝ : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) n = ⟨f b... | [] | group | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 198,
"column": 20
} | {
"line": 198,
"column": 25
} | {
"line": 200,
"column": 0
} | [
{
"pp": "G : Type u_2\nG' : Type u_3\ninst✝² : Group G\ninst✝¹ : Group G'\ninst✝ : IsCyclic G'\nf : G →* G'\nhf : f.ker ≤ center G\na b : G\nx : G'\ny : G\nhxy : f y = x\nhx : ∀ (a : ↥f.range), a ∈ zpowers ⟨x, ⋯⟩\nm : ℤ\nhm✝ : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) m = ⟨f a, ⋯⟩\nn : ℤ\nhn✝ : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) n = ⟨f b... | [] | group | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 373,
"column": 6
} | {
"line": 373,
"column": 32
} | {
"line": 373,
"column": 32
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝ : Group α\nthis✝ : Finite α\nthis : Nontrivial α\nh_cyc : IsCyclic α\nhp : Nat.Prime 1\nhα : Nat.card α = 1 ^ 2\nh_exp : exponent α = 1\n⊢ False",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Nat.not_prime_one"
],
"usedFVars": [
"... | [] | exact Nat.not_prime_one hp | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 373,
"column": 6
} | {
"line": 373,
"column": 32
} | {
"line": 373,
"column": 32
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝ : Group α\nthis✝ : Finite α\nthis : Nontrivial α\nh_cyc : IsCyclic α\nhp : Nat.Prime 1\nhα : Nat.card α = 1 ^ 2\nh_exp : exponent α = 1\n⊢ False",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Nat.not_prime_one"
],
"usedFVars": [
"... | [] | exact Nat.not_prime_one hp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 373,
"column": 6
} | {
"line": 373,
"column": 32
} | {
"line": 373,
"column": 32
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝ : Group α\nthis✝ : Finite α\nthis : Nontrivial α\nh_cyc : IsCyclic α\nhp : Nat.Prime 1\nhα : Nat.card α = 1 ^ 2\nh_exp : exponent α = 1\n⊢ False",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Nat.not_prime_one"
],
"usedFVars": [
"... | [] | exact Nat.not_prime_one hp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 697,
"column": 2
} | {
"line": 697,
"column": 36
} | {
"line": 698,
"column": 2
} | [
{
"pp": "G : Type u_2\nG' : Type u_3\ninst✝¹ : Group G\ninst✝ : Group G'\ng : G\nhg : ∀ (x : G), x ∈ zpowers g\nf₁ f₂ : G →* G'\n⊢ (∀ (x : G), f₁ x = f₂ x) ↔ f₁ g = f₂ g",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"MonoidHom.instFunLike",
"MonoidHom",
"Monoid.toMulOne... | [
"G : Type u_2\nG' : Type u_3\ninst✝¹ : Group G\ninst✝ : Group G'\ng : G\nhg : ∀ (x : G), x ∈ zpowers g\nf₁ f₂ : G →* G'\nH : f₁ g = f₂ g\nx : G\n⊢ f₁ x = f₂ x"
] | refine ⟨fun H ↦ H g, fun H x ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.GroupTheory.Sylow | {
"line": 463,
"column": 21
} | {
"line": 465,
"column": 41
} | {
"line": 467,
"column": 0
} | [
{
"pp": "p : ℕ\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fact (Nat.Prime p)\nP : Sylow p G\ninst✝ : (↑P).FiniteIndex\n⊢ ¬p ∣ (↑P).index",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Sylow.toSubgroup",
"Sylow.not_dvd_index'",
"Sylow.instSetLike",
"Subgroup.subgroup... | [] | by
have := P.finite_of_finiteIndex
exact P.not_dvd_index' Nat.card_pos.ne' | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Coprime.Ideal | {
"line": 43,
"column": 6
} | {
"line": 43,
"column": 43
} | {
"line": 44,
"column": 2
} | [
{
"pp": "case refine_1.refine_2\nι : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nI : ι → Ideal R\nthis : DecidableEq ι\na : ι\n⊢ ↑((fun i ↦ if h : i = a then ⟨1, ⋯⟩ else 0) a) = 1",
"ppTerm": "?refine_1.refine_2",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOn... | [] | simp only [dif_pos, Submodule.coe_mk] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Coprime.Ideal | {
"line": 43,
"column": 6
} | {
"line": 43,
"column": 43
} | {
"line": 44,
"column": 2
} | [
{
"pp": "case refine_1.refine_2\nι : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nI : ι → Ideal R\nthis : DecidableEq ι\na : ι\n⊢ ↑((fun i ↦ if h : i = a then ⟨1, ⋯⟩ else 0) a) = 1",
"ppTerm": "?refine_1.refine_2",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOn... | [] | simp only [dif_pos, Submodule.coe_mk] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Coprime.Ideal | {
"line": 43,
"column": 6
} | {
"line": 43,
"column": 43
} | {
"line": 44,
"column": 2
} | [
{
"pp": "case refine_1.refine_2\nι : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nI : ι → Ideal R\nthis : DecidableEq ι\na : ι\n⊢ ↑((fun i ↦ if h : i = a then ⟨1, ⋯⟩ else 0) a) = 1",
"ppTerm": "?refine_1.refine_2",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOn... | [] | simp only [dif_pos, Submodule.coe_mk] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Control.Bifunctor | {
"line": 145,
"column": 39
} | {
"line": 146,
"column": 77
} | {
"line": 148,
"column": 0
} | [
{
"pp": "F : Type u₀ → Type u₁ → Type u₂\ninst✝⁵ : Bifunctor F\nG : Type u_1 → Type u₀\nH : Type u_2 → Type u₁\ninst✝⁴ : Functor G\ninst✝³ : Functor H\ninst✝² : LawfulFunctor G\ninst✝¹ : LawfulFunctor H\ninst✝ : LawfulBifunctor F\n⊢ LawfulBifunctor (bicompl F G H)",
"ppTerm": "?m.10",
"assigned": true,
... | [] | by
constructor <;> intros <;> simp [bimap, map_id, map_comp_map, functor_norm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Sylow | {
"line": 802,
"column": 12
} | {
"line": 802,
"column": 34
} | {
"line": 802,
"column": 35
} | [
{
"pp": "case inr\nG : Type u\ninst✝³ : Group G\ninst✝² : Finite G\nhnc : ∀ (H : Subgroup G), IsCoatom H → H.Normal\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nK : Subgroup G\nhK : IsCoatom K\nhNK : normalizer ↑P ≤ K\nthis : K.Normal\nhPK : ↑P ≤ K\n⊢ K = ⊤",
"ppTerm": "?i... | [
"case inr\nG : Type u\ninst✝³ : Group G\ninst✝² : Finite G\nhnc : ∀ (H : Subgroup G), IsCoatom H → H.Normal\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nK : Subgroup G\nhK : IsCoatom K\nhNK : normalizer ↑P ≤ K\nthis : K.Normal\nhPK : ↑P ≤ K\n⊢ normalizer ↑P ⊔ K = ⊤"
