module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.LinearAlgebra.Matrix.SemiringInverse | {
"line": 242,
"column": 11
} | {
"line": 242,
"column": 17
} | {
"line": 242,
"column": 17
} | [
{
"pp": "n : Type u_1\nR : Type u_3\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommSemiring R\nA B : Matrix n n R\nd : n → R\nhAB : A * B = diagonal d\nh : ∀ {s t : ℤˣ}, s ≠ t → IsAddUnit (detp s A • (B * adjp t B))\n⊢ 1 ≠ -1",
"ppTerm": "?m.93",
"assigned": true,
"usedConstants": [
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.SemiringInverse | {
"line": 242,
"column": 11
} | {
"line": 242,
"column": 17
} | {
"line": 242,
"column": 17
} | [
{
"pp": "n : Type u_1\nR : Type u_3\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommSemiring R\nA B : Matrix n n R\nd : n → R\nhAB : A * B = diagonal d\nh : ∀ {s t : ℤˣ}, s ≠ t → IsAddUnit (detp s A • (B * adjp t B))\n⊢ 1 ≠ -1",
"ppTerm": "?m.93",
"assigned": true,
"usedConstants": [
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.SemiringInverse | {
"line": 242,
"column": 31
} | {
"line": 242,
"column": 37
} | {
"line": 242,
"column": 37
} | [
{
"pp": "n : Type u_1\nR : Type u_3\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommSemiring R\nA B : Matrix n n R\nd : n → R\nhAB : A * B = diagonal d\nh : ∀ {s t : ℤˣ}, s ≠ t → IsAddUnit (detp s A • (B * adjp t B))\n⊢ -1 ≠ 1",
"ppTerm": "?m.100",
"assigned": true,
"usedConstants": [
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.LinearAlgebra.Matrix.SemiringInverse | {
"line": 242,
"column": 31
} | {
"line": 242,
"column": 37
} | {
"line": 242,
"column": 37
} | [
{
"pp": "n : Type u_1\nR : Type u_3\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommSemiring R\nA B : Matrix n n R\nd : n → R\nhAB : A * B = diagonal d\nh : ∀ {s t : ℤˣ}, s ≠ t → IsAddUnit (detp s A • (B * adjp t B))\n⊢ -1 ≠ 1",
"ppTerm": "?m.100",
"assigned": true,
"usedConstants": [
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.SemiringInverse | {
"line": 242,
"column": 31
} | {
"line": 242,
"column": 37
} | {
"line": 242,
"column": 37
} | [
{
"pp": "n : Type u_1\nR : Type u_3\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommSemiring R\nA B : Matrix n n R\nd : n → R\nhAB : A * B = diagonal d\nh : ∀ {s t : ℤˣ}, s ≠ t → IsAddUnit (detp s A • (B * adjp t B))\n⊢ -1 ≠ 1",
"ppTerm": "?m.100",
"assigned": true,
"usedConstants": [
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Multilinear.Basic | {
"line": 588,
"column": 4
} | {
"line": 588,
"column": 25
} | {
"line": 589,
"column": 2
} | [
{
"pp": "R : Type uR\nι : Type uι\nM₁ : ι → Type v₁\nM₂ : Type v₂\ninst✝⁶ : Semiring R\ninst✝⁵ : (i : ι) → AddCommMonoid (M₁ i)\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : (i : ι) → Module R (M₁ i)\ninst✝² : Module R M₂\nf : MultilinearMap R M₁ M₂\nα : ι → Type u_1\ng : (i : ι) → α i → M₁ i\ninst✝¹ : DecidableEq ι\nin... | [] | exact IH _ this C rfl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Category.Ring.Colimits | {
"line": 566,
"column": 6
} | {
"line": 566,
"column": 49
} | {
"line": 567,
"column": 6
} | [
{
"pp": "J : Type v\ninst✝ : SmallCategory J\nF : J ⥤ CommRingCat\ns : Cocone F\nx y : ColimitType F\n⊢ descFun F s (x + y) = descFun F s x + descFun F s y",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"CommRingCat.carrier",
"C... | [
"J : Type v\ninst✝ : SmallCategory J\nF : J ⥤ CommRingCat\ns : Cocone F\nx y : ColimitType F\na b : Prequotient F\n⊢ descFun F s (Quot.mk (⇑(colimitSetoid F)) a + Quot.mk (⇑(colimitSetoid F)) b) =\n descFun F s (Quot.mk (⇑(colimitSetoid F)) a) + descFun F s (Quot.mk (⇑(colimitSetoid F)) b)"
] | refine Quot.induction_on₂ x y fun a b => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Filtered.Basic | {
"line": 228,
"column": 4
} | {
"line": 233,
"column": 78
} | {
"line": 235,
"column": 0
} | [
{
"pp": "case insert\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : IsFiltered C\nX : C\nO' : Finset C\nnm : X ∉ O'\nh : ∃ S, ∀ {X : C}, X ∈ O' → Nonempty (X ⟶ S)\n⊢ ∃ S, ∀ {X_1 : C}, X_1 ∈ insert X O' → Nonempty (X_1 ⟶ S)",
"ppTerm": "?insert",
"assigned": true,
"usedConstants": [
"Category... | [] | obtain ⟨S', w'⟩ := h
use max X S'
rintro Y mY
obtain rfl | h := eq_or_ne Y X
· exact ⟨leftToMax _ _⟩
· exact ⟨(w' (Finset.mem_of_mem_insert_of_ne mY h)).some ≫ rightToMax _ _⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Filtered.Basic | {
"line": 228,
"column": 4
} | {
"line": 233,
"column": 78
} | {
"line": 235,
"column": 0
} | [
{
"pp": "case insert\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : IsFiltered C\nX : C\nO' : Finset C\nnm : X ∉ O'\nh : ∃ S, ∀ {X : C}, X ∈ O' → Nonempty (X ⟶ S)\n⊢ ∃ S, ∀ {X_1 : C}, X_1 ∈ insert X O' → Nonempty (X_1 ⟶ S)",
"ppTerm": "?insert",
"assigned": true,
"usedConstants": [
"Category... | [] | obtain ⟨S', w'⟩ := h
use max X S'
rintro Y mY
obtain rfl | h := eq_or_ne Y X
· exact ⟨leftToMax _ _⟩
· exact ⟨(w' (Finset.mem_of_mem_insert_of_ne mY h)).some ≫ rightToMax _ _⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Multilinear.Basic | {
"line": 601,
"column": 8
} | {
"line": 601,
"column": 43
} | {
"line": 602,
"column": 6
} | [
{
"pp": "case neg\nR : Type uR\nι : Type uι\nM₁ : ι → Type v₁\nM₂ : Type v₂\ninst✝⁶ : Semiring R\ninst✝⁵ : (i : ι) → AddCommMonoid (M₁ i)\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : (i : ι) → Module R (M₁ i)\ninst✝² : Module R M₂\nf : MultilinearMap R M₁ M₂\nα : ι → Type u_1\ng : (i : ι) → α i → M₁ i\ninst✝¹ : Decidab... | [] | · simp [C, hi, mem_piFinset.1 hr i] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Filtered.Basic | {
"line": 346,
"column": 4
} | {
"line": 346,
"column": 16
} | {
"line": 347,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nh : ∀ {J : Type w} [inst : SmallCategory J] [FinCategory J] (F : J ⥤ C), Nonempty (Cocone F)\nc : Cocone (Functor.empty C)\n⊢ Nonempty C",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"PEmpty",
"Nonempty.intro",
"CategoryTh... | [] | exact ⟨c.pt⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.Matrix.Determinant.Basic | {
"line": 573,
"column": 2
} | {
"line": 573,
"column": 89
} | {
"line": 574,
"column": 2
} | [
{
"pp": "case refine_2\nR : Type v\ninst✝ : CommRing R\nn : ℕ\nk✝ : Fin (n + 1)\nk : Fin n\nih :\n ∀ (c : Fin n → R),\n (∀ (i : Fin n), k.castSucc < i.succ → c i = 0) →\n ∀ {M N : Matrix (Fin n.succ) (Fin n.succ) R},\n (∀ (j : Fin n.succ), M 0 j = N 0 j) →\n (∀ (i : Fin n) (j : Fin n.su... | [
"case refine_2._hc\nR : Type v\ninst✝ : CommRing R\nn : ℕ\nk✝ : Fin (n + 1)\nk : Fin n\nih :\n ∀ (c : Fin n → R),\n (∀ (i : Fin n), k.castSucc < i.succ → c i = 0) →\n ∀ {M N : Matrix (Fin n.succ) (Fin n.succ) R},\n (∀ (j : Fin n.succ), M 0 j = N 0 j) →\n (∀ (i : Fin n) (j : Fin n.succ), M i... | rw [hM, M_k, det_updateRow_add_smul_self M' k_ne_succ.symm, ih (Function.update c k 0)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Limits.Types.Colimits | {
"line": 228,
"column": 2
} | {
"line": 230,
"column": 32
} | {
"line": 231,
"column": 2
} | [
{
"pp": "J : Type v\ninst✝ : Category.{w, v} J\nF : J ⥤ Type u\nt : Cocone F\nh : IsColimit t\nx : t.pt\nhx : ∀ (x_1 : J) (x_2 : F.obj x_1), ¬(hom (t.ι.app x_1)) x_2 = x\n⊢ False",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [],
"usedFVars": [],
"usedGoals": [
{
"new"... | [
"J : Type v\ninst✝ : Category.{w, v} J\nF : J ⥤ Type u\nt : Cocone F\nh : IsColimit t\nx : t.pt\nhx : ∀ (x_1 : J) (x_2 : F.obj x_1), ¬(hom (t.ι.app x_1)) x_2 = x\n⊢ (↾fun x ↦ { down := True }) = ↾fun y ↦ { down := y ≠ x }",
"J : Type v\ninst✝ : Category.{w, v} J\nF : J ⥤ Type u\nt : Cocone F\nh : IsColimit t\nx :... | apply (_ : (↾fun _ ↦ ULift.up True :
t.pt ⟶ (ULift.{u} Prop)) ≠
(↾fun y ↦ ULift.up (y ≠ x))) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Filtered.Basic | {
"line": 836,
"column": 2
} | {
"line": 836,
"column": 6
} | {
"line": 837,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : IsCofiltered C\nJ : Type w\ninst✝¹ : SmallCategory J\ninst✝ : FinCategory J\nF : J ⥤ C\nO : Finset C := Finset.image F.obj Finset.univ\nH : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y)) :=\n Finset.univ.biUnion fun X ↦\n Finset... | [
