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.UniformSpace.Cauchy | {
"line": 875,
"column": 8
} | {
"line": 875,
"column": 31
} | {
"line": 876,
"column": 8
} | [
{
"pp": "case refine_1\nα : Type u\nuniformSpace : UniformSpace α\nf : Filter α\nhf : Cauchy f\nU : ℕ → SetRel α α\nU_mem : ∀ (n : ℕ), U n ∈ 𝓤 α\nU_le : ∀ s ∈ 𝓤 α, ∃ n, U n ⊆ s\na : α\nha : Tendsto (seq hf U_mem) atTop (𝓝 a)\ns : Set (α × α)\nhs : s ∈ 𝓤 α\nm : ℕ\nhm : U m ⊆ s\nn : ℕ\nhn : ∀ (b : ℕ), n ≤ b →... | [
"case refine_1\nα : Type u\nuniformSpace : UniformSpace α\nf : Filter α\nhf : Cauchy f\nU : ℕ → SetRel α α\nU_mem : ∀ (n : ℕ), U n ∈ 𝓤 α\nU_le : ∀ s ∈ 𝓤 α, ∃ n, U n ⊆ s\na : α\nha : Tendsto (seq hf U_mem) atTop (𝓝 a)\ns : Set (α × α)\nhs : s ∈ 𝓤 α\nm : ℕ\nhm : U m ⊆ s\nn : ℕ\nhn : ∀ (b : ℕ), n ≤ b → seq hf U_me... | have := le_max_left m n | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Algebra.Module.ModuleTopology | {
"line": 219,
"column": 4
} | {
"line": 219,
"column": 35
} | {
"line": 220,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nτR : TopologicalSpace R\nτS : TopologicalSpace S\ninst✝⁸ : Semiring R\ninst✝⁷ : Semiring S\nσ : R →+* S\nσ' : S →+* R\ninst✝⁶ : RingHomInvPair σ σ'\ninst✝⁵ : RingHomInvPair σ' σ\nA : Type u_3\ninst✝⁴ : AddCommMonoid A\ninst✝³ : Module R A\nτA : TopologicalSpace A\ninst✝² : I... | [
"R : Type u_1\nS : Type u_2\nτR : TopologicalSpace R\nτS : TopologicalSpace S\ninst✝⁸ : Semiring R\ninst✝⁷ : Semiring S\nσ : R →+* S\nσ' : S →+* R\ninst✝⁶ : RingHomInvPair σ σ'\ninst✝⁵ : RingHomInvPair σ' σ\nA : Type u_3\ninst✝⁴ : AddCommMonoid A\ninst✝³ : Module R A\nτA : TopologicalSpace A\ninst✝² : IsModuleTopol... | let g' : B' →ₛₗ[σ'] A := e.symm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Topology.Algebra.Module.ModuleTopology | {
"line": 547,
"column": 2
} | {
"line": 547,
"column": 43
} | {
"line": 548,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁹ : TopologicalSpace R\ninst✝⁸ : Semiring R\nM : Type u_2\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : IsModuleTopology R M\nN : Type u_3\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : TopologicalSpace N\ninst✝ : IsModuleTopology R N... | [
"R : Type u_1\ninst✝⁹ : TopologicalSpace R\ninst✝⁸ : Semiring R\nM : Type u_2\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : IsModuleTopology R M\nN : Type u_3\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : TopologicalSpace N\ninst✝ : IsModuleTopology R N\nthis✝² : C... | let i₂ : N →ₗ[R] P := LinearMap.inr R M N | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Algebra.Homology.ShortComplex.RightHomology | {
"line": 867,
"column": 2
} | {
"line": 868,
"column": 99
} | {
"line": 870,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\nS : ShortComplex C\nh : S.RightHomologyData\ninst✝ : S.HasRightHomology\n⊢ h.rightHomologyIso.hom ≫ h.ι = S.rightHomologyι ≫ h.opcyclesIso.hom",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Category... | [] | simp only [← cancel_mono h.opcyclesIso.inv, ← cancel_epi h.rightHomologyIso.inv,
assoc, Iso.inv_hom_id_assoc, Iso.hom_inv_id, comp_id, rightHomologyIso_inv_comp_rightHomologyι] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.ShortComplex.RightHomology | {
"line": 867,
"column": 2
} | {
"line": 868,
"column": 99
} | {
"line": 870,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\nS : ShortComplex C\nh : S.RightHomologyData\ninst✝ : S.HasRightHomology\n⊢ h.rightHomologyIso.hom ≫ h.ι = S.rightHomologyι ≫ h.opcyclesIso.hom",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Category... | [] | simp only [← cancel_mono h.opcyclesIso.inv, ← cancel_epi h.rightHomologyIso.inv,
assoc, Iso.inv_hom_id_assoc, Iso.hom_inv_id, comp_id, rightHomologyIso_inv_comp_rightHomologyι] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.RightHomology | {
"line": 867,
"column": 2
} | {
"line": 868,
"column": 99
} | {
"line": 870,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\nS : ShortComplex C\nh : S.RightHomologyData\ninst✝ : S.HasRightHomology\n⊢ h.rightHomologyIso.hom ≫ h.ι = S.rightHomologyι ≫ h.opcyclesIso.hom",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Category... | [] | simp only [← cancel_mono h.opcyclesIso.inv, ← cancel_epi h.rightHomologyIso.inv,
assoc, Iso.inv_hom_id_assoc, Iso.hom_inv_id, comp_id, rightHomologyIso_inv_comp_rightHomologyι] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.Homology | {
"line": 1102,
"column": 2
} | {
"line": 1103,
"column": 77
} | {
"line": 1105,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\nS : ShortComplex C\nA : C\ninst✝ : S.HasHomology\nk : A ⟶ S.X₂\nx : A ⟶ S.X₁\nhx : k = x ≫ S.f\n⊢ S.liftCycles k ⋯ ≫ S.homologyπ = 0",
"ppTerm": "?m.84",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Category... | [] | dsimp only [homologyπ]
rw [S.liftCycles_leftHomologyπ_eq_zero_of_boundary_assoc k x hx, zero_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.Homology | {
"line": 1102,
"column": 2
} | {
"line": 1103,
"column": 77
} | {
"line": 1105,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\nS : ShortComplex C\nA : C\ninst✝ : S.HasHomology\nk : A ⟶ S.X₂\nx : A ⟶ S.X₁\nhx : k = x ≫ S.f\n⊢ S.liftCycles k ⋯ ≫ S.homologyπ = 0",
"ppTerm": "?m.84",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Category... | [] | dsimp only [homologyπ]
rw [S.liftCycles_leftHomologyπ_eq_zero_of_boundary_assoc k x hx, zero_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.Homology | {
"line": 1207,
"column": 4
} | {
"line": 1207,
"column": 18
} | {
"line": 1208,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\ninst✝¹ : S₁.HasHomology\ninst✝ : S₂.HasHomology\nh : Epi (cyclesMap φ)\n⊢ Epi (cyclesMap φ ≫ S₂.homologyπ)",
"ppTerm": "?m.79",
"assigned": true,
"usedConstants": [
"CategoryTheor... | [] | apply epi_comp | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Homology.ShortComplex.Preadditive | {
"line": 443,
"column": 14
} | {
"line": 443,
"column": 55
} | {
"line": 445,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nh₁₂ : Homotopy φ₁ φ₂\nh₂₃ : Homotopy φ₂ φ₃\n⊢ φ₁.τ₃ = h₁₂.h₃ + h₂₃.h₃ + (h₁₂.h₂ + h₂₃.h₂) ≫ S₂.g + φ₃.τ₃",
"ppTerm": "?m.181",
"assigned": true,
"usedConstants": [
"E... | [] | rw [h₁₂.comm₃, h₂₃.comm₃, add_comp]; abel | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.Preadditive | {
"line": 443,
"column": 14
} | {
"line": 443,
"column": 55
} | {
"line": 445,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nh₁₂ : Homotopy φ₁ φ₂\nh₂₃ : Homotopy φ₂ φ₃\n⊢ φ₁.τ₃ = h₁₂.h₃ + h₂₃.h₃ + (h₁₂.h₂ + h₂₃.h₂) ≫ S₂.g + φ₃.τ₃",
"ppTerm": "?m.181",
"assigned": true,
"usedConstants": [
"E... | [] | rw [h₁₂.comm₃, h₂₃.comm₃, add_comp]; abel | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.Preadditive | {
"line": 454,
"column": 14
} | {
"line": 454,
"column": 62
} | {
"line": 454,
"column": 62
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nh : Homotopy φ₁ φ₂\nh' : Homotopy φ₃ φ₄\n⊢ (φ₁ + φ₃).τ₃ = h.h₃ + h'.h₃ + (h.h₂ + h'.h₂) ≫ S₂.g + (φ₂ + φ₄).τ₃",
"ppTerm": "?m.221",
"assigned": true,
"usedConstants": [
... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nh : Homotopy φ₁ φ₂\nh' : Homotopy φ₃ φ₄\n⊢ h.h₃ + h.h₂ ≫ S₂.g + φ₂.τ₃ + (h'.h₃ + h'.h₂ ≫ S₂.g + φ₄.τ₃) =\n h.h₃ + h'.h₃ + (h.h₂ ≫ S₂.g + h'.h₂ ≫ S₂.g) + (φ₂.τ₃ + φ₄.τ₃)"