] | ← sup_of_le_right hNK, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.SimpleModule.Basic | {
"line": 285,
"column": 2
} | {
"line": 285,
"column": 33
} | {
"line": 286,
"column": 2
} | [
{
"pp": "R : Type u_2\ninst✝³ : Ring R\nM : Type u_4\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsSemisimpleModule R M\n⊢ ∃ s, sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R ↥m",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Submodule",
"AddCommGroup.toAddCom... | [
"R : Type u_2\ninst✝³ : Ring R\nM : Type u_4\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsSemisimpleModule R M\nthis : sSup {m | IsSimpleModule R ↥m} = ⊤\n⊢ ∃ s, sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R ↥m"
] | have := sSup_simples_eq_top R M | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Polynomial.Module.AEval | {
"line": 112,
"column": 16
} | {
"line": 112,
"column": 43
} | {
"line": 113,
"column": 2
} | [
{
"pp": "R : Type ?u.5\nA : Type ?u.7\nM : Type ?u.13\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : Semiring A\na : A\ninst✝⁸ : Algebra R A\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module A M\ninst✝⁵ : Module R M\ninst✝⁴ : IsScalarTower R A M\nN : Type ?u.26\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : Module R[X]... | [] | by simp [LinearMap.ofAEval] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Module.AEval | {
"line": 113,
"column": 17
} | {
"line": 113,
"column": 44
} | {
"line": 115,
"column": 0
} | [
{
"pp": "R : Type ?u.5\nA : Type ?u.7\nM : Type ?u.13\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : Semiring A\na : A\ninst✝⁸ : Algebra R A\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module A M\ninst✝⁵ : Module R M\ninst✝⁴ : IsScalarTower R A M\nN : Type ?u.26\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : Module R[X]... | [] | by simp [LinearMap.ofAEval] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Finiteness.Nakayama | {
"line": 35,
"column": 4
} | {
"line": 36,
"column": 22
} | {
"line": 37,
"column": 4
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝² : CommRing R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nN : Submodule R M\nhin : N ≤ I • N\ns : Set M\nhfs : s.Finite\nhs : span R s = N\n⊢ 1 - 1 ∈ I",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [
"case refine_2\nR : Type u_1\ninst✝² : CommRing R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nN : Submodule R M\nhin : N ≤ I • N\ns : Set M\nhfs : s.Finite\nhs : span R s = N\n⊢ N ≤ comap ((LinearMap.lsmul R M) 1) (I • span R s)",
"case refine_3\nR : Type u_1\ninst✝² : CommRing R\nM :... | · rw [sub_self]
exact I.zero_mem | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Module.Torsion.Basic | {
"line": 880,
"column": 4
} | {
"line": 880,
"column": 12
} | {
"line": 881,
"column": 2
} | [
{
"pp": "case h\nR : Type u_1\nM : Type u_2\ninst✝² : Monoid R\ninst✝¹ : AddCommMonoid M\ninst✝ : DistribMulAction R M\np : R\nh : IsTorsion' M ↥(Submonoid.powers p)\nx : M\na : R\nn : ℕ\nhn : p ^ n = a\nhx : ⟨a, ⋯⟩ • x = 0\n⊢ a • x = 0",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [],
"u... | [] | apply hx | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Module.Torsion.Basic | {
"line": 1029,
"column": 2
} | {
"line": 1029,
"column": 41
} | {
"line": 1031,
"column": 0
} | [
{
"pp": "M : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : IsAddTorsionFree M\nx y : M\nh : y ≠ 0\nr s : ℕ\nhrs : ↑r • y = ↑s • y\n⊢ r = s",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Int.instIsCancelMulZero",
"AddGroupWithOne.toAddMonoidWithOne",
"Eq.mp",
"smul_le... | [] | simpa using smul_left_injective _ h hrs | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Category.ModuleCat.Presheaf | {
"line": 125,
"column": 34
} | {
"line": 127,
"column": 49
} | {
"line": 127,
"column": 50
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nR : Cᵒᵖ ⥤ RingCat\nM M₁ M₂ : PresheafOfModules R\napp : (X : Cᵒᵖ) → M₁.obj X ≅ M₂.obj X\nnaturality :\n ∀ ⦃X Y : Cᵒᵖ⦄ (f : X ⟶ Y),\n M₁.map f ≫ (ModuleCat.restrictScalars (RingCat.Hom.hom (R.map f))).map (app Y).hom = (app X).hom ≫ M₂.map f\nX Y : Cᵒᵖ\nf : ... | [] | by
rw [← cancel_epi (app X).hom, ← reassoc_of% (naturality f), Iso.map_hom_inv_id,
Category.comp_id, Iso.hom_inv_id_assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.HomologicalComplexBiprod | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 39
} | {
"line": 88,
"column": 0
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nc : ComplexShape ι\nK L : HomologicalComplex C c\ninst✝ : ∀ (i : ι), HasBinaryBiproduct (K.X i) (L.X i)\ni : ι\n⊢ biprod.inr.f i ≫ biprod.fst.f i = 0",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [... | [] | rw [← comp_f, biprod.inr_fst, zero_f] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.HomologicalComplexBiprod | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 39
} | {
"line": 88,
"column": 0
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nc : ComplexShape ι\nK L : HomologicalComplex C c\ninst✝ : ∀ (i : ι), HasBinaryBiproduct (K.X i) (L.X i)\ni : ι\n⊢ biprod.inr.f i ≫ biprod.fst.f i = 0",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [... | [] | rw [← comp_f, biprod.inr_fst, zero_f] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.HomologicalComplexBiprod | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 39
} | {
"line": 88,
"column": 0
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nc : ComplexShape ι\nK L : HomologicalComplex C c\ninst✝ : ∀ (i : ι), HasBinaryBiproduct (K.X i) (L.X i)\ni : ι\n⊢ biprod.inr.f i ≫ biprod.fst.f i = 0",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [... | [] | rw [← comp_f, biprod.inr_fst, zero_f] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.HomotopyCofiber | {
"line": 300,
"column": 4
} | {
"line": 300,
"column": 61
} | {
"line": 301,
"column": 4
} | [
{
"pp": "C : 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 ≫ α).f (c.next j) = (... | [