"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : IsCofiltered C\nJ : Type w\ninst✝¹ : SmallCategory J\ninst✝ : FinCategory J\nF : J ⥤ C\nO : Finset C := ⋯\nH : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y)) := ⋯\nZ : C\nf : {X : C} → X ∈ O → (Z ⟶ X)\nw : ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) ... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Filtered.Basic | {
"line": 893,
"column": 4
} | {
"line": 893,
"column": 16
} | {
"line": 894,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nh : ∀ {J : Type w} [inst : SmallCategory J] [FinCategory J] (F : J ⥤ C), Nonempty (Cone F)\nc : Cone (Functor.empty C)\n⊢ Nonempty C",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"PEmpty",
"CategoryTheory.Limits.Cone.pt",
... | [] | exact ⟨c.pt⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.Multilinear.Basic | {
"line": 1001,
"column": 6
} | {
"line": 1002,
"column": 39
} | {
"line": 1002,
"column": 40
} | [
{
"pp": "R : Type uR\nS : Type uS\nι : Type uι\nn : ℕ\nM : Fin n.succ → Type v\nM₁ : ι → Type v₁\nM₁' : ι → Type v₁'\nM₁'' : ι → Type v₁''\nM₂ : Type v₂\nM₃ : Type v₃\nM₄ : Type v₄\nM' : Type v'\ninst✝¹¹ : Semiring R\ninst✝¹⁰ : (i : ι) → AddCommMonoid (M₁ i)\ninst✝⁹ : (i : ι) → Module R (M₁ i)\ninst✝⁸ : AddComm... | [] | ext
simp [MultilinearMap.domDomCongr] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Multilinear.Basic | {
"line": 1001,
"column": 6
} | {
"line": 1002,
"column": 39
} | {
"line": 1002,
"column": 40
} | [
{
"pp": "R : Type uR\nS : Type uS\nι : Type uι\nn : ℕ\nM : Fin n.succ → Type v\nM₁ : ι → Type v₁\nM₁' : ι → Type v₁'\nM₁'' : ι → Type v₁''\nM₂ : Type v₂\nM₃ : Type v₃\nM₄ : Type v₄\nM' : Type v'\ninst✝¹¹ : Semiring R\ninst✝¹⁰ : (i : ι) → AddCommMonoid (M₁ i)\ninst✝⁹ : (i : ι) → Module R (M₁ i)\ninst✝⁸ : AddComm... | [] | ext
simp [MultilinearMap.domDomCongr] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Multilinear.Basic | {
"line": 1000,
"column": 28
} | {
"line": 1002,
"column": 39
} | {
"line": 1002,
"column": 40
} | [
{
"pp": "R : Type uR\nS : Type uS\nι : Type uι\nn : ℕ\nM : Fin n.succ → Type v\nM₁ : ι → Type v₁\nM₁' : ι → Type v₁'\nM₁'' : ι → Type v₁''\nM₂ : Type v₂\nM₃ : Type v₃\nM₄ : Type v₄\nM' : Type v'\ninst✝¹¹ : Semiring R\ninst✝¹⁰ : (i : ι) → AddCommMonoid (M₁ i)\ninst✝⁹ : (i : ι) → Module R (M₁ i)\ninst✝⁸ : AddComm... | [] | by
ext
simp [MultilinearMap.domDomCongr] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Category.MonCat.FilteredColimits | {
"line": 223,
"column": 23
} | {
"line": 223,
"column": 27
} | {
"line": 223,
"column": 27
} | [
{
"pp": "J : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj : J\nx y : ↑(F.1 j)\n⊢ (ConcreteCategory.hom ((Types.TypeMax.colimitCocone (F ⋙ forget MonCat)).ι.app j)) (x * y) =\n (ConcreteCategory.hom ((Types.TypeMax.colimitCocone (F ⋙ forget MonCat)).ι.app j)) x *\n (ConcreteC... | [
"J : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj : J\nx y : ↑(F.1 j)\n⊢ (ConcreteCategory.hom ((Types.TypeMax.colimitCocone (F ⋙ forget MonCat)).ι.app j)) x *\n (ConcreteCategory.hom ((Types.TypeMax.colimitCocone (F ⋙ forget MonCat)).ι.app j)) y =\n (ConcreteCategory.hom ((Type... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.LinearAlgebra.Multilinear.Basic | {
"line": 1338,
"column": 6
} | {
"line": 1338,
"column": 30
} | {
"line": 1338,
"column": 31
} | [
{
"pp": "case insert\nR : Type uR\nι : Type uι\nM₁ : ι → Type v₁\nM₂ : Type v₂\ninst✝⁵ : Semiring R\ninst✝⁴ : (i : ι) → AddCommGroup (M₁ i)\ninst✝³ : AddCommGroup M₂\ninst✝² : (i : ι) → Module R (M₁ i)\ninst✝¹ : Module R M₂\nf : MultilinearMap R M₁ M₂\ninst✝ : LinearOrder ι\na b : (i : ι) → M₁ i\nk : ι\ns : Fin... | [
"case insert\nR : Type uR\nι : Type uι\nM₁ : ι → Type v₁\nM₂ : Type v₂\ninst✝⁵ : Semiring R\ninst✝⁴ : (i : ι) → AddCommGroup (M₁ i)\ninst✝³ : AddCommGroup M₂\ninst✝² : (i : ι) → Module R (M₁ i)\ninst✝¹ : Module R M₂\nf : MultilinearMap R M₁ M₂\ninst✝ : LinearOrder ι\na b : (i : ι) → M₁ i\nk : ι\ns : Finset ι\nhk : ... | Finset.piecewise_insert, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Category.Ring.FilteredColimits | {
"line": 77,
"column": 25
} | {
"line": 84,
"column": 9
} | {
"line": 85,
"column": 4
} | [
{
"pp": "J : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ SemiRingCat\ninst✝ : IsFiltered J\nx : ↑(R F)\n⊢ 0 * x = 0",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SemiRingCat.instConcreteCategoryRingHomCarrier",
"Semigroup.toMul",
"AddMonCat.instConcreteCa... | [] | by
refine Quot.inductionOn x ?_; clear x; intro x
obtain ⟨j, x⟩ := x
erw [colimit_zero_eq _ j, colimit_mul_mk_eq _ ⟨j, _⟩ ⟨j, _⟩ j (𝟙 j) (𝟙 j)]
rw [CategoryTheory.Functor.map_id]
dsimp
rw [zero_mul x]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.Determinant.Basic | {
"line": 693,
"column": 8
} | {
"line": 693,
"column": 26
} | {
"line": 694,
"column": 6
} | [
{
"pp": "case e'_3.refine_2.snd\nm : Type u_1\nn : Type u_2\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\nσ₁ : Perm m × Perm n\na✝¹ : σ₁ ∈ ↑univ\nσ₂ : Perm m × Perm n\na✝ : σ₂ ∈ ↑univ... | [] | · exact h2.right x | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Matrix.Determinant.Basic | {
"line": 718,
"column": 16
} | {
"line": 718,
"column": 35
} | {
"line": 718,
"column": 36
} | [
{
"pp": "case convert_2.inr\nm : Type u_1\nn : Type u_2\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\nσ : Perm (m ⊕ n)\nhσn : σ ∉ (↑(sumCongrHom m n).range).toFinset\nh1 : ¬∀ (x : m),... | [
"case convert_2.inr\nm : Type u_1\nn : Type u_2\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\nσ : Perm (m ⊕ n)\nhσn : σ ∉ (↑(sumCongrHom m n).range).toFinset\nh1 : ¬∀ (x : m), ∃ y, Sum.in... | fromBlocks_apply₂₁, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Functor.Currying | {
"line": 66,
"column": 31
} | {
"line": 66,
"column": 47
} | {
"line": 67,
"column": 2
} | [
{
"pp": "B : Type u₁\ninst✝⁴ : Category.{v₁, u₁} B\nC : Type u₂\ninst✝³ : Category.{v₂, u₂} C\nD : Type u₃\ninst✝² : Category.{v₃, u₃} D\nE : Type u₄\ninst✝¹ : Category.{v₄, u₄} E\nH : Type u₅\ninst✝ : Category.{v₅, u₅} H\nF : C × D ⥤ E\nX : C\nY : D\n⊢ { app := fun Y ↦ F.map (𝟙 X ×ₘ 𝟙 Y), naturality := ⋯ }.a... | [] | exact F.map_id _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Limits.Preserves.Ulift | {
"line": 70,
"column": 32
} | {
"line": 74,
"column": 66
} | {
"line": 74,
"column": 67
} | [
{
"pp": "J✝ : Type u_1\ninst✝ : Category.{v_1, u_1} J✝\nK : J✝ ⥤ Type u\nc✝ : Cocone K\nhc✝ : IsColimit c✝\nlc : Cocone (K ⋙ uliftFunctor.{v, u})\nJ : Type w\nx✝ : Category.{w', w} J\nF : J ⥤ Type u\nc : Cocone F\nhc : IsColimit c\n⊢ Nonempty (IsColimit (uliftFunctor.{v, u}.mapCocone c))",
"ppTerm": "?m.51"... | [] | by
rw [isColimit_iff_coconeTypesIsColimit]
exact (((isColimit_iff_coconeTypesIsColimit _).1 ⟨hc⟩).precompose
(G := F ⋙ uliftFunctor.{v}) (fun _ ↦ Equiv.ulift)
(fun _ ↦ rfl)).of_equiv Equiv.ulift.symm (fun _ _ ↦ rfl) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Category.ModuleCat.Limits | {
"line": 221,
"column": 4
} | {
"line": 221,
"column": 8
} | {
"line": 222,
"column": 4
} | [
{
"pp": "R : Type u\ninst✝⁶ : Ring R\nJ : Type v\ninst✝⁵ : Category.{t, v} J\nF : J ⥤ ModuleCat R\nι : Type v\ninst✝⁴ : DecidableEq ι\ninst✝³ : Preorder ι\nG : ι → Type v\ninst✝² : (i : ι) → AddCommGroup (G i)\ninst✝¹ : (i : ι) → Module R (G i)\nf : (i j : ι) → i ≤ j → G i →ₗ[R] G j\ninst✝ : DirectedSystem G fu... | [
"R : Type u\ninst✝⁶ : Ring R\nJ : Type v\ninst✝⁵ : Category.{t, v} J\nF : J ⥤ ModuleCat R\nι : Type v\ninst✝⁴ : DecidableEq ι\ninst✝³ : Preorder ι\nG : ι → Type v\ninst✝² : (i : ι) → AddCommGroup (G i)\ninst✝¹ : (i : ι) → Module R (G i)\nf : (i j : ι) → i ≤ j → G i →ₗ[R] G j\ninst✝ : DirectedSystem G fun i j h ↦ ⇑(... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Monoidal.Functor | {
"line": 554,
"column": 2
} | {
"line": 554,
"column": 14
} | {