] | rw [add_τ₃, add_τ₃, h.comm₃, h'.comm₃, add_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.ShortComplex.Abelian | {
"line": 275,
"column": 2
} | {
"line": 276,
"column": 27
} | {
"line": 278,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nS : ShortComplex C\nkf : KernelFork S.g\nhkf : IsLimit kf\n⊢ hkf.lift (KernelFork.ofι S.f ⋯) = S.toCycles ≫ (S.isoCyclesOfIsLimit hkf).inv",
"ppTerm": "?m.114",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPr... | [] | have := Fork.IsLimit.mono hkf
simp [← cancel_mono kf.ι] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.Abelian | {
"line": 275,
"column": 2
} | {
"line": 276,
"column": 27
} | {
"line": 278,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nS : ShortComplex C\nkf : KernelFork S.g\nhkf : IsLimit kf\n⊢ hkf.lift (KernelFork.ofι S.f ⋯) = S.toCycles ≫ (S.isoCyclesOfIsLimit hkf).inv",
"ppTerm": "?m.114",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPr... | [] | have := Fork.IsLimit.mono hkf
simp [← cancel_mono kf.ι] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.ModuleCat.Topology.Homology | {
"line": 53,
"column": 67
} | {
"line": 55,
"column": 89
} | {
"line": 55,
"column": 89
} | [
{
"pp": "R : Type u\ninst✝¹ : Ring R\ninst✝ : TopologicalSpace R\nM N : TopModuleCat R\nφ : M ⟶ N\ns : KernelFork φ\nm : ↑s.pt.toModuleCat\n⊢ (Hom.hom (Fork.ι s)) m ∈ (↑(Hom.hom φ)).ker",
"ppTerm": "?m.120",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"TopModuleCat.instCategory",
... | [] | by
rw [LinearMap.mem_ker, ContinuousLinearMap.coe_coe,
← ConcreteCategory.comp_apply (Fork.ι s) φ, KernelFork.condition, hom_zero_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.ShortComplex.Preadditive | {
"line": 508,
"column": 35
} | {
"line": 508,
"column": 60
} | {
"line": 508,
"column": 60
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nh : Homotopy φ₁ φ₂\n⊢ (opMap φ₁).τ₁.unop = (S₂.op.f ≫ h.h₂.op + h.h₃.op + (opMap φ₂).τ₁).unop",
"ppTerm": "?m.134",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | dsimp; rw [h.comm₃]; abel | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.Preadditive | {
"line": 508,
"column": 35
} | {
"line": 508,
"column": 60
} | {
"line": 508,
"column": 60
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nh : Homotopy φ₁ φ₂\n⊢ (opMap φ₁).τ₁.unop = (S₂.op.f ≫ h.h₂.op + h.h₃.op + (opMap φ₂).τ₁).unop",
"ppTerm": "?m.134",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | dsimp; rw [h.comm₃]; abel | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.Preadditive | {
"line": 525,
"column": 33
} | {
"line": 525,
"column": 58
} | {
"line": 525,
"column": 58
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nS₁✝ S₂✝ S₃ : ShortComplex C\nφ₁✝ φ₂✝ φ₃ φ₄ : S₁✝ ⟶ S₂✝\nS₁ S₂ : ShortComplex Cᵒᵖ\nφ₁ φ₂ : S₁ ⟶ S₂\nh : Homotopy φ₁ φ₂\n⊢ (unopMap φ₁).τ₁.op = (S₂.unop.f ≫ h.h₂.unop + h.h₃.unop + (unopMap φ₂).τ₁).op",
"ppTerm": "?m.142",
"assi... | [] | dsimp; rw [h.comm₃]; abel | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.Preadditive | {
"line": 525,
"column": 33
} | {
"line": 525,
"column": 58
} | {
"line": 525,
"column": 58
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nS₁✝ S₂✝ S₃ : ShortComplex C\nφ₁✝ φ₂✝ φ₃ φ₄ : S₁✝ ⟶ S₂✝\nS₁ S₂ : ShortComplex Cᵒᵖ\nφ₁ φ₂ : S₁ ⟶ S₂\nh : Homotopy φ₁ φ₂\n⊢ (unopMap φ₁).τ₁.op = (S₂.unop.f ≫ h.h₂.unop + h.h₃.unop + (unopMap φ₂).τ₁).op",
"ppTerm": "?m.142",
"assi... | [] | dsimp; rw [h.comm₃]; abel | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.Preadditive | {
"line": 549,
"column": 4
} | {
"line": 549,
"column": 29
} | {
"line": 551,
"column": 0
} | [
{
"pp": "case h₃\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ : ShortComplex C\nφ₁ φ₂ : S₁ ⟶ S₂\nh : Homotopy φ₁ φ₂\n⊢ φ₁.τ₃ = (φ₂ + S₁.nullHomotopic S₂ h.h₀ ⋯ h.h₁ h.h₂ h.h₃ ⋯).τ₃",
"ppTerm": "?h₃",
"assigned": true,
"usedConstants": [
"CategoryTheory.ShortCompl... | [] | dsimp; rw [h.comm₃]; abel | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.Preadditive | {
"line": 549,
"column": 4
} | {
"line": 549,
"column": 29
} | {
"line": 551,
"column": 0
} | [
{
"pp": "case h₃\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ : ShortComplex C\nφ₁ φ₂ : S₁ ⟶ S₂\nh : Homotopy φ₁ φ₂\n⊢ φ₁.τ₃ = (φ₂ + S₁.nullHomotopic S₂ h.h₀ ⋯ h.h₁ h.h₂ h.h₃ ⋯).τ₃",
"ppTerm": "?h₃",
"assigned": true,
"usedConstants": [
"CategoryTheory.ShortCompl... | [] | dsimp; rw [h.comm₃]; abel | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Embedding.Basic | {
"line": 243,
"column": 44
} | {
"line": 243,
"column": 87
} | {
"line": 245,
"column": 0
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\np : ℤ\n⊢ (embeddingUpIntGE p).IsRelIff",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"ComplexShape.Embedding.instIsRelIffMk'",
"Nat.instOne",
"AddGroupWithOne.toAddMonoidWithOne",
"AddC... | [] | by dsimp [embeddingUpIntGE]; infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Subobject.Basic | {
"line": 677,
"column": 24
} | {
"line": 687,
"column": 30
} | {
"line": 689,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX Y Z : C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\ne : X ≅ Y\nA B : Subobject X\n⊢ { toFun := (map e.hom).obj, invFun := (map e.inv).obj, left_inv := ⋯, right_inv := ⋯ } A ≤\n { toFun := (map e.hom).obj, invFun := (map e.inv).obj, left_inv := ⋯, right_i... | [] | by
dsimp
constructor
· intro h
apply_fun (map e.inv).obj at h
· simpa only [← map_comp, e.hom_inv_id, map_id] using h
· apply Functor.monotone
· intro h
apply_fun (map e.hom).obj at h
· exact h
· apply Functor.monotone | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Subobject.Limits | {
"line": 291,
"column": 8
} | {
"line": 293,
"column": 87
} | {
"line": 294,
"column": 8
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝³ : Category.{v, u} C\nX✝ Y Z : C\ninst✝² : HasZeroMorphisms C\nf : X✝ ⟶ Y\ninst✝¹ : HasKernel f\ninst✝ : HasKernels C\nX : C\nA B : Cᵒᵖ\ng : B ⟶ op X\nhg : Mono g\ni : A ≅ B\nhf : Mono (i.hom ≫ g)\n⊢ kernel (i.hom ≫ g).unop ≅ kernel g.unop",
"ppTerm": "?refine_1",
... | [
"case refine_2\nC : Type u\ninst✝³ : Category.{v, u} C\nX✝ Y Z : C\ninst✝² : HasZeroMorphisms C\nf : X✝ ⟶ Y\ninst✝¹ : HasKernel f\ninst✝ : HasKernels C\nX : C\nA B : Cᵒᵖ\ng : B ⟶ op X\nhg : Mono g\ni : A ≅ B\nhf : Mono (i.hom ≫ g)\n⊢ ((limit.isLimit (parallelPair (g.unop ≫ i.unop.hom) 0)).conePointUniqueUpToIso\n ... | · exact
IsLimit.conePointUniqueUpToIso (limit.isLimit _)
(isKernelCompMono (limit.isLimit (parallelPair g.unop 0)) i.unop.hom rfl) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Subobject.Limits | {
"line": 334,
"column": 2
} | {
"line": 334,
"column": 16
} | {
"line": 336,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nX Y Z : C\nf : X ⟶ Y\ninst✝¹ : HasImage f\ninst✝ : HasEqualizers C\n⊢ Epi (factorThruImage f ≫ (imageSubobjectIso f).inv)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.factorThruImage",
"CategoryTheory.Su... | [] | apply epi_comp | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Subobject.Limits | {
"line": 392,
"column": 82
} | {
"line": 395,
"column": 8