"C : 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.next j) = (d... | simp only [comp_f, zero_f, add_zero, prevD_eq _ hjk] at H | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.HomotopyCategory.MappingCone | {
"line": 183,
"column": 4
} | {
"line": 183,
"column": 19
} | {
"line": 184,
"column": 4
} | [
{
"pp": "case mpr\nC : 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\ni j : ℤ\nhij : i + 1 = j\nA : C\nf g : A ⟶ (mappingCone φ).X i\n⊢ f ≫ (↑(fst φ)).v i j hij = g ≫ (↑(fst φ)).v i j hij ∧ f ≫ (snd φ).v i i ⋯ = g ≫ (snd φ).v i ... | [
"case mpr\nC : 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\ni j : ℤ\nhij : i + 1 = j\nA : C\nf g : A ⟶ (mappingCone φ).X i\nh₁ : f ≫ (↑(fst φ)).v i j hij = g ≫ (↑(fst φ)).v i j hij\nh₂ : f ≫ (snd φ).v i i ⋯ = g ≫ (snd φ).v i i ⋯\n... | rintro ⟨h₁, h₂⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.Homology.HomotopyCategory.MappingCone | {
"line": 198,
"column": 4
} | {
"line": 198,
"column": 19
} | {
"line": 199,
"column": 4
} | [
{
"pp": "case mpr\nC : 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\ni j : ℤ\nhij : j + 1 = i\nA : C\nf g : (mappingCone φ).X j ⟶ A\n⊢ (inl φ).v i j ⋯ ≫ f = (inl φ).v i j ⋯ ≫ g ∧ (inr φ).f j ≫ f = (inr φ).f j ≫ g → f = g",
... | [
"case mpr\nC : 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\ni j : ℤ\nhij : j + 1 = i\nA : C\nf g : (mappingCone φ).X j ⟶ A\nh₁ : (inl φ).v i j ⋯ ≫ f = (inl φ).v i j ⋯ ≫ g\nh₂ : (inr φ).f j ≫ f = (inr φ).f j ≫ g\n⊢ f = g"
] | rintro ⟨h₁, h₂⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplex | {
"line": 377,
"column": 2
} | {
"line": 378,
"column": 76
} | {
"line": 380,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nF G : CochainComplex C ℤ\nn : ℤ\nz₁ : Cochain F G n\n⊢ z₁.comp (ofHom (𝟙 G)) ⋯ = z₁",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"HomologicalComplex.instCategory",
"CategoryTheory.CategoryStruct.toQuive... | [] | ext p q hpq
simp only [comp_zero_cochain_v, ofHom_v, HomologicalComplex.id_f, comp_id] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplex | {
"line": 377,
"column": 2
} | {
"line": 378,
"column": 76
} | {
"line": 380,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nF G : CochainComplex C ℤ\nn : ℤ\nz₁ : Cochain F G n\n⊢ z₁.comp (ofHom (𝟙 G)) ⋯ = z₁",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"HomologicalComplex.instCategory",
"CategoryTheory.CategoryStruct.toQuive... | [] | ext p q hpq
simp only [comp_zero_cochain_v, ofHom_v, HomologicalComplex.id_f, comp_id] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.HomotopyCategory.MappingCone | {
"line": 219,
"column": 4
} | {
"line": 219,
"column": 19
} | {
"line": 220,
"column": 4
} | [
{
"pp": "case mpr\nC : 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\ni j : ℤ\nhij : i + 1 = j\nK : CochainComplex C ℤ\nγ₁ γ₂ : Cochain K (mappingCone φ) i\n⊢ γ₁.comp (↑(fst φ)) hij = γ₂.comp (↑(fst φ)) hij ∧ γ₁.comp (snd φ) ⋯ =... | [
"case mpr\nC : 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\ni j : ℤ\nhij : i + 1 = j\nK : CochainComplex C ℤ\nγ₁ γ₂ : Cochain K (mappingCone φ) i\nh₁ : γ₁.comp (↑(fst φ)) hij = γ₂.comp (↑(fst φ)) hij\nh₂ : γ₁.comp (snd φ) ⋯ = γ₂.c... | rintro ⟨h₁, h₂⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.RingTheory.Kaehler.Basic | {
"line": 234,
"column": 2
} | {
"line": 234,
"column": 48
} | {
"line": 235,
"column": 2
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S ⊗[R] S\nhx✝ : x ∈ ideal R S\nhx : x ∈ Submodule.span S (Set.range fun s ↦ 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1)\n⊢ ∃ (hx : x ∈ ideal R S), (fromIdeal R S) ⟨x, hx⟩ ∈ Submodule.span S (Set.range ⇑(D R S))",
"ppTerm": "?m.10... | [
"case refine_1\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S ⊗[R] S\nhx✝ : x ∈ ideal R S\nhx : x ∈ Submodule.span S (Set.range fun s ↦ 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1)\n⊢ ∀ x ∈ Set.range fun s ↦ 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1,\n ∃ (hx : x ∈ ideal R S), (fromIdeal R S) ⟨x, hx⟩ ∈ Submo... | refine Submodule.span_induction ?_ ?_ ?_ ?_ hx | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Homology.HomotopyCategory.MappingCone | {
"line": 237,
"column": 4
} | {
"line": 237,
"column": 19
} | {
"line": 238,
"column": 4
} | [
{
"pp": "case mpr\nC : 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\ni j : ℤ\nhij : i + 1 = j\nK : CochainComplex C ℤ\nγ₁ γ₂ : Cochain (mappingCone φ) K j\n⊢ (inl φ).comp γ₁ ⋯ = (inl φ).comp γ₂ ⋯ ∧ (Cochain.ofHom (inr φ)).comp ... | [
"case mpr\nC : 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\ni j : ℤ\nhij : i + 1 = j\nK : CochainComplex C ℤ\nγ₁ γ₂ : Cochain (mappingCone φ) K j\nh₁ : (inl φ).comp γ₁ ⋯ = (inl φ).comp γ₂ ⋯\nh₂ : (Cochain.ofHom (inr φ)).comp γ₁ ⋯ ... | rintro ⟨h₁, h₂⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.CategoryTheory.Quotient.Preadditive | {
"line": 53,
"column": 6
} | {
"line": 53,
"column": 39
} | {
"line": 54,
"column": 6
} | [
{
"pp": "C : Type ?u.2\ninst✝² : Category.{v_1, ?u.2} C\ninst✝¹ : Preadditive C\nr : HomRel C\ninst✝ : Congruence r\nhr : ∀ ⦃X Y : C⦄ (f₁ f₂ g₁ g₂ : X ⟶ Y), r f₁ f₂ → r g₁ g₂ → r (f₁ + g₁) (f₂ + g₂)\nX Y : Quotient r\nf✝ : X ⟶ Y\nf g : X.as ⟶ Y.as\nhfg : r f g\n⊢ r (-f) (-g)",
"ppTerm": "?m.62",
"assign... | [
"C : Type ?u.2\ninst✝² : Category.{v_1, ?u.2} C\ninst✝¹ : Preadditive C\nr : HomRel C\ninst✝ : Congruence r\nhr : ∀ ⦃X Y : C⦄ (f₁ f₂ g₁ g₂ : X ⟶ Y), r f₁ f₂ → r g₁ g₂ → r (f₁ + g₁) (f₂ + g₂)\nX Y : Quotient r\nf✝ : X ⟶ Y\nf g : X.as ⟶ Y.as\nhfg : r f g\n⊢ r (-g) (-f)"
] | apply Congruence.equivalence.symm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Homology.HomotopyCategory | {
"line": 264,
"column": 4
} | {
"line": 264,
"column": 30
} | {
"line": 265,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : Semiring R\nι : Type u_2\nV : Type u\ninst✝⁵ : Category.{v, u} V\ninst✝⁴ : Preadditive V\nc✝ : ComplexShape ι\nW : Type u_3\ninst✝³ : Category.{v_1, u_3} W\ninst✝² : Preadditive W\nF G : V ⥤ W\ninst✝¹ : F.Additive\ninst✝ : G.Additive\nα : F ⟶ G\nc : ComplexShape ι\n⊢ ∀ ⦃X Y : Hom... | [
