module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.Category.ModuleCat.Adjunctions | {
"line": 333,
"column": 8
} | {
"line": 333,
"column": 38
} | {
"line": 334,
"column": 8
} | [
{
"pp": "case single.add.h_add\nR : Type u_1\ninst✝⁴ : CommRing R\nC : Type u\ninst✝³ : Category.{v, u} C\nD : Type u\ninst✝² : Category.{v, u} D\ninst✝¹ : Preadditive D\ninst✝ : Linear R D\nF : C ⥤ D\nX Y Z : Free R C\nf' : X ⟶ Y\nr : R\nf₁ f₂ : (Y ⟶ Z) →₀ R\nw₁ :\n (sum (single f' r ≫ f₁) fun f' r ↦ r • F.ma... | [
"case single.add.h_zero\nR : Type u_1\ninst✝⁴ : CommRing R\nC : Type u\ninst✝³ : Category.{v, u} C\nD : Type u\ninst✝² : Category.{v, u} D\ninst✝¹ : Preadditive D\ninst✝ : Linear R D\nF : C ⥤ D\nX Y Z : Free R C\nf' : X ⟶ Y\nr : R\nf₁ f₂ : (Y ⟶ Z) →₀ R\nw₁ :\n (sum (single f' r ≫ f₁) fun f' r ↦ r • F.map f') =\n ... | · intros; simp only [add_smul] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Category.ModuleCat.Adjunctions | {
"line": 335,
"column": 8
} | {
"line": 335,
"column": 38
} | {
"line": 336,
"column": 6
} | [
{
"pp": "case single.add.h_add\nR : Type u_1\ninst✝⁴ : CommRing R\nC : Type u\ninst✝³ : Category.{v, u} C\nD : Type u\ninst✝² : Category.{v, u} D\ninst✝¹ : Preadditive D\ninst✝ : Linear R D\nF : C ⥤ D\nX Y Z : Free R C\nf' : X ⟶ Y\nr : R\nf₁ f₂ : (Y ⟶ Z) →₀ R\nw₁ :\n (sum (single f' r ≫ f₁) fun f' r ↦ r • F.ma... | [] | · intros; simp only [add_smul] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Homology.ShortComplex.SnakeLemma | {
"line": 204,
"column": 47
} | {
"line": 208,
"column": 29
} | {
"line": 210,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\nS : SnakeInput C\ninst✝ : Epi S.L₂.g\n⊢ Epi S.L₃.g",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"Eq.mpr",
"CategoryTheory.ShortComplex.SnakeInput.L₃",
... | [] | by
have : Epi (S.v₂₃.τ₂ ≫ S.L₃.g) := by
rw [S.v₂₃.comm₂₃]
apply epi_comp
exact epi_of_epi S.v₂₃.τ₂ _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.ShortComplex.SnakeLemma | {
"line": 275,
"column": 45
} | {
"line": 275,
"column": 62
} | {
"line": 275,
"column": 63
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nS : SnakeInput C\nA : C\nx₂ : A ⟶ S.P\nhx₂ : x₂ ≫ pullback.snd S.L₁.g S.v₀₁.τ₃ = 0\nA' : C\nπ : A' ⟶ A\nhπ : Epi π\nx₁ : A' ⟶ S.L₁.X₁\nfac : π ≫ x₂ ≫ pullback.fst S.L₁.g S.v₀₁.τ₃ = x₁ ≫ S.L₁.f\n⊢ (π ≫ x₂) ≫ pullback.fst S.L₁.g S.v₀₁.τ₃ = ... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nS : SnakeInput C\nA : C\nx₂ : A ⟶ S.P\nhx₂ : x₂ ≫ pullback.snd S.L₁.g S.v₀₁.τ₃ = 0\nA' : C\nπ : A' ⟶ A\nhπ : Epi π\nx₁ : A' ⟶ S.L₁.X₁\nfac : π ≫ x₂ ≫ pullback.fst S.L₁.g S.v₀₁.τ₃ = x₁ ≫ S.L₁.f\n⊢ π ≫ x₂ ≫ pullback.fst S.L₁.g S.v₀₁.τ₃ = x₁ ≫ S.L₁.f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.ShortComplex.ConcreteCategory | {
"line": 55,
"column": 2
} | {
"line": 56,
"column": 18
} | {
"line": 57,
"column": 2
} | [
{
"pp": "case mp\nC : Type u\ninst✝⁷ : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type w\ninst✝⁶ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝⁵ : ConcreteCategory C FC\ninst✝⁴ : HasForget₂ C Ab\ninst✝³ : Preadditive C\ninst✝² : (forget₂ C Ab).Additive\ninst✝¹ : (forget₂ C Ab).PreservesHomology\nin... | [
"case mpr\nC : Type u\ninst✝⁷ : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type w\ninst✝⁶ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝⁵ : ConcreteCategory C FC\ninst✝⁴ : HasForget₂ C Ab\ninst✝³ : Preadditive C\ninst✝² : (forget₂ C Ab).Additive\ninst✝¹ : (forget₂ C Ab).PreservesHomology\ninst✝ : HasZe... | · intro
infer_instance | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Homology.ShortComplex.ConcreteCategory | {
"line": 70,
"column": 2
} | {
"line": 71,
"column": 18
} | {
"line": 72,
"column": 2
} | [
{
"pp": "case mp\nC : Type u\ninst✝⁷ : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type w\ninst✝⁶ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝⁵ : ConcreteCategory C FC\ninst✝⁴ : HasForget₂ C Ab\ninst✝³ : Preadditive C\ninst✝² : (forget₂ C Ab).Additive\ninst✝¹ : (forget₂ C Ab).PreservesHomology\nin... | [
"case mpr\nC : Type u\ninst✝⁷ : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type w\ninst✝⁶ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝⁵ : ConcreteCategory C FC\ninst✝⁴ : HasForget₂ C Ab\ninst✝³ : Preadditive C\ninst✝² : (forget₂ C Ab).Additive\ninst✝¹ : (forget₂ C Ab).PreservesHomology\ninst✝ : HasZe... | · intro
infer_instance | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Homology.ShortComplex.ConcreteCategory | {
"line": 155,
"column": 4
} | {
"line": 155,
"column": 29
} | {
"line": 155,
"column": 30
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type v\ninst✝⁵ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝⁴ : ConcreteCategory C FC\ninst✝³ : HasForget₂ C Ab\ninst✝² : Abelian C\ninst✝¹ : (forget₂ C Ab).Additive\ninst✝ : (forget₂ C Ab).PreservesHomology\nD : SnakeInput C... | [
"C : Type u\ninst✝⁶ : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type v\ninst✝⁵ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝⁴ : ConcreteCategory C FC\ninst✝³ : HasForget₂ C Ab\ninst✝² : Abelian C\ninst✝¹ : (forget₂ C Ab).Additive\ninst✝ : (forget₂ C Ab).PreservesHomology\nD : SnakeInput C\nx₃ : ToTyp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory | {
"line": 375,
"column": 2
} | {
"line": 375,
"column": 13
} | {
"line": 375,
"column": 14
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : HasZeroMorphisms C\nFC : C → C → Type u_2\nCC : C → Type u_3\ninst✝² : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝¹ : ConcreteCategory C FC\nM N K : C\nf : M ⟶ N\ninst✝ : HasCokernel f\ng h : cokernel f ⟶ K\nw : ∀ (n : ToType N), (hom g) ((ho... | [
"C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : HasZeroMorphisms C\nFC : C → C → Type u_2\nCC : C → Type u_3\ninst✝² : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝¹ : ConcreteCategory C FC\nM N K : C\nf : M ⟶ N\ninst✝ : HasCokernel f\ng h : cokernel f ⟶ K\nw : ∀ (n : ToType N), (hom g) ((hom (cokernel.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.ShortComplex.ModuleCat | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 53
} | {
"line": 66,
"column": 54
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nS : ShortComplex (ModuleCat R)\nhS : S.Exact\n⊢ (ModuleCat.Hom.hom S.f).range = (ModuleCat.Hom.hom S.g).ker",
"ppTerm": "?m.45",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝ : Ring R\nS : ShortComplex (ModuleCat R)\nhS : S.Exact\n⊢ (ModuleCat.Hom.hom S.f).range = (ModuleCat.Hom.hom S.g).ker"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.ShortComplex.ModuleCat | {
"line": 165,
"column": 25
} | {
"line": 165,
"column": 36
} | {
"line": 165,
"column": 37
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nS : ShortComplex (ModuleCat R)\nx y : ↑S.X₂\nh : (ConcreteCategory.hom S.pOpcycles) x = (ConcreteCategory.hom S.pOpcycles) y\n⊢ Submodule.Quotient.mk x = Submodule.Quotient.mk y",
"ppTerm": "?m.55",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"R : Type u\ninst✝ : Ring R\nS : ShortComplex (ModuleCat R)\nx y : ↑S.X₂\nh : (ConcreteCategory.hom S.pOpcycles) x = (ConcreteCategory.hom S.pOpcycles) y\n⊢ Submodule.Quotient.mk x = Submodule.Quotient.mk y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.ShortComplex.ModuleCat | {
"line": 171,
"column": 2
} | {
"line": 171,
"column": 13
} | {
"line": 171,
"column": 14
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nS : ShortComplex (ModuleCat R)\nx : ↑S.X₂\n⊢ (ConcreteCategory.hom S.pOpcycles) x = 0 ↔ x ∈ (ModuleCat.Hom.hom S.f).range",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"CategoryTheory.ShortComplex.opcycles",
"Eq.mpr",
"Submodule",
... | [
"R : Type u\ninst✝ : Ring R\nS : ShortComplex (ModuleCat R)\nx : ↑S.X₂\n⊢ (ConcreteCategory.hom S.pOpcycles) x = 0 ↔ ∃ y, (ModuleCat.Hom.hom S.f) y = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.ShortComplex.ModuleCat | {
"line": 265,
"column": 6
} | {
"line": 265,
"column": 17
} | {
"line": 265,
"column": 18
} | [
{
"pp": "R : Type u\ninst✝¹² : Ring R\nM : Type v\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : Module R M\nN : Type v\ninst✝⁹ : AddCommGroup N\ninst✝⁸ : Module R N\nL : Type v\ninst✝⁷ : AddCommGroup L\ninst✝⁶ : Module R L\nM' : Type u_1\nN' : Type u_2\nL' : Type u_3\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : AddCommGroup N'\n... | [