} | {
"line": 397,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\nX Y Z : C\nf✝ : X ⟶ Y\ninst✝⁴ : HasImage f✝\ninst✝³ : HasEqualizers C\nX' : C\nh : X' ⟶ X\ninst✝² : Epi h\nf : X ⟶ Y\ninst✝¹ : HasImage f\ninst✝ : HasImage (h ≫ f)\n⊢ Epi ((imageSubobject (h ≫ f)).ofLE (imageSubobject f) ⋯)",
"ppTerm": "?m.45",
"assigned"... | [] | by
rw [ofLE_mk_le_mk_of_comm (image.preComp h f)]
· infer_instance
· simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Subobject.Lattice | {
"line": 421,
"column": 23
} | {
"line": 421,
"column": 32
} | {
"line": 424,
"column": 0
} | [
{
"pp": "case insert\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : HasPullbacks C\nI : Type u_1\nA B : C\nP : I → Subobject B\nf : A ⟶ B\na✝¹ : I\ns✝ : Finset I\na✝ : a✝¹ ∉ s✝\nih : (s✝.inf P).Factors f ↔ ∀ i ∈ s✝, (P i).Factors f\n⊢ ((insert a✝¹ s✝).inf P).Factors f ↔ ∀ i ∈ insert a✝¹ s✝, (P i).Factors f... | [] | simp [ih] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Subobject.Lattice | {
"line": 421,
"column": 23
} | {
"line": 421,
"column": 32
} | {
"line": 424,
"column": 0
} | [
{
"pp": "case insert\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : HasPullbacks C\nI : Type u_1\nA B : C\nP : I → Subobject B\nf : A ⟶ B\na✝¹ : I\ns✝ : Finset I\na✝ : a✝¹ ∉ s✝\nih : (s✝.inf P).Factors f ↔ ∀ i ∈ s✝, (P i).Factors f\n⊢ ((insert a✝¹ s✝).inf P).Factors f ↔ ∀ i ∈ insert a✝¹ s✝, (P i).Factors f... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Subobject.Lattice | {
"line": 421,
"column": 23
} | {
"line": 421,
"column": 32
} | {
"line": 424,
"column": 0
} | [
{
"pp": "case insert\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : HasPullbacks C\nI : Type u_1\nA B : C\nP : I → Subobject B\nf : A ⟶ B\na✝¹ : I\ns✝ : Finset I\na✝ : a✝¹ ∉ s✝\nih : (s✝.inf P).Factors f ↔ ∀ i ∈ s✝, (P i).Factors f\n⊢ ((insert a✝¹ s✝).inf P).Factors f ↔ ∀ i ∈ insert a✝¹ s✝, (P i).Factors f... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.GradedObject | {
"line": 136,
"column": 2
} | {
"line": 137,
"column": 36
} | {
"line": 139,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nE : Type u_3\nJ : Type u_4\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\ninst✝ : Category.{v_3, u_3} E\nX Y✝ : GradedObject J C\ne : X ≅ Y✝\nF : C ⥤ D ⥤ E\nj : J\nY : D\n⊢ (F.map (e.hom j)).app Y ≫ (F.map (e.inv j)).app Y = 𝟙 ((F.obj (X j)).obj Y)",
"... | [] | rw [← NatTrans.comp_app, ← F.map_comp, hom_inv_id_eval,
Functor.map_id, NatTrans.id_app] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.GradedObject | {
"line": 136,
"column": 2
} | {
"line": 137,
"column": 36
} | {
"line": 139,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nE : Type u_3\nJ : Type u_4\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\ninst✝ : Category.{v_3, u_3} E\nX Y✝ : GradedObject J C\ne : X ≅ Y✝\nF : C ⥤ D ⥤ E\nj : J\nY : D\n⊢ (F.map (e.hom j)).app Y ≫ (F.map (e.inv j)).app Y = 𝟙 ((F.obj (X j)).obj Y)",
"... | [] | rw [← NatTrans.comp_app, ← F.map_comp, hom_inv_id_eval,
Functor.map_id, NatTrans.id_app] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.GradedObject | {
"line": 136,
"column": 2
} | {
"line": 137,
"column": 36
} | {
"line": 139,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nE : Type u_3\nJ : Type u_4\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\ninst✝ : Category.{v_3, u_3} E\nX Y✝ : GradedObject J C\ne : X ≅ Y✝\nF : C ⥤ D ⥤ E\nj : J\nY : D\n⊢ (F.map (e.hom j)).app Y ≫ (F.map (e.inv j)).app Y = 𝟙 ((F.obj (X j)).obj Y)",
"... | [] | rw [← NatTrans.comp_app, ← F.map_comp, hom_inv_id_eval,
Functor.map_id, NatTrans.id_app] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.GradedObject | {
"line": 442,
"column": 62
} | {
"line": 443,
"column": 60
} | {
"line": 443,
"column": 60
} | [
{
"pp": "I : Type u_1\nJ : Type u_2\nK : Type u_3\nC : Type u_4\ninst✝² : Category.{v_1, u_4} C\nX Y Z : GradedObject I C\nφ : X ⟶ Y\ne : X ≅ Y\nψ : Y ⟶ Z\np : I → J\nj : J\ninst✝¹ : X.HasMap p\ninst✝ : Y.HasMap p\nq : J → K\nr : I → K\nhpqr : ∀ (i : I), q (p i) = r i\nk : K\nc : (j : J) → q j = k → X.CofanMapO... | [] | by
rw [Set.mem_preimage, Set.mem_singleton_iff, hpqr, hi] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.HomologicalComplex | {
"line": 487,
"column": 2
} | {
"line": 488,
"column": 33
} | {
"line": 490,
"column": 0
} | [
{
"pp": "ι : Type u_1\nV : Type u\ninst✝² : Category.{v, u} V\ninst✝¹ : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\ninst✝ : HasKernels V\ni j : ι\nr : c.Rel i j\n⊢ kernelSubobject (C.dFrom i) = kernelSubobject (C.d i j)",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": ... | [] | rw [C.dFrom_eq r]
apply kernelSubobject_comp_mono | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.HomologicalComplex | {
"line": 487,
"column": 2
} | {
"line": 488,
"column": 33
} | {
"line": 490,
"column": 0
} | [
{
"pp": "ι : Type u_1\nV : Type u\ninst✝² : Category.{v, u} V\ninst✝¹ : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\ninst✝ : HasKernels V\ni j : ι\nr : c.Rel i j\n⊢ kernelSubobject (C.dFrom i) = kernelSubobject (C.d i j)",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": ... | [] | rw [C.dFrom_eq r]
apply kernelSubobject_comp_mono | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.HomologicalComplex | {
"line": 732,
"column": 2
} | {
"line": 733,
"column": 35
} | {
"line": 735,
"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\n⊢ (mk X₀ X₁ X₂ d₀ d₁ s succ).d 2 1 = d₁",
"ppTerm": "?m.47",
"assigned": true,
"usedCons... | [] | change ite (2 = 1 + 1) (𝟙 X₂ ≫ d₁) 0 = d₁
rw [if_pos rfl, Category.id_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.HomologicalComplex | {
"line": 732,
"column": 2
} | {
"line": 733,
"column": 35
} | {
"line": 735,
"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\n⊢ (mk X₀ X₁ X₂ d₀ d₁ s succ).d 2 1 = d₁",
"ppTerm": "?m.47",
"assigned": true,
"usedCons... | [] | change ite (2 = 1 + 1) (𝟙 X₂ ≫ d₁) 0 = d₁
rw [if_pos rfl, Category.id_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Shift.Basic | {
"line": 423,
"column": 4
} | {
"line": 433,
"column": 9
} | {
"line": 435,
"column": 0
} | [
{
"pp": "C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddGroup A\ninst✝ : HasShift C A\ni j : A\nh : i + j = 0\nX : C\n⊢ (shiftFunctor C i).map ((shiftFunctorCompIsoId C i j h).symm.hom.app X) ≫\n (shiftFunctorCompIsoId C j i ⋯).hom.app ((shiftFunctor C i).obj X) =\n 𝟙 ((shiftFunctor ... | [] | convert!
(equivOfTensorIsoUnit (shiftMonoidalFunctor C A) ⟨i⟩ ⟨j⟩ (Discrete.eqToIso h)
(Discrete.eqToIso (by dsimp; rw [← add_left_inj j, add_assoc, h, zero_add, add_zero]))
(Subsingleton.elim _ _)).functor_unitIso_comp
X
all_goals
ext X
dsimp [shiftFunctorCompIsoId... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Shift.Basic | {
"line": 423,
"column": 4
} | {
"line": 433,
"column": 9
} | {
"line": 435,
"column": 0
} | [
{
"pp": "C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddGroup A\ninst✝ : HasShift C A\ni j : A\nh : i + j = 0\nX : C\n⊢ (shiftFunctor C i).map ((shiftFunctorCompIsoId C i j h).symm.hom.app X) ≫\n (shiftFunctorCompIsoId C j i ⋯).hom.app ((shiftFunctor C i).obj X) =\n 𝟙 ((shiftFunctor ... | [] | convert!
