module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.CategoryTheory.Limits.Presheaf | {
"line": 801,
"column": 4
} | {
"line": 803,
"column": 50
} | {
"line": 805,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : LocallySmall.{w, v₁, u₁} C\nF : C ⥤ Type w\nu v : F.Elementsᵒᵖ\ng : u ⟶ v\n⊢ ((CategoryOfElements.π F).op ⋙ shrinkYoneda.{w, v₁, u₁}.flip).map g ≫\n { app := fun X ↦ (coconeπOpCompShrinkYonedaObj F X).ι.app v, naturality := ⋯ } =\n { app := fun... | [] | ext X x
obtain ⟨x, rfl⟩ := shrinkYonedaObjObjEquiv.symm.surjective x
simp [← shrinkYonedaObjObjEquiv_symm_comp.{w}] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.QuotientGroup.ModEq | {
"line": 28,
"column": 11
} | {
"line": 28,
"column": 34
} | {
"line": 28,
"column": 35
} | [
{
"pp": "G : Type u_1\ninst✝ : AddCommGroup G\na b p : G\n⊢ b ≡ a [PMOD p] ↔ ↑a = ↑b",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AddCommGroup.ModEq",
"instHSMul",
"_private.Mathlib.GroupTheory.QuotientGroup.ModEq.0.AddCommGroup.modEq_iff_eq_mod_zmulti... | [
"G : Type u_1\ninst✝ : AddCommGroup G\na b p : G\n⊢ (∃ z, a = b + z • p) ↔ ↑a = ↑b"
] | modEq_iff_eq_add_zsmul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Data.Nat.Cast.Field | {
"line": 34,
"column": 33
} | {
"line": 34,
"column": 43
} | {
"line": 35,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝ : DivisionSemiring K\nk : ℕ\nhn : ↑0 ≠ 0\n⊢ False",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"False",
"congrArg",
"False.elim",
"AddMonoid.toAddZeroClass",
"AddZeroClass.toAddZe... | [] | simp at hn | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Nat.Totient | {
"line": 119,
"column": 6
} | {
"line": 120,
"column": 61
} | {
"line": 121,
"column": 6
} | [
{
"pp": "m : ℕ\ninst✝¹ : NeZero (m + 1)\ninst✝ : Fintype (ZMod (m + 1))ˣ\n⊢ Fintype.card { x // x.val.Coprime (m + 1) } = φ (m + 1)",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.Coprime",
"Finset.univ",
"Finset.card_eq_sum_ones",
"Iff.of_eq",
... | [
"m : ℕ\ninst✝¹ : NeZero (m + 1)\ninst✝ : Fintype (ZMod (m + 1))ˣ\n⊢ (∑ a, if a.val.Coprime (m + 1) then 1 else 0) = ∑ i, if (↑i).Coprime (m + 1) then 1 else 0"
] | simp only [totient, Finset.card_eq_sum_ones, Fintype.card_subtype, Finset.sum_filter, ←
Fin.sum_univ_eq_sum_range, @Nat.coprime_comm (m + 1)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Nat.Totient | {
"line": 164,
"column": 2
} | {
"line": 172,
"column": 77
} | {
"line": 174,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ n.divisors.sum φ = n",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Nat.gcd",
"Iff.mpr",
"Nat.gcd_dvd_left",
"Eq.mpr",
"Dvd.dvd",
"instHDiv",
"congrArg",
"Finset",
"Nat.mem_divisors",
"AddMonoid.toAddZeroClass",... | [] | rcases n.eq_zero_or_pos with (rfl | hn)
· simp
rw [← sum_div_divisors n φ]
have : n = ∑ d ∈ n.divisors, #{k ∈ range n | n.gcd k = d} := by
nth_rw 1 [← card_range n]
refine card_eq_sum_card_fiberwise fun x _ => mem_divisors.2 ⟨?_, hn.ne'⟩
apply gcd_dvd_left
nth_rw 3 [this]
exact sum_congr rfl fun x... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Totient | {
"line": 164,
"column": 2
} | {
"line": 172,
"column": 77
} | {
"line": 174,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ n.divisors.sum φ = n",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Nat.gcd",
"Iff.mpr",
"Nat.gcd_dvd_left",
"Eq.mpr",
"Dvd.dvd",
"instHDiv",
"congrArg",
"Finset",
"Nat.mem_divisors",
"AddMonoid.toAddZeroClass",... | [] | rcases n.eq_zero_or_pos with (rfl | hn)
· simp
rw [← sum_div_divisors n φ]
have : n = ∑ d ∈ n.divisors, #{k ∈ range n | n.gcd k = d} := by
nth_rw 1 [← card_range n]
refine card_eq_sum_card_fiberwise fun x _ => mem_divisors.2 ⟨?_, hn.ne'⟩
apply gcd_dvd_left
nth_rw 3 [this]
exact sum_congr rfl fun x... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UniformSpace.Equicontinuity | {