"line": 555,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : MonoidalCategory D\nF : C ⥤ D\n⊢ Function.Injective (@toLaxMonoidal C inst✝³ inst✝² D inst✝¹ inst✝ F)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"C... | [
"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : MonoidalCategory D\nF : C ⥤ D\na b : F.Monoidal\neq : a.toLaxMonoidal = b.toLaxMonoidal\n⊢ a = b"
] | intro a b eq | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.Monoidal.Functor | {
"line": 568,
"column": 2
} | {
"line": 568,
"column": 14
} | {
"line": 569,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : MonoidalCategory D\nF : C ⥤ D\n⊢ Function.Injective (@toOplaxMonoidal C inst✝³ inst✝² D inst✝¹ inst✝ F)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
... | [
"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : MonoidalCategory D\nF : C ⥤ D\na b : F.Monoidal\neq : a.toOplaxMonoidal = b.toOplaxMonoidal\n⊢ a = b"
] | intro a b eq | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.Monoidal.Braided.Basic | {
"line": 626,
"column": 2
} | {
"line": 628,
"column": 96
} | {
"line": 629,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX₁ X₂ Y₁ Y₂ U₁ U₂ V₁ V₂ : C\nf₁ : X₁ ⟶ Y₁\nf₂ : X₂ ⟶ Y₂\ng₁ : U₁ ⟶ V₁\ng₂ : U₂ ⟶ V₂\n⊢ (α_ X₁ X₂ (U₁ ⊗ U₂)).hom ≫\n ((((𝟙 X₁ ⊗ₘ (α_ X₂ U₁ U₂).inv) ≫ (f₁ ⊗ₘ (f₂ ⊗ₘ g₁) ⊗ₘ g₂)) ≫ (𝟙 Y₁ ⊗ₘ (β_ Y₂ V₁).... | [
"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX₁ X₂ Y₁ Y₂ U₁ U₂ V₁ V₂ : C\nf₁ : X₁ ⟶ Y₁\nf₂ : X₂ ⟶ Y₂\ng₁ : U₁ ⟶ V₁\ng₂ : U₂ ⟶ V₂\n⊢ (α_ X₁ X₂ (U₁ ⊗ U₂)).hom ≫\n (𝟙 X₁ ⊗ₘ (α_ X₂ U₁ U₂).inv) ≫\n (((𝟙 X₁ ⊗ₘ (β_ X₂ U₁).hom ⊗ₘ 𝟙 U₂) ≫ (f₁ ⊗ₘ (g₁ ⊗ₘ f₂... | slice_lhs 3 4 =>
rw [tensorHom_comp_tensorHom, tensorHom_comp_tensorHom, comp_id f₁, ← id_comp f₁, comp_id g₂,
← id_comp g₂, braiding_naturality, ← tensorHom_comp_tensorHom, ← tensorHom_comp_tensorHom] | Mathlib.Tactic.Slice._aux_Mathlib_Tactic_CategoryTheory_Slice___macroRules_Mathlib_Tactic_Slice_sliceLHS_1 | Mathlib.Tactic.Slice.sliceLHS |
Mathlib.Algebra.Category.Grp.Colimits | {
"line": 168,
"column": 4
} | {
"line": 172,
"column": 7
} | {
"line": 173,
"column": 2
} | [
{
"pp": "J : Type u\ninst✝¹ : Category.{v, u} J\nF : J ⥤ AddCommGrpCat\nc : Cocone F\ninst✝ : DecidableEq J\nx : Quot (F ⋙ uliftFunctor)\n⊢ (quotToQuotUlift F) ((quotUliftToQuot F) x) = x",
"ppTerm": "?m.113",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AddCommGrpCat.uliftFunctor",
... | [] | conv_rhs => rw [← AddMonoidHom.id_apply _ x]
rw [← AddMonoidHom.comp_apply, Quot.addMonoidHom_ext _ (f := (quotToQuotUlift F).comp
(quotUliftToQuot F)) (fun j a ↦ ?_)]
rw [AddMonoidHom.comp_apply, AddMonoidHom.id_apply, quotUliftToQuot_ι, quotToQuotUlift_ι]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.Grp.Colimits | {
"line": 168,
"column": 4
} | {
"line": 172,
"column": 7
} | {
"line": 173,
"column": 2
} | [
{
"pp": "J : Type u\ninst✝¹ : Category.{v, u} J\nF : J ⥤ AddCommGrpCat\nc : Cocone F\ninst✝ : DecidableEq J\nx : Quot (F ⋙ uliftFunctor)\n⊢ (quotToQuotUlift F) ((quotUliftToQuot F) x) = x",
"ppTerm": "?m.113",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AddCommGrpCat.uliftFunctor",
... | [] | conv_rhs => rw [← AddMonoidHom.id_apply _ x]
rw [← AddMonoidHom.comp_apply, Quot.addMonoidHom_ext _ (f := (quotToQuotUlift F).comp
(quotUliftToQuot F)) (fun j a ↦ ?_)]
rw [AddMonoidHom.comp_apply, AddMonoidHom.id_apply, quotUliftToQuot_ι, quotToQuotUlift_ι]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Adjunction.Mates | {
"line": 109,
"column": 4
} | {
"line": 110,
"column": 39
} | {
"line": 111,
"column": 4
} | [
{
"pp": "C : Type u₁\nD : Type u₂\nE : Type u₃\nF : Type u₄\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} E\ninst✝ : Category.{v₄, u₄} F\nG : C ⥤ E\nH : D ⥤ F\nL₁ : C ⥤ D\nR₁ : D ⥤ C\nL₂ : E ⥤ F\nR₂ : F ⥤ E\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nα : TwoSquare G L₁ L₂ H\nX✝ : ... | [
"C : Type u₁\nD : Type u₂\nE : Type u₃\nF : Type u₄\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : Category.{v₃, u₃} E\ninst✝ : Category.{v₄, u₄} F\nG : C ⥤ E\nH : D ⥤ F\nL₁ : C ⥤ D\nR₁ : D ⥤ C\nL₂ : E ⥤ F\nR₂ : F ⥤ E\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nα : TwoSquare G L₁ L₂ H\nX✝ : C\n⊢ α.natTr... | rw [← assoc, ← Functor.comp_map, α.natTrans.naturality, Functor.comp_map, assoc, ← H.map_comp,
left_triangle_components, map_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.TensorAlgebra.Basic | {
"line": 282,
"column": 60
} | {
"line": 282,
"column": 74
} | {
"line": 282,
"column": 75
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx : M\n⊢ (ι R) x = 0 ↔ x = 0",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"CommSemiring.toSemiring",
"AddMonoid.toAddZeroClass",
... | [
"R : Type u_1\ninst✝² : CommSemiring R\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx : M\n⊢ (ι R) x = 0 ↔ (ι R) x = (ι R) 0"
] | ← ι_inj R x 0, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Coalgebra.Basic | {
"line": 465,
"column": 2
} | {
"line": 465,
"column": 52
} | {
"line": 467,
"column": 0
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\nA : n → Type u_3\ninst✝² : (i : n) → AddCommMonoid (A i)\ninst✝¹ : (i : n) → Module R (A i)\ninst✝ : (i : n) → CoalgebraStruct R (A i)\ni j : n\nx✝ : A j\nthis : i = j ∨ i ≠ j\n⊢ ((comul ∘ₗ proj i) ∘ₗ Linea... | [] | aesop (add simp [map_map, proj_comp_single, diag]) | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Category.CoalgCat.Basic | {
"line": 187,
"column": 4
} | {
"line": 189,
"column": 36
} | {
"line": 190,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nX Y : CoalgCat R\nf : X ⟶ Y\nx✝ : IsIso ((forget (CoalgCat R)).map f)\n⊢ IsIso f",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"CoalgCat.toModuleCat",
"Coalgebra.toCoalgebraStruct",
"CoalgCat",
"CoalgEquiv.mk",
"Equiv... | [] | let i := asIso ((forget (CoalgCat.{v} R)).map f)
let e : X ≃ₗc[R] Y := { f.toCoalgHom, i.toEquiv with }
exact ⟨e.toCoalgIso.isIso_hom.1⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.CoalgCat.Basic | {
"line": 187,
"column": 4
} | {
"line": 189,
"column": 36
} | {
"line": 190,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nX Y : CoalgCat R\nf : X ⟶ Y\nx✝ : IsIso ((forget (CoalgCat R)).map f)\n⊢ IsIso f",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"CoalgCat.toModuleCat",
"Coalgebra.toCoalgebraStruct",
"CoalgCat",
"CoalgEquiv.mk",
"Equiv... | [] | let i := asIso ((forget (CoalgCat.{v} R)).map f)
let e : X ≃ₗc[R] Y := { f.toCoalgHom, i.toEquiv with }
exact ⟨e.toCoalgIso.isIso_hom.1⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Bialgebra.Hom | {
"line": 74,
"column": 4
} | {
"line": 74,
"column": 65
} | {
"line": 76,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\nF : Type u_4\ninst✝⁸ : CommSemiring R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : Semiring B\ninst✝⁴ : Algebra R B\ninst✝³ : CoalgebraStruct R A\ninst✝² : CoalgebraStruct R B\ninst✝¹ : FunLike F A B\ninst✝ : BialgHomClass F R A B\nc : F\nr : R\n⊢ c ((a... | [] | simp only [Algebra.algebraMap_eq_smul_one, map_smul, map_one] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Bialgebra.Hom | {
"line": 74,
"column": 4
} | {
"line": 74,
"column": 65
} | {
"line": 76,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\nF : Type u_4\ninst✝⁸ : CommSemiring R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : Semiring B\ninst✝⁴ : Algebra R B\ninst✝³ : CoalgebraStruct R A\ninst✝² : CoalgebraStruct R B\ninst✝¹ : FunLike F A B\ninst✝ : BialgHomClass F R A B\nc : F\nr : R\n⊢ c ((a... | [] | simp only [Algebra.algebraMap_eq_smul_one, map_smul, map_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Bialgebra.Hom | {
"line": 74,
"column": 4
} | {
"line": 74,
"column": 65
} | {
"line": 76,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\nF : Type u_4\ninst✝⁸ : CommSemiring R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : Semiring B\ninst✝⁴ : Algebra R B\ninst✝³ : CoalgebraStruct R A\ninst✝² : CoalgebraStruct R B\ninst✝¹ : FunLike F A B\ninst✝ : BialgHomClass F R A B\nc : F\nr : R\n⊢ c ((a... | [] | simp only [Algebra.algebraMap_eq_smul_one, map_smul, map_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.TrivSqZeroExt.Basic | {