"case mk\nR : Type u_1\ninst✝⁶ : Semiring R\nι : Type u_2\nV : Type u\ninst✝⁵ : Category.{v, u} V\ninst✝⁴ : Preadditive V\nc✝ : ComplexShape ι\nW : Type u_3\ninst✝³ : Category.{v_1, u_3} W\ninst✝² : Preadditive W\nF G : V ⥤ W\ninst✝¹ : F.Additive\ninst✝ : G.Additive\nα : F ⟶ G\nc : ComplexShape ι\nC D : Homological... | rintro ⟨C⟩ ⟨D⟩ ⟨f : C ⟶ D⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.CategoryTheory.Shift.Induced | {
"line": 124,
"column": 24
} | {
"line": 124,
"column": 42
} | {
"line": 124,
"column": 42
} | [
{
"pp": "C : Type ?u.2\nD : Type ?u.4\ninst✝⁵ : Category.{v_1, ?u.2} C\ninst✝⁴ : Category.{v_2, ?u.4} D\nF : C ⥤ D\nA : Type ?u.15\ninst✝³ : AddMonoid A\ninst✝² : HasShift C A\ns : A → D ⥤ D\ni : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F\ninst✝¹ : ((whiskeringLeft C D D).obj F).Full\ninst✝ : ((whiskeringLeft C D... | [] | rw [add_zero]; rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Shift.Induced | {
"line": 124,
"column": 24
} | {
"line": 124,
"column": 42
} | {
"line": 124,
"column": 42
} | [
{
"pp": "C : Type ?u.2\nD : Type ?u.4\ninst✝⁵ : Category.{v_1, ?u.2} C\ninst✝⁴ : Category.{v_2, ?u.4} D\nF : C ⥤ D\nA : Type ?u.15\ninst✝³ : AddMonoid A\ninst✝² : HasShift C A\ns : A → D ⥤ D\ni : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F\ninst✝¹ : ((whiskeringLeft C D D).obj F).Full\ninst✝ : ((whiskeringLeft C D... | [] | rw [add_zero]; rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Shift.CommShift | {
"line": 417,
"column": 48
} | {
"line": 417,
"column": 61
} | {
"line": 418,
"column": 2
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁸ : Category.{v_1, u_1} C\ninst✝⁷ : Category.{v_2, u_2} D\nF₁ F₂ : C ⥤ D\nτ : F₁ ⟶ F₂\nA : Type u_5\ninst✝⁶ : AddMonoid A\ninst✝⁵ : HasShift C A\ninst✝⁴ : HasShift D A\ninst✝³ : F₁.CommShift A\ninst✝² : F₂.CommShift A\ninst✝¹ : IsIso τ\ninst✝ : CommShift τ A\n⊢ CommShif... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Shift.CommShift | {
"line": 485,
"column": 18
} | {
"line": 485,
"column": 59
} | {
"line": 485,
"column": 59
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Category.{v_2, u_2} D\nF G : C ⥤ D\ne : F ≅ G\nA : Type u_4\ninst✝³ : AddMonoid A\ninst✝² : HasShift C A\ninst✝¹ : HasShift D A\ninst✝ : F.CommShift A\nthis : G.CommShift A := ofIso e A\na : A\n⊢ (commShiftIso F a).hom ≫ whiskerRight ... | [] | by ext; simp [ofIso_commShiftIso_hom_app] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Triangulated.Triangulated | {
"line": 181,
"column": 4
} | {
"line": 182,
"column": 52
} | {
"line": 183,
"column": 2
} | [
{
"pp": "case refine_2\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nX₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C\nu₁₂ : X₁ ⟶ X₂\nu₂₃ : X₂ ⟶ X₃\nu₁₃ : X₁ ⟶ X₃\ncomm : u₁₂ ≫ u₂₃ = u₁₃... | [] | rw [← cancel_mono (e₁.hom⟦(1 : ℤ)⟧'), eq₁₂', assoc, assoc, assoc, eq₁₃',
iso₁₃.inv_hom_id_triangle_hom₃_assoc, ← rel₁₃] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Triangulated.Triangulated | {
"line": 181,
"column": 4
} | {
"line": 182,
"column": 52
} | {
"line": 183,
"column": 2
} | [
{
"pp": "case refine_2\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nX₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C\nu₁₂ : X₁ ⟶ X₂\nu₂₃ : X₂ ⟶ X₃\nu₁₃ : X₁ ⟶ X₃\ncomm : u₁₂ ≫ u₂₃ = u₁₃... | [] | rw [← cancel_mono (e₁.hom⟦(1 : ℤ)⟧'), eq₁₂', assoc, assoc, assoc, eq₁₃',
iso₁₃.inv_hom_id_triangle_hom₃_assoc, ← rel₁₃] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Triangulated.Triangulated | {
"line": 181,
"column": 4
} | {
"line": 182,
"column": 52
} | {
"line": 183,
"column": 2
} | [
{
"pp": "case refine_2\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nX₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C\nu₁₂ : X₁ ⟶ X₂\nu₂₃ : X₂ ⟶ X₃\nu₁₃ : X₁ ⟶ X₃\ncomm : u₁₂ ≫ u₂₃ = u₁₃... | [] | rw [← cancel_mono (e₁.hom⟦(1 : ℤ)⟧'), eq₁₂', assoc, assoc, assoc, eq₁₃',
iso₁₃.inv_hom_id_triangle_hom₃_assoc, ← rel₁₃] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Triangulated.Triangulated | {
"line": 157,
"column": 75
} | {
"line": 194,
"column": 77
} | {
"line": 196,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nX₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C\nu₁₂ : X₁ ⟶ X₂\nu₂₃ : X₂ ⟶ X₃\nu₁₃ : X₁ ⟶ X₃\ncomm : u₁₂ ≫ u₂₃ = u₁₃\nv₁₂ : X₂ ⟶ Z₁... | [] | by
let iso₁₂ := isoTriangleOfIso₁₂ _ _ h₁₂ h₁₂' e₁ e₂ comm₁₂
let iso₂₃ := isoTriangleOfIso₁₂ _ _ h₂₃ h₂₃' e₂ e₃ comm₂₃
let iso₁₃ := isoTriangleOfIso₁₂ _ _ h₁₃ h₁₃' e₁ e₃ (by
dsimp; rw [← comm, assoc, ← comm', ← reassoc_of% comm₁₂, comm₂₃])
have eq₁₂ := iso₁₂.hom.comm₂
have eq₁₂' := iso₁₂.hom.comm₃
have ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Triangulated.Triangulated | {
"line": 357,
"column": 6
} | {
"line": 357,
"column": 42
} | {
"line": 357,
"column": 43
} | [
{
"pp": "case refine_2\nC : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\ninst✝ : IsTriangulated C\nX₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C\nu₁₂ : X₁ ⟶ X₂\nu₂₃ : X₂ ⟶ X₃\nu₁₃ : X₁ ⟶... | [
"case refine_2\nC : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\ninst✝ : IsTriangulated C\nX₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C\nu₁₂ : X₁ ⟶ X₂\nu₂₃ : X₂ ⟶ X₃\nu₁₃ : X₁ ⟶ X₃\ncomm : ... | shift_shiftFunctorCompIsoId_inv_app, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Triangulated.Triangulated | {
"line": 360,
"column": 6
} | {
"line": 360,
"column": 42
} | {
"line": 360,
"column": 43
} | [
{
"pp": "case refine_3\nC : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\ninst✝ : IsTriangulated C\nX₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C\nu₁₂ : X₁ ⟶ X₂\nu₂₃ : X₂ ⟶ X₃\nu₁₃ : X₁ ⟶... | [