"R : Type u\ninst✝¹² : Ring R\nM : Type v\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : Module R M\nN : Type v\ninst✝⁹ : AddCommGroup N\ninst✝⁸ : Module R N\nL : Type v\ninst✝⁷ : AddCommGroup L\ninst✝⁶ : Module R L\nM' : Type u_1\nN' : Type u_2\nL' : Type u_3\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : AddCommGroup N'\ninst✝³ : Add... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Category.ModuleCat.Biproducts | {
"line": 186,
"column": 8
} | {
"line": 186,
"column": 19
} | {
"line": 186,
"column": 20
} | [
{
"pp": "R : Type u\nA : Type uA\nM : Type uM\nB : Type uB\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup A\ninst✝⁴ : AddCommGroup B\ninst✝³ : AddCommGroup M\ninst✝² : Module R A\ninst✝¹ : Module R B\ninst✝ : Module R M\nj : A →ₗ[R] M\ng : M →ₗ[R] B\nf : B →ₗ[R] M\nhj : Function.Injective ⇑j\nexac : j.range = g.ker\nh... | [
"R : Type u\nA : Type uA\nM : Type uM\nB : Type uB\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup A\ninst✝⁴ : AddCommGroup B\ninst✝³ : AddCommGroup M\ninst✝² : Module R A\ninst✝¹ : Module R B\ninst✝ : Module R M\nj : A →ₗ[R] M\ng : M →ₗ[R] B\nf : B →ₗ[R] M\nhj : Function.Injective ⇑j\nexac : j.range = g.ker\nh : g ∘ₗ f = ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Category.ModuleCat.Biproducts | {
"line": 188,
"column": 15
} | {
"line": 188,
"column": 26
} | {
"line": 188,
"column": 27
} | [
{
"pp": "R : Type u\nA : Type uA\nM : Type uM\nB : Type uB\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup A\ninst✝⁴ : AddCommGroup B\ninst✝³ : AddCommGroup M\ninst✝² : Module R A\ninst✝¹ : Module R B\ninst✝ : Module R M\nj : A →ₗ[R] M\ng : M →ₗ[R] B\nf : B →ₗ[R] M\nhj : Function.Injective ⇑j\nexac : j.range = g.ker\nh... | [
"R : Type u\nA : Type uA\nM : Type uM\nB : Type uB\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup A\ninst✝⁴ : AddCommGroup B\ninst✝³ : AddCommGroup M\ninst✝² : Module R A\ninst✝¹ : Module R B\ninst✝ : Module R M\nj : A →ₗ[R] M\ng : M →ₗ[R] B\nf : B →ₗ[R] M\nhj : Function.Injective ⇑j\nexac : j.range = g.ker\nh : g ∘ₗ f = ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Category.ModuleCat.Biproducts | {
"line": 202,
"column": 8
} | {
"line": 202,
"column": 19
} | {
"line": 202,
"column": 20
} | [
{
"pp": "R : Type u\nA : Type uA\nM : Type uM\nB : Type uB\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup A\ninst✝⁴ : AddCommGroup B\ninst✝³ : AddCommGroup M\ninst✝² : Module R A\ninst✝¹ : Module R B\ninst✝ : Module R M\nj : A →ₗ[R] M\ng : M →ₗ[R] B\nf : M →ₗ[R] A\nhg : Function.Surjective ⇑g\nexac : j.range = g.ker\n... | [
"R : Type u\nA : Type uA\nM : Type uM\nB : Type uB\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup A\ninst✝⁴ : AddCommGroup B\ninst✝³ : AddCommGroup M\ninst✝² : Module R A\ninst✝¹ : Module R B\ninst✝ : Module R M\nj : A →ₗ[R] M\ng : M →ₗ[R] B\nf : M →ₗ[R] A\nhg : Function.Surjective ⇑g\nexac : j.range = g.ker\nh : f ∘ₗ j =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Category.ModuleCat.Biproducts | {
"line": 204,
"column": 15
} | {
"line": 204,
"column": 26
} | {
"line": 204,
"column": 27
} | [
{
"pp": "R : Type u\nA : Type uA\nM : Type uM\nB : Type uB\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup A\ninst✝⁴ : AddCommGroup B\ninst✝³ : AddCommGroup M\ninst✝² : Module R A\ninst✝¹ : Module R B\ninst✝ : Module R M\nj : A →ₗ[R] M\ng : M →ₗ[R] B\nf : M →ₗ[R] A\nhg : Function.Surjective ⇑g\nexac : j.range = g.ker\n... | [
"R : Type u\nA : Type uA\nM : Type uM\nB : Type uB\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup A\ninst✝⁴ : AddCommGroup B\ninst✝³ : AddCommGroup M\ninst✝² : Module R A\ninst✝¹ : Module R B\ninst✝ : Module R M\nj : A →ₗ[R] M\ng : M →ₗ[R] B\nf : M →ₗ[R] A\nhg : Function.Surjective ⇑g\nexac : j.range = g.ker\nh : f ∘ₗ j =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer | {
"line": 607,
"column": 8
} | {
"line": 607,
"column": 17
} | {
"line": 608,
"column": 6
} | [
{
"pp": "case left\nC : Type u\ninst✝ : Category.{v, u} C\nJ : MulticospanShape\nI : MulticospanIndex J C\nK : Multifork I\nlift : (E : Multifork I) → E.pt ⟶ K.pt\nfac : ∀ (E : Multifork I) (i : J.L), lift E ≫ K.ι i = E.ι i\nuniq : ∀ (E : Multifork I) (m : E.pt ⟶ K.pt), (∀ (i : J.L), m ≫ K.ι i = E.ι i) → m = li... | [] | apply fac | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer | {
"line": 607,
"column": 8
} | {
"line": 607,
"column": 17
} | {
"line": 608,
"column": 6
} | [
{
"pp": "case left\nC : Type u\ninst✝ : Category.{v, u} C\nJ : MulticospanShape\nI : MulticospanIndex J C\nK : Multifork I\nlift : (E : Multifork I) → E.pt ⟶ K.pt\nfac : ∀ (E : Multifork I) (i : J.L), lift E ≫ K.ι i = E.ι i\nuniq : ∀ (E : Multifork I) (m : E.pt ⟶ K.pt), (∀ (i : J.L), m ≫ K.ι i = E.ι i) → m = li... | [] | apply fac | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer | {
"line": 607,
"column": 8
} | {
"line": 607,
"column": 17
} | {
"line": 608,
"column": 6
} | [
{
"pp": "case left\nC : Type u\ninst✝ : Category.{v, u} C\nJ : MulticospanShape\nI : MulticospanIndex J C\nK : Multifork I\nlift : (E : Multifork I) → E.pt ⟶ K.pt\nfac : ∀ (E : Multifork I) (i : J.L), lift E ≫ K.ι i = E.ι i\nuniq : ∀ (E : Multifork I) (m : E.pt ⟶ K.pt), (∀ (i : J.L), m ≫ K.ι i = E.ι i) → m = li... | [] | apply fac | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer | {
"line": 611,
"column": 8
} | {
"line": 611,
"column": 17
} | {
"line": 612,
"column": 4
} | [
{
"pp": "case right.e_a\nC : Type u\ninst✝ : Category.{v, u} C\nJ : MulticospanShape\nI : MulticospanIndex J C\nK : Multifork I\nlift : (E : Multifork I) → E.pt ⟶ K.pt\nfac : ∀ (E : Multifork I) (i : J.L), lift E ≫ K.ι i = E.ι i\nuniq : ∀ (E : Multifork I) (m : E.pt ⟶ K.pt), (∀ (i : J.L), m ≫ K.ι i = E.ι i) → m... | [] | apply fac | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Monad.Adjunction | {
"line": 194,
"column": 19
} | {
"line": 194,
"column": 30
} | {
"line": 194,
"column": 31
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\nT : Monad C\nX : T.adj.toMonad.Algebra\n⊢ T.η.app X.A ≫ X.a = 𝟙 X.A",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"CategoryTheory.Monad.forget",
"CategoryTheor... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\nT : Monad C\nX : T.adj.toMonad.Algebra\n⊢ T.η.app X.A ≫ X.a = 𝟙 X.A"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monad.Adjunction | {
"line": 195,
"column": 20
} | {
"line": 195,
"column": 31
} | {
"line": 195,
"column": 32
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\nT : Monad C\nX : T.adj.toMonad.Algebra\n⊢ T.μ.app X.A ≫ X.a = T.map X.a ≫ X.a",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"CategoryTheory.Monad.forget",
"Cate... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\nT : Monad C\nX : T.adj.toMonad.Algebra\n⊢ T.μ.app X.A ≫ X.a = T.map X.a ≫ X.a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monad.Adjunction | {
"line": 243,
"column": 21
} | {
"line": 243,
"column": 32
} | {
"line": 243,
"column": 33
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\nG : Comonad C\nX : G.adj.toComonad.Coalgebra\n⊢ X.a ≫ G.ε.app X.A = 𝟙 X.A",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"CategoryTheory.Comonad.Coalgebra.a",
"... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\nG : Comonad C\nX : G.adj.toComonad.Coalgebra\n⊢ X.a ≫ G.ε.app X.A = 𝟙 X.A"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monad.Adjunction | {
"line": 244,
"column": 22
} | {
"line": 244,
"column": 33
} | {
"line": 244,
"column": 34
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\nG : Comonad C\nX : G.adj.toComonad.Coalgebra\n⊢ X.a ≫ G.δ.app X.A = X.a ≫ G.map X.a",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"CategoryTheory.Comonad.Coalgebra.a"... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\nG : Comonad C\nX : G.adj.toComonad.Coalgebra\n⊢ X.a ≫ G.δ.app X.A = X.a ≫ G.map X.a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer | {
"line": 878,
"column": 8
} | {
"line": 878,
"column": 17
} | {
"line": 879,
"column": 6
} | [
{