"line": 582,
"column": 2
} | {
"line": 582,
"column": 54
} | {
"line": 583,
"column": 2
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nX' : Type u_4\nα : Type u_6\nuα : UniformSpace α\nt : κ → TopologicalSpace X'\nF : ι → X' → α\nx₀ : X'\nk : κ\nhk : EquicontinuousAt F x₀\n⊢ EquicontinuousAt F x₀",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"iInf",
"Equic... | [
"ι : Type u_1\nκ : Type u_2\nX' : Type u_4\nα : Type u_6\nuα : UniformSpace α\nt : κ → TopologicalSpace X'\nF : ι → X' → α\nx₀ : X'\nk : κ\nhk : EquicontinuousWithinAt F univ x₀\n⊢ EquicontinuousWithinAt F univ x₀"
] | rw [← equicontinuousWithinAt_univ (tX := _)] at hk ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.UniformSpace.Equicontinuity | {
"line": 774,
"column": 65
} | {
"line": 781,
"column": 44
} | {
"line": 783,
"column": 0
} | [
{
"pp": "X : Type u_3\nY : Type u_5\nα : Type u_6\ntX : TopologicalSpace X\ntY : TopologicalSpace Y\nuα : UniformSpace α\nA : Set Y\nu : Y → X → α\nS : Set X\nx₀ : X\nhA : EquicontinuousWithinAt (u ∘ Subtype.val) S x₀\nhu₁ : Continuous[tY, Pi.topologicalSpace] (S.restrict ∘ u)\nhu₂ : Continuous[tY, uα.toTopolog... | [] | by
intro U hU
rcases mem_uniformity_isClosed hU with ⟨V, hV, hVclosed, hVU⟩
filter_upwards [hA V hV, eventually_mem_nhdsWithin] with x hx hxS
rw [SetCoe.forall] at *
change A ⊆ (fun f => (u f x₀, u f x)) ⁻¹' V at hx
refine (closure_minimal hx <| hVclosed.preimage <| hu₂.prodMk ?_).trans (preimage_mono hVU)
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.ToIntervalMod | {
"line": 117,
"column": 90
} | {
"line": 118,
"column": 24
} | {
"line": 120,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : AddCommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedAddMonoid α\nhα : Archimedean α\np : α\nhp : 0 < p\na b : α\n⊢ toIocDiv hp a b • p - b = -toIocMod hp a b",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
... | [] | by
rw [toIocMod, neg_sub] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.ToIntervalMod | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 31
} | {
"line": 140,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : AddCommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedAddMonoid α\nhα : Archimedean α\np : α\nhp : 0 < p\na b : α\n⊢ toIcoMod hp a b + toIcoDiv hp a b • p = b",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"toIcoM... | [] | rw [toIcoMod, sub_add_cancel] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Order.ToIntervalMod | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 31
} | {
"line": 140,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : AddCommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedAddMonoid α\nhα : Archimedean α\np : α\nhp : 0 < p\na b : α\n⊢ toIcoMod hp a b + toIcoDiv hp a b • p = b",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"toIcoM... | [] | rw [toIcoMod, sub_add_cancel] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.ToIntervalMod | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 31
} | {
"line": 140,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : AddCommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedAddMonoid α\nhα : Archimedean α\np : α\nhp : 0 < p\na b : α\n⊢ toIcoMod hp a b + toIcoDiv hp a b • p = b",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"toIcoM... | [] | rw [toIcoMod, sub_add_cancel] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.ToIntervalMod | {
"line": 196,
"column": 2
} | {
"line": 196,
"column": 46
} | {
"line": 198,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : AddCommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedAddMonoid α\nhα : Archimedean α\np : α\nhp : 0 < p\na : α\n⊢ a < a + p ∧ ∃ z, a + p = a + z • p",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"lt_add_of_pos_right",