"line": 775,
"column": 2
} | {
"line": 775,
"column": 34
} | {
"line": 776,
"column": 2
} | [
{
"pp": "case h1\nR : Type u\nM : Type v\ninst✝³ : DivisionSemiring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module Rᵐᵒᵖ M\ninst✝ : Module R M\nr : R\n⊢ (inl r)⁻¹.fst = (inl r⁻¹).fst",
"ppTerm": "?h1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"TrivSqZeroExt"... | [
"case h2\nR : Type u\nM : Type v\ninst✝³ : DivisionSemiring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module Rᵐᵒᵖ M\ninst✝ : Module R M\nr : R\n⊢ (inl r)⁻¹.snd = (inl r⁻¹).snd"
] | · rw [fst_inv, fst_inl, fst_inl] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Category.ModuleCat.ChangeOfRings | {
"line": 704,
"column": 6
} | {
"line": 704,
"column": 57
} | {
"line": 705,
"column": 6
} | [
{
"pp": "R✝ : Type u₁\nS✝ : Type u₂\ninst✝³ : CommRing R✝\ninst✝² : CommRing S✝\nf✝ : R✝ →+* S✝\nR : Type u₁\nS : Type u₂\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (extendScalars f).obj X ⟶ Y\nr : R\ns : ↑X\n⊢ (ConcreteCategory.hom g) (1 ⊗ₜ[R] (r • s)) = f r • ... | [
"R✝ : Type u₁\nS✝ : Type u₂\ninst✝³ : CommRing R✝\ninst✝² : CommRing S✝\nf✝ : R✝ →+* S✝\nR : Type u₁\nS : Type u₂\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (extendScalars f).obj X ⟶ Y\nr : R\ns : ↑X\n⊢ (ConcreteCategory.hom g)\n (1 ⊗ₜ[R] (RestrictScalars.addEq... | rw [RestrictScalars.smul_def, ← LinearMap.map_smul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Monoidal.Comon_ | {
"line": 413,
"column": 26
} | {
"line": 413,
"column": 38
} | {
"line": 413,
"column": 39
} | [
{
"pp": "C : Type u₁\ninst✝⁸ : Category.{v₁, u₁} C\ninst✝⁷ : MonoidalCategory C\nM N O : C\ninst✝⁶ : ComonObj M\ninst✝⁵ : ComonObj N\ninst✝⁴ : ComonObj O\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\ninst✝² : MonoidalCategory D\nA : C\ninst✝¹ : ComonObj A\nF : C ⥤ D\ninst✝ : F.OplaxMonoidal\n⊢ F.map (Δ ≫ A ◁ Δ) ≫... | [
"C : Type u₁\ninst✝⁸ : Category.{v₁, u₁} C\ninst✝⁷ : MonoidalCategory C\nM N O : C\ninst✝⁶ : ComonObj M\ninst✝⁵ : ComonObj N\ninst✝⁴ : ComonObj O\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\ninst✝² : MonoidalCategory D\nA : C\ninst✝¹ : ComonObj A\nF : C ⥤ D\ninst✝ : F.OplaxMonoidal\n⊢ F.map (Δ ≫ Δ ▷ A ≫ (α_ A A A).h... | comul_assoc, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Localization.Away.Basic | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 6
} | {
"line": 65,
"column": 2
} | [
{
"pp": "case e'_5\nR : Type u_1\ninst✝³ : CommSemiring R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nx : R\ninst✝ : Away x S\n⊢ (algebraMap R S) x = mk' S (↑⟨x, ⋯⟩) 1",
"ppTerm": "?e'_5",
"assigned": true,
"usedConstants": [
"Algebra.algebraMap",
"Monoid.toMulOneClass"... | [
"case e'_5\nR : Type u_1\ninst✝³ : CommSemiring R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nx : R\ninst✝ : Away x S\n⊢ mk' S (↑⟨x, ⋯⟩) 1 = (algebraMap R S) x"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity | {
"line": 156,
"column": 2
} | {
"line": 169,
"column": 46
} | {
"line": 171,
"column": 0
} | [
{
"pp": "R : Type u_2\ninst✝¹ : CommMonoidWithZero R\ninst✝ : UniqueFactorizationMonoid R\na b : R\nha : a ≠ 0\n⊢ a ∣ b ↔ ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"UniqueFactorizationMonoid.normalizedFactors",
"P... | [] | classical
refine ⟨fun h _ _ ↦ emultiplicity_le_emultiplicity_of_dvd_right h, fun h ↦ ?_⟩
by_cases hb : b = 0
· simp_all
letI : StrongNormalizationMonoid R := UniqueFactorizationMonoid.strongNormalizationMonoid
rw [dvd_iff_normalizedFactors_le_normalizedFactors ha hb, Multiset.le_iff_count]
intro q
by_case... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity | {
"line": 156,
"column": 2
} | {
"line": 169,
"column": 46
} | {
"line": 171,
"column": 0
} | [
{
"pp": "R : Type u_2\ninst✝¹ : CommMonoidWithZero R\ninst✝ : UniqueFactorizationMonoid R\na b : R\nha : a ≠ 0\n⊢ a ∣ b ↔ ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"UniqueFactorizationMonoid.normalizedFactors",
"P... | [] | classical
refine ⟨fun h _ _ ↦ emultiplicity_le_emultiplicity_of_dvd_right h, fun h ↦ ?_⟩
by_cases hb : b = 0
· simp_all
letI : StrongNormalizationMonoid R := UniqueFactorizationMonoid.strongNormalizationMonoid
rw [dvd_iff_normalizedFactors_le_normalizedFactors ha hb, Multiset.le_iff_count]
intro q
by_case... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity | {
"line": 156,
"column": 2
} | {
"line": 169,
"column": 46
} | {
"line": 171,
"column": 0
} | [
{
"pp": "R : Type u_2\ninst✝¹ : CommMonoidWithZero R\ninst✝ : UniqueFactorizationMonoid R\na b : R\nha : a ≠ 0\n⊢ a ∣ b ↔ ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"UniqueFactorizationMonoid.normalizedFactors",
"P... | [] | classical
refine ⟨fun h _ _ ↦ emultiplicity_le_emultiplicity_of_dvd_right h, fun h ↦ ?_⟩
by_cases hb : b = 0
· simp_all
letI : StrongNormalizationMonoid R := UniqueFactorizationMonoid.strongNormalizationMonoid
rw [dvd_iff_normalizedFactors_le_normalizedFactors ha hb, Multiset.le_iff_count]
intro q
by_case... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.TwoSidedIdeal.Operations | {
"line": 233,
"column": 6
} | {
"line": 233,
"column": 70
} | {
"line": 234,
"column": 6
} | [
{
"pp": "case refine_1.inl.inr\nR : Type u_1\ninst✝ : NonUnitalRing R\ns : Set R\nz x r y : R\nhy : y ∈ s\n⊢ x * (fun x1 x2 ↦ x1 * x2) r y ∈ s ∪ s * univ ∪ univ * s ∪ univ * s * univ",
"ppTerm": "?refine_1.inl.inr",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"Set.univ",
"mul... | [
"case refine_1.inr\nR : Type u_1\ninst✝ : NonUnitalRing R\ns : Set R\nz x r' y : R\nhy : y ∈ s\nr : R\n⊢ x * (fun x1 x2 ↦ x1 * x2) ((fun x1 x2 ↦ x1 * x2) r' y) r ∈ s ∪ s * univ ∪ univ * s ∪ univ * s * univ"
] | · exact .inl <| .inr <| ⟨x * r, mem_univ _, y, hy, mul_assoc ..⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Localization.Away.Basic | {
"line": 691,
"column": 4
} | {
"line": 691,
"column": 18
} | {
"line": 693,
"column": 0
} | [
{
"pp": "case e_y\nR : Type u_1\ninst✝³ : CommSemiring R\nx : R\nB : Type u_2\ninst✝² : CommSemiring B\ninst✝¹ : Algebra R B\ninst✝ : IsLocalization.Away x B\nn m : ℤ\nhn : n < 0\nhm : m < 0\n⊢ ↑(Submonoid.pow x (n.natAbs + m.natAbs)) = ↑(Submonoid.pow x n.natAbs * Submonoid.pow x m.natAbs)",
"ppTerm": "?e_... | [] | simp [pow_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Monoidal.Mon | {
"line": 375,
"column": 2
} | {
"line": 375,
"column": 6
} | {
"line": 376,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : BraidedCategory C\nM N : C\ninst✝¹ : MonObj M\ninst✝ : MonObj N\n⊢ (λ_ (𝟙_ C)).inv ▷ (M ⊗ N) ≫ tensorμ (𝟙_ C) (𝟙_ C) M N ≫ ((λ_ M).hom ⊗ₘ (λ_ N).hom) = (λ_ (M ⊗ N)).hom",
"ppTerm": "?m.184",
"assigned": true,
... | [
"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : BraidedCategory C\nM N : C\ninst✝¹ : MonObj M\ninst✝ : MonObj N\n⊢ (λ_ (M ⊗ N)).hom = (λ_ (𝟙_ C)).inv ▷ (M ⊗ N) ≫ tensorμ (𝟙_ C) (𝟙_ C) M N ≫ ((λ_ M).hom ⊗ₘ (λ_ N).hom)"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Monoidal.Mon | {
"line": 386,
"column": 2
} | {
"line": 386,
"column": 6
} | {
"line": 387,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : BraidedCategory C\nM N : C\ninst✝¹ : MonObj M\ninst✝ : MonObj N\n⊢ (M ⊗ N) ◁ (λ_ (𝟙_ C)).inv ≫ tensorμ M N (𝟙_ C) (𝟙_ C) ≫ ((ρ_ M).hom ⊗ₘ (ρ_ N).hom) = (ρ_ (M ⊗ N)).hom",
"ppTerm": "?m.182",
"assigned": true,
... | [
"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : BraidedCategory C\nM N : C\ninst✝¹ : MonObj M\ninst✝ : MonObj N\n⊢ (ρ_ (M ⊗ N)).hom = (M ⊗ N) ◁ (λ_ (𝟙_ C)).inv ≫ tensorμ M N (𝟙_ C) (𝟙_ C) ≫ ((ρ_ M).hom ⊗ₘ (ρ_ N).hom)"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.RingTheory.Localization.Integer | {
"line": 82,
"column": 80
} | {
"line": 84,
"column": 32
} | {
"line": 86,
"column": 0
} | [
{
"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\na : S\n⊢ ∃ b, IsInteger R (↑b • a)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"IsLocalization.IsInteger",
"Eq.mpr",
... | [] | by
simp_rw [Algebra.smul_def, mul_comm _ a]
apply exists_integer_multiple' | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monoidal.Mon | {