"pp": "case e_a\nC : Type u\ninst✝ : Category.{v, u} C\nJ : MultispanShape\nI : MultispanIndex J C\nK : Multicofork I\ndesc : (E : Multicofork I) → K.pt ⟶ E.pt\nfac : ∀ (E : Multicofork I) (i : J.R), K.π i ≫ desc E = E.π i\nuniq : ∀ (E : Multicofork I) (m : K.pt ⟶ E.pt), (∀ (i : J.R), K.π i ≫ m = E.π i) → m =... | [] | apply fac | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Monad.Adjunction | {
"line": 339,
"column": 8
} | {
"line": 339,
"column": 19
} | {
"line": 339,
"column": 20
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\ninst✝ : Reflective R\nX : (reflectorAdjunction R).toMonad.Algebra\n⊢ R.map ((reflector R).map ((reflectorAdjunction R).unit.app X.A ≫ X.a)) = 𝟙 (R.obj ((reflector R).obj X.A))",
"ppTerm": "?... | [
"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\ninst✝ : Reflective R\nX : (reflectorAdjunction R).toMonad.Algebra\n⊢ R.map ((reflector R).map ((reflectorAdjunction R).unit.app X.A)) ≫ R.map ((reflector R).map X.a) =\n 𝟙 (R.obj ((reflector R).obj X.A))... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer | {
"line": 879,
"column": 8
} | {
"line": 879,
"column": 17
} | {
"line": 880,
"column": 4
} | [
{
"pp": "case right\nC : Type u\ninst✝ : Category.{v, u} C\nJ : MultispanShape\nI : MultispanIndex J C\nK : Multicofork I\ndesc : (E : Multicofork I) → K.pt ⟶ E.pt\nfac : ∀ (E : Multicofork I) (i : J.R), K.π i ≫ desc E = E.π i\nuniq : ∀ (E : Multicofork I) (m : K.pt ⟶ E.pt), (∀ (i : J.R), K.π i ≫ m = E.π i) → m... | [] | apply fac | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer | {
"line": 879,
"column": 8
} | {
"line": 879,
"column": 17
} | {
"line": 880,
"column": 4
} | [
{
"pp": "case right\nC : Type u\ninst✝ : Category.{v, u} C\nJ : MultispanShape\nI : MultispanIndex J C\nK : Multicofork I\ndesc : (E : Multicofork I) → K.pt ⟶ E.pt\nfac : ∀ (E : Multicofork I) (i : J.R), K.π i ≫ desc E = E.π i\nuniq : ∀ (E : Multicofork I) (m : K.pt ⟶ E.pt), (∀ (i : J.R), K.π i ≫ m = E.π i) → m... | [] | apply fac | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer | {
"line": 879,
"column": 8
} | {
"line": 879,
"column": 17
} | {
"line": 880,
"column": 4
} | [
{
"pp": "case right\nC : Type u\ninst✝ : Category.{v, u} C\nJ : MultispanShape\nI : MultispanIndex J C\nK : Multicofork I\ndesc : (E : Multicofork I) → K.pt ⟶ E.pt\nfac : ∀ (E : Multicofork I) (i : J.R), K.π i ≫ desc E = E.π i\nuniq : ∀ (E : Multicofork I) (m : K.pt ⟶ E.pt), (∀ (i : J.R), K.π i ≫ m = E.π i) → m... | [] | apply fac | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monad.Adjunction | {
"line": 375,
"column": 8
} | {
"line": 375,
"column": 19
} | {
"line": 375,
"column": 20
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\ninst✝ : Coreflective R\nX : (coreflectorAdjunction R).toComonad.Coalgebra\n⊢ R.map ((coreflector R).map (X.a ≫ (coreflectorAdjunction R).counit.app X.A)) = 𝟙 (R.obj ((coreflector R).obj X.A))",
... | [
"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\ninst✝ : Coreflective R\nX : (coreflectorAdjunction R).toComonad.Coalgebra\n⊢ R.map ((coreflector R).map X.a) ≫ R.map ((coreflector R).map ((coreflectorAdjunction R).counit.app X.A)) =\n 𝟙 (R.obj ((corefl... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monad.Adjunction | {
"line": 387,
"column": 2
} | {
"line": 387,
"column": 13
} | {
"line": 387,
"column": 14
} | [
{
"pp": "case refine_2\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\ninst✝ : Coreflective R\nX : (coreflectorAdjunction R).toComonad.Coalgebra\n⊢ ((Comonad.comparison (coreflectorAdjunction R)).obj ((coreflector R).obj X.A)).a ≫\n (coreflectorA... | [
"case refine_2\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\ninst✝ : Coreflective R\nX : (coreflectorAdjunction R).toComonad.Coalgebra\n⊢ (coreflectorAdjunction R).counit.app X.A =\n (coreflectorAdjunction R).counit.app X.A ≫ X.a ≫ (coreflectorAdjunc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monad.Limits | {
"line": 90,
"column": 8
} | {
"line": 90,
"column": 19
} | {
"line": 90,
"column": 20
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nT : Monad C\nJ : Type u\ninst✝ : Category.{v, u} J\nD : J ⥤ T.Algebra\nc : Cone (D ⋙ T.forget)\nt : IsLimit c\nX Y : J\nf : X ⟶ Y\n⊢ (((const J).obj (conePoint D c t)).map f ≫ { f := c.π.app Y, h := ⋯ }).f = ({ f := c.π.app X, h := ⋯ } ≫ D.map f).f",
"ppTe... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nT : Monad C\nJ : Type u\ninst✝ : Category.{v, u} J\nD : J ⥤ T.Algebra\nc : Cone (D ⋙ T.forget)\nt : IsLimit c\nX Y : J\nf : X ⟶ Y\n⊢ c.π.app Y = c.π.app X ≫ (D.map f).f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.ComposableArrows.Basic | {
"line": 255,
"column": 35
} | {
"line": 255,
"column": 46
} | {
"line": 255,
"column": 47
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nn m : ℕ\nF✝ G✝ : ComposableArrows C n\nF G : ComposableArrows C 1\nleft : F.obj' 0 homMk₁._proof_4 ⟶ G.obj' 0 homMk₁._proof_4\nright : F.obj' 1 homMk₁._proof_5 ⟶ G.obj' 1 homMk₁._proof_5\nw : F.map' 0 1 homMk₁._proof_4 homMk₁._proof_5 ≫ right = left ≫ G.map'... | [
"C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nn m : ℕ\nF✝ G✝ : ComposableArrows C n\nF G : ComposableArrows C 1\nleft : F.obj' 0 homMk₁._proof_4 ⟶ G.obj' 0 homMk₁._proof_4\nright : F.obj' 1 homMk₁._proof_5 ⟶ G.obj' 1 homMk₁._proof_5\nw : F.map' 0 1 homMk₁._proof_4 homMk₁._proof_5 ≫ right = left ≫ G.map' 0 1 homMk₁.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.ComposableArrows.Basic | {
"line": 261,
"column": 14
} | {
"line": 265,
"column": 19
} | {
"line": 267,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nF G : ComposableArrows C 1\nφ φ' : F ⟶ G\nh₀ : app' φ 0 homMk₁._proof_4 = app' φ' 0 homMk₁._proof_4\nh₁ : app' φ 1 homMk₁._proof_5 = app' φ' 1 homMk₁._proof_5\n⊢ φ = φ'",
"ppTerm": "?m.70",
"assigned": true,
"usedConstants": [
"CategoryTheo... | [] | by
ext i
match i with
| 0 => exact h₀
| 1 => exact h₁ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monad.Limits | {
"line": 108,
"column": 4
} | {
"line": 108,
"column": 48
} | {
"line": 108,
"column": 49
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nT : Monad C\nJ✝ : Type u\ninst✝ : Category.{v, u} J✝\nD : J✝ ⥤ T.Algebra\nc : Cone (D ⋙ T.forget)\nt : IsLimit c\ns : Cone D\nm : s.pt ⟶ (liftedCone D c t).pt\nJ : ∀ (j : J✝), m ≫ (liftedCone D c t).π.app j = s.π.app j\nj : J✝\n⊢ m.f ≫ c.π.app j = { f := t.lif... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nT : Monad C\nJ✝ : Type u\ninst✝ : Category.{v, u} J✝\nD : J✝ ⥤ T.Algebra\nc : Cone (D ⋙ T.forget)\nt : IsLimit c\ns : Cone D\nm : s.pt ⟶ (liftedCone D c t).pt\nJ : ∀ (j : J✝), m ≫ (liftedCone D c t).π.app j = s.π.app j\nj : J✝\n⊢ m.f ≫ c.π.app j = (s.π.app j).f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.ComposableArrows.Basic | {
"line": 348,
"column": 52
} | {
"line": 348,
"column": 63
} | {
"line": 348,
"column": 64
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nn m : ℕ\nF G : ComposableArrows C n\nX : C\nf : X ⟶ F.left\ni : ℕ\nhi : i + 1 < n + 1 + 1\nj : ℕ\nhj : j + 1 < n + 1 + 1\nhij : ⟨i + 1, hi⟩ ≤ ⟨j + 1, hj⟩\n⊢ i ≤ j",
"ppTerm": "?m.254",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nn m : ℕ\nF G : ComposableArrows C n\nX : C\nf : X ⟶ F.left\ni : ℕ\nhi : i + 1 < n + 1 + 1\nj : ℕ\nhj : j + 1 < n + 1 + 1\nhij : ⟨i + 1, hi⟩ ≤ ⟨j + 1, hj⟩\n⊢ i ≤ j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monad.Limits | {
"line": 226,
"column": 4
} | {
"line": 226,
"column": 15
} | {
"line": 226,
"column": 16
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nT : Monad C\nJ✝ : Type u\ninst✝² : Category.{v, u} J✝\nD : J✝ ⥤ T.Algebra\nc : Cocone (D ⋙ T.forget)\nt : IsColimit c\ninst✝¹ : PreservesColimit (D ⋙ T.forget) T.toFunctor\ninst✝ : PreservesColimit ((D ⋙ T.forget) ⋙ T.toFunctor) T.toFunctor\ns : Cocone D\nm : ... | [
"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nT : Monad C\nJ✝ : Type u\ninst✝² : Category.{v, u} J✝\nD : J✝ ⥤ T.Algebra\nc : Cocone (D ⋙ T.forget)\nt : IsColimit c\ninst✝¹ : PreservesColimit (D ⋙ T.forget) T.toFunctor\ninst✝ : PreservesColimit ((D ⋙ T.forget) ⋙ T.toFunctor) T.toFunctor\ns : Cocone D\nm : (liftedCocon... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monad.Limits | {
"line": 248,
"column": 16
} | {
"line": 248,
"column": 27
} | {
"line": 248,
"column": 28