(equivOfTensorIsoUnit (shiftMonoidalFunctor C A) ⟨i⟩ ⟨j⟩ (Discrete.eqToIso h)
(Discrete.eqToIso (by dsimp; rw [← add_left_inj j, add_assoc, h, zero_add, add_zero]))
(Subsingleton.elim _ _)).functor_unitIso_comp
X
all_goals
ext X
dsimp [shiftFunctorCompIsoId... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.Exact | {
"line": 295,
"column": 4
} | {
"line": 295,
"column": 18
} | {
"line": 296,
"column": 2
} | [
{
"pp": "case mp\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nS : ShortComplex C\ninst✝ : HasZeroObject C\nhg : S.g = 0\nthis✝¹ : S.HasHomology\nh : IsZero S.homology\nthis✝ : IsIso S.iCycles\nthis : Epi S.toCycles\n⊢ Epi (S.toCycles ≫ S.iCycles)",
"ppTerm": "?mp",
"assigned": ... | [] | apply epi_comp | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex | {
"line": 158,
"column": 2
} | {
"line": 159,
"column": 16
} | {
"line": 161,
"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 L M : HomologicalComplex C c\nφ : K ⟶ L\niso : K ≅ L\nψ : L ⟶ M\ni j k : ι\ninst✝ : K.HasHomology i\n⊢ Epi (K.homologyπ i)",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [... | [] | dsimp only [homologyπ]
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex | {
"line": 158,
"column": 2
} | {
"line": 159,
"column": 16
} | {
"line": 161,
"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 L M : HomologicalComplex C c\nφ : K ⟶ L\niso : K ≅ L\nψ : L ⟶ M\ni j k : ι\ninst✝ : K.HasHomology i\n⊢ Epi (K.homologyπ i)",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [... | [] | dsimp only [homologyπ]
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex | {
"line": 302,
"column": 54
} | {
"line": 302,
"column": 71
} | {
"line": 302,
"column": 72
} | [
{
"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 L M : HomologicalComplex C c\nφ : K ⟶ L\niso : K ≅ L\nψ : L ⟶ M\ni j✝ k✝ : ι\ninst✝ : K.HasHomology i\nA : C\nk : K.X i ⟶ A\nj : ι\nhj : c.prev i = j\ni' : ι\nx : K.X i' ⟶ A\nhx : k = K.d i i'... | [
"C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\nι : Type u_2\nc : ComplexShape ι\nK L M : HomologicalComplex C c\nφ : K ⟶ L\niso : K ≅ L\nψ : L ⟶ M\ni j✝ k✝ : ι\ninst✝ : K.HasHomology i\nA : C\nk : K.X i ⟶ A\nj : ι\nhj : c.prev i = j\ni' : ι\nx : K.X i' ⟶ A\nhx : k = K.d i i' ≫ x\n⊢ 0 ≫ ... | K.d_comp_d_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex | {
"line": 306,
"column": 45
} | {
"line": 306,
"column": 62
} | {
"line": 306,
"column": 63
} | [
{
"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 : ι\ninst✝ : K.HasHomology i\nA : C\nk : K.X i ⟶ A\nj : ι\nhj : c.prev i = j\ni' : ι\nx : K.X i' ⟶ A\nhx : k = K.d i i' ≫ x\nh : ¬c.Rel i i'\n⊢ K.d j i ≫ K.d i i' ≫... | [
"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 : ι\ninst✝ : K.HasHomology i\nA : C\nk : K.X i ⟶ A\nj : ι\nhj : c.prev i = j\ni' : ι\nx : K.X i' ⟶ A\nhx : k = K.d i i' ≫ x\nh : ¬c.Rel i i'\n⊢ 0 ≫ x = 0"
] | K.d_comp_d_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.ShortComplex.Exact | {
"line": 547,
"column": 2
} | {
"line": 547,
"column": 18
} | {
"line": 548,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nS : ShortComplex C\ns s' : S.Splitting\nh : s.s = s'.s\n⊢ s = s'",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"CategoryTheory.ShortComplex.Splitting.mono_f",
"CategoryTheory.Mono",
"CategoryT... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nS : ShortComplex C\ns s' : S.Splitting\nh : s.s = s'.s\nthis : Mono S.f\n⊢ s = s'"
] | have := s.mono_f | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex | {
"line": 918,
"column": 2
} | {
"line": 918,
"column": 35
} | {
"line": 920,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : HasZeroMorphisms C\nι : Type u_2\nc : ComplexShape ι\nK : HomologicalComplex C c\nj : ι\ninst✝¹ : K.HasHomology j\ninst✝ : (K.sc' (c.prev j) j (c.next j)).HasHomology\n⊢ (K.homologyIsoSc' (c.prev j) j (c.next j) ⋯ ⋯).hom = (Iso.refl (K.homology j))... | [] | apply ShortComplex.homologyMap_id | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Homology.ShortComplex.Exact | {
"line": 901,
"column": 4
} | {
"line": 901,
"column": 19
} | {
"line": 902,
"column": 4
} | [
{
"pp": "case mpr\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nhf₁ : S₁.f = 0\nhg₁ : S₁.g = 0\nhf₂ : S₂.f = 0\nw : φ.τ₂ ≫ S₂.g = 0\n⊢ { X₁ := S₁.X₂, X₂ := S₂.X₂, X₃ := S₂.X₃, f := φ.τ₂, g := S₂.g, zero := ⋯ }.Exact ∧ Mono φ.τ₂ →\n IsIso (S₂.liftCycles... | [
"case mpr\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nhf₁ : S₁.f = 0\nhg₁ : S₁.g = 0\nhf₂ : S₂.f = 0\nw : φ.τ₂ ≫ S₂.g = 0\nh₁ : { X₁ := S₁.X₂, X₂ := S₂.X₂, X₃ := S₂.X₃, f := φ.τ₂, g := S₂.g, zero := ⋯ }.Exact\nh₂ : Mono φ.τ₂\n⊢ IsIso (S₂.liftCycles φ.τ₂ ⋯)"... | rintro ⟨h₁, h₂⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.Homology.Homotopy | {
"line": 319,
"column": 2
} | {
"line": 319,
"column": 19
} | {
"line": 321,
"column": 0
} | [
{
"pp": "case neg\nι : Type u_1\nV : Type u\ninst✝⁴ : Category.{v, u} V\ninst✝³ : Preadditive V\nc : ComplexShape ι\nC D : HomologicalComplex V c\nW : Type u_2\ninst✝² : Category.{v_1, u_2} W\ninst✝¹ : Preadditive W\nG : V ⥤ W\ninst✝ : G.Additive\nhom : (i j : ι) → c.Rel j i → (C.X i ⟶ D.X j)\ni j : ι\nh✝ : ¬c.... | [] | · rw [G.map_zero] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.Filter.IsBounded | {
"line": 596,
"column": 70
} | {
"line": 603,
"column": 15
} | {
"line": 605,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝ : LinearOrder β\nf : Filter α\nF : ι → α → β\ns : Finset ι\nhs : s.Nonempty\nh : ∃ i ∈ s, IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f (F i)\n⊢ IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f fun a ↦ s.sup' hs fun i ↦ F i a",
"ppTerm": "?m.30",
"assigned": t... | [] | by
rcases h with ⟨i, i_s, b, hb⟩
use b
refine fun c hc ↦ hb c ?_
rw [eventually_map] at hc ⊢
refine hc.mono fun a h ↦ ?_
simp only [sup'_le_iff] at h ⊢
exact h i i_s | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Filter.IsBounded | {
"line": 638,
"column": 2
} | {
"line": 638,
"column": 28
} | {
"line": 639,
"column": 2
} | [
{
"pp": "R : Type u_5\nS : Type u_6\nF : Filter R\ninst✝¹ : LinearOrder R\ninst✝ : LinearOrder S\nf : R → S\nf_incr : Monotone f\nl : R\nfreq_ge : ∃ᶠ (x : R) in F, l ≤ x\n⊢ ∃ᶠ (x' : S) in map f F, f l ≤ x'",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Filter.map",
"Filter.Ev... | [
"R : Type u_5\nS : Type u_6\nF : Filter R\ninst✝¹ : LinearOrder R\ninst✝ : LinearOrder S\nf : R → S\nf_incr : Monotone f\nl : R\nfreq_ge : ∃ᶠ (x : R) in F, l ≤ x\nev : ∀ᶠ (x : S) in map f F, ¬(fun x' ↦ f l ≤ x') x\n⊢ ∀ᶠ (x : R) in F, ¬(fun x ↦ l ≤ x) x"
] | refine fun ev ↦ freq_ge ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Algebra.InfiniteSum.SummationFilter | {
"line": 86,
"column": 2
} | {
"line": 87,
"column": 69
} | {
"line": 88,
"column": 2
} | [
{
"pp": "β : Type u_2\nL : SummationFilter β\nhL : ¬L.NeBot\n⊢ L.LeAtTop",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"SummationFilter.neBot_or_eq_bot",
"congrArg",
"Finset",
"Set.univ",
"Filter.Eventually",
"setOf",
"Membership.mem",
"Or.r... | [
"β : Type u_2\nL : SummationFilter β\nhL : ¬L.NeBot\nhLs : L.support = univ\n⊢ L.LeAtTop"
] | have hLs : L.support = Set.univ := by
simp [SummationFilter.support, L.neBot_or_eq_bot.resolve_left hL] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Order.Filter.AtTopBot.BigOperators | {
"line": 40,
"column": 4
} | {
"line": 40,
"column": 25
} | {
"line": 41,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nM : Type u_3\ninst✝ : CommMonoid M\nf : α → M\ng : β → M\nh_eq : ∀ (u : Finset β), ∃ v, ∀ (v' : Finset α), v ⊆ v' → ∃ u', u ⊆ u' ∧ ∏ x ∈ u', g x = ∏ b ∈ v', f b\nb : Finset β\nx✝ : True\n⊢ ∃ i, True ∧ (fun s ↦ ∏ b ∈ s, f b) '' Set.Ici i ⊆ (fun s ↦ ∏ x ∈ s, g x) '' Set.Ici b"... | [