"case refine_3\nC : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\ninst✝ : IsTriangulated C\nX₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C\nu₁₂ : X₁ ⟶ X₂\nu₂₃ : X₂ ⟶ X₃\nu₁₃ : X₁ ⟶ X₃\ncomm : ... | shift_shiftFunctorCompIsoId_inv_app, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Triangulated.Triangulated | {
"line": 372,
"column": 44
} | {
"line": 372,
"column": 80
} | {
"line": 373,
"column": 6
} | [
{
"pp": "case refine_5\nC : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\ninst✝ : IsTriangulated C\nX₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C\nu₁₂ : X₁ ⟶ X₂\nu₂₃ : X₂ ⟶ X₃\nu₁₃ : X₁ ⟶... | [
"case refine_5\nC : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Preadditive C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\ninst✝ : IsTriangulated C\nX₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C\nu₁₂ : X₁ ⟶ X₂\nu₂₃ : X₂ ⟶ X₃\nu₁₃ : X₁ ⟶ X₃\ncomm : ... | shift_shiftFunctorCompIsoId_inv_app, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Shift.ShiftSequence | {
"line": 236,
"column": 33
} | {
"line": 236,
"column": 69
} | {
"line": 236,
"column": 69
} | [
{
"pp": "C : Type u_1\nA : Type u_2\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Category.{v_2, u_2} A\nF : C ⥤ A\nM : Type u_3\ninst✝⁴ : AddMonoid M\ninst✝³ : HasShift C M\nG : Type u_4\ninst✝² : AddGroup G\ninst✝¹ : HasShift C G\ninst✝ : F.ShiftSequence M\nX Y : C\nm : M\nf : X ⟶ (shiftFunctor C m).obj Y\nn mn :... | [] | rw [← ha'', ← ha', ← hnm, add_assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Shift.ShiftSequence | {
"line": 236,
"column": 33
} | {
"line": 236,
"column": 69
} | {
"line": 236,
"column": 69
} | [
{
"pp": "C : Type u_1\nA : Type u_2\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Category.{v_2, u_2} A\nF : C ⥤ A\nM : Type u_3\ninst✝⁴ : AddMonoid M\ninst✝³ : HasShift C M\nG : Type u_4\ninst✝² : AddGroup G\ninst✝¹ : HasShift C G\ninst✝ : F.ShiftSequence M\nX Y : C\nm : M\nf : X ⟶ (shiftFunctor C m).obj Y\nn mn :... | [] | rw [← ha'', ← ha', ← hnm, add_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Shift.ShiftSequence | {
"line": 236,
"column": 33
} | {
"line": 236,
"column": 69
} | {
"line": 236,
"column": 69
} | [
{
"pp": "C : Type u_1\nA : Type u_2\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Category.{v_2, u_2} A\nF : C ⥤ A\nM : Type u_3\ninst✝⁴ : AddMonoid M\ninst✝³ : HasShift C M\nG : Type u_4\ninst✝² : AddGroup G\ninst✝¹ : HasShift C G\ninst✝ : F.ShiftSequence M\nX Y : C\nm : M\nf : X ⟶ (shiftFunctor C m).obj Y\nn mn :... | [] | rw [← ha'', ← ha', ← hnm, add_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.PathCategory.Basic | {
"line": 227,
"column": 19
} | {
"line": 227,
"column": 28
} | {
"line": 229,
"column": 0
} | [
{
"pp": "case cons\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y Z : C\nf : Path X Y\nb✝ c✝ : C\ng : Path Y b✝\ne : b✝ ⟶ c✝\nih : composePath (f.comp g) = composePath f ≫ composePath g\n⊢ composePath (f.comp (g.cons e)) = composePath f ≫ composePath (g.cons e)",
"ppTerm": "?cons",
"assigned": true,
... | [] | simp [ih] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.PathCategory.Basic | {
"line": 227,
"column": 19
} | {
"line": 227,
"column": 28
} | {
"line": 229,
"column": 0
} | [
{
"pp": "case cons\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y Z : C\nf : Path X Y\nb✝ c✝ : C\ng : Path Y b✝\ne : b✝ ⟶ c✝\nih : composePath (f.comp g) = composePath f ≫ composePath g\n⊢ composePath (f.comp (g.cons e)) = composePath f ≫ composePath (g.cons e)",
"ppTerm": "?cons",
"assigned": true,
... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.PathCategory.Basic | {
"line": 227,
"column": 19
} | {
"line": 227,
"column": 28
} | {
"line": 229,
"column": 0
} | [
{
"pp": "case cons\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y Z : C\nf : Path X Y\nb✝ c✝ : C\ng : Path Y b✝\ne : b✝ ⟶ c✝\nih : composePath (f.comp g) = composePath f ≫ composePath g\n⊢ composePath (f.comp (g.cons e)) = composePath f ≫ composePath (g.cons e)",
"ppTerm": "?cons",
"assigned": true,
... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Localization.CalculusOfFractions.Preadditive | {
"line": 284,
"column": 2
} | {
"line": 284,
"column": 87
} | {
"line": 286,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\ninst✝² : Preadditive C\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\ninst✝ : W.HasLeftCalculusOfFractions\nX Y Z : C\nX' Y' Z' : D\neX : L.obj X ≅ X'\neY : L.obj Y ≅ Y'\neZ : L.obj Z ≅ Z'\nthi... | [] | exact Equiv.addCommGroup (homEquiv (L.objObjPreimageIso X') (L.objObjPreimageIso Y')) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Shift.Localization | {
"line": 302,
"column": 2
} | {
"line": 303,
"column": 38
} | {
"line": 305,
"column": 0
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\ninst✝¹² : Category.{v_1, u_1} C₁\ninst✝¹¹ : Category.{v_2, u_2} C₂\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nΦ : LocalizerMorphism W₁ W₂\nM : Type u_3\ninst✝¹⁰ : AddMonoid M\ninst✝⁹ : HasShift C₁ M\ninst✝⁸ : HasShift C₂ M\ninst✝⁷ : Φ.functor.CommShift M\nD₁ : Ty... | [] | simp [Functor.commShiftOfLocalization_iso_inv_app,
Functor.commShiftIso_comp_inv_app] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Shift.Localization | {
"line": 302,
"column": 2
} | {
"line": 303,
"column": 38
} | {
"line": 305,
"column": 0
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\ninst✝¹² : Category.{v_1, u_1} C₁\ninst✝¹¹ : Category.{v_2, u_2} C₂\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nΦ : LocalizerMorphism W₁ W₂\nM : Type u_3\ninst✝¹⁰ : AddMonoid M\ninst✝⁹ : HasShift C₁ M\ninst✝⁸ : HasShift C₂ M\ninst✝⁷ : Φ.functor.CommShift M\nD₁ : Ty... | [] | simp [Functor.commShiftOfLocalization_iso_inv_app,
Functor.commShiftIso_comp_inv_app] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Shift.Localization | {
"line": 302,
"column": 2
} | {
"line": 303,
"column": 38
} | {
"line": 305,
"column": 0