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nT : Monad C\nJ : Type u\ninst✝² : Category.{v, u} J\nD : J ⥤ T.Algebra\ninst✝¹ : PreservesColimit (D ⋙ T.forget) T.toFunctor\ninst✝ : PreservesColimit ((D ⋙ T.forget) ⋙ T.toFunctor) T.toFunctor\nc : Cocone (D ⋙ T.forget)\nt : IsColimit c\nA B : J\nf : A ⟶ B\n⊢... | [
"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nT : Monad C\nJ : Type u\ninst✝² : Category.{v, u} J\nD : J ⥤ T.Algebra\ninst✝¹ : PreservesColimit (D ⋙ T.forget) T.toFunctor\ninst✝ : PreservesColimit ((D ⋙ T.forget) ⋙ T.toFunctor) T.toFunctor\nc : Cocone (D ⋙ T.forget)\nt : IsColimit c\nA B : J\nf : A ⟶ B\n⊢ (D.map f).f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monad.Limits | {
"line": 434,
"column": 8
} | {
"line": 434,
"column": 19
} | {
"line": 434,
"column": 20
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD✝ : Type u₂\ninst✝¹ : Category.{v₂, u₂} D✝\nJ : Type u\ninst✝ : Category.{v, u} J\nT : Comonad C\nD : J ⥤ T.Coalgebra\nc : Cocone (D ⋙ T.forget)\nt : IsColimit c\nX Y : J\nf : X ⟶ Y\n⊢ (D.map f ≫ { f := c.ι.app Y, h := ⋯ }).f = ({ f := c.ι.app X, h := ⋯ } ≫ (... | [
"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD✝ : Type u₂\ninst✝¹ : Category.{v₂, u₂} D✝\nJ : Type u\ninst✝ : Category.{v, u} J\nT : Comonad C\nD : J ⥤ T.Coalgebra\nc : Cocone (D ⋙ T.forget)\nt : IsColimit c\nX Y : J\nf : X ⟶ Y\n⊢ (D.map f).f ≫ c.ι.app Y = c.ι.app X"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monad.Limits | {
"line": 452,
"column": 4
} | {
"line": 452,
"column": 50
} | {
"line": 452,
"column": 51
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD✝ : Type u₂\ninst✝¹ : Category.{v₂, u₂} D✝\nJ✝ : Type u\ninst✝ : Category.{v, u} J✝\nT : Comonad C\nD : J✝ ⥤ T.Coalgebra\nc : Cocone (D ⋙ T.forget)\nt : IsColimit c\ns : Cocone D\nm : (liftedCocone D c t).pt ⟶ s.pt\nJ : ∀ (j : J✝), (liftedCocone D c t).ι.app ... | [
"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD✝ : Type u₂\ninst✝¹ : Category.{v₂, u₂} D✝\nJ✝ : Type u\ninst✝ : Category.{v, u} J✝\nT : Comonad C\nD : J✝ ⥤ T.Coalgebra\nc : Cocone (D ⋙ T.forget)\nt : IsColimit c\ns : Cocone D\nm : (liftedCocone D c t).pt ⟶ s.pt\nJ : ∀ (j : J✝), (liftedCocone D c t).ι.app j ≫ m = s.ι.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.ComposableArrows.Basic | {
"line": 483,
"column": 59
} | {
"line": 483,
"column": 70
} | {
"line": 483,
"column": 71
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nn✝ m : ℕ\nF G : ComposableArrows C n✝\nn k l : ℕ\nh : k + l ≤ n\nx✝¹ x✝ : Fin (l + 1)\nhij : x✝¹ ⟶ x✝\n⊢ ↑(match x✝¹ with\n | ⟨i, isLt⟩ => ⟨k + i, ⋯⟩) ≤\n ↑(match x✝ with\n | ⟨i, isLt⟩ => ⟨k + i, ⋯⟩)",
"ppTerm": "?m.65",
"assigned": true... | [
"C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nn✝ m : ℕ\nF G : ComposableArrows C n✝\nn k l : ℕ\nh : k + l ≤ n\nx✝¹ x✝ : Fin (l + 1)\nhij : x✝¹ ⟶ x✝\n⊢ x✝¹ ≤ x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monad.Limits | {
"line": 553,
"column": 10
} | {
"line": 554,
"column": 45
} | {
"line": 555,
"column": 10
} | [
{
"pp": "C : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\nD✝ : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D✝\nJ : Type u\ninst✝³ : Category.{v, u} J\nT : Comonad C\nD : J ⥤ T.Coalgebra\nc : Cone (D ⋙ T.forget)\nt : IsLimit c\ninst✝² : PreservesLimit (D ⋙ T.forget) T.toFunctor\ninst✝¹ : PreservesLimit ((D ⋙ T.forget) ⋙ T.toF... | [
"C : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\nD✝ : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D✝\nJ : Type u\ninst✝³ : Category.{v, u} J\nT : Comonad C\nD : J ⥤ T.Coalgebra\nc : Cone (D ⋙ T.forget)\nt : IsLimit c\ninst✝² : PreservesLimit (D ⋙ T.forget) T.toFunctor\ninst✝¹ : PreservesLimit ((D ⋙ T.forget) ⋙ T.toFunctor) T.to... | rw [Category.assoc, ← t.fac, Category.assoc, t.fac, commuting, ← assoc, ← assoc, t.fac,
assoc, ← Functor.map_comp, t.fac] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Monad.Limits | {
"line": 560,
"column": 4
} | {
"line": 560,
"column": 15
} | {
"line": 560,
"column": 16
} | [
{
"pp": "C : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\nD✝ : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D✝\nJ✝ : Type u\ninst✝³ : Category.{v, u} J✝\nT : Comonad C\nD : J✝ ⥤ T.Coalgebra\nc : Cone (D ⋙ T.forget)\nt : IsLimit c\ninst✝² : PreservesLimit (D ⋙ T.forget) T.toFunctor\ninst✝¹ : PreservesLimit ((D ⋙ T.forget) ⋙ T.... | [
"C : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\nD✝ : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D✝\nJ✝ : Type u\ninst✝³ : Category.{v, u} J✝\nT : Comonad C\nD : J✝ ⥤ T.Coalgebra\nc : Cone (D ⋙ T.forget)\nt : IsLimit c\ninst✝² : PreservesLimit (D ⋙ T.forget) T.toFunctor\ninst✝¹ : PreservesLimit ((D ⋙ T.forget) ⋙ T.toFunctor) T... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monad.Limits | {
"line": 582,
"column": 16
} | {
"line": 582,
"column": 27
} | {
"line": 582,
"column": 28
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD✝ : Type u₂\ninst✝³ : Category.{v₂, u₂} D✝\nJ : Type u\ninst✝² : Category.{v, u} J\nT : Comonad C\nD : J ⥤ T.Coalgebra\ninst✝¹ : PreservesLimit (D ⋙ T.forget) T.toFunctor\ninst✝ : PreservesLimit ((D ⋙ T.forget) ⋙ T.toFunctor) T.toFunctor\nc : Cone (D ⋙ T.forg... | [
"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD✝ : Type u₂\ninst✝³ : Category.{v₂, u₂} D✝\nJ : Type u\ninst✝² : Category.{v, u} J\nT : Comonad C\nD : J ⥤ T.Coalgebra\ninst✝¹ : PreservesLimit (D ⋙ T.forget) T.toFunctor\ninst✝ : PreservesLimit ((D ⋙ T.forget) ⋙ T.toFunctor) T.toFunctor\nc : Cone (D ⋙ T.forget)\nt : IsL... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Reflexive | {
"line": 578,
"column": 2
} | {
"line": 579,
"column": 9
} | {
"line": 579,
"column": 10
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : WalkingReflexivePair ⥤ C\n⊢ HasColimit F ↔ HasCoequalizer (F.map left) (F.map right)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.Cocone",
"_private.Mathlib.CategoryTheory.Limits.Shape... | [
"C : Type u\ninst✝ : Category.{v, u} C\nF : WalkingReflexivePair ⥤ C\n⊢ HasInitial (Cocone F) ↔ HasInitial (Cocone (parallelPair (F.map left) (F.map right)))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Pointwise | {
"line": 131,
"column": 15
} | {
"line": 131,
"column": 26
} | {
"line": 131,
"column": 27
} | [
{
"pp": "M : Type u_1\nR : Type u_3\ninst✝² : Group M\ninst✝¹ : Semiring R\ninst✝ : MulSemiringAction M R\na : M\nS : Ideal R\nx : R\nh : a • x ∈ a • S\n⊢ x ∈ S",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"M : Type u_1\nR : Type u_3\ninst✝² : Group M\ninst✝¹ : Semiring R\ninst✝ : MulSemiringAction M R\na : M\nS : Ideal R\nx : R\nh : a • x ∈ a • S\n⊢ x ∈ S"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Pointwise | {
"line": 131,
"column": 15
} | {
"line": 131,
"column": 60
} | {
"line": 131,
"column": 60
} | [
{
"pp": "M : Type u_1\nR : Type u_3\ninst✝² : Group M\ninst✝¹ : Semiring R\ninst✝ : MulSemiringAction M R\na : M\nS : Ideal R\nx : R\nh : a • x ∈ a • S\n⊢ x ∈ S",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Submodule.instAddCommMono... | [] | simpa using smul_mem_pointwise_smul a⁻¹ _ _ h | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.Ideal.Pointwise | {
"line": 131,
"column": 15
} | {
"line": 131,
"column": 60
} | {
"line": 131,
"column": 60
} | [
{
"pp": "M : Type u_1\nR : Type u_3\ninst✝² : Group M\ninst✝¹ : Semiring R\ninst✝ : MulSemiringAction M R\na : M\nS : Ideal R\nx : R\nh : a • x ∈ a • S\n⊢ x ∈ S",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Submodule.instAddCommMono... | [] | simpa using smul_mem_pointwise_smul a⁻¹ _ _ h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Pointwise | {
"line": 131,
"column": 15
} | {
"line": 131,
"column": 60
} | {
"line": 131,
"column": 60
} | [
{
"pp": "M : Type u_1\nR : Type u_3\ninst✝² : Group M\ninst✝¹ : Semiring R\ninst✝ : MulSemiringAction M R\na : M\nS : Ideal R\nx : R\nh : a • x ∈ a • S\n⊢ x ∈ S",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Submodule.instAddCommMono... | [] | simpa using smul_mem_pointwise_smul a⁻¹ _ _ h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Pointwise | {
"line": 135,
"column": 15
} | {
"line": 135,
"column": 26
} | {
"line": 135,
"column": 27
} | [
{