"line": 436,
"column": 19
} | {
"line": 436,
"column": 46
} | {
"line": 437,
"column": 2
} | [
{
"pp": "case a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nM : C\ninst✝ : MonObj M\n| μ ▷ 𝟙_ C ≫ (ρ_ M).hom",
"ppTerm": "?a.a",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
... | [
"case a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nM : C\ninst✝ : MonObj M\n| (ρ_ (M ⊗ M)).hom ≫ μ",
"case a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nM : C\ninst✝ : MonObj M\n| tensorμ M (𝟙_ C)... | rw [rightUnitor_naturality] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.CategoryTheory.Monoidal.Mon | {
"line": 436,
"column": 19
} | {
"line": 436,
"column": 46
} | {
"line": 437,
"column": 2
} | [
{
"pp": "case a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nM : C\ninst✝ : MonObj M\n| μ ▷ 𝟙_ C ≫ (ρ_ M).hom",
"ppTerm": "?a.a",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
... | [
"case a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nM : C\ninst✝ : MonObj M\n| (ρ_ (M ⊗ M)).hom ≫ μ",
"case a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nM : C\ninst✝ : MonObj M\n| tensorμ M (𝟙_ C)... | rw [rightUnitor_naturality] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Mon | {
"line": 436,
"column": 19
} | {
"line": 436,
"column": 46
} | {
"line": 437,
"column": 2
} | [
{
"pp": "case a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nM : C\ninst✝ : MonObj M\n| μ ▷ 𝟙_ C ≫ (ρ_ M).hom",
"ppTerm": "?a.a",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
... | [
"case a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nM : C\ninst✝ : MonObj M\n| (ρ_ (M ⊗ M)).hom ≫ μ",
"case a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nM : C\ninst✝ : MonObj M\n| tensorμ M (𝟙_ C)... | rw [rightUnitor_naturality] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.CategoryTheory.Monoidal.Mon | {
"line": 648,
"column": 4
} | {
"line": 648,
"column": 44
} | {
"line": 650,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nM N O X : C\ninst✝² : MonObj M\ninst✝¹ : MonObj N\ninst✝ : MonObj O\nA : Mon C\nf : trivial C ⟶ A\n⊢ 𝟙 (𝟙_ C) ≫ f.hom = η",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory... | [] | rw [← trivial_mon_one, IsMonHom.one_hom] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Localization.Module | {
"line": 99,
"column": 20
} | {
"line": 99,
"column": 23
} | {
"line": 100,
"column": 2
} | [
{
"pp": "case h\nR : Type u_1\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type u_3\nMₛ : Type u_4\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid Mₛ\ninst✝¹ : Module R Mₛ\nf : M →ₗ[R] Mₛ\ninst✝ : IsLocalizedModule S f\nι : Type u_5\nv : ι → Mₛ\nhf : ∀ (s : Finset ι) (f g : ι → R), ∑ i ... | [
"case h\nR : Type u_1\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type u_3\nMₛ : Type u_4\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid Mₛ\ninst✝¹ : Module R Mₛ\nf : M →ₗ[R] Mₛ\ninst✝ : IsLocalizedModule S f\nι : Type u_5\nv : ι → Mₛ\nhf : ∀ (s : Finset ι) (f g : ι → R), ∑ i ∈ s, f i • v... | hit | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts | {
"line": 182,
"column": 4
} | {
"line": 182,
"column": 80
} | {
"line": 183,
"column": 2
} | [
{
"pp": "case right.left\nC : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nF : C ⥤ D\nW X Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\nh : W ⟶ X\nk : W ⟶ Y\nH₁ : IsInitial W\nH₂ : IsColimit (BinaryCofan.mk f g)\ns : PushoutCocone h k\nm : (PushoutCocone.mk f g ⋯).pt ⟶ s.pt\nh₁ : (PushoutCocon... | [] | exact h₁.trans (H₂.fac (BinaryCofan.mk s.inl s.inr) ⟨WalkingPair.left⟩).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts | {
"line": 182,
"column": 4
} | {
"line": 182,
"column": 80
} | {
"line": 183,
"column": 2
} | [
{
"pp": "case right.left\nC : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nF : C ⥤ D\nW X Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\nh : W ⟶ X\nk : W ⟶ Y\nH₁ : IsInitial W\nH₂ : IsColimit (BinaryCofan.mk f g)\ns : PushoutCocone h k\nm : (PushoutCocone.mk f g ⋯).pt ⟶ s.pt\nh₁ : (PushoutCocon... | [] | exact h₁.trans (H₂.fac (BinaryCofan.mk s.inl s.inr) ⟨WalkingPair.left⟩).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts | {
"line": 182,
"column": 4
} | {
"line": 182,
"column": 80
} | {
"line": 183,
"column": 2
} | [
{
"pp": "case right.left\nC : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nF : C ⥤ D\nW X Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\nh : W ⟶ X\nk : W ⟶ Y\nH₁ : IsInitial W\nH₂ : IsColimit (BinaryCofan.mk f g)\ns : PushoutCocone h k\nm : (PushoutCocone.mk f g ⋯).pt ⟶ s.pt\nh₁ : (PushoutCocon... | [] | exact h₁.trans (H₂.fac (BinaryCofan.mk s.inl s.inr) ⟨WalkingPair.left⟩).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Pasting | {
"line": 68,
"column": 54
} | {
"line": 68,
"column": 69
} | {
"line": 68,
"column": 70
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX₃ Y₁ Y₂ Y₃ : C\ng₁ : Y₁ ⟶ Y₂\ng₂ : Y₂ ⟶ Y₃\ni₃ : X₃ ⟶ Y₃\nt₂ : PullbackCone g₂ i₃\ni₂ : t₂.pt ⟶ Y₂\nt₁ : PullbackCone g₁ i₂\nhi₂ : i₂ = t₂.fst\n⊢ t₁.snd ≫ i₂ ≫ g₂ = t₁.snd ≫ t₂.snd ≫ i₃",
"ppTerm": "?m.104",
"assigned": true,
"usedConstants": [
... | [
"C : Type u\ninst✝ : Category.{v, u} C\nX₃ Y₁ Y₂ Y₃ : C\ng₁ : Y₁ ⟶ Y₂\ng₂ : Y₂ ⟶ Y₃\ni₃ : X₃ ⟶ Y₃\nt₂ : PullbackCone g₂ i₃\ni₂ : t₂.pt ⟶ Y₂\nt₁ : PullbackCone g₁ i₂\nhi₂ : i₂ = t₂.fst\n⊢ t₁.snd ≫ i₂ ≫ g₂ = t₁.snd ≫ t₂.fst ≫ g₂"
] | ← t₂.condition, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Defs | {
"line": 287,
"column": 19
} | {
"line": 287,
"column": 27
} | {
"line": 287,
"column": 27
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nD : WalkingSpan ⥤ C\nc : Cocone D\nhc : IsColimit c\n⊢ D.map WalkingSpan.Hom.fst ≫ c.ι.app WalkingSpan.left = D.map WalkingSpan.Hom.snd ≫ c.ι.app WalkingSpan.right",
"ppTerm": "?m.70",
"assigned": true,
"usedConstants": [
"CategoryTheory.Funct... | [] | Cocone.w | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Pasting | {
"line": 124,
"column": 28
} | {
"line": 124,
"column": 43
} | {
"line": 124,
"column": 44
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX₃ Y₁ Y₂ Y₃ : C\ng₁ : Y₁ ⟶ Y₂\ng₂ : Y₂ ⟶ Y₃\ni₃ : X₃ ⟶ Y₃\nt₂ : PullbackCone g₂ i₃\ni₂ : t₂.pt ⟶ Y₂\nt₁ : PullbackCone g₁ i₂\nhi₂ : i₂ = t₂.fst\nH : IsLimit t₂\nH' : IsLimit (t₂.pasteHoriz t₁ hi₂)\ns : PullbackCone g₁ i₂\n⊢ s.fst ≫ g₁ ≫ g₂ = s.snd ≫ t₂.snd ≫ i₃",
... | [
"C : Type u\ninst✝ : Category.{v, u} C\nX₃ Y₁ Y₂ Y₃ : C\ng₁ : Y₁ ⟶ Y₂\ng₂ : Y₂ ⟶ Y₃\ni₃ : X₃ ⟶ Y₃\nt₂ : PullbackCone g₂ i₃\ni₂ : t₂.pt ⟶ Y₂\nt₁ : PullbackCone g₁ i₂\nhi₂ : i₂ = t₂.fst\nH : IsLimit t₂\nH' : IsLimit (t₂.pasteHoriz t₁ hi₂)\ns : PullbackCone g₁ i₂\n⊢ s.fst ≫ g₁ ≫ g₂ = s.snd ≫ t₂.fst ≫ g₂"
] | ← t₂.condition, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Pasting | {
"line": 257,
"column": 54
} | {
"line": 257,
"column": 69
} | {
"line": 257,
"column": 70
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX₁ X₂ X₃ Y₁ : C\nf₁ : X₁ ⟶ X₂\nf₂ : X₂ ⟶ X₃\ni₁ : X₁ ⟶ Y₁\nt₁ : PushoutCocone i₁ f₁\ni₂ : X₂ ⟶ t₁.pt\nt₂ : PushoutCocone i₂ f₂\nhi₂ : i₂ = t₁.inr\n⊢ f₁ ≫ t₁.inr ≫ t₂.inl = f₁ ≫ f₂ ≫ t₂.inr",
"ppTerm": "?m.104",
"assigned": true,
"usedConstants": [
... | [
"C : Type u\ninst✝ : Category.{v, u} C\nX₁ X₂ X₃ Y₁ : C\nf₁ : X₁ ⟶ X₂\nf₂ : X₂ ⟶ X₃\ni₁ : X₁ ⟶ Y₁\nt₁ : PushoutCocone i₁ f₁\ni₂ : X₂ ⟶ t₁.pt\nt₂ : PushoutCocone i₂ f₂\nhi₂ : i₂ = t₁.inr\n⊢ f₁ ≫ t₁.inr ≫ t₂.inl = f₁ ≫ i₂ ≫ t₂.inl"
] | ← t₂.condition, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Localization.BaseChange | {
"line": 54,
"column": 2
} | {
"line": 54,
"column": 84
} | {
"line": 55,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁹ : CommSemiring R\nS : Submonoid R\nA : Type u_2\ninst✝⁸ : CommSemiring A\ninst✝⁷ : Algebra R A\ninst✝⁶ : IsLocalization S A\nM : Type u_3\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nM' : Type u_4\ninst✝³ : AddCommMonoid M'\ninst✝² : Module R M'\ninst✝¹ : Module A M'\ninst✝ : Is... | [