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\ninst✝¹² : Category.{v_1, u_1} C₁\ninst✝¹¹ : Category.{v_2, u_2} C₂\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nΦ : LocalizerMorphism W₁ W₂\nM : Type u_3\ninst✝¹⁰ : AddMonoid M\ninst✝⁹ : HasShift C₁ M\ninst✝⁸ : HasShift C₂ M\ninst✝⁷ : Φ.functor.CommShift M\nD₁ : Ty... | [] | simp [Functor.commShiftOfLocalization_iso_inv_app,
Functor.commShiftIso_comp_inv_app] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.ObjectProperty.LimitsClosure | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 24
} | {
"line": 89,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nP : ObjectProperty C\nα : Type t\nJ : α → Type u'\ninst✝ : (a : α) → Category.{v', u'} (J a)\n⊢ P.isoClosure ≤ P.limitsClosure J",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Prop.le",
"id",
... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nP : ObjectProperty C\nα : Type t\nJ : α → Type u'\ninst✝ : (a : α) → Category.{v', u'} (J a)\n⊢ P ≤ P.limitsClosure J"
] | rw [isoClosure_le_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.ObjectProperty.ColimitsClosure | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 24
} | {
"line": 91,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nP : ObjectProperty C\nα : Type t\nJ : α → Type u'\ninst✝ : (a : α) → Category.{v', u'} (J a)\n⊢ P.isoClosure ≤ P.colimitsClosure J",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Prop.le",
"Cate... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nP : ObjectProperty C\nα : Type t\nJ : α → Type u'\ninst✝ : (a : α) → Category.{v', u'} (J a)\n⊢ P ≤ P.colimitsClosure J"
] | rw [isoClosure_le_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.ObjectProperty.LimitsClosure | {
"line": 219,
"column": 4
} | {
"line": 219,
"column": 26
} | {
"line": 220,
"column": 4
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝² : Category.{v, u} C\nP : ObjectProperty C\nα : Type t\nJ : α → Type u'\ninst✝¹ : (a : α) → Category.{v', u'} (J a)\nκ : Cardinal.{w}\ninst✝ : Fact κ.IsRegular\nh : ∀ (a : α), HasCardinalLT (J a) κ\n⊢ (P.strictLimitsClosureIter J κ.ord).isoClosure ≤ P.limitsClosure J",
... | [
"case refine_1\nC : Type u\ninst✝² : Category.{v, u} C\nP : ObjectProperty C\nα : Type t\nJ : α → Type u'\ninst✝¹ : (a : α) → Category.{v', u'} (J a)\nκ : Cardinal.{w}\ninst✝ : Fact κ.IsRegular\nh : ∀ (a : α), HasCardinalLT (J a) κ\n⊢ P.strictLimitsClosureIter J κ.ord ≤ P.limitsClosure J"
] | rw [isoClosure_le_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Localization.CalculusOfFractions | {
"line": 624,
"column": 4
} | {
"line": 624,
"column": 55
} | {
"line": 624,
"column": 55
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\nW : MorphismProperty C\ninst✝¹ : W.HasLeftCalculusOfFractions\nE : Type u_3\ninst✝ : Category.{v_3, u_3} E\nF : C ⥤ E\nhF : W.IsInvertedBy F\nX Y : C\nf : X ⟶ Y\n⊢ Hom.map ((Q W).map f) F hF =\n 𝟙 ((Q W ⋙ { obj := fun X ↦ F.obj X, map := fun {x x_1} f ↦... | [] | rw [Q_map, Hom.map_mk, id_comp, comp_id, map_ofHom] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Localization.CalculusOfFractions | {
"line": 881,
"column": 4
} | {
"line": 881,
"column": 40
} | {
"line": 882,
"column": 2
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\nL : C ⥤ D\nW✝ : MorphismProperty C\ninst✝ : L.IsLocalization W✝\nW : MorphismProperty Cᵒᵖ\nh : W.HasLeftCalculusOfFractions\nX Y : C\nφ : W.unop.LeftFraction X Y\nψ : W.LeftFraction (Opposite.op Y) (Opposite.op ... | [] | exact ⟨ψ.unop, Quiver.Hom.op_inj eq⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Localization.CalculusOfFractions | {
"line": 890,
"column": 4
} | {
"line": 890,
"column": 40
} | {
"line": 891,
"column": 2
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\nL : C ⥤ D\nW✝ : MorphismProperty C\ninst✝ : L.IsLocalization W✝\nW : MorphismProperty Cᵒᵖ\nh : W.HasRightCalculusOfFractions\nX Y : C\nφ : W.unop.RightFraction X Y\nψ : W.RightFraction (Opposite.op Y) (Opposite.... | [] | exact ⟨ψ.unop, Quiver.Hom.op_inj eq⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Triangulated.Subcategory | {
"line": 449,
"column": 2
} | {
"line": 452,
"column": 89
} | {
"line": 454,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : Preadditive C\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nP : ObjectProperty C\nn : ℕ\n⊢ (P.retractClosure.extensionProductIter n).retractClosure = (P.extensionProduc... | [] | apply le_antisymm
· rw [retractClosure_le_iff]
exact extensionProductIter_retractClosure_le P
· exact monotone_retractClosure (monotone_extensionProductIter (le_retractClosure P) n) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Triangulated.Subcategory | {
"line": 449,
"column": 2
} | {
"line": 452,
"column": 89
} | {
"line": 454,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : Preadditive C\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nP : ObjectProperty C\nn : ℕ\n⊢ (P.retractClosure.extensionProductIter n).retractClosure = (P.extensionProduc... | [] | apply le_antisymm
· rw [retractClosure_le_iff]
exact extensionProductIter_retractClosure_le P
· exact monotone_retractClosure (monotone_extensionProductIter (le_retractClosure P) n) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Triangulated.Subcategory | {
"line": 490,
"column": 4
} | {
"line": 492,
"column": 61
} | {
"line": 493,
"column": 2
} | [
{
"pp": "case mp\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : Preadditive C\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nP : ObjectProperty C\nX Y : C\nf : X ⟶ Y\n⊢ P.isoClosure.trW f → P.trW f",
"ppTerm": "?mp",
... | [] | rintro ⟨Z, g, h, mem, ⟨Z', hZ', ⟨e⟩⟩⟩
refine ⟨Z', g ≫ e.hom, e.inv ≫ h, isomorphic_distinguished _ mem _ ?_, hZ'⟩
exact Triangle.isoMk _ _ (Iso.refl _) (Iso.refl _) e.symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Triangulated.Subcategory | {
"line": 490,
"column": 4
} | {
"line": 492,
"column": 61
} | {
"line": 493,
"column": 2
} | [
{
"pp": "case mp\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : Preadditive C\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nP : ObjectProperty C\nX Y : C\nf : X ⟶ Y\n⊢ P.isoClosure.trW f → P.trW f",