"pp": "M : Type u_1\nR : Type u_3\ninst✝² : Group M\ninst✝¹ : Semiring R\ninst✝ : MulSemiringAction M R\na : M\nS : Ideal R\nx : R\nh : x ∈ a • S\n⊢ a⁻¹ • x ∈ S",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"M : Type u_1\nR : Type u_3\ninst✝² : Group M\ninst✝¹ : Semiring R\ninst✝ : MulSemiringAction M R\na : M\nS : Ideal R\nx : R\nh : x ∈ a • S\n⊢ a⁻¹ • x ∈ S"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Pointwise | {
"line": 135,
"column": 15
} | {
"line": 135,
"column": 60
} | {
"line": 135,
"column": 60
} | [
{
"pp": "M : Type u_1\nR : Type u_3\ninst✝² : Group M\ninst✝¹ : Semiring R\ninst✝ : MulSemiringAction M R\na : M\nS : Ideal R\nx : R\nh : x ∈ a • S\n⊢ a⁻¹ • x ∈ S",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Submodule.instAddCommMo... | [] | simpa using smul_mem_pointwise_smul a⁻¹ _ _ h | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.Ideal.Pointwise | {
"line": 135,
"column": 15
} | {
"line": 135,
"column": 60
} | {
"line": 135,
"column": 60
} | [
{
"pp": "M : Type u_1\nR : Type u_3\ninst✝² : Group M\ninst✝¹ : Semiring R\ninst✝ : MulSemiringAction M R\na : M\nS : Ideal R\nx : R\nh : x ∈ a • S\n⊢ a⁻¹ • x ∈ S",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Submodule.instAddCommMo... | [] | simpa using smul_mem_pointwise_smul a⁻¹ _ _ h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Pointwise | {
"line": 135,
"column": 15
} | {
"line": 135,
"column": 60
} | {
"line": 135,
"column": 60
} | [
{
"pp": "M : Type u_1\nR : Type u_3\ninst✝² : Group M\ninst✝¹ : Semiring R\ninst✝ : MulSemiringAction M R\na : M\nS : Ideal R\nx : R\nh : x ∈ a • S\n⊢ a⁻¹ • x ∈ S",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Submodule.instAddCommMo... | [] | simpa using smul_mem_pointwise_smul a⁻¹ _ _ h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Pointwise | {
"line": 136,
"column": 16
} | {
"line": 136,
"column": 27
} | {
"line": 136,
"column": 28
} | [
{
"pp": "M : Type u_1\nR : Type u_3\ninst✝² : Group M\ninst✝¹ : Semiring R\ninst✝ : MulSemiringAction M R\na : M\nS : Ideal R\nx : R\nh : a⁻¹ • x ∈ S\n⊢ x ∈ a • S",
"ppTerm": "?m.26",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"M : Type u_1\nR : Type u_3\ninst✝² : Group M\ninst✝¹ : Semiring R\ninst✝ : MulSemiringAction M R\na : M\nS : Ideal R\nx : R\nh : a⁻¹ • x ∈ S\n⊢ x ∈ a • S"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Pointwise | {
"line": 143,
"column": 15
} | {
"line": 143,
"column": 26
} | {
"line": 143,
"column": 27
} | [
{
"pp": "M : Type u_1\nR : Type u_3\ninst✝² : Group M\ninst✝¹ : Semiring R\ninst✝ : MulSemiringAction M R\na : M\nS T : Ideal R\nh : a • S ≤ a • T\n⊢ S ≤ T",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"M : Type u_1\nR : Type u_3\ninst✝² : Group M\ninst✝¹ : Semiring R\ninst✝ : MulSemiringAction M R\na : M\nS T : Ideal R\nh : a • S ≤ a • T\n⊢ S ≤ T"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.Action.Pointwise.Set | {
"line": 54,
"column": 2
} | {
"line": 54,
"column": 29
} | {
"line": 54,
"column": 30
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : Semiring α\ninst✝¹ : AddCommMonoid β\ninst✝ : Module α β\na b : α\ns : Set β\nx : β\nhx : x ∈ s\n⊢ (fun x ↦ (a + b) • x) x ∈ a • s + b • s",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"congrArg",
... | [
"α : Type u_1\nβ : Type u_2\ninst✝² : Semiring α\ninst✝¹ : AddCommMonoid β\ninst✝ : Module α β\na b : α\ns : Set β\nx : β\nhx : x ∈ s\n⊢ a • x + b • x ∈ a • s + b • s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IsPrimary | {
"line": 84,
"column": 6
} | {
"line": 84,
"column": 22
} | {
"line": 84,
"column": 23
} | [
{
"pp": "case insert.inl\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nι : Type u_3\nf : ι → Submodule R M\na i : ι\nha : a ∉ ∅\nIH :\n ∀ {i : ι},\n i ∈ ∅ →\n (∀ ⦃y : ι⦄, y ∈ ∅ → (f y).IsPrimary) →\n (∀ ⦃y : ι⦄, y ∈ ∅ → ((f y).colon Set.univ)... | [
"case insert.inl\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nι : Type u_3\nf : ι → Submodule R M\na i : ι\nha : a ∉ ∅\nIH :\n ∀ {i : ι},\n i ∈ ∅ →\n (∀ ⦃y : ι⦄, y ∈ ∅ → (f y).IsPrimary) →\n (∀ ⦃y : ι⦄, y ∈ ∅ → ((f y).colon Set.univ).radical = (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Colon | {
"line": 177,
"column": 4
} | {
"line": 177,
"column": 25
} | {
"line": 177,
"column": 26
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nN : Submodule R M\nr : R\n⊢ ⊤ = map N.mkQ ⊤",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"RingHomSurjective.ids",
"Submodule.Quotient.addComm... | [
"R : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nN : Submodule R M\nr : R\n⊢ ⊤ = N.mkQ.range"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.LocalizedModule.Submodule | {
"line": 235,
"column": 23
} | {
"line": 235,
"column": 52
} | {
"line": 236,
"column": 6
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : AddCommMonoid N\ninst✝⁶ : Module R M\ninst✝⁵ : Module R N\ninst✝⁴ : Algebra R S\ninst✝³ : Module S N\ninst✝² : IsScalarTower R S N\np : Submonoid R\ninst✝¹ : IsL... | [
"R : Type u_1\nS : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : AddCommMonoid N\ninst✝⁶ : Module R M\ninst✝⁵ : Module R N\ninst✝⁴ : Algebra R S\ninst✝³ : Module S N\ninst✝² : IsScalarTower R S N\np : Submonoid R\ninst✝¹ : IsLocalization ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.MinimalPrime.Basic | {
"line": 119,
"column": 4
} | {
"line": 123,
"column": 21
} | {
"line": 124,
"column": 2
} | [
{
"pp": "case a\nR : Type u_1\ninst✝ : CommSemiring R\nI : Ideal R\n⊢ sInf I.minimalPrimes ≤ sInf {J | I ≤ J ∧ J.IsPrime}",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"Semiring.toModule",
"Ideal.minimalPrimes",
"congrArg",
"CommSe... | [] | intro x hx
rw [Ideal.mem_sInf] at hx ⊢
rintro J ⟨e, hJ⟩
obtain ⟨p, hp, hp'⟩ := Ideal.exists_minimalPrimes_le e
exact hp' (hx hp) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.MinimalPrime.Basic | {
"line": 119,
"column": 4
} | {
"line": 123,
"column": 21
} | {
"line": 124,
"column": 2
} | [
{
"pp": "case a\nR : Type u_1\ninst✝ : CommSemiring R\nI : Ideal R\n⊢ sInf I.minimalPrimes ≤ sInf {J | I ≤ J ∧ J.IsPrime}",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"Semiring.toModule",
"Ideal.minimalPrimes",
"congrArg",
"CommSe... | [] | intro x hx
rw [Ideal.mem_sInf] at hx ⊢
rintro J ⟨e, hJ⟩
obtain ⟨p, hp, hp'⟩ := Ideal.exists_minimalPrimes_le e
exact hp' (hx hp) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.MinimalPrime.Basic | {
"line": 183,
"column": 20
} | {
"line": 183,
"column": 37
} | {
"line": 183,
"column": 38
} | [
{
"pp": "R : Type u_2\ninst✝¹ : CommRing R\np I J : Ideal R\ninst✝ : p.IsPrime\nhle : I ≤ p\nh : map (Quotient.mk I) p ∈ (map (Quotient.mk I) J).minimalPrimes\n⊢ J ≤ p",
"ppTerm": "?m.73",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_2\ninst✝¹ : CommRing R\np I J : Ideal R\ninst✝ : p.IsPrime\nhle : I ≤ p\nh : map (Quotient.mk I) p ∈ (map (Quotient.mk I) J).minimalPrimes\n⊢ J ≤ p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.MinimalPrime.Basic | {
"line": 187,
"column": 4
} | {
"line": 187,
"column": 56
} | {
"line": 187,
"column": 57
} | [
{
"pp": "case refine_2\nR : Type u_2\ninst✝¹ : CommRing R\np I J : Ideal R\ninst✝ : p.IsPrime\nhle : I ≤ p\nh : map (Quotient.mk I) p ∈ (map (Quotient.mk I) J).minimalPrimes\nq : Ideal R\nx✝ : (fun q ↦ q.IsPrime ∧ I ⊔ J ≤ q) q\nhqp : q ≤ p\nleft✝ : q.IsPrime\nhq : I ≤ q ∧ J ≤ q\nh2 : map (Quotient.mk I) p ≤ map... | [
"case refine_2\nR : Type u_2\ninst✝¹ : CommRing R\np I J : Ideal R\ninst✝ : p.IsPrime\nhle : I ≤ p\nh : map (Quotient.mk I) p ∈ (map (Quotient.mk I) J).minimalPrimes\nq : Ideal R\nx✝ : (fun q ↦ q.IsPrime ∧ I ⊔ J ≤ q) q\nhqp : q ≤ p\nleft✝ : q.IsPrime\nhq : I ≤ q ∧ J ≤ q\nh2 : map (Quotient.mk I) p ≤ map (Quotient.m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.Ideal | {
"line": 155,
"column": 4
} | {
"line": 155,
"column": 42
} | {
"line": 155,
"column": 43
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nhI : I.IsPrimary\nhM : Disjoint ↑M ↑I\nkey : Disjoint ↑M ↑I.radical\na : R\nb : ↥I\ns : ↥M\nh : (algebraMap R S) a * (algebraMap R S) ↑(b, s).2 =... | [
"R : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nhI : I.IsPrimary\nhM : Disjoint ↑M ↑I\nkey : Disjoint ↑M ↑I.radical\na : R\nb : ↥I\ns : ↥M\nh : (algebraMap R S) a * (algebraMap R S) ↑(b, s).2 = (algebraMap... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.MinimalPrime.Basic | {