"R : Type u_1\ninst✝⁹ : CommSemiring R\nS : Submonoid R\nA : Type u_2\ninst✝⁸ : CommSemiring A\ninst✝⁷ : Algebra R A\ninst✝⁶ : IsLocalization S A\nM : Type u_3\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nM' : Type u_4\ninst✝³ : AddCommMonoid M'\ninst✝² : Module R M'\ninst✝¹ : Module A M'\ninst✝ : IsScalarTower ... | letI : Module A (LocalizedModule S M) := LocalizedModule.moduleOfIsLocalization .. | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLetI___1 | Lean.Parser.Tactic.tacticLetI__ |
Mathlib.Control.Bifunctor | {
"line": 135,
"column": 18
} | {
"line": 135,
"column": 46
} | {
"line": 137,
"column": 0
} | [
{
"pp": "F : Type u₀ → Type u₁ → Type u₂\ninst✝¹ : Bifunctor F\ninst✝ : LawfulBifunctor F\nα : Type u₀\n⊢ ∀ {α_1 β : Type u₁}, mapConst = map ∘ Function.const β",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Function.comp",
"True",
"Bifunctor.snd",
"eq_self",
... | [] | simp [mapConst, Functor.map] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Control.Bifunctor | {
"line": 135,
"column": 18
} | {
"line": 135,
"column": 46
} | {
"line": 137,
"column": 0
} | [
{
"pp": "F : Type u₀ → Type u₁ → Type u₂\ninst✝¹ : Bifunctor F\ninst✝ : LawfulBifunctor F\nα : Type u₀\n⊢ ∀ {α_1 β : Type u₁}, mapConst = map ∘ Function.const β",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Function.comp",
"True",
"Bifunctor.snd",
"eq_self",
... | [] | simp [mapConst, Functor.map] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Control.Bifunctor | {
"line": 135,
"column": 18
} | {
"line": 135,
"column": 46
} | {
"line": 137,
"column": 0
} | [
{
"pp": "F : Type u₀ → Type u₁ → Type u₂\ninst✝¹ : Bifunctor F\ninst✝ : LawfulBifunctor F\nα : Type u₀\n⊢ ∀ {α_1 β : Type u₁}, mapConst = map ∘ Function.const β",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Function.comp",
"True",
"Bifunctor.snd",
"eq_self",
... | [] | simp [mapConst, Functor.map] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic | {
"line": 778,
"column": 75
} | {
"line": 778,
"column": 79
} | {
"line": 778,
"column": 80
} | [
{
"pp": "case uniq.h₀\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y Z : C\nf f' : X ⟶ Y\ng g' : Y ⟶ Z\nH : IsPullback f f g g'\ns : Fork g g'\nm : s.pt ⟶ (Fork.ofι f ⋯).pt\ne : m ≫ (Fork.ofι f ⋯).ι = s.ι\n⊢ s.ι = H.isLimit.lift (PullbackCone.mk s.ι s.ι ⋯) ≫ H.cone.fst",
"ppTerm": "?uniq.h₀",
"assigned"... | [
"case uniq.h₀\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y Z : C\nf f' : X ⟶ Y\ng g' : Y ⟶ Z\nH : IsPullback f f g g'\ns : Fork g g'\nm : s.pt ⟶ (Fork.ofι f ⋯).pt\ne : m ≫ (Fork.ofι f ⋯).ι = s.ι\n⊢ H.isLimit.lift (PullbackCone.mk s.ι s.ι ⋯) ≫ H.cone.fst = s.ι"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic | {
"line": 778,
"column": 75
} | {
"line": 778,
"column": 79
} | {
"line": 778,
"column": 80
} | [
{
"pp": "case uniq.h₁\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y Z : C\nf f' : X ⟶ Y\ng g' : Y ⟶ Z\nH : IsPullback f f g g'\ns : Fork g g'\nm : s.pt ⟶ (Fork.ofι f ⋯).pt\ne : m ≫ (Fork.ofι f ⋯).ι = s.ι\n⊢ s.ι = H.isLimit.lift (PullbackCone.mk s.ι s.ι ⋯) ≫ H.cone.snd",
"ppTerm": "?uniq.h₁",
"assigned"... | [
"case uniq.h₁\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y Z : C\nf f' : X ⟶ Y\ng g' : Y ⟶ Z\nH : IsPullback f f g g'\ns : Fork g g'\nm : s.pt ⟶ (Fork.ofι f ⋯).pt\ne : m ≫ (Fork.ofι f ⋯).ι = s.ι\n⊢ H.isLimit.lift (PullbackCone.mk s.ι s.ι ⋯) ≫ H.cone.snd = s.ι"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic | {
"line": 789,
"column": 80
} | {
"line": 789,
"column": 84
} | {
"line": 789,
"column": 85
} | [
{
"pp": "case uniq.h₀\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y Z : C\nf f' : X ⟶ Y\ng g' : Y ⟶ Z\nH : IsPushout f f' g g\ns : Cofork f f'\nm : (Cofork.ofπ g ⋯).pt ⟶ s.pt\ne : (Cofork.ofπ g ⋯).π ≫ m = s.π\n⊢ s.π = H.cocone.inl ≫ H.isColimit.desc (PushoutCocone.mk s.π s.π ⋯)",
"ppTerm": "?uniq.h₀",
... | [
"case uniq.h₀\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y Z : C\nf f' : X ⟶ Y\ng g' : Y ⟶ Z\nH : IsPushout f f' g g\ns : Cofork f f'\nm : (Cofork.ofπ g ⋯).pt ⟶ s.pt\ne : (Cofork.ofπ g ⋯).π ≫ m = s.π\n⊢ H.cocone.inl ≫ H.isColimit.desc (PushoutCocone.mk s.π s.π ⋯) = s.π"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.RingTheory.FreeCommRing | {
"line": 250,
"column": 6
} | {
"line": 250,
"column": 12
} | {
"line": 251,
"column": 6
} | [
{
"pp": "case refine_2\nα : Type u\np : α\ns : Set α\nhps : (of p).IsSupported s\nthis : DecidablePred fun x ↦ x ∈ s\nx : FreeCommRing α\nhx : x.IsSupported s\n⊢ ∃ n, (lift fun a ↦ if a ∈ s then 0 else X) (-1) = ↑n",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Int.cast",
... | [
"case h\nα : Type u\np : α\ns : Set α\nhps : (of p).IsSupported s\nthis : DecidablePred fun x ↦ x ∈ s\nx : FreeCommRing α\nhx : x.IsSupported s\n⊢ (lift fun a ↦ if a ∈ s then 0 else X) (-1) = ↑(-1)"
] | use -1 | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic | {
"line": 789,
"column": 80
} | {
"line": 789,
"column": 84
} | {
"line": 789,
"column": 85
} | [
{
"pp": "case uniq.h₁\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y Z : C\nf f' : X ⟶ Y\ng g' : Y ⟶ Z\nH : IsPushout f f' g g\ns : Cofork f f'\nm : (Cofork.ofπ g ⋯).pt ⟶ s.pt\ne : (Cofork.ofπ g ⋯).π ≫ m = s.π\n⊢ s.π = H.cocone.inr ≫ H.isColimit.desc (PushoutCocone.mk s.π s.π ⋯)",
"ppTerm": "?uniq.h₁",
... | [
"case uniq.h₁\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y Z : C\nf f' : X ⟶ Y\ng g' : Y ⟶ Z\nH : IsPushout f f' g g\ns : Cofork f f'\nm : (Cofork.ofπ g ⋯).pt ⟶ s.pt\ne : (Cofork.ofπ g ⋯).π ≫ m = s.π\n⊢ H.cocone.inr ≫ H.isColimit.desc (PushoutCocone.mk s.π s.π ⋯) = s.π"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.RingTheory.Localization.BaseChange | {
"line": 176,
"column": 2
} | {
"line": 176,
"column": 23
} | {
"line": 177,
"column": 2
} | [
{
"pp": "M : Type u_7\nN : Type u_8\ninst✝³ : Monoid M\ninst✝² : Monoid N\nF : Type u_9\ninst✝¹ : FunLike F M N\ninst✝ : MonoidHomClass F M N\nf : F\n⊢ map f (IsUnit.submonoid M) ≤ IsUnit.submonoid N",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Monoid.toMulOneClass",
"Membe... | [
"M : Type u_7\nN : Type u_8\ninst✝³ : Monoid M\ninst✝² : Monoid N\nF : Type u_9\ninst✝¹ : FunLike F M N\ninst✝ : MonoidHomClass F M N\nf : F\ny : M\nhy : y ∈ ↑(IsUnit.submonoid M)\n⊢ f y ∈ IsUnit.submonoid N"
] | rintro x ⟨y, hy, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.RingTheory.Localization.BaseChange | {
"line": 182,
"column": 2
} | {
"line": 182,
"column": 23
} | {
"line": 183,
"column": 2
} | [
{
"pp": "R : Type u_7\nS : Type u_8\ninst✝² : CommSemiring R\ninst✝¹ : Semiring S\ninst✝ : Algebra R S\n⊢ algebraMapSubmonoid S (IsUnit.submonoid R) ≤ IsUnit.submonoid S",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Algebra.algebraMap",
"CommSemiring.toSemiring",
"Ring... | [
"R : Type u_7\nS : Type u_8\ninst✝² : CommSemiring R\ninst✝¹ : Semiring S\ninst✝ : Algebra R S\ny : R\nhy : y ∈ ↑(IsUnit.submonoid R)\n⊢ (algebraMap R S) y ∈ IsUnit.submonoid S"
] | rintro x ⟨y, hy, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.RingTheory.FreeCommRing | {
"line": 353,
"column": 15
} | {
"line": 353,
"column": 21
} | {
"line": 353,
"column": 21
} | [
{
"pp": "case neg_one\nα : Type u\n⊢ ∃ a, ↑a = -1",
"ppTerm": "?neg_one",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"FreeCommRing",
"AddGroupWithOne.toAddMonoidWithOne",
"instCommRingFreeCommRing",
"instRingFreeRing",
"SubtractionMonoid.toSubNegZe... | [
"case h\nα : Type u\n⊢ ↑(-1) = -1"
] | use -1 | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.RingTheory.Localization.BaseChange | {
"line": 246,
"column": 2
} | {
"line": 246,
"column": 26
} | {
"line": 247,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝²¹ : CommSemiring R\nS✝ : Submonoid R\nA : Type u_2\ninst✝²⁰ : CommSemiring A\ninst✝¹⁹ : Algebra R A\ninst✝¹⁸ : IsLocalization S✝ A\nM : Type u_3\ninst✝¹⁷ : AddCommMonoid M\ninst✝¹⁶ : Module R M\nM' : Type u_4\ninst✝¹⁵ : AddCommMonoid M'\ninst✝¹⁴ : Module R M'\ninst✝¹³ : Module A M'\... | [