"ppTerm": "?mp",
... | [] | rintro ⟨Z, g, h, mem, ⟨Z', hZ', ⟨e⟩⟩⟩
refine ⟨Z', g ≫ e.hom, e.inv ≫ h, isomorphic_distinguished _ mem _ ?_, hZ'⟩
exact Triangle.isoMk _ _ (Iso.refl _) (Iso.refl _) e.symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Embedding.IsSupported | {
"line": 139,
"column": 4
} | {
"line": 139,
"column": 19
} | {
"line": 140,
"column": 4
} | [
{
"pp": "case mpr\nι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝¹ : Category.{v_1, u_3} C\ninst✝ : HasZeroMorphisms C\nK : HomologicalComplex C c'\ne : c.Embedding c'\n⊢ K.IsStrictlySupported e ∧ K.IsStrictlySupportedOutside e → IsZero K",
"ppTerm": "?mpr",
"... | [
"case mpr\nι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝¹ : Category.{v_1, u_3} C\ninst✝ : HasZeroMorphisms C\nK : HomologicalComplex C c'\ne : c.Embedding c'\nh₁ : K.IsStrictlySupported e\nh₂ : K.IsStrictlySupportedOutside e\n⊢ IsZero K"
] | rintro ⟨h₁, h₂⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.Homology.Embedding.Extend | {
"line": 137,
"column": 34
} | {
"line": 137,
"column": 51
} | {
"line": 137,
"column": 52
} | [
{
"pp": "case inr.inr.inr\nι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝² : Category.{v_1, u_3} C\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nK L M : HomologicalComplex C c\nφ : K ⟶ L\nφ' : L ⟶ M\ne : c.Embedding c'\ni' j' k' : ι'\nx✝¹ : c'.Rel i' j'\nx✝ :... | [
"case inr.inr.inr\nι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝² : Category.{v_1, u_3} C\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nK L M : HomologicalComplex C c\nφ : K ⟶ L\nφ' : L ⟶ M\ne : c.Embedding c'\ni' j' k' : ι'\nx✝¹ : c'.Rel i' j'\nx✝ : c'.Rel j' k... | K.d_comp_d_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.Embedding.Boundary | {
"line": 94,
"column": 91
} | {
"line": 96,
"column": 46
} | {
"line": 98,
"column": 0
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\ne : c.Embedding c'\nj : ι\nhj : e.BoundaryGE j\ninst✝ : e.IsTruncLE\n⊢ False",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"ComplexShape.Embedding.mem_prev",
"Eq.mpr",
"False",
"congrA... | [] | by
obtain ⟨i, hi⟩ := e.mem_prev hj.1
exact hj.2 i (by simpa only [hi] using hj.1) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.Embedding.Extend | {
"line": 207,
"column": 2
} | {
"line": 207,
"column": 19
} | {
"line": 208,
"column": 2
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝² : Category.{v_1, u_3} C\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nK L : HomologicalComplex C c\nφ : K ⟶ L\ne : c.Embedding c'\ni : ι\ni' : ι'\nh : e.f i = i'\n⊢ (extendMap φ e).f i' = (K.extendXIso ... | [
"ι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝² : Category.{v_1, u_3} C\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nK L : HomologicalComplex C c\nφ : K ⟶ L\ne : c.Embedding c'\ni : ι\ni' : ι'\nh : e.f i = i'\n⊢ extend.mapX φ (e.r i') = (K.extendXIso e h).hom ≫... | dsimp [extendMap] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.Algebra.Homology.Embedding.Extend | {
"line": 214,
"column": 2
} | {
"line": 214,
"column": 19
} | {
"line": 215,
"column": 2
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝² : Category.{v_1, u_3} C\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nK L : HomologicalComplex C c\nφ : K ⟶ L\ne : c.Embedding c'\ni' : ι'\nhi' : ∀ (i : ι), e.f i ≠ i'\n⊢ (extendMap φ e).f i' = 0",
... | [
"ι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝² : Category.{v_1, u_3} C\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nK L : HomologicalComplex C c\nφ : K ⟶ L\ne : c.Embedding c'\ni' : ι'\nhi' : ∀ (i : ι), e.f i ≠ i'\n⊢ extend.mapX φ (e.r i') = 0"
] | dsimp [extendMap] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.Algebra.Homology.DerivedCategory.FullyFaithful | {
"line": 53,
"column": 4
} | {
"line": 53,
"column": 85
} | {
"line": 54,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasDerivedCategory C\nn : ℤ\nA B : C\nf : Q.obj ((CochainComplex.singleFunctor C n).obj A) ⟶ Q.obj ((CochainComplex.singleFunctor C n).obj B)\n⊢ ∃ f', f = Q.map f'",
"ppTerm": "?m.75",
"assigned": true,
"usedConstants": [
... | [
"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasDerivedCategory C\nn : ℤ\nA B : C\nX : CochainComplex C ℤ\nw✝² : X.IsStrictlyGE n\nw✝¹ : X.IsStrictlyLE n\ns : X ⟶ (CochainComplex.singleFunctor C n).obj A\nw✝ : IsIso (Q.map s)\ng : X ⟶ (CochainComplex.singleFunctor C n).obj B\n⊢ ∃ f', inv (Q.... | obtain ⟨X, _, _, s, _, g, rfl⟩ := right_fac_of_isStrictlyLE_of_isStrictlyGE n n f | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic | {
"line": 70,
"column": 4
} | {
"line": 70,
"column": 19
} | {
"line": 71,
"column": 4
} | [
{
"pp": "case mpr\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasDerivedCategory C\n⊢ (∀ (X Y : C) (n : ℤ),\n 0 ≤ n →\n Small.{w, w'}\n ((singleFunctor C 0).obj X ⟶ (shiftFunctor (DerivedCategory C) n).obj ((singleFunctor C 0).obj Y))) →\n ∀ (X Y : C) (a b : ℤ),\... | [
"case mpr\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasDerivedCategory C\nh :\n ∀ (X Y : C) (n : ℤ),\n 0 ≤ n →\n Small.{w, w'} ((singleFunctor C 0).obj X ⟶ (shiftFunctor (DerivedCategory C) n).obj ((singleFunctor C 0).obj Y))\nX Y : C\na b : ℤ\n⊢ Small.{w, w'}\n ((shiftFunctor... | intro h X Y a b | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic | {
"line": 344,
"column": 13
} | {
"line": 351,
"column": 22
} | {
"line": 353,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nY : C\nn : ℕ\nX₁ X₂ : C\nα β : Ext (X₁ ⊞ X₂) Y n\nh₁ : (mk₀ biprod.inl).comp α ⋯ = (mk₀ biprod.inl).comp β ⋯\nh₂ : (mk₀ biprod.inr).comp α ⋯ = (mk₀ biprod.inr).comp β ⋯\n⊢ α = β",
"ppTerm": "?m.119",
"assigned": true,... | [] | by
letI := HasDerivedCategory.standard C
rw [Ext.ext_iff] at h₁ h₂ ⊢
simp only [comp_hom, mk₀_hom, ShiftedHom.mk₀_comp] at h₁ h₂
apply BinaryCofan.IsColimit.hom_ext
(isBinaryBilimitOfPreserves (singleFunctor C 0)