"α : Type u_1\nβ : Type u_2\nM : Type u_3\ninst✝ : CommMonoid M\nf : α → M\ng : β → M\nh_eq : ∀ (u : Finset β), ∃ v, ∀ (v' : Finset α), v ⊆ v' → ∃ u', u ⊆ u' ∧ ∏ x ∈ u', g x = ∏ b ∈ v', f b\nb : Finset β\nx✝ : True\nv : Finset α\nhv : ∀ (v' : Finset α), v ⊆ v' → ∃ u', b ⊆ u' ∧ ∏ x ∈ u', g x = ∏ b ∈ v', f b\n⊢ ∃ i, ... | let ⟨v, hv⟩ := h_eq b | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Order.LiminfLimsup | {
"line": 273,
"column": 2
} | {
"line": 273,
"column": 84
} | {
"line": 275,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : ConditionallyCompleteLattice α\nf : Filter β\nu v : β → α\np : β → Prop\nh : ∀ᶠ (a : β) in f, p a → u a = v a\n⊢ blimsup u f p = blimsup v f p",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Filter.eventually_inf_principal",
"Iff.mp... | [] | simpa only [blimsup_eq_limsup] using! limsup_congr <| eventually_inf_principal.2 h | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.Order.LiminfLimsup | {
"line": 273,
"column": 2
} | {
"line": 273,
"column": 84
} | {
"line": 275,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : ConditionallyCompleteLattice α\nf : Filter β\nu v : β → α\np : β → Prop\nh : ∀ᶠ (a : β) in f, p a → u a = v a\n⊢ blimsup u f p = blimsup v f p",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Filter.eventually_inf_principal",
"Iff.mp... | [] | simpa only [blimsup_eq_limsup] using! limsup_congr <| eventually_inf_principal.2 h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.LiminfLimsup | {
"line": 273,
"column": 2
} | {
"line": 273,
"column": 84
} | {
"line": 275,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : ConditionallyCompleteLattice α\nf : Filter β\nu v : β → α\np : β → Prop\nh : ∀ᶠ (a : β) in f, p a → u a = v a\n⊢ blimsup u f p = blimsup v f p",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Filter.eventually_inf_principal",
"Iff.mp... | [] | simpa only [blimsup_eq_limsup] using! limsup_congr <| eventually_inf_principal.2 h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.LiminfLimsup | {
"line": 856,
"column": 49
} | {
"line": 859,
"column": 15
} | {
"line": 861,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : ConditionallyCompleteLinearOrder β\nf : Filter α\nu : α → β\nb : β\nhu : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u\nh : b < limsup u f\n⊢ ∃ᶠ (x : α) in f, b < u x",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.t... | [] | by
contrapose! h
apply limsSup_le_of_le hu
simpa using h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.UniformSpace.Compact | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 73
} | {
"line": 80,
"column": 2
} | [
{
"pp": "α : Type ua\ninst✝ : UniformSpace α\nK : Set α\nι' : Sort u_2\nι : Sort u_3\np : ι' → Prop\nV : ι' → Set (α × α)\nU : ι → Set α\nhbasis : (𝓤 α).HasBasis p V\nhK : IsCompact K\nhopen : ∀ (j : ι), IsOpen[inst✝.toTopologicalSpace] (U j)\nhcover : K ⊆ ⋃ j, U j\n⊢ ∃ i, p i ∧ ∀ x ∈ K, ∃ j, ball x (V i) ⊆ U ... | [
"α : Type ua\ninst✝ : UniformSpace α\nK : Set α\nι' : Sort u_2\nι : Sort u_3\np : ι' → Prop\nV : ι' → Set (α × α)\nU : ι → Set α\nhbasis : (𝓤 α).HasBasis p V\nhK : IsCompact K\nhopen : ∀ (j : ι), IsOpen[inst✝.toTopologicalSpace] (U j)\nhcover : K ⊆ ⋃ j, U j\n⊢ ∀ ⦃s t : Set (α × α)⦄, s ⊆ t → (∀ x ∈ K, ∃ i, ball x t... | refine (hbasis.exists_iff ?_).1 (lebesgue_number_lemma hK hopen hcover) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 297,
"column": 8
} | {
"line": 297,
"column": 71
} | {
"line": 298,
"column": 8
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁶ : CommMonoid α\ninst✝⁵ : TopologicalSpace α\nL : SummationFilter β\ninst✝⁴ : CommMonoid γ\ninst✝³ : TopologicalSpace γ\ninst✝² : T2Space γ\nG : Type u_4\ninst✝¹ : FunLike G α γ\ninst✝ : MonoidHomClass G α γ\ng : G\nhg : IsInducing ⇑g\nf : β → α... | [
"case neg\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁶ : CommMonoid α\ninst✝⁵ : TopologicalSpace α\nL : SummationFilter β\ninst✝⁴ : CommMonoid γ\ninst✝³ : TopologicalSpace γ\ninst✝² : T2Space γ\nG : Type u_4\ninst✝¹ : FunLike G α γ\ninst✝ : MonoidHomClass G α γ\ng : G\nhg : IsInducing ⇑g\nf : β → α\nhf : Multi... | · simp [tprod_bot hL, finprod_eq_prod _ hfs, ← _root_.map_prod] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.LiminfLimsup | {
"line": 1182,
"column": 4
} | {
"line": 1182,
"column": 70
} | {
"line": 1183,
"column": 2
} | [
{
"pp": "case a\nα : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝ : ConditionallyCompleteLinearOrder β\nf : Filter α\nF : ι → α → β\ns : Finset ι\nhs : s.Nonempty\nh₁ : ∀ i ∈ s, IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f (F i)\nh₂ : ∀ i ∈ s, IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f (F i)\nbddsup : IsBoundedUnder (fu... | [] | exact fun i i_s ↦ eventually_lt_of_limsup_lt (hb i i_s) (h₂ i i_s) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 522,
"column": 2
} | {
"line": 522,
"column": 33
} | {
"line": 524,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : CommMonoid α\ninst✝² : TopologicalSpace α\nb : β\ninst✝¹ : DecidablePred fun x ↦ b = x\na : β → α\nL : SummationFilter β\ninst✝ : L.LeAtTop\n⊢ ∀ (b' : β), b' ≠ b → (if b = b' then a b' else 1) = 1",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [... | [] | · intro b' hb'; simp [hb'.symm] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.UniformSpace.CompactConvergence | {
"line": 247,
"column": 2
} | {
"line": 247,
"column": 35
} | {
"line": 249,
"column": 0
} | [
{
"pp": "α : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nf : C(α, β)\nι : Type u₃\np : Filter ι\nF : ι → C(α, β)\nh : TendstoLocallyUniformly (fun i a ↦ (F i) a) (⇑f) p\nK : Set α\nhK : IsCompact K\n⊢ TendstoLocallyUniformlyOn (fun i a ↦ (F i) a) (⇑f) p K",
"ppTerm": "?m.41",
... | [] | exact h.tendstoLocallyUniformlyOn | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.UniformSpace.CompactConvergence | {
"line": 385,
"column": 2
} | {
"line": 385,
"column": 83
} | {
"line": 386,
"column": 2
} | [
{
"pp": "α : Type u₁\nβ : Type u₂\ninst✝³ : TopologicalSpace α\ninst✝² : UniformSpace β\nδ₁ : Type u_1\nδ₂ : Type u_2\ninst✝¹ : TopologicalSpace δ₁\ninst✝ : TopologicalSpace δ₂\nφ₁ : C(δ₁, α)\nφ₂ : C(δ₂, α)\nh_proper₁ : IsProperMap ⇑φ₁\nh_proper₂ : IsProperMap ⇑φ₂\nh_cover : range ⇑φ₁ ∪ range ⇑φ₂ = univ\n𝔖 : S... | [
"α : Type u₁\nβ : Type u₂\ninst✝³ : TopologicalSpace α\ninst✝² : UniformSpace β\nδ₁ : Type u_1\nδ₂ : Type u_2\ninst✝¹ : TopologicalSpace δ₁\ninst✝ : TopologicalSpace δ₂\nφ₁ : C(δ₁, α)\nφ₂ : C(δ₂, α)\nh_proper₁ : IsProperMap ⇑φ₁\nh_proper₂ : IsProperMap ⇑φ₂\nh_cover : range ⇑φ₁ ∪ range ⇑φ₂ = univ\n𝔖 : Set (Set α) :... | have h_preimage₁ : MapsTo (φ₁ ⁻¹' ·) 𝔖 𝔗₁ := fun K ↦ h_proper₁.isCompact_preimage | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.UniformSpace.UniformConvergence | {
"line": 563,
"column": 2
} | {
"line": 563,
"column": 42
} | {
"line": 564,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝ : UniformSpace β\nF : ι → α → β\ns : Set α\nx : α\np : Filter ι\nhp : p.NeBot\nhf : UniformCauchySeqOn F p s\nhx : x ∈ s\n⊢ Cauchy (map (fun i ↦ F i x) p)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cauchy"... | [
"α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝ : UniformSpace β\nF : ι → α → β\ns : Set α\nx : α\np : Filter ι\nhp : p.NeBot\nhf : UniformCauchySeqOn F p s\nhx : x ∈ s\n⊢ Tendsto (fun p ↦ (F p.1 x, F p.2 x)) (p ×ˢ p) (𝓤 β)"
] | simp only [cauchy_map_iff, hp, true_and] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.UniformSpace.CompactConvergence | {