"instHSMul",
"Pr... | [] | exact ⟨lt_add_of_pos_right _ hp, 1, by simp⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Algebra.IsUniformGroup.Basic | {
"line": 467,
"column": 15
} | {
"line": 474,
"column": 50
} | {
"line": 476,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nhom : Type u_3\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : Group α\ninst✝⁴ : IsTopologicalGroup α\ninst✝³ : TopologicalSpace β\ninst✝² : Group β\ninst✝¹ : FunLike hom β α\ninst✝ : MonoidHomClass hom β α\ne : hom\nde : IsDenseInducing ⇑e\nx₀ : α\n⊢ Tendsto (fun t ↦ t.2 / t.1) (com... | [] | by
have comm : ((fun x : α × α => x.2 / x.1) ∘ fun t : β × β => (e t.1, e t.2)) =
e ∘ fun t : β × β => t.2 / t.1 := by
ext t
simp
have lim : Tendsto (fun x : α × α => x.2 / x.1) (𝓝 (x₀, x₀)) (𝓝 (e 1)) := by
simpa using! (continuous_div'.comp (@continuous_swap α α _ _)).tendsto (x₀, x₀)
simpa u... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.ToIntervalMod | {
"line": 1361,
"column": 2
} | {
"line": 1361,
"column": 47
} | {
"line": 1362,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝³ : Field α\ninst✝² : LinearOrder α\ninst✝¹ : IsStrictOrderedRing α\ninst✝ : FloorRing α\np : α\nhp : 0 < p\na b : α\n⊢ toIocDiv hp a b = -⌊(a + p - b) / p⌋",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"instHDiv",
"Int.floor",
"toIocDiv_eq_o... | [
"α : Type u_1\ninst✝³ : Field α\ninst✝² : LinearOrder α\ninst✝¹ : IsStrictOrderedRing α\ninst✝ : FloorRing α\np : α\nhp : 0 < p\na b : α\n⊢ b - -⌊(a + p - b) / p⌋ • p ∈ Set.Ioc a (a + p)"
] | refine toIocDiv_eq_of_sub_zsmul_mem_Ioc hp ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Algebra.Order.Field | {
"line": 71,
"column": 2
} | {
"line": 72,
"column": 17
} | {
"line": 74,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nα : Type u_2\ninst✝⁴ : Semifield 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nl : Filter α\nf : 𝕜 → α\nh : Tendsto (fun x ↦ f x⁻¹) atTop l\n⊢ Tendsto f (𝓝[>] 0) l",
"ppTerm": "?m.22",
"assigned": true,
... | [] | convert! h.comp tendsto_inv_nhdsGT_zero
grind [inv_inv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Order.Field | {
"line": 71,
"column": 2
} | {
"line": 72,
"column": 17
} | {
"line": 74,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nα : Type u_2\ninst✝⁴ : Semifield 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nl : Filter α\nf : 𝕜 → α\nh : Tendsto (fun x ↦ f x⁻¹) atTop l\n⊢ Tendsto f (𝓝[>] 0) l",
"ppTerm": "?m.22",
"assigned": true,
... | [] | convert! h.comp tendsto_inv_nhdsGT_zero
grind [inv_inv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.IsUniformGroup.Basic | {
"line": 727,
"column": 4
} | {
"line": 727,
"column": 59
} | {
"line": 728,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝³ : Group G\nus : UniformSpace G\ninst✝² : IsRightUniformGroup G\ninst✝¹ : FirstCountableTopology G\nN : Subgroup G\ninst✝ : N.Normal\nhG : CompleteSpace G\n⊢ 𝓤 G = 𝓤 G",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InvOneClass.toOne",
... | [
"G : Type u_1\ninst✝³ : Group G\nus : UniformSpace G\ninst✝² : IsRightUniformGroup G\ninst✝¹ : FirstCountableTopology G\nN : Subgroup G\ninst✝ : N.Normal\nhG : CompleteSpace G\n⊢ 𝓤 G = comap (fun x ↦ x.2 * x.1⁻¹) (𝓝 1)"
] | rw [@IsRightUniformGroup.uniformity_eq (G := G) us _ _] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Category.ModuleCat.Subobject | {
"line": 50,
"column": 8
} | {
"line": 59,
"column": 48
} | {
"line": 60,
"column": 6
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nM : ModuleCat R\nN : Submodule R ↑M\n⊢ (fun S ↦ (Hom.hom S.arrow).range) ((fun N ↦ mk (↟N.subtype)) N) = N",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"CategoryTheory.Subobject.arrow",
"RingHomSurjectiv... | [] | convert!