(BinaryBiproduct.isBilimit X₁ X₂)).isColimit
all_goals assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Shift.Adjunction | {
"line": 475,
"column": 6
} | {
"line": 475,
"column": 42
} | {
"line": 475,
"column": 43
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Category.{v_2, u_2} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nA : Type u_3\ninst✝³ : AddGroup A\ninst✝² : HasShift C A\ninst✝¹ : HasShift D A\ninst✝ : G.CommShift A\na : A\nX✝ : C\n⊢ (shiftFunctor C a).map (adj.unit.app X✝) =\n (shiftF... | [
"C : Type u_1\nD : Type u_2\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Category.{v_2, u_2} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nA : Type u_3\ninst✝³ : AddGroup A\ninst✝² : HasShift C A\ninst✝¹ : HasShift D A\ninst✝ : G.CommShift A\na : A\nX✝ : C\n⊢ (shiftFunctor C a).map (adj.unit.app X✝) =\n (shiftFunctor C a).... | shift_shiftFunctorCompIsoId_inv_app, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Linear | {
"line": 45,
"column": 2
} | {
"line": 45,
"column": 38
} | {
"line": 46,
"column": 2
} | [
{
"pp": "R : Type t\ninst✝⁴ : Ring R\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : Linear R C\ninst✝ : HasExt C\nX Y : C\nn : ℕ\nx : Ext X Y n\nr : R\n⊢ r • x = x.comp (mk₀ (r • 𝟙 Y)) ⋯",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"HasDerivedCategory.stand... | [
"R : Type t\ninst✝⁴ : Ring R\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : Linear R C\ninst✝ : HasExt C\nX Y : C\nn : ℕ\nx : Ext X Y n\nr : R\nthis : HasDerivedCategory C := HasDerivedCategory.standard C\n⊢ r • x = x.comp (mk₀ (r • 𝟙 Y)) ⋯"
] | let := HasDerivedCategory.standard C | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Linear | {
"line": 61,
"column": 2
} | {
"line": 61,
"column": 38
} | {
"line": 62,
"column": 2
} | [
{
"pp": "R : Type t\ninst✝⁴ : Ring R\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : Linear R C\ninst✝ : HasExt C\nX Y Z : C\na b : ℕ\nα : Ext X Y a\nβ : Ext Y Z b\nc : ℕ\nh : a + b = c\nr : R\n⊢ α.comp (r • β) h = r • α.comp β h",
"ppTerm": "?m.49",
"assigned": true,
"usedCons... | [
"R : Type t\ninst✝⁴ : Ring R\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : Linear R C\ninst✝ : HasExt C\nX Y Z : C\na b : ℕ\nα : Ext X Y a\nβ : Ext Y Z b\nc : ℕ\nh : a + b = c\nr : R\nthis : HasDerivedCategory C := HasDerivedCategory.standard C\n⊢ α.comp (r • β) h = r • α.comp β h"
] | let := HasDerivedCategory.standard C | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Linear | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 38
} | {
"line": 69,
"column": 2
} | [
{
"pp": "R : Type t\ninst✝⁴ : Ring R\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : Linear R C\ninst✝ : HasExt C\nX Y Z : C\na b : ℕ\nα : Ext X Y a\nβ : Ext Y Z b\nc : ℕ\nh : a + b = c\nr : R\n⊢ (r • α).comp β h = r • α.comp β h",
"ppTerm": "?m.49",
"assigned": true,
"usedCons... | [
"R : Type t\ninst✝⁴ : Ring R\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : Linear R C\ninst✝ : HasExt C\nX Y Z : C\na b : ℕ\nα : Ext X Y a\nβ : Ext Y Z b\nc : ℕ\nh : a + b = c\nr : R\nthis : HasDerivedCategory C := HasDerivedCategory.standard C\n⊢ (r • α).comp β h = r • α.comp β h"
] | let := HasDerivedCategory.standard C | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Linear | {
"line": 83,
"column": 2
} | {
"line": 83,
"column": 38
} | {
"line": 84,
"column": 2
} | [
{
"pp": "R : Type t\ninst✝⁴ : Ring R\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : Linear R C\ninst✝ : HasExt C\nX Y : C\nr : R\nf : X ⟶ Y\n⊢ mk₀ (r • f) = r • mk₀ f",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"HasDerivedCategory.standard",
"HasDeriv... | [
"R : Type t\ninst✝⁴ : Ring R\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : Linear R C\ninst✝ : HasExt C\nX Y : C\nr : R\nf : X ⟶ Y\nthis : HasDerivedCategory C := HasDerivedCategory.standard C\n⊢ mk₀ (r • f) = r • mk₀ f"
] | let := HasDerivedCategory.standard C | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Algebra.Category.ModuleCat.Ext.Basic | {
"line": 37,
"column": 2
} | {
"line": 41,
"column": 36
} | {
"line": 43,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : Small.{v, u} R\nM N : ModuleCat R\nr : R\nmem_ann : r ∈ Module.annihilator R ↑N\nn : ℕ\n⊢ AddCommGrpCat.ofHom ((mk₀ (r • 𝟙 M)).postcomp N ⋯) = 0",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.Ext.zero_comp",
... | [] | ext h
have : r • 𝟙 N = 0 := by
simp [← ModuleCat.lsmul_eq_smul_id, Module.mem_annihilator_iff_lsmul_eq_zero.mp mem_ann]
have smul_eq : r • h = (Ext.mk₀ (r • 𝟙 N)).comp h (zero_add n) := by simp [Ext.mk₀_smul]
simp [Ext.mk₀_smul, this, smul_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.Ext.Basic | {
"line": 37,
"column": 2
} | {
"line": 41,
"column": 36
} | {
"line": 43,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : Small.{v, u} R\nM N : ModuleCat R\nr : R\nmem_ann : r ∈ Module.annihilator R ↑N\nn : ℕ\n⊢ AddCommGrpCat.ofHom ((mk₀ (r • 𝟙 M)).postcomp N ⋯) = 0",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.Ext.zero_comp",
... | [] | ext h
have : r • 𝟙 N = 0 := by
simp [← ModuleCat.lsmul_eq_smul_id, Module.mem_annihilator_iff_lsmul_eq_zero.mp mem_ann]
have smul_eq : r • h = (Ext.mk₀ (r • 𝟙 N)).comp h (zero_add n) := by simp [Ext.mk₀_smul]
simp [Ext.mk₀_smul, this, smul_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.RingQuot | {
"line": 513,
"column": 4
} | {
"line": 513,
"column": 64
} | {
"line": 513,
"column": 65
} | [
{
"pp": "R : Type uR\ninst✝⁶ : Semiring R\nS : Type uS\ninst✝⁵ : CommSemiring S\nT : Type uT\nA : Type uA\ninst✝⁴ : Semiring A\ninst✝³ : Algebra S A\nr : R → R → Prop\ninst✝² : Semiring T\nB : Type u₄\ninst✝¹ : Semiring B\ninst✝ : Algebra S B\ns : A → A → Prop\nf : A →ₐ[S] B\nh : ∀ ⦃x y : A⦄, s x y → f x = f y\... | [] | simp [← one_quot, smul_quot, Algebra.algebraMap_eq_smul_one] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
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