"line": 214,
"column": 4
} | {
"line": 214,
"column": 20
} | {
"line": 214,
"column": 21
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝⁴ : CommSemiring R\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nI p : Ideal R\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : P.LiesOver p\nhI : p ∈ I.minimalPrimes\nJ : Ideal S\nhJP : J ≤ P\nhJ : map (Quotient.mk (map (algebraMap R S) p)) P ∈ (map (Quotient.mk ... | [
"case refine_2\nR : Type u_1\ninst✝⁴ : CommSemiring R\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nI p : Ideal R\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : P.LiesOver p\nhI : p ∈ I.minimalPrimes\nJ : Ideal S\nhJP : J ≤ P\nhJ : map (Quotient.mk (map (algebraMap R S) p)) P ∈ (map (Quotient.mk (map (algebr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalProperties.Submodule | {
"line": 49,
"column": 20
} | {
"line": 49,
"column": 35
} | {
"line": 49,
"column": 36
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nMₚ : (P : Ideal R) → [P.IsMaximal] → Type u_5\ninst✝² : (P : Ideal R) → [inst : P.IsMaximal] → AddCommMonoid (Mₚ P)\ninst✝¹ : (P : Ideal R) → [inst : P.IsMaximal] → Module R (Mₚ P)\nf : (P : Ideal R) → [... | [
"R : Type u_1\nM : Type u_2\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nMₚ : (P : Ideal R) → [P.IsMaximal] → Type u_5\ninst✝² : (P : Ideal R) → [inst : P.IsMaximal] → AddCommMonoid (Mₚ P)\ninst✝¹ : (P : Ideal R) → [inst : P.IsMaximal] → Module R (Mₚ P)\nf : (P : Ideal R) → [inst : P.IsM... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalProperties.Submodule | {
"line": 78,
"column": 59
} | {
"line": 78,
"column": 70
} | {
"line": 78,
"column": 71
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nMₚ : (P : Ideal R) → [P.IsMaximal] → Type u_5\ninst✝² : (P : Ideal R) → [inst : P.IsMaximal] → AddCommMonoid (Mₚ P)\ninst✝¹ : (P : Ideal R) → [inst : P.IsMaximal] → Module R (Mₚ P)\nf : (P : Ideal R) → [... | [
"R : Type u_1\nM : Type u_2\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nMₚ : (P : Ideal R) → [P.IsMaximal] → Type u_5\ninst✝² : (P : Ideal R) → [inst : P.IsMaximal] → AddCommMonoid (Mₚ P)\ninst✝¹ : (P : Ideal R) → [inst : P.IsMaximal] → Module R (Mₚ P)\nf : (P : Ideal R) → [inst : P.IsM... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Quotient.Nilpotent | {
"line": 94,
"column": 29
} | {
"line": 98,
"column": 46
} | {
"line": 99,
"column": 0
} | [
{
"pp": "S : Type u_1\ninst✝¹ : CommRing S\nI : Ideal S\ninst✝ : I.IsMaximal\nn : ℕ\nx : S\nhx : x ∉ I\n⊢ IsUnit ((mk (I ^ n)) x)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Ideal.Quotient.instUniqueQuotientTop",
"Iff.mpr",
"Eq.mpr",
"Semiring.toModule",
... | [] | by
by_cases! hn : n = 0
· rw [pow_eq_top_iff.mpr (Or.inr hn)]
exact isUnit_of_subsingleton _
exact (isUnit_mk_pow_iff_notMem I hn).mpr hx | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.LocalProperties.Submodule | {
"line": 83,
"column": 59
} | {
"line": 83,
"column": 70
} | {
"line": 83,
"column": 71
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nMₚ : (P : Ideal R) → [P.IsMaximal] → Type u_5\ninst✝² : (P : Ideal R) → [inst : P.IsMaximal] → AddCommMonoid (Mₚ P)\ninst✝¹ : (P : Ideal R) → [inst : P.IsMaximal] → Module R (Mₚ P)\nf : (P : Ideal R) → [... | [
"R : Type u_1\nM : Type u_2\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nMₚ : (P : Ideal R) → [P.IsMaximal] → Type u_5\ninst✝² : (P : Ideal R) → [inst : P.IsMaximal] → AddCommMonoid (Mₚ P)\ninst✝¹ : (P : Ideal R) → [inst : P.IsMaximal] → Module R (Mₚ P)\nf : (P : Ideal R) → [inst : P.IsM... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalProperties.Submodule | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 50
} | {
"line": 105,
"column": 51
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nM₁ : Type u_3\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : Module R M₁\nMₚ : (P : Ideal R) → [P.IsMaximal] → Type u_5\ninst✝⁵ : (P : Ideal R) → [inst : P.IsMaximal] → AddCommMonoid (Mₚ P)\ninst✝⁴ : (P : Ideal R... | [
"R : Type u_1\nM : Type u_2\nM₁ : Type u_3\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : Module R M₁\nMₚ : (P : Ideal R) → [P.IsMaximal] → Type u_5\ninst✝⁵ : (P : Ideal R) → [inst : P.IsMaximal] → AddCommMonoid (Mₚ P)\ninst✝⁴ : (P : Ideal R) → [inst : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalProperties.Submodule | {
"line": 124,
"column": 61
} | {
"line": 124,
"column": 72
} | {
"line": 124,
"column": 73
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Module R M\nRₚ : (P : Ideal R) → [P.IsMaximal] → Type u_4\ninst✝⁷ : (P : Ideal R) → [inst : P.IsMaximal] → CommSemiring (Rₚ P)\ninst✝⁶ : (P : Ideal R) → [inst : P.IsMaximal] → Algebra R (Rₚ P)\ninst✝⁵ : ∀ (P : Idea... | [
"R : Type u_1\nM : Type u_2\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Module R M\nRₚ : (P : Ideal R) → [P.IsMaximal] → Type u_4\ninst✝⁷ : (P : Ideal R) → [inst : P.IsMaximal] → CommSemiring (Rₚ P)\ninst✝⁶ : (P : Ideal R) → [inst : P.IsMaximal] → Algebra R (Rₚ P)\ninst✝⁵ : ∀ (P : Ideal R) [inst :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalProperties.Submodule | {
"line": 129,
"column": 61
} | {
"line": 129,
"column": 72
} | {
"line": 129,
"column": 73
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Module R M\nRₚ : (P : Ideal R) → [P.IsMaximal] → Type u_4\ninst✝⁷ : (P : Ideal R) → [inst : P.IsMaximal] → CommSemiring (Rₚ P)\ninst✝⁶ : (P : Ideal R) → [inst : P.IsMaximal] → Algebra R (Rₚ P)\ninst✝⁵ : ∀ (P : Idea... | [
"R : Type u_1\nM : Type u_2\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Module R M\nRₚ : (P : Ideal R) → [P.IsMaximal] → Type u_4\ninst✝⁷ : (P : Ideal R) → [inst : P.IsMaximal] → CommSemiring (Rₚ P)\ninst✝⁶ : (P : Ideal R) → [inst : P.IsMaximal] → Algebra R (Rₚ P)\ninst✝⁵ : ∀ (P : Ideal R) [inst :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalProperties.Submodule | {
"line": 154,
"column": 39
} | {
"line": 154,
"column": 67
} | {
"line": 154,
"column": 68
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ns : Set R\nspan_eq : span s = ⊤\nMₚ : ↑s → Type u_5\ninst✝² : (r : ↑s) → AddCommMonoid (Mₚ r)\ninst✝¹ : (r : ↑s) → Module R (Mₚ r)\nf : (r : ↑s) → M →ₗ[R] Mₚ r\ninst✝ : ∀ (r : ↑s), IsLocalizedModule.Away... | [
"R : Type u_1\nM : Type u_2\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ns : Set R\nspan_eq : span s = ⊤\nMₚ : ↑s → Type u_5\ninst✝² : (r : ↑s) → AddCommMonoid (Mₚ r)\ninst✝¹ : (r : ↑s) → Module R (Mₚ r)\nf : (r : ↑s) → M →ₗ[R] Mₚ r\ninst✝ : ∀ (r : ↑s), IsLocalizedModule.Away (↑r) (f r)\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalProperties.Submodule | {
"line": 162,
"column": 49
} | {
"line": 162,
"column": 76
} | {
"line": 162,
"column": 77
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ns : Set R\nspan_eq : span s = ⊤\nMₚ : ↑s → Type u_5\ninst✝² : (r : ↑s) → AddCommMonoid (Mₚ r)\ninst✝¹ : (r : ↑s) → Module R (Mₚ r)\nf : (r : ↑s) → M →ₗ[R] Mₚ r\ninst✝ : ∀ (r : ↑s), IsLocalizedModule.Away... | [
"R : Type u_1\nM : Type u_2\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ns : Set R\nspan_eq : span s = ⊤\nMₚ : ↑s → Type u_5\ninst✝² : (r : ↑s) → AddCommMonoid (Mₚ r)\ninst✝¹ : (r : ↑s) → Module R (Mₚ r)\nf : (r : ↑s) → M →ₗ[R] Mₚ r\ninst✝ : ∀ (r : ↑s), IsLocalizedModule.Away (↑r) (f r)\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.Ideal | {
"line": 326,
"column": 25
} | {
"line": 326,
"column": 71
} | {
"line": 326,
"column": 71
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsLocalization M S\nI : Ideal S\ninst✝ : I.IsPrime\nJ : Ideal R\nH : J ≤ Ideal.under R I\nhI : (Ideal.under R I).IsMaximal\nr m : R\nhm : m ∈ M\nhM : (Ideal.Quotient.mk (Ideal.comap (al... | [] | by simp [Ideal.Quotient.eq_zero_iff_mem, this] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.LocalProperties.Submodule | {
"line": 167,
"column": 20
} | {
"line": 167,
"column": 35
} | {
"line": 167,
"column": 36
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ns : Set R\nspan_eq : Ideal.span s = ⊤\nMₚ : ↑s → Type u_5\ninst✝² : (r : ↑s) → AddCommMonoid (Mₚ r)\ninst✝¹ : (r : ↑s) → Module R (Mₚ r)\nf : (r : ↑s) → M →ₗ[R] Mₚ r\ninst✝ : ∀ (r : ↑s), IsLocalizedModul... | [