"R : Type u_1\ninst✝²¹ : CommSemiring R\nS✝ : Submonoid R\nA : Type u_2\ninst✝²⁰ : CommSemiring A\ninst✝¹⁹ : Algebra R A\ninst✝¹⁸ : IsLocalization S✝ A\nM : Type u_3\ninst✝¹⁷ : AddCommMonoid M\ninst✝¹⁶ : Module R M\nM' : Type u_4\ninst✝¹⁵ : AddCommMonoid M'\ninst✝¹⁴ : Module R M'\ninst✝¹³ : Module A M'\ninst✝¹² : I... | let Aₚ := Localization S | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Algebra.Module.LocalizedModule.Basic | {
"line": 634,
"column": 4
} | {
"line": 634,
"column": 89
} | {
"line": 635,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝¹¹ : CommSemiring R\nS : Submonoid R\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : AddCommMonoid M'\ninst✝⁸ : AddCommMonoid M''\nA : Type u_5\ninst✝⁷ : CommSemiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : Module A M'\ninst✝⁴ : IsLocalization S A\ninst✝³ ... | [
"R : Type u_1\ninst✝¹¹ : CommSemiring R\nS : Submonoid R\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : AddCommMonoid M'\ninst✝⁸ : AddCommMonoid M''\nA : Type u_5\ninst✝⁷ : CommSemiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : Module A M'\ninst✝⁴ : IsLocalization S A\ninst✝³ : Module R M... | have : Function.Injective (h c).unit.inv := ((Module.End.isUnit_iff _).1 (by simp)).1 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Module.LocalizedModule.Basic | {
"line": 638,
"column": 4
} | {
"line": 638,
"column": 8
} | {
"line": 639,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝¹¹ : CommSemiring R\nS : Submonoid R\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : AddCommMonoid M'\ninst✝⁸ : AddCommMonoid M''\nA : Type u_5\ninst✝⁷ : CommSemiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : Module A M'\ninst✝⁴ : IsLocalization S A\ninst✝³ ... | [
"R : Type u_1\ninst✝¹¹ : CommSemiring R\nS : Submonoid R\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : AddCommMonoid M'\ninst✝⁸ : AddCommMonoid M''\nA : Type u_5\ninst✝⁷ : CommSemiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : Module A M'\ninst✝⁴ : IsLocalization S A\ninst✝³ : Module R M... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Algebra.Category.Ring.Constructions | {
"line": 167,
"column": 70
} | {
"line": 171,
"column": 93
} | {
"line": 173,
"column": 0
} | [
{
"pp": "R S Rₘ Sₘ : Type u\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing Rₘ\ninst✝⁵ : Algebra R Rₘ\ninst✝⁴ : CommRing S\ninst✝³ : CommRing Sₘ\ninst✝² : Algebra S Sₘ\nf : R →+* S\nfₘ : Rₘ →+* Sₘ\nH : fₘ.comp (algebraMap R Rₘ) = (algebraMap S Sₘ).comp f\nM : Submonoid R\ninst✝¹ : IsLocalization M Rₘ\ninst✝ : IsLocaliz... | [] | by
algebraize [f, fₘ, fₘ.comp (algebraMap R Rₘ)]
have : IsScalarTower R S Sₘ := .of_algebraMap_eq' H
have : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ := ‹_›
exact CommRingCat.isPushout_iff_isPushout.mpr (Algebra.isPushout_of_isLocalization M _ _ _) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.LocalizedModule.Basic | {
"line": 657,
"column": 18
} | {
"line": 657,
"column": 22
} | {
"line": 658,
"column": 6
} | [
{
"pp": "case e_a\nR : Type u_1\ninst✝⁴ : CommSemiring R\nS : Submonoid R\nM : Type u_2\nM'' : Type u_4\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M''\ninst✝¹ : Module R M\ninst✝ : Module R M''\ng : M →ₗ[R] M''\nh : ∀ (x : ↥S), IsUnit ((algebraMap R (End R M'')) ↑x)\nx y : LocalizedModule S M\na a' : M\n... | [
"case e_a\nR : Type u_1\ninst✝⁴ : CommSemiring R\nS : Submonoid R\nM : Type u_2\nM'' : Type u_4\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M''\ninst✝¹ : Module R M\ninst✝ : Module R M''\ng : M →ₗ[R] M''\nh : ∀ (x : ↥S), IsUnit ((algebraMap R (End R M'')) ↑x)\nx y : LocalizedModule S M\na a' : M\nb b' : ↥S\n⊢... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Algebra.Module.LocalizedModule.Basic | {
"line": 657,
"column": 18
} | {
"line": 657,
"column": 22
} | {
"line": 658,
"column": 6
} | [
{
"pp": "case e_a\nR : Type u_1\ninst✝⁴ : CommSemiring R\nS : Submonoid R\nM : Type u_2\nM'' : Type u_4\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M''\ninst✝¹ : Module R M\ninst✝ : Module R M''\ng : M →ₗ[R] M''\nh : ∀ (x : ↥S), IsUnit ((algebraMap R (End R M'')) ↑x)\nx y : LocalizedModule S M\na a' : M\n... | [
"case e_a\nR : Type u_1\ninst✝⁴ : CommSemiring R\nS : Submonoid R\nM : Type u_2\nM'' : Type u_4\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M''\ninst✝¹ : Module R M\ninst✝ : Module R M''\ng : M →ₗ[R] M''\nh : ∀ (x : ↥S), IsUnit ((algebraMap R (End R M'')) ↑x)\nx y : LocalizedModule S M\na a' : M\nb b' : ↥S\n⊢... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Algebra.Category.Ring.Constructions | {
"line": 253,
"column": 4
} | {
"line": 253,
"column": 52
} | {
"line": 254,
"column": 4
} | [
{
"pp": "case refine_2\nX : CommRingCat\nf : of PUnit.{u + 1} ⟶ X\nx : ↑X\ne : 0 = 1\n⊢ (Hom.hom (ofHom { toMonoidHom := 1, map_zero' := ⋯, map_add' := ⋯ } ≫ f)) x = (Hom.hom (𝟙 X)) x",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"CommRingCat.carrier",
... | [
"case refine_2\nX : CommRingCat\nf : of PUnit.{u + 1} ⟶ X\nx : ↑X\ne : 0 * x = 1 * x\n⊢ (Hom.hom (ofHom { toMonoidHom := 1, map_zero' := ⋯, map_add' := ⋯ } ≫ f)) x = (Hom.hom (𝟙 X)) x"
] | replace e : 0 * x = 1 * x := congr_arg (· * x) e | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.Algebra.Module.LocalizedModule.Basic | {
"line": 900,
"column": 2
} | {
"line": 900,
"column": 57
} | {
"line": 901,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type u_2\nM' : Type u_3\ninst✝⁴ : AddCommMonoid M\ninst✝³ : AddCommMonoid M'\ninst✝² : Module R M\ninst✝¹ : Module R M'\nf : M →ₗ[R] M'\ninst✝ : IsLocalizedModule S f\nm : M'\n⊢ (iso S f).symm m = LocalizedModule.mk ⋯.choose.1 ⋯.choose.2",
... | [
"R : Type u_1\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type u_2\nM' : Type u_3\ninst✝⁴ : AddCommMonoid M\ninst✝³ : AddCommMonoid M'\ninst✝² : Module R M\ninst✝¹ : Module R M'\nf : M →ₗ[R] M'\ninst✝ : IsLocalizedModule S f\nm : M'\n⊢ (iso S f) ((iso S f).symm m) = (iso S f) (LocalizedModule.mk ⋯.choose.1 ⋯.cho... | apply_fun iso S f using LinearEquiv.injective (iso S f) | Mathlib.Tactic._aux_Mathlib_Tactic_ApplyFun___elabRules_Mathlib_Tactic_applyFun_1 | Mathlib.Tactic.applyFun |
Mathlib.Algebra.Category.Ring.Constructions | {
"line": 407,
"column": 4
} | {
"line": 407,
"column": 14
} | {
"line": 408,
"column": 4
} | [
{
"pp": "case property.right\nA B : CommRingCat\nf g : A ⟶ B\ns : Fork f g\n⊢ ∀ {m : s.pt ⟶ (equalizerFork f g).pt},\n m ≫ (equalizerFork f g).ι = s.ι → m = ofHom ((Hom.hom s.ι).codRestrict ((Hom.hom f).eqLocus (Hom.hom g)) ⋯)",
"ppTerm": "?property.right",
"assigned": true,
"usedConstants": [
... | [
"case property.right\nA B : CommRingCat\nf g : A ⟶ B\ns : Fork f g\nm : s.pt ⟶ (equalizerFork f g).pt\nhm : m ≫ (equalizerFork f g).ι = s.ι\n⊢ m = ofHom ((Hom.hom s.ι).codRestrict ((Hom.hom f).eqLocus (Hom.hom g)) ⋯)"
] | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Algebra.Module.LocalizedModule.Basic | {
"line": 1103,
"column": 2
} | {
"line": 1103,
"column": 6
} | {
"line": 1104,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type u_6\nM' : Type u_7\ninst✝⁴ : NonUnitalNonAssocSemiring M\ninst✝³ : Semiring M'\ninst✝² : Module R M\ninst✝¹ : Algebra R M'\nf : M →ₗ[R] M'\nhf : ∀ (m₁ m₂ : M), f (m₁ * m₂) = f m₁ * f m₂\ninst✝ : IsLocalizedModule S f\nm₁ m₂ : M\ns₁ s₂ : ↥... | [
"R : Type u_1\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type u_6\nM' : Type u_7\ninst✝⁴ : NonUnitalNonAssocSemiring M\ninst✝³ : Semiring M'\ninst✝² : Module R M\ninst✝¹ : Algebra R M'\nf : M →ₗ[R] M'\nhf : ∀ (m₁ m₂ : M), f (m₁ * m₂) = f m₁ * f m₂\ninst✝ : IsLocalizedModule S f\nm₁ m₂ : M\ns₁ s₂ : ↥S\n⊢ mk' f (... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Algebra.Module.LocalizedModule.Basic | {
"line": 1373,
"column": 2
} | {
"line": 1373,
"column": 47
} | {
"line": 1375,
"column": 0
} | [
{
"pp": "R : Type u_6\nM : Type u_8\nM' : Type u_9\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M'\nS : Submonoid R\ng : M →ₗ[R] M'\ninst✝ : IsLocalizedModule S g\nm : M\n⊢ m ∈ g.ker ↔ ∃ r ∈ S, r • m = 0",
"ppTerm": "?m.161",
"assi... | [] | simpa using IsLocalizedModule.eq_zero_iff S g | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Module.LocalizedModule.Basic | {
"line": 1373,