"line": 449,
"column": 4
} | {
"line": 449,
"column": 100
} | {
"line": 450,
"column": 4
} | [
{
"pp": "α : Type u₁\nβ : Type u₂\ninst✝² : TopologicalSpace α\ninst✝¹ : UniformSpace β\ninst✝ : CompleteSpace C(α, β)\nf : α → β\ns : Set α\nl : Filter C(α, β)\nhlc : Cauchy l\nhlf : l ≤ 𝓟 {g | EqOn (⇑g) f s}\nf' : C(α, β)\nhf' : l ≤ 𝓝 f'\nthis : l.NeBot\nx : α\nhx : x ∈ s\n⊢ Inseparable (f x) (f' x)",
"... | [
"α : Type u₁\nβ : Type u₂\ninst✝² : TopologicalSpace α\ninst✝¹ : UniformSpace β\ninst✝ : CompleteSpace C(α, β)\nf : α → β\ns : Set α\nl : Filter C(α, β)\nhlc : Cauchy l\nhlf : l ≤ 𝓟 {g | EqOn (⇑g) f s}\nf' : C(α, β)\nhf' : l ≤ 𝓝 f'\nthis : l.NeBot\nx : α\nhx : x ∈ s\n⊢ Tendsto (fun f ↦ f x) l (𝓝 (f x))"
] | refine tendsto_nhds_unique_inseparable ?_ ((continuous_eval_const x).continuousAt.mono_left hf') | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Adjunction.Parametrized | {
"line": 100,
"column": 48
} | {
"line": 104,
"column": 58
} | {
"line": 106,
"column": 0
} | [
{
"pp": "C₁ : Type u₁\nC₂ : Type u₂\nC₃ : Type u₃\ninst✝² : Category.{v₁, u₁} C₁\ninst✝¹ : Category.{v₂, u₂} C₂\ninst✝ : Category.{v₃, u₃} C₃\nF : C₁ ⥤ C₂ ⥤ C₃\nG : C₁ᵒᵖ ⥤ C₃ ⥤ C₂\nadj₂ : F ⊣₂ G\nX₁ Y₁ : C₁\nX₂ : C₂\nX₃ : C₃\nf₁ : X₁ ⟶ Y₁\ng : (F.obj Y₁).obj X₂ ⟶ X₃\n⊢ adj₂.homEquiv ((F.map f₁).app X₂ ≫ g) = ad... | [] | by
have := NatTrans.congr_app (adj₂.unit_whiskerRight_map f₁) X₂
dsimp at this
simp only [homEquiv_eq, Adjunction.homEquiv_unit, Functor.comp_obj, Functor.map_comp,
Category.assoc, NatTrans.naturality, reassoc_of% this] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.UniformSpace.UniformConvergenceTopology | {
"line": 1159,
"column": 4
} | {
"line": 1160,
"column": 60
} | {
"line": 1161,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\nδ : ι → Type u_5\nφ : (i : ι) → δ i → α\n𝔗 : (i : ι) → Set (Set (δ i))\nh_image : ∀ (i : ι), MapsTo (fun x ↦ φ i '' x) (𝔗 i) 𝔖\nh_preimage : ∀ (i : ι), MapsTo (fun x ↦ φ i ⁻¹' x) 𝔖 (𝔗 i)\nh_cover : ∀... | [] | rw [← uniformContinuous_iff_le_comap]
exact UniformOnFun.precomp_uniformContinuous (h_image i) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.UniformConvergenceTopology | {
"line": 1159,
"column": 4
} | {
"line": 1160,
"column": 60
} | {
"line": 1161,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\nδ : ι → Type u_5\nφ : (i : ι) → δ i → α\n𝔗 : (i : ι) → Set (Set (δ i))\nh_image : ∀ (i : ι), MapsTo (fun x ↦ φ i '' x) (𝔗 i) 𝔖\nh_preimage : ∀ (i : ι), MapsTo (fun x ↦ φ i ⁻¹' x) 𝔖 (𝔗 i)\nh_cover : ∀... | [] | rw [← uniformContinuous_iff_le_comap]
exact UniformOnFun.precomp_uniformContinuous (h_image i) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.W.Basic | {
"line": 107,
"column": 8
} | {
"line": 107,
"column": 41
} | {
"line": 108,
"column": 8
} | [
{
"pp": "case succ.succ\nα : Type u_1\nβ : α → Type u_2\na b : α\nha : Nonempty (β a)\nhe : IsEmpty (β b)\nhf : Finite (WType β)\nhba : b ≠ a\nn : ℕ\nih :\n ∀ ⦃m : ℕ⦄,\n (fun n ↦\n have this := Nat.recOn n (mk b he.elim') fun x ih ↦ mk a fun x ↦ ih;\n this)\n n =\n (fun... | [
"case succ.succ\nα : Type u_1\nβ : α → Type u_2\na b : α\nha : Nonempty (β a)\nhe : IsEmpty (β b)\nhf : Finite (WType β)\nhba : b ≠ a\nn : ℕ\nih :\n ∀ ⦃m : ℕ⦄,\n (fun n ↦\n have this := Nat.recOn n (mk b he.elim') fun x ih ↦ mk a fun x ↦ ih;\n this)\n n =\n (fun n ↦\n ... | refine congr_arg Nat.succ (ih ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Monoidal.Rigid.Basic | {
"line": 135,
"column": 30
} | {
"line": 135,
"column": 51
} | {
"line": 136,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : MonoidalCategory C\n⊢ 𝟙_ C ◁ (ρ_ (𝟙_ C)).inv ≫ (α_ (𝟙_ C) (𝟙_ C) (𝟙_ C)).inv ≫ (ρ_ (𝟙_ C)).hom ▷ 𝟙_ C = (ρ_ (𝟙_ C)).hom ≫ (λ_ (𝟙_ C)).inv",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"CategoryTheory.MonoidalCateg... | [] | by monoidal_coherence | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monoidal.Rigid.Basic | {
"line": 136,
"column": 30
} | {
"line": 136,
"column": 51
} | {
"line": 138,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : MonoidalCategory C\n⊢ (ρ_ (𝟙_ C)).inv ▷ 𝟙_ C ≫ (α_ (𝟙_ C) (𝟙_ C) (𝟙_ C)).hom ≫ 𝟙_ C ◁ (ρ_ (𝟙_ C)).hom = (λ_ (𝟙_ C)).hom ≫ (ρ_ (𝟙_ C)).inv",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"CategoryTheory.MonoidalCateg... | [] | by monoidal_coherence | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Dual.Basis | {
"line": 142,
"column": 61
} | {
"line": 148,
"column": 18
} | {
"line": 150,
"column": 0
} | [
{
"pp": "R : Type uR\nM : Type uM\nι : Type uι\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : DecidableEq ι\nb : Basis ι R M\ninst✝ : Finite ι\nf : ι →₀ R\ni : ι\n⊢ ((Finsupp.linearCombination R ⇑b.dualBasis) f) (b i) = f i",
"ppTerm": "?m.26",
"assigned": true,
"u... | [] | by
cases nonempty_fintype ι
rw [Finsupp.linearCombination_apply, Finsupp.sum_fintype, LinearMap.sum_apply]
· simp_rw [LinearMap.smul_apply, smul_eq_mul, dualBasis_apply_self, mul_boole,
Finset.sum_ite_eq, if_pos (Finset.mem_univ i)]
· intro
rw [zero_smul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Dual.Basis | {
"line": 270,
"column": 6
} | {
"line": 270,
"column": 21
} | {
"line": 270,
"column": 22
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nι : Type u_3\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ne : ι → M\nε : ι → Dual R M\nh : DualBases e ε\nl : ι →₀ R\ni : ι\n⊢ (h.coeffs (lc e l)) i = l i",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Finsupp.instFunL... | [
"R : Type u_1\nM : Type u_2\nι : Type u_3\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ne : ι → M\nε : ι → Dual R M\nh : DualBases e ε\nl : ι →₀ R\ni : ι\n⊢ (ε i) (lc e l) = l i"
] | h.coeffs_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Dimension.RankNullity | {
"line": 87,
"column": 2
} | {
"line": 87,
"column": 60
} | {
"line": 89,
"column": 0
} | [
{
"pp": "R : Type u_1\nM M₁ : Type u\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup M₁\ninst✝² : Module R M\ninst✝¹ : Module R M₁\ninst✝ : HasRankNullity.{u, u_1} R\nf : M →ₗ[R] M₁\nthis : (p : Submodule R M) → DecidableEq (M ⧸ p)\n⊢ Module.rank R ↥f.range + Module.rank R ↥f.ker = Module.rank ... | [] | rw [← f.quotKerEquivRange.rank_eq, rank_quotient_add_rank] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Monoidal.Rigid.Basic | {
"line": 490,
"column": 70
} | {
"line": 491,
"column": 43
} | {
"line": 493,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\nX X' Y Y' Z Z' : C\ninst✝ : ExactPairing Y Y'\nf : X ⟶ Y ⊗ Z\ng : X' ⟶ Z'\n⊢ (tensorLeftHomEquiv (X ⊗ X') Y Y' (Z ⊗ Z')).symm ((f ⊗ₘ g) ≫ (α_ Y Z Z').hom) =\n (α_ Y' X X').inv ≫ ((tensorLeftHomEquiv X Y Y' Z).symm f ⊗ₘ g)",
... | [] | by
simp [tensorLeftHomEquiv, tensorHom_def'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 108,
"column": 10
} | {
"line": 108,
"column": 24
} | {
"line": 108,
"column": 24
} | [
{
"pp": "case mpr.inl\nK : Type u_13\nK₁ : Type u_14\nK₂ : Type u_15\nV : Type u_16\nV₁ : Type u_17\nV₂ : Type u_18\ninst✝⁸ : Field K\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module K V\ninst✝⁵ : Field K₁\ninst✝⁴ : AddCommGroup V₁\ninst✝³ : Module K₁ V₁\ninst✝² : Field K₂\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K₂ ... | [
"case mpr.inl\nK : Type u_13\nK₁ : Type u_14\nK₂ : Type u_15\nV : Type u_16\nV₁ : Type u_17\nV₂ : Type u_18\ninst✝⁸ : Field K\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module K V\ninst✝⁵ : Field K₁\ninst✝⁴ : AddCommGroup V₁\ninst✝³ : Module K₁ V₁\ninst✝² : Field K₂\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K₂ V₂\nI₁ : K₁ ... | map_eq_zero I₁ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 548,
"column": 2
} | {