"R : Type u_1\nM : Type u_2\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ns : Set R\nspan_eq : Ideal.span s = ⊤\nMₚ : ↑s → Type u_5\ninst✝² : (r : ↑s) → AddCommMonoid (Mₚ r)\ninst✝¹ : (r : ↑s) → Module R (Mₚ r)\nf : (r : ↑s) → M →ₗ[R] Mₚ r\ninst✝ : ∀ (r : ↑s), IsLocalizedModule.Away (↑r) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PolynomialAlgebra | {
"line": 131,
"column": 4
} | {
"line": 131,
"column": 13
} | {
"line": 132,
"column": 4
} | [
{
"pp": "case refine_2\nR : Type u_1\nA : Type u_3\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : A ⊗[R] R[X]\n⊢ ∀ (x : A) (y : R[X]), invFun R A ((toFunAlgHom R A) (x ⊗ₜ[R] y)) = x ⊗ₜ[R] y",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"CommSemiring.toS... | [
"case refine_2\nR : Type u_1\nA : Type u_3\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : A ⊗[R] R[X]\na : A\np : R[X]\n⊢ invFun R A ((toFunAlgHom R A) (a ⊗ₜ[R] p)) = a ⊗ₜ[R] p"
] | intro a p | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.Localization.Ideal | {
"line": 391,
"column": 4
} | {
"line": 391,
"column": 46
} | {
"line": 391,
"column": 47
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : IsLocalization M S\nR' : Type u_3\nS' : Type u_4\ninst✝² : CommRing R'\ninst✝¹ : CommRing S'\ninst✝ : Algebra R' S'\nf : R →+* R'\nhf : Function.Surjective ⇑f\ng : S →+* S'\nhg : Functi... | [
"R : Type u_1\ninst✝⁶ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : IsLocalization M S\nR' : Type u_3\nS' : Type u_4\ninst✝² : CommRing R'\ninst✝¹ : CommRing S'\ninst✝ : Algebra R' S'\nf : R →+* R'\nhf : Function.Surjective ⇑f\ng : S →+* S'\nhg : Function.Surjectiv... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.Ideal | {
"line": 397,
"column": 4
} | {
"line": 397,
"column": 55
} | {
"line": 397,
"column": 56
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : IsLocalization M S\nR' : Type u_3\nS' : Type u_4\ninst✝² : CommRing R'\ninst✝¹ : CommRing S'\ninst✝ : Algebra R' S'\nf : R →+* R'\nhf : Function.Surjective ⇑f\ng : S →+* S'\nhg : Functi... | [
"R : Type u_1\ninst✝⁶ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : IsLocalization M S\nR' : Type u_3\nS' : Type u_4\ninst✝² : CommRing R'\ninst✝¹ : CommRing S'\ninst✝ : Algebra R' S'\nf : R →+* R'\nhf : Function.Surjective ⇑f\ng : S →+* S'\nhg : Function.Surjectiv... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.Ideal | {
"line": 405,
"column": 31
} | {
"line": 405,
"column": 65
} | {
"line": 405,
"column": 66
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : IsLocalization M S\nR' : Type u_3\nS' : Type u_4\ninst✝² : CommRing R'\ninst✝¹ : CommRing S'\ninst✝ : Algebra R' S'\nf : R →+* R'\nhf : Function.Surjective ⇑f\ng : S →+* S'\nhg : Functi... | [
"R : Type u_1\ninst✝⁶ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : IsLocalization M S\nR' : Type u_3\nS' : Type u_4\ninst✝² : CommRing R'\ninst✝¹ : CommRing S'\ninst✝ : Algebra R' S'\nf : R →+* R'\nhf : Function.Surjective ⇑f\ng : S →+* S'\nhg : Function.Surjectiv... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.Algebra | {
"line": 44,
"column": 10
} | {
"line": 44,
"column": 40
} | {
"line": 44,
"column": 41
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nP : Type u_3\nQ : Type u_4\ninst✝⁷ : CommSemiring R\ninst✝⁶ : CommSemiring S\ninst✝⁵ : CommSemiring P\ninst✝⁴ : CommSemiring Q\nM : Submonoid R\nT : Submonoid P\ninst✝³ : Algebra R S\ninst✝² : Algebra P Q\ninst✝¹ : IsLocalization M S\ninst✝ : IsLocalization T Q\ng : R →+* P\... | [
"R : Type u_1\nS : Type u_2\nP : Type u_3\nQ : Type u_4\ninst✝⁷ : CommSemiring R\ninst✝⁶ : CommSemiring S\ninst✝⁵ : CommSemiring P\ninst✝⁴ : CommSemiring Q\nM : Submonoid R\nT : Submonoid P\ninst✝³ : Algebra R S\ninst✝² : Algebra P Q\ninst✝¹ : IsLocalization M S\ninst✝ : IsLocalization T Q\ng : R →+* P\nI : Ideal R... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.Localization | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 45
} | {
"line": 72,
"column": 46
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : CommSemiring S\ninst✝⁹ : Algebra R S\nM : Type u_3\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\ninst✝⁶ : Module S M\ninst✝⁵ : IsScalarTower R S M\nMₚ : (P : Ideal S) → [P.IsMaximal] → Type u_4\ninst✝⁴ : (P : Ideal S) → [inst : P.IsMaxim... | [
"R : Type u_1\nS : Type u_2\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : CommSemiring S\ninst✝⁹ : Algebra R S\nM : Type u_3\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\ninst✝⁶ : Module S M\ninst✝⁵ : IsScalarTower R S M\nMₚ : (P : Ideal S) → [P.IsMaximal] → Type u_4\ninst✝⁴ : (P : Ideal S) → [inst : P.IsMaximal] → AddCom... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.Localization | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 45
} | {
"line": 97,
"column": 46
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : CommSemiring S\ninst✝⁹ : Algebra R S\nM : Type u_3\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\ninst✝⁶ : Module S M\ninst✝⁵ : IsScalarTower R S M\ns : Set S\nspn : Ideal.span s = ⊤\nMₛ : ↑s → Type u_5\ninst✝⁴ : (r : ↑s) → AddCommMonoid ... | [
"R : Type u_1\nS : Type u_2\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : CommSemiring S\ninst✝⁹ : Algebra R S\nM : Type u_3\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\ninst✝⁶ : Module S M\ninst✝⁵ : IsScalarTower R S M\ns : Set S\nspn : Ideal.span s = ⊤\nMₛ : ↑s → Type u_5\ninst✝⁴ : (r : ↑s) → AddCommMonoid (Mₛ r)\ninst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.EssentialFiniteness | {
"line": 86,
"column": 19
} | {
"line": 86,
"column": 30
} | {
"line": 86,
"column": 31
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nσ : Finset S\nhσ : ∀ (s : S), ∃ t ∈ adjoin R ↑σ, IsUnit t ∧ s * t ∈ adjoin R ↑σ\nx y : ↥(adjoin R ↑σ)\ne : (algebraMap (↥(adjoin R ↑σ)) S) x = (algebraMap (↥(adjoin R ↑σ)) S) y\n⊢ ↑1 * x = ↑1 * y",
"ppTerm": ... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nσ : Finset S\nhσ : ∀ (s : S), ∃ t ∈ adjoin R ↑σ, IsUnit t ∧ s * t ∈ adjoin R ↑σ\nx y : ↥(adjoin R ↑σ)\ne : (algebraMap (↥(adjoin R ↑σ)) S) x = (algebraMap (↥(adjoin R ↑σ)) S) y\n⊢ x = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.EssentialFiniteness | {
"line": 98,
"column": 23
} | {
"line": 98,
"column": 73
} | {
"line": 98,
"column": 74
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : CommRing T\ninst✝² : Algebra R S\ninst✝¹ : Algebra R T\ninst✝ : FiniteType R S\ns : Finset S\nhs : adjoin R ↑s = ⊤\nx✝ : S\n⊢ ∃ t ∈ adjoin R ↑s, IsUnit t ∧ x✝ * t ∈ adjoin R ↑s",
"ppTerm": "?m.42",
"ass... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : CommRing T\ninst✝² : Algebra R S\ninst✝¹ : Algebra R T\ninst✝ : FiniteType R S\ns : Finset S\nhs : adjoin R ↑s = ⊤\nx✝ : S\n⊢ ∃ t, IsUnit t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.AtPrime.Basic | {
"line": 66,
"column": 33
} | {
"line": 66,
"column": 44
} | {
"line": 66,
"column": 45
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommSemiring R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nhze : (algebraMap R S) 0 = (algebraMap R S) 1\nt : ↥P.primeCompl\nht : ↑t * 0 = ↑t * 1\n⊢ ↑t = 0",
"ppTerm": "?m.66",
"assigned": ... | [
"R : Type u_1\ninst✝³ : CommSemiring R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nhze : (algebraMap R S) 0 = (algebraMap R S) 1\nt : ↥P.primeCompl\nht : ↑t * 0 = ↑t * 1\n⊢ ↑t = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.AtPrime.Basic | {
"line": 171,
"column": 24
} | {
"line": 171,
"column": 58
} | {
"line": 171,
"column": 59
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommSemiring R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nI : Ideal R\nhI : I.IsPrime\ninst✝ : IsLocalization.AtPrime S I\nh : IsLocalRing S\nx : R\n⊢ x ∈ Ideal.under R (IsLocalRing.maximalIdeal S) ↔ x ∈ I",
"ppTerm": "?m.31",
"assigned": true,
"use... | [
"R : Type u_1\ninst✝³ : CommSemiring R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nI : Ideal R\nhI : I.IsPrime\ninst✝ : IsLocalization.AtPrime S I\nh : IsLocalRing S\nx : R\n⊢ (algebraMap R S) x ∈ IsLocalRing.maximalIdeal S ↔ x ∈ I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.AtPrime.Basic | {
"line": 218,
"column": 4
} | {
"line": 218,
"column": 52