congr_arg LinearMap.range
(ModuleCat.hom_ext_iff.mp (underlyingIso_arrow (ofHom N.subtype))) using 1
· have :
(underlyingIso (ofHom N.subtype)).inv =
ofHom (underlyingIso (ofHom N.subtype)).symm.toLinearEquiv.toLinearMap := by
ext x
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.Subobject | {
"line": 50,
"column": 8
} | {
"line": 59,
"column": 48
} | {
"line": 60,
"column": 6
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nM : ModuleCat R\nN : Submodule R ↑M\n⊢ (fun S ↦ (Hom.hom S.arrow).range) ((fun N ↦ mk (↟N.subtype)) N) = N",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"CategoryTheory.Subobject.arrow",
"RingHomSurjectiv... | [] | convert!
congr_arg LinearMap.range
(ModuleCat.hom_ext_iff.mp (underlyingIso_arrow (ofHom N.subtype))) using 1
· have :
(underlyingIso (ofHom N.subtype)).inv =
ofHom (underlyingIso (ofHom N.subtype)).symm.toLinearEquiv.toLinearMap := by
ext x
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.ModuleCat.Adjunctions | {
"line": 175,
"column": 39
} | {
"line": 175,
"column": 54
} | {
"line": 176,
"column": 10
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nX Y Z : Type u\nz : Z\nx : X\ny : Y\n⊢ (Hom.hom ((free R).map (α_ X Y Z).hom)) (freeMk ((((x, y), z).1.1, ((x, y), z).1.2), ((x, y), z).2)) =\n (Hom.hom (μIso R X (Y ⊗ Z)).hom)\n ((Hom.hom ((free R).obj X ◁ (μIso R Y Z).hom))\n ((Hom.hom (α_ ((free R).obj ... | [
"R : Type u\ninst✝ : CommRing R\nX Y Z : Type u\nz : Z\nx : X\ny : Y\n⊢ freeMk ((ConcreteCategory.hom (α_ X Y Z).hom) ((((x, y), z).1.1, ((x, y), z).1.2), ((x, y), z).2)) =\n (Hom.hom (μIso R X (Y ⊗ Z)).hom)\n ((Hom.hom ((free R).obj X ◁ (μIso R Y Z).hom))\n ((Hom.hom (α_ ((free R).obj X) ((free R).o... | free_map_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Category.ModuleCat.Adjunctions | {
"line": 252,
"column": 4
} | {
"line": 252,
"column": 50
} | {
"line": 253,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nC : Type u\ninst✝ : Category.{v, u} C\nX Y Z : Free R C\nf f' : X ⟶ Y\ng : Y ⟶ Z\n⊢ (f + f') ≫ g = f ≫ g + f' ≫ g",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"AddCommGr... | [
"R : Type u_1\ninst✝¹ : CommRing R\nC : Type u\ninst✝ : Category.{v, u} C\nX Y Z : Free R C\nf f' : X ⟶ Y\ng : Y ⟶ Z\n⊢ ((f + f').sum fun f' s ↦ sum g fun g' t ↦ single (f' ≫ g') (s * t)) =\n (sum f fun f' s ↦ sum g fun g' t ↦ single (f' ≫ g') (s * t)) +\n sum f' fun f' s ↦ sum g fun g' t ↦ single (f' ≫ g')... | dsimp +instances [CategoryTheory.categoryFree] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.Algebra.Category.ModuleCat.Adjunctions | {
"line": 255,
"column": 4
} | {
"line": 255,
"column": 50
} | {
"line": 256,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nC : Type u\ninst✝ : Category.{v, u} C\nX Y Z : Free R C\nf : X ⟶ Y\ng g' : Y ⟶ Z\n⊢ f ≫ (g + g') = f ≫ g + f ≫ g'",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"AddCommGr... | [