"line": 548,
"column": 20
} | {
"line": 549,
"column": 2
} | [
{
"pp": "R : Type u_20\nM : Type u_21\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nB : LinearMap.BilinForm R M\nF : Type u_22\ninst✝¹ : FunLike F M M\ninst✝ : LinearMapClass F R M M\nf : F\nh : IsLeftRegular 2\nhB : IsSymm B\nhf : ∀ (x : M), (B (f x)) (f x) = (B x) x\nx y : M\n⊢ 2 * (B (f... | [
"R : Type u_20\nM : Type u_21\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nB : LinearMap.BilinForm R M\nF : Type u_22\ninst✝¹ : FunLike F M M\ninst✝ : LinearMapClass F R M M\nf : F\nh : IsLeftRegular 2\nhB : IsSymm B\nhf : ∀ (x : M), (B (f x)) (f x) = (B x) x\nx y : M\nthis : (B (f (x + y))) ... | have := hf (x + y) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 826,
"column": 2
} | {
"line": 826,
"column": 34
} | {
"line": 828,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_5\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nB : M →ₗ[R] M →ₗ[R] R\nhB : B.IsSymm\nW : Submodule R M\nhW : IsCompl W B.ker\nhB' : (B.domRestrict₁₂ W W).IsRefl\nx : M\nhx : x ∈ W\nhx' : x ∈ W ⊓ B.ker\n⊢ x = 0",
"ppTerm": "?m.351",
"assigned": true... | [] | simpa [hW.inf_eq_bot] using! hx' | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.Algebra.Homology.ShortComplex.ShortExact | {
"line": 58,
"column": 4
} | {
"line": 58,
"column": 18
} | {
"line": 60,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS₁ S₂ : ShortComplex C\ne : S₁ ≅ S₂\nh : S₁.ShortExact\nthis : Epi S₁.g\n⊢ Epi (S₁.g ≫ e.hom.τ₃)",
"ppTerm": "?m.154",
"assigned": true,
"usedConstants": [
"CategoryTheory.epi_comp",
"CategoryTheory.ShortC... | [] | apply epi_comp | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Preadditive.LeftExact | {
"line": 58,
"column": 2
} | {
"line": 60,
"column": 78
} | {
"line": 62,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\ninst✝⁴ : Preadditive C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\ninst✝² : Preadditive D\nF : C ⥤ D\ninst✝¹ : F.PreservesZeroMorphisms\nX Y Z : C\nπ₁ : Z ⟶ X\nπ₂ : Z ⟶ Y\ninst✝ : PreservesLimit (parallelPair π₂ 0) F\ni : IsLimit (BinaryFan.mk π₁ π₂)\nbc : Bin... | [] | exact (isLimitMapConeBinaryFanEquiv F π₁ π₂).invFun
(BinaryBicone.isBilimitOfKernelInl (F.mapBinaryBicone bc)
(isLimitMapConeForkEquiv' F bc.inl_snd (isLimitOfPreserves F hf))).isLimit | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Abelian.Exact | {
"line": 64,
"column": 4
} | {
"line": 64,
"column": 18
} | {
"line": 66,
"column": 0
} | [
{
"pp": "case mpr\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : Abelian C\nS : ShortComplex C\nthis : factorThruImage S.f ≫ imageToKernel' S.f S.g ⋯ = kernel.lift S.g S.f ⋯\na✝ : Epi (imageToKernel' S.f S.g ⋯)\n⊢ Epi (factorThruImage S.f ≫ imageToKernel' S.f S.g ⋯)",
"ppTerm": "?mpr",
"assigned": ... | [] | apply epi_comp | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.LinearAlgebra.Dual.Lemmas | {
"line": 542,
"column": 63
} | {
"line": 543,
"column": 92
} | {
"line": 545,
"column": 0
} | [
{
"pp": "K : Type u_1\nV : Type u_2\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nW : Subspace K V\n⊢ finrank K ↥W + finrank K ↥(dualAnnihilator W) = finrank K V",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodu... | [] | by
rw [← W.quotEquivAnnihilator.finrank_eq, add_comm, Submodule.finrank_quotient_add_finrank] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Dual.Lemmas | {
"line": 932,
"column": 2
} | {
"line": 932,
"column": 78
} | {
"line": 933,
"column": 2
} | [
{
"pp": "K : Type u_1\nV₁ : Type u_2\nV₂ : Type u_3\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nf : V₁ →ₗ[K] V₂\n⊢ Function.Injective ⇑f.dualMap ↔ Function.Surjective ⇑f",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
... | [
"K : Type u_1\nV₁ : Type u_2\nV₂ : Type u_3\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nf : V₁ →ₗ[K] V₂\nnot_surj : ¬Function.Surjective ⇑f\ninj : Function.Injective ⇑f.dualMap\n⊢ False"
] | refine ⟨Function.mtr fun not_surj inj ↦ ?_, dualMap_injective_of_surjective⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Comma.Presheaf.Basic | {
"line": 96,
"column": 45
} | {
"line": 96,
"column": 48
} | {
"line": 96,
"column": 49
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nA F G : Cᵒᵖ ⥤ Type v\nη : F ⟶ A\nμ : G ⟶ A\nε : F ⟶ G\nhε : ε ≫ μ = η\nX : C\ns : yoneda.obj X ⟶ A\nu : F.obj (op X)\nh : MakesOverArrow η s u\n⊢ (ConcreteCategory.hom ((ε ≫ μ).app (op X))) u = yonedaEquiv s",
"ppTerm": "?m.82",
"assigned": true,
"used... | [
"C : Type u\ninst✝ : Category.{v, u} C\nA F G : Cᵒᵖ ⥤ Type v\nη : F ⟶ A\nμ : G ⟶ A\nε : F ⟶ G\nhε : ε ≫ μ = η\nX : C\ns : yoneda.obj X ⟶ A\nu : F.obj (op X)\nh : MakesOverArrow η s u\n⊢ (ConcreteCategory.hom (η.app (op X))) u = yonedaEquiv s"
] | hε, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Constructions.Filtered | {
"line": 217,
"column": 8
} | {
"line": 225,
"column": 14
} | {
"line": 225,
"column": 15
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nα : Type w\ninst✝¹ : HasFiniteProducts C\ninst✝ : HasLimitsOfShape (Finset (Discrete α))ᵒᵖ C\nF : Discrete α ⥤ C\ns : Cone F\nm :\n s.pt ⟶\n { pt := limit (liftToFinsetObj F),\n π := Discrete.natTrans fun j ↦ limit.π (liftToFinsetObj F) (op {j}) ≫ Pi.π... | [] | apply limit.hom_ext
rintro t
dsimp [liftToFinsetObj]
apply limit.hom_ext
rintro ⟨⟨j, hj⟩⟩
convert! h j using 1
· simp [← limit.w (liftToFinsetObj F) ⟨⟨⟨Finset.singleton_subset_iff.2 hj⟩⟩⟩]
rfl
· simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Constructions.Filtered | {
"line": 217,
"column": 8
} | {
"line": 225,
"column": 14
} | {
"line": 225,
"column": 15
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nα : Type w\ninst✝¹ : HasFiniteProducts C\ninst✝ : HasLimitsOfShape (Finset (Discrete α))ᵒᵖ C\nF : Discrete α ⥤ C\ns : Cone F\nm :\n s.pt ⟶\n { pt := limit (liftToFinsetObj F),\n π := Discrete.natTrans fun j ↦ limit.π (liftToFinsetObj F) (op {j}) ≫ Pi.π... | [] | apply limit.hom_ext
rintro t
dsimp [liftToFinsetObj]
apply limit.hom_ext
rintro ⟨⟨j, hj⟩⟩
convert! h j using 1
· simp [← limit.w (liftToFinsetObj F) ⟨⟨⟨Finset.singleton_subset_iff.2 hj⟩⟩⟩]
rfl
· simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Functor.KanExtension.Pointwise | {
"line": 547,
"column": 58
} | {
"line": 549,
"column": 83
} | {
"line": 549,
"column": 83
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nD' : Type u_3\nH : Type u_4\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\ninst✝¹ : Category.{v_3, u_3} D'\ninst✝ : Category.{v_4, u_4} H\nL : C ⥤ D\nL' : C ⥤ D'\nF : C ⥤ H\nE E' : L.RightExtension F\nh : E.IsPointwiseRightKanExtension\nG : L.RightExtension... | [] | by
rw [assoc, (h Y₂).fac (coneAt G Y₂) X]
simpa using ((h Y₁).fac (coneAt G Y₁) ((StructuredArrow.map φ).obj X)).symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Functor.KanExtension.Basic | {
"line": 730,
"column": 2
} | {
"line": 730,
"column": 51
} | {
"line": 731,
"column": 2
} | [
{
"pp": "C : Type u_1\nH : Type u_3\nD : Type u_4\nD' : Type u_5\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_3, u_3} H\ninst✝² : Category.{v_4, u_4} D\ninst✝¹ : Category.{v_5, u_5} D'\nL : C ⥤ D\nL' : D ⥤ D'\nF₀ : C ⥤ H\nF₁ : D ⥤ H\nα : F₀ ⟶ L ⋙ F₁\ninst✝ : F₁.IsLeftKanExtension α\nF₂ : D' ⥤ H\nL'' : ... | [
"case refine_1\nC : Type u_1\nH : Type u_3\nD : Type u_4\nD' : Type u_5\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_3, u_3} H\ninst✝² : Category.{v_4, u_4} D\ninst✝¹ : Category.{v_5, u_5} D'\nL : C ⥤ D\nL' : D ⥤ D'\nF₀ : C ⥤ H\nF₁ : D ⥤ H\nα : F₀ ⟶ L ⋙ F₁\ninst✝ : F₁.IsLeftKanExtension α\nF₂ : D' ⥤ H\nL''... | refine ⟨fun ⟨⟨h⟩⟩ => ⟨⟨?_⟩⟩, fun ⟨⟨h⟩⟩ => ⟨⟨?_⟩⟩⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 214,