} | {
"line": 218,
"column": 53
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nI : Ideal R\nhI : I.IsPrime\nJ : Ideal (Localization.AtPrime I)\nh : Ideal.under R J = I\n⊢ IsLocalRing.maximalIdeal (Localization.AtPrime I) ≤ J",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
... | [
"R : Type u_1\ninst✝ : CommSemiring R\nI : Ideal R\nhI : I.IsPrime\nJ : Ideal (Localization.AtPrime I)\nh : Ideal.under R J = I\n⊢ Ideal.map (algebraMap R (Localization.AtPrime I)) (Ideal.under R J) ≤ J"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.AtPrime.Basic | {
"line": 342,
"column": 6
} | {
"line": 345,
"column": 77
} | {
"line": 345,
"column": 78
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommSemiring R\nS : Type u_2\ninst✝⁴ : CommSemiring S\ninst✝³ : Algebra R S\nP : Ideal S\ninst✝² : P.IsPrime\ns : Subalgebra R S\nH : s.saturation (P.primeCompl ⊓ s.toSubmonoid) ⋯ = ⊤\np : Ideal ↥s\ninst✝¹ : p.IsPrime\ninst✝ : P.LiesOver p\nthis : ∀ (y z : S), z ∉ P → ∃ y' ∈ s, ∃... | [
"R : Type u_1\ninst✝⁵ : CommSemiring R\nS : Type u_2\ninst✝⁴ : CommSemiring S\ninst✝³ : Algebra R S\nP : Ideal S\ninst✝² : P.IsPrime\ns : Subalgebra R S\nH : s.saturation (P.primeCompl ⊓ s.toSubmonoid) ⋯ = ⊤\np : Ideal ↥s\ninst✝¹ : p.IsPrime\ninst✝ : P.LiesOver p\nthis : ∀ (y z : S), z ∉ P → ∃ y' ∈ s, ∃ z' ∉ P, z' ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.EssentialFiniteness | {
"line": 186,
"column": 6
} | {
"line": 186,
"column": 42
} | {
"line": 186,
"column": 43
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : CommRing T\ninst✝³ : Algebra R S\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nσ : Finset S\nhσ : ∀ (s : S), ∃ t ∈ adjoin R ↑σ, IsUnit t ∧ s * t ∈ adjoin R ↑σ\nx✝ : T\ny t : S\nh₁ : ... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : CommRing T\ninst✝³ : Algebra R S\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nσ : Finset S\nhσ : ∀ (s : S), ∃ t ∈ adjoin R ↑σ, IsUnit t ∧ s * t ∈ adjoin R ↑σ\nx✝ : T\ny t : S\nh₁ : t ∈ adjoin R... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.Submodule | {
"line": 77,
"column": 6
} | {
"line": 77,
"column": 24
} | {
"line": 77,
"column": 25
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\nS : Type u_2\ninst✝¹ : CommSemiring S\ninst✝ : Algebra R S\nx : R\n⊢ coeSubmodule S (Ideal.span {x}) = R ∙ (algebraMap R S) x",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"IsLocalization.coeSubmodul... | [
"R : Type u_1\ninst✝² : CommSemiring R\nS : Type u_2\ninst✝¹ : CommSemiring S\ninst✝ : Algebra R S\nx : R\n⊢ Submodule.span R (⇑(algebraMap R S) '' {x}) = R ∙ (algebraMap R S) x"
] | coeSubmodule_span, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Localization.AtPrime.Basic | {
"line": 587,
"column": 4
} | {
"line": 587,
"column": 55
} | {
"line": 587,
"column": 56
} | [
{
"pp": "R : Type u_7\ninst✝⁵ : CommRing R\np : Ideal R\ninst✝⁴ : p.IsMaximal\nRₚ : Type u_8\ninst✝³ : CommRing Rₚ\ninst✝² : Algebra R Rₚ\ninst✝¹ : IsLocalization.AtPrime Rₚ p\ninst✝ : IsLocalRing Rₚ\nx : R\ns : ↥p.primeCompl\n⊢ (Ideal.Quotient.mk p) ↑s ≠ 0",
"ppTerm": "?m.93",
"assigned": true,
"us... | [
"R : Type u_7\ninst✝⁵ : CommRing R\np : Ideal R\ninst✝⁴ : p.IsMaximal\nRₚ : Type u_8\ninst✝³ : CommRing Rₚ\ninst✝² : Algebra R Rₚ\ninst✝¹ : IsLocalization.AtPrime Rₚ p\ninst✝ : IsLocalRing Rₚ\nx : R\ns : ↥p.primeCompl\n⊢ ↑s ∉ p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.Regular | {
"line": 66,
"column": 4
} | {
"line": 66,
"column": 36
} | {
"line": 66,
"column": 37
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝ : Ring α\nh : ∀ {a b : α}, a * b = 1 → b * a = 1\nx y z : α\nhxy : x * y = 1\nhxz : x * z = 1\n⊢ y = z",
"ppTerm": "?refine_1",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case refine_1\nα : Type u_1\ninst✝ : Ring α\nh : ∀ {a b : α}, a * b = 1 → b * a = 1\nx y z : α\nhxy : x * y = 1\nhxz : x * z = 1\n⊢ y = z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.Regular | {
"line": 75,
"column": 4
} | {
"line": 75,
"column": 34
} | {
"line": 75,
"column": 35
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝ : Ring α\nh : ∀ {a b : α}, a * b = 1 → b * a = 1\nx y z : α\nhxz : x * z = 1\nhyz : y * z = 1\n⊢ x = y",
"ppTerm": "?refine_1",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case refine_1\nα : Type u_1\ninst✝ : Ring α\nh : ∀ {a b : α}, a * b = 1 → b * a = 1\nx y z : α\nhxz : x * z = 1\nhyz : y * z = 1\n⊢ x = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.AtPrime.Basic | {
"line": 608,
"column": 4
} | {
"line": 608,
"column": 52
} | {
"line": 608,
"column": 53
} | [
{
"pp": "case refine_2\nR✝ : Type u_1\ninst✝¹⁸ : CommSemiring R✝\nS✝ : Type u_2\ninst✝¹⁷ : CommSemiring S✝\ninst✝¹⁶ : Algebra R✝ S✝\nP✝ : Type u_3\ninst✝¹⁵ : CommSemiring P✝\np✝ : Ideal R✝\ninst✝¹⁴ : p✝.IsPrime\nRₚ✝ : Type u_4\ninst✝¹³ : CommSemiring Rₚ✝\ninst✝¹² : Algebra R✝ Rₚ✝\ninst✝¹¹ : IsLocalization.AtPri... | [
"case refine_2\nR✝ : Type u_1\ninst✝¹⁸ : CommSemiring R✝\nS✝ : Type u_2\ninst✝¹⁷ : CommSemiring S✝\ninst✝¹⁶ : Algebra R✝ S✝\nP✝ : Type u_3\ninst✝¹⁵ : CommSemiring P✝\np✝ : Ideal R✝\ninst✝¹⁴ : p✝.IsPrime\nRₚ✝ : Type u_4\ninst✝¹³ : CommSemiring Rₚ✝\ninst✝¹² : Algebra R✝ Rₚ✝\ninst✝¹¹ : IsLocalization.AtPrime Rₚ✝ p✝\ni... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.TensorProduct.Quotient | {
"line": 176,
"column": 53
} | {
"line": 176,
"column": 80
} | {
"line": 176,
"column": 81
} | [
{
"pp": "R : Type u_1\nR' : Type u_2\nR'' : Type u_3\nS : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing R'\ninst✝⁶ : CommRing R''\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R R'\ninst✝³ : Algebra R R''\ninst✝² : Algebra R' R''\ninst✝¹ : IsScalarTower R R' R''\ninst✝ : Algebra R S\ne : R' ⊗[R] S\nφ : R' ⊗[R] S →ₐ... | [
"R : Type u_1\nR' : Type u_2\nR'' : Type u_3\nS : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing R'\ninst✝⁶ : CommRing R''\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R R'\ninst✝³ : Algebra R R''\ninst✝² : Algebra R' R''\ninst✝¹ : IsScalarTower R R' R''\ninst✝ : Algebra R S\ne : R' ⊗[R] S\nφ : R' ⊗[R] S →ₐ[R'] R'' ⊗[R... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalProperties.Basic | {
"line": 579,
"column": 50
} | {
"line": 579,
"column": 61
} | {
"line": 579,
"column": 62
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nI : Ideal R\nh : ∀ (J : Ideal R) (x : J.IsMaximal), Ideal.map (algebraMap R (Localization.AtPrime J)) I = ⊥\nP : Ideal R\nhP : P.IsMaximal\n⊢ Ideal.map (algebraMap R (Localization.AtPrime P)) I = Ideal.map (algebraMap R (Localization.AtPrime P)) ⊥",
"ppTerm": "... | [
"R : Type u_1\ninst✝ : CommSemiring R\nI : Ideal R\nh : ∀ (J : Ideal R) (x : J.IsMaximal), Ideal.map (algebraMap R (Localization.AtPrime J)) I = ⊥\nP : Ideal R\nhP : P.IsMaximal\n⊢ Ideal.map (algebraMap R (Localization.AtPrime P)) I = ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Symmetric | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 13
} | {
"line": 147,
"column": 14
} | [
{
"pp": "α : Type u_1\nn : Type u_3\nm : Type u_4\nA : Matrix n n α\nf : n ≃ m\nh : ((reindex f f) A).IsSymm\n⊢ A.IsSymm",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nn : Type u_3\nm : Type u_4\nA : Matrix n n α\nf : n ≃ m\nh : ((reindex f f) A).IsSymm\n⊢ A.IsSymm"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Laurent | {
"line": 180,
"column": 2
} | {
"line": 180,
"column": 32
} | {
"line": 184,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nr : R\nn : ℤ\n⊢ AddMonoidAlgebra.single n r = C r * T n",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"LaurentPolynomial.T",
"NonAssocSemiring.toAddCommMonoidWithOne",
"AddMonoidAlgebra.instAddMonoid",
"HMul.hMul",
... | [] | simp [C, T, single_mul_single] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Polynomial.Laurent | {
"line": 180,
"column": 2
} | {
"line": 180,
"column": 32
} | {
"line": 184,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nr : R\nn : ℤ\n⊢ AddMonoidAlgebra.single n r = C r * T n",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"LaurentPolynomial.T",
"NonAssocSemiring.toAddCommMonoidWithOne",
"AddMonoidAlgebra.instAddMonoid",
"HMul.hMul",
... | [] | simp [C, T, single_mul_single] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.