"column": 2
} | {
"line": 1373,
"column": 47
} | {
"line": 1375,
"column": 0
} | [
{
"pp": "R : Type u_6\nM : Type u_8\nM' : Type u_9\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M'\nS : Submonoid R\ng : M →ₗ[R] M'\ninst✝ : IsLocalizedModule S g\nm : M\n⊢ m ∈ g.ker ↔ ∃ r ∈ S, r • m = 0",
"ppTerm": "?m.161",
"assi... | [] | simpa using IsLocalizedModule.eq_zero_iff S g | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.LocalizedModule.Basic | {
"line": 1373,
"column": 2
} | {
"line": 1373,
"column": 47
} | {
"line": 1375,
"column": 0
} | [
{
"pp": "R : Type u_6\nM : Type u_8\nM' : Type u_9\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M'\nS : Submonoid R\ng : M →ₗ[R] M'\ninst✝ : IsLocalizedModule S g\nm : M\n⊢ m ∈ g.ker ↔ ∃ r ∈ S, r • m = 0",
"ppTerm": "?m.161",
"assi... | [] | simpa using IsLocalizedModule.eq_zero_iff S g | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.FinitePresentation | {
"line": 219,
"column": 39
} | {
"line": 219,
"column": 50
} | {
"line": 219,
"column": 51
} | [
{
"pp": "case refine_1\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R A\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : Algebra A B\ninst✝² : IsScalarTower R A B\ninst✝¹ : FinitePresentation R B\ninst✝ : FiniteType R A\nn : ℕ\nf : MvPolynomial (Fin n) R... | [
"case refine_1\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R A\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : Algebra A B\ninst✝² : IsScalarTower R A B\ninst✝¹ : FinitePresentation R B\ninst✝ : FiniteType R A\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] B\nhf... | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.MorphismProperty.Comma | {
"line": 564,
"column": 2
} | {
"line": 566,
"column": 12
} | {
"line": 568,
"column": 0
} | [
{
"pp": "T : Type u_1\ninst✝² : Category.{v_1, u_1} T\nP Q W : MorphismProperty T\ninst✝¹ : Q.IsMultiplicative\ninst✝ : W.IsMultiplicative\nA B : P.Arrow Q W\nf g : A ⟶ B\nhl : f.left = g.left\nhr : f.right = g.right\n⊢ f = g",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"CategoryT... | [] | ext
· exact hl
· exact hr | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.MorphismProperty.Comma | {
"line": 564,
"column": 2
} | {
"line": 566,
"column": 12
} | {
"line": 568,
"column": 0
} | [
{
"pp": "T : Type u_1\ninst✝² : Category.{v_1, u_1} T\nP Q W : MorphismProperty T\ninst✝¹ : Q.IsMultiplicative\ninst✝ : W.IsMultiplicative\nA B : P.Arrow Q W\nf g : A ⟶ B\nhl : f.left = g.left\nhr : f.right = g.right\n⊢ f = g",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"CategoryT... | [] | ext
· exact hl
· exact hr | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.WithTerminal.Basic | {
"line": 225,
"column": 6
} | {
"line": 225,
"column": 27
} | {
"line": 226,
"column": 4
} | [
{
"pp": "case of\nC : Type u\ninst✝ : Category.{v, u} C\na✝¹ b✝ : Cat\nf✝ : a✝¹ ⟶ b✝\na✝ : ↑a✝¹\n⊢ (prelaxfunctor.map₂ (Bicategory.rightUnitor f✝).hom).toNatTrans.app (of a✝) =\n ((Cat.Hom.isoMk (mapComp f✝.toFunctor (𝟙 b✝).toFunctor)).hom ≫\n Bicategory.whiskerLeft (prelaxfunctor.map f✝) (Cat.Ho... | [] | simpa using! (refl _) | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.CategoryTheory.WithTerminal.Basic | {
"line": 225,
"column": 6
} | {
"line": 225,
"column": 27
} | {
"line": 226,
"column": 4
} | [
{
"pp": "case of\nC : Type u\ninst✝ : Category.{v, u} C\na✝¹ b✝ : Cat\nf✝ : a✝¹ ⟶ b✝\na✝ : ↑a✝¹\n⊢ (prelaxfunctor.map₂ (Bicategory.rightUnitor f✝).hom).toNatTrans.app (of a✝) =\n ((Cat.Hom.isoMk (mapComp f✝.toFunctor (𝟙 b✝).toFunctor)).hom ≫\n Bicategory.whiskerLeft (prelaxfunctor.map f✝) (Cat.Ho... | [] | simpa using! (refl _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.WithTerminal.Basic | {
"line": 225,
"column": 6
} | {
"line": 225,
"column": 27
} | {
"line": 226,
"column": 4
} | [
{
"pp": "case of\nC : Type u\ninst✝ : Category.{v, u} C\na✝¹ b✝ : Cat\nf✝ : a✝¹ ⟶ b✝\na✝ : ↑a✝¹\n⊢ (prelaxfunctor.map₂ (Bicategory.rightUnitor f✝).hom).toNatTrans.app (of a✝) =\n ((Cat.Hom.isoMk (mapComp f✝.toFunctor (𝟙 b✝).toFunctor)).hom ≫\n Bicategory.whiskerLeft (prelaxfunctor.map f✝) (Cat.Ho... | [] | simpa using! (refl _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Grothendieck | {
"line": 423,
"column": 8
} | {
"line": 423,
"column": 27
} | {
"line": 424,
"column": 8
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u₁\ninst✝ : Category.{v₁, u₁} D\nF : C ⥤ Cat\nG : C ⥤ Type w\n⊢ ∀ {X Y : G.Elements} (f : X ⟶ Y),\n (grothendieckTypeToCatInverse G ⋙ grothendieckTypeToCatFunctor G).map f ≫\n (Sigma.casesOn (motive := fun t ↦\n Y = t → ((grothendiec... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u₁\ninst✝ : Category.{v₁, u₁} D\nF : C ⥤ Cat\nG : C ⥤ Type w\nfst✝¹ : C\nsnd✝¹ : G.obj fst✝¹\nfst✝ : C\nsnd✝ : G.obj fst✝\nf : ⟨fst✝¹, snd✝¹⟩.fst ⟶ ⟨fst✝, snd✝⟩.fst\ne : (ConcreteCategory.hom (G.map f)) ⟨fst✝¹, snd✝¹⟩.snd = ⟨fst✝, snd✝⟩.snd\n⊢ (grothendieckTypeToCat... | rintro ⟨⟩ ⟨⟩ ⟨f, e⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.CategoryTheory.Limits.Connected | {
"line": 123,
"column": 2
} | {
"line": 124,
"column": 42
} | {
"line": 125,
"column": 2
} | [
{
"pp": "J : Type u₁\ninst✝³ : Category.{v₁, u₁} J\nC : Type u₂\ninst✝² : Category.{v₂, u₂} C\ninst✝¹ : IsConnected J\nF : J ⥤ C\ninst✝ : HasLimit F\nc : Cone F\nhc : IsLimit c\nj : J\n⊢ limMap c.π ≫ limit.π F j =\n ((limit.isLimit ((Functor.const J).obj c.pt)).conePointUniqueUpToIso (isLimitConstCone J c.pt... | [
"J : Type u₁\ninst✝³ : Category.{v₁, u₁} J\nC : Type u₂\ninst✝² : Category.{v₂, u₂} C\ninst✝¹ : IsConnected J\nF : J ⥤ C\ninst✝ : HasLimit F\nc : Cone F\nhc : IsLimit c\nj : J\n⊢ limit.π ((Functor.const J).obj c.pt) j ≫ c.π.app j =\n ((limit.isLimit ((Functor.const J).obj c.pt)).conePointUniqueUpToIso (isLimitCo... | simp only [limMap_π, limit.cone_x, Iso.trans_hom, assoc,
limit.conePointUniqueUpToIso_hom_comp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Limits.Final | {
"line": 239,
"column": 14
} | {
"line": 241,
"column": 20
} | {
"line": 242,
"column": 12
} | [
{
"pp": "case h₂\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝¹ : F.Final\nE : Type u₃\ninst✝ : Category.{v₃, u₃} E\nG : D ⥤ E\nc : Cocone (F ⋙ G)\nX Y : D\nf : X ⟶ Y\n⊢ ∀ (X₁ X₂ : C) (k₁ : X ⟶ F.obj X₁) (k₂ : X ⟶ F.obj X₂) (f_1 : X₁ ⟶ X₂),\n k₁ ≫ F.m... | [] | intro Z₁ Z₂ k₁ k₂ g a z
rw [← a, Functor.map_comp, Category.assoc, ← Functor.comp_map, c.w] at z
rw [z] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Final | {
"line": 239,
"column": 14
} | {
"line": 241,
"column": 20
} | {
"line": 242,
"column": 12
} | [
{
"pp": "case h₂\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝¹ : F.Final\nE : Type u₃\ninst✝ : Category.{v₃, u₃} E\nG : D ⥤ E\nc : Cocone (F ⋙ G)\nX Y : D\nf : X ⟶ Y\n⊢ ∀ (X₁ X₂ : C) (k₁ : X ⟶ F.obj X₁) (k₂ : X ⟶ F.obj X₂) (f_1 : X₁ ⟶ X₂),\n k₁ ≫ F.m... | [] | intro Z₁ Z₂ k₁ k₂ g a z
rw [← a, Functor.map_comp, Category.assoc, ← Functor.comp_map, c.w] at z
rw [z] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.WithTerminal.Basic | {
"line": 635,
"column": 6
} | {
"line": 635,
"column": 27
} | {
"line": 636,
"column": 4
} | [
{
"pp": "case of\nC : Type u\ninst✝ : Category.{v, u} C\na✝¹ b✝ : Cat\nf✝ : a✝¹ ⟶ b✝\na✝ : ↑a✝¹\n⊢ (prelaxfunctor.map₂ (Bicategory.rightUnitor f✝).hom).toNatTrans.app (of a✝) =\n ((Cat.Hom.isoMk (mapComp f✝.toFunctor (𝟙 b✝).toFunctor)).hom ≫\n Bicategory.whiskerLeft (prelaxfunctor.map f✝) (Cat.Ho... | [] | simpa using! (refl _) | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.CategoryTheory.WithTerminal.Basic | {
"line": 635,
"column": 6
} | {
"line": 635,
"column": 27
} | {
"line": 636,
"column": 4
} | [
{
"pp": "case of\nC : Type u\ninst✝ : Category.{v, u} C\na✝¹ b✝ : Cat\nf✝ : a✝¹ ⟶ b✝\na✝ : ↑a✝¹\n⊢ (prelaxfunctor.map₂ (Bicategory.rightUnitor f✝).hom).toNatTrans.app (of a✝) =\n ((Cat.Hom.isoMk (mapComp f✝.toFunctor (𝟙 b✝).toFunctor)).hom ≫\n Bicategory.whiskerLeft (prelaxfunctor.map f✝) (Cat.Ho... | [] | simpa using! (refl _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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