"column": 4
} | {
"line": 214,
"column": 44
} | {
"line": 216,
"column": 0
} | [
{
"pp": "case refine_2\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nP : ObjectProperty C\ninst✝ : Balanced C\nhP : P.IsSeparating\nX Y : C\nf : X ⟶ Y\nhf : ∀ (G : C), P G → ∀ (h : G ⟶ Y), ∃! h', h' ≫ f = h\nZ✝ : C\ng h : Y ⟶ Z✝\nhgh : f ≫ g = f ≫ h\nG : C\nhG : P G\nt : G ⟶ X\n⊢ (t ≫ f) ≫ g = (t ≫ f) ≫ h",
"... | [] | rw [Category.assoc, hgh, Category.assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Limits.Presheaf | {
"line": 204,
"column": 2
} | {
"line": 205,
"column": 55
} | {
"line": 207,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nℰ : Type u₂\ninst✝² : Category.{v₂, u₂} ℰ\nA : C ⥤ ℰ\ninst✝¹ : uliftYoneda.{max w v₂, v₁, u₁}.HasPointwiseLeftKanExtension A\nL : (Cᵒᵖ ⥤ Type (max w v₁ v₂)) ⥤ ℰ\nα : A ⟶ uliftYoneda.{max w v₂, v₁, u₁} ⋙ L\ninst✝ : L.IsLeftKanExtension α\nP : Cᵒᵖ ⥤ Type (max w ... | [] | simp [uliftYonedaAdjunction, restrictedULiftYonedaHomEquiv,
restrictedULiftYonedaHomEquiv', IsColimit.homEquiv] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Limits.Presheaf | {
"line": 204,
"column": 2
} | {
"line": 205,
"column": 55
} | {
"line": 207,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nℰ : Type u₂\ninst✝² : Category.{v₂, u₂} ℰ\nA : C ⥤ ℰ\ninst✝¹ : uliftYoneda.{max w v₂, v₁, u₁}.HasPointwiseLeftKanExtension A\nL : (Cᵒᵖ ⥤ Type (max w v₁ v₂)) ⥤ ℰ\nα : A ⟶ uliftYoneda.{max w v₂, v₁, u₁} ⋙ L\ninst✝ : L.IsLeftKanExtension α\nP : Cᵒᵖ ⥤ Type (max w ... | [] | simp [uliftYonedaAdjunction, restrictedULiftYonedaHomEquiv,
restrictedULiftYonedaHomEquiv', IsColimit.homEquiv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Presheaf | {
"line": 204,
"column": 2
} | {
"line": 205,
"column": 55
} | {
"line": 207,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nℰ : Type u₂\ninst✝² : Category.{v₂, u₂} ℰ\nA : C ⥤ ℰ\ninst✝¹ : uliftYoneda.{max w v₂, v₁, u₁}.HasPointwiseLeftKanExtension A\nL : (Cᵒᵖ ⥤ Type (max w v₁ v₂)) ⥤ ℰ\nα : A ⟶ uliftYoneda.{max w v₂, v₁, u₁} ⋙ L\ninst✝ : L.IsLeftKanExtension α\nP : Cᵒᵖ ⥤ Type (max w ... | [] | simp [uliftYonedaAdjunction, restrictedULiftYonedaHomEquiv,
restrictedULiftYonedaHomEquiv', IsColimit.homEquiv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 783,
"column": 6
} | {
"line": 783,
"column": 74
} | {
"line": 784,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nG : C\nh : (coyoneda.obj (op G)).ReflectsIsomorphisms\nX Y : C\nf : X ⟶ Y\nhf : ∀ (h : G ⟶ Y), ∃! h', h' ≫ f = h\nthis : IsIso ((coyoneda.obj (op G)).map f)\n⊢ IsIso f",
"ppTerm": "?m.88",
"assigned": true,
"usedConstants": [
"CategoryTheory.F... | [] | exact @isIso_of_reflects_iso _ _ _ _ _ _ _ (coyoneda.obj (op G)) _ h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 783,
"column": 6
} | {
"line": 783,
"column": 74
} | {
"line": 784,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nG : C\nh : (coyoneda.obj (op G)).ReflectsIsomorphisms\nX Y : C\nf : X ⟶ Y\nhf : ∀ (h : G ⟶ Y), ∃! h', h' ≫ f = h\nthis : IsIso ((coyoneda.obj (op G)).map f)\n⊢ IsIso f",
"ppTerm": "?m.88",
"assigned": true,
"usedConstants": [
"CategoryTheory.F... | [] | exact @isIso_of_reflects_iso _ _ _ _ _ _ _ (coyoneda.obj (op G)) _ h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 783,
"column": 6
} | {
"line": 783,
"column": 74
} | {
"line": 784,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nG : C\nh : (coyoneda.obj (op G)).ReflectsIsomorphisms\nX Y : C\nf : X ⟶ Y\nhf : ∀ (h : G ⟶ Y), ∃! h', h' ≫ f = h\nthis : IsIso ((coyoneda.obj (op G)).map f)\n⊢ IsIso f",
"ppTerm": "?m.88",
"assigned": true,
"usedConstants": [
"CategoryTheory.F... | [] | exact @isIso_of_reflects_iso _ _ _ _ _ _ _ (coyoneda.obj (op G)) _ h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.Grp.Zero | {
"line": 32,
"column": 2
} | {
"line": 33,
"column": 16
} | {
"line": 35,
"column": 0
} | [
{
"pp": "case refine_2\nG : GrpCat\ninst✝ : Subsingleton ↑G\nX : GrpCat\nf : X ⟶ G\n⊢ f = default",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Inhabited.default",
"GrpCat",
"MonoidHom.instFunLike",
"CategoryTheory.CategoryStruct.toQuiver",
"MonoidHom",... | [] | · ext
subsingleton | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Category.Grp.Zero | {
"line": 63,
"column": 2
} | {
"line": 64,
"column": 16
} | {
"line": 66,
"column": 0
} | [
{
"pp": "case refine_2\nG : CommGrpCat\ninst✝ : Subsingleton ↑G\nX : CommGrpCat\nf : X ⟶ G\n⊢ f = default",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Inhabited.default",
"MonoidHom.instFunLike",
"CategoryTheory.CategoryStruct.toQuiver",
"MonoidHom",
"... | [] | · ext
subsingleton | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Limits.Types.Coproducts | {
"line": 128,
"column": 11
} | {
"line": 128,
"column": 13
} | {
"line": 128,
"column": 14
} | [
{
"pp": "C : Type u\nF : C → Type v\nc : CofanTypes F\nhc : Functor.CoconeTypes.IsColimit c\ni : C\ny₁ : F i\n⊢ ∀ ⦃a₂ : F i⦄, c.inj i y₁ = c.inj i a₂ → y₁ = a₂",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"F",
"i"
],
"usedGoals": [
{
... | [
"C : Type u\nF : C → Type v\nc : CofanTypes F\nhc : Functor.CoconeTypes.IsColimit c\ni : C\ny₁ y₂ : F i\n⊢ c.inj i y₁ = c.inj i y₂ → y₁ = y₂"
] | y₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Limits.Presheaf | {
"line": 769,
"column": 4
} | {
"line": 769,
"column": 47
} | {
"line": 770,
"column": 4
} | [
{
"pp": "case refine_1\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : LocallySmall.{w, v₁, u₁} C\nF : C ⥤ Type w\nX : C\nG : F.Elementsᵒᵖ ⥤ Type w := (CategoryOfElements.π F).op ⋙ shrinkYoneda.{w, v₁, u₁}.obj X\nc : G.CoconeTypes := G.coconeTypesEquiv.symm (coconeπOpCompShrinkYonedaObj F X)\nthis :\n ∀ (u... | [
"case refine_1\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : LocallySmall.{w, v₁, u₁} C\nF : C ⥤ Type w\nX : C\nG : F.Elementsᵒᵖ ⥤ Type w := (CategoryOfElements.π F).op ⋙ shrinkYoneda.{w, v₁, u₁}.obj X\nc : G.CoconeTypes := G.coconeTypesEquiv.symm (coconeπOpCompShrinkYonedaObj F X)\nthis :\n ∀ (u : G.Colimit... | generalize hx₁ : G.descColimitType c u₁ = x | Lean.Elab.Tactic.evalGeneralize | Lean.Parser.Tactic.generalize |
Mathlib.CategoryTheory.Limits.Types.Coproducts | {
"line": 293,
"column": 62
} | {
"line": 293,
"column": 72
} | {
"line": 293,
"column": 72
} | [
{
"pp": "case mp\nX Y : Type u\nc : BinaryCofan X Y\nh : IsColimit c\n⊢ ⇑(ConcreteCategory.hom (h.coconePointUniqueUpToIso (binaryCoproductColimit X Y)).inv) ''\n Set.range ⇑(ConcreteCategory.hom (↾Sum.inl)) =\n (Set.range ⇑(ConcreteCategory.hom (↾Sum.inr ≫ (h.coconePointUniqueUpToIso (binaryCoproductCo... | [
"case mp\nX Y : Type u\nc : BinaryCofan X Y\nh : IsColimit c\n⊢ ⇑(ConcreteCategory.hom (h.coconePointUniqueUpToIso (binaryCoproductColimit X Y)).inv) ''\n Set.range ⇑(ConcreteCategory.hom (↾Sum.inl)) =\n (Set.range\n (⇑(ConcreteCategory.hom (h.coconePointUniqueUpToIso (binaryCoproductColimit X Y)).in... | types_comp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Presheaf | {
"line": 801,
"column": 4
} | {
"line": 803,
"column": 50
} | {
"line": 805,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : LocallySmall.{w, v₁, u₁} C\nF : C ⥤ Type w\nu v : F.Elementsᵒᵖ\ng : u ⟶ v\n⊢ ((CategoryOfElements.π F).op ⋙ shrinkYoneda.{w, v₁, u₁}.flip).map g ≫\n { app := fun X ↦ (coconeπOpCompShrinkYonedaObj F X).ι.app v, naturality := ⋯ } =\n { app := fun... | [] | ext X x
obtain ⟨x, rfl⟩ := shrinkYonedaObjObjEquiv.symm.surjective x
simp [← shrinkYonedaObjObjEquiv_symm_comp.{w}] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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