"R : Type u_1\ninst✝¹ : CommRing R\nC : Type u\ninst✝ : Category.{v, u} C\nX Y Z : Free R C\nf : X ⟶ Y\ng g' : Y ⟶ Z\n⊢ (sum f fun f' s ↦ (g + g').sum fun g' t ↦ single (f' ≫ g') (s * t)) =\n (sum f fun f' s ↦ sum g fun g' t ↦ single (f' ≫ g') (s * t)) +\n sum f fun f' s ↦ sum g' fun g' t ↦ single (f' ≫ g')... | dsimp +instances [CategoryTheory.categoryFree] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.Algebra.Category.ModuleCat.Adjunctions | {
"line": 263,
"column": 4
} | {
"line": 263,
"column": 50
} | {
"line": 264,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nC : Type u\ninst✝ : Category.{v, u} C\nX Y Z : Free R C\nr : R\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ (r • f) ≫ g = r • f ≫ g",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"instHSMul",
"Semiring.toModule",
"Finsupp.module",
"CategoryT... | [
"R : Type u_1\ninst✝¹ : CommRing R\nC : Type u\ninst✝ : Category.{v, u} C\nX Y Z : Free R C\nr : R\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ ((r • f).sum fun f' s ↦ sum g fun g' t ↦ single (f' ≫ g') (s * t)) =\n r • sum f fun f' s ↦ sum g fun g' t ↦ single (f' ≫ g') (s * t)"
] | dsimp +instances [CategoryTheory.categoryFree] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.Algebra.Category.ModuleCat.Adjunctions | {
"line": 266,
"column": 4
} | {
"line": 266,
"column": 50
} | {
"line": 267,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nC : Type u\ninst✝ : Category.{v, u} C\nX Y Z : Free R C\nf : X ⟶ Y\nr : R\ng : Y ⟶ Z\n⊢ f ≫ (r • g) = r • f ≫ g",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"instHSMul",
"Semiring.toModule",
"Finsupp.module",
"CategoryT... | [
"R : Type u_1\ninst✝¹ : CommRing R\nC : Type u\ninst✝ : Category.{v, u} C\nX Y Z : Free R C\nf : X ⟶ Y\nr : R\ng : Y ⟶ Z\n⊢ (sum f fun f' s ↦ (r • g).sum fun g' t ↦ single (f' ≫ g') (s * t)) =\n r • sum f fun f' s ↦ sum g fun g' t ↦ single (f' ≫ g') (s * t)"
] | dsimp +instances [CategoryTheory.categoryFree] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.Algebra.Category.ModuleCat.Adjunctions | {
"line": 274,
"column": 2
} | {
"line": 274,
"column": 48
} | {
"line": 275,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nC : Type u\ninst✝ : Category.{v, u} C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nr s : R\n⊢ single f r ≫ single g s = single (f ≫ g) (r * s)",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"CategoryTheory.CategoryStruct.toQuiver",... | [
"R : Type u_1\ninst✝¹ : CommRing R\nC : Type u\ninst✝ : Category.{v, u} C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nr s : R\n⊢ ((single f r).sum fun f' s_1 ↦ (single g s).sum fun g' t ↦ single (f' ≫ g') (s_1 * t)) = single (f ≫ g) (r * s)"
] | dsimp +instances [CategoryTheory.categoryFree] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.Algebra.Category.ModuleCat.Adjunctions | {
"line": 305,
"column": 4
} | {
"line": 305,
"column": 50
} | {
"line": 306,
"column": 4
} | [
{
"pp": "R : 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\n⊢ ∀ (X : Free R C), (sum (𝟙 X) fun f' r ↦ r • F.map f') = 𝟙 (F.obj X)",
"ppTerm": "?m.46",
"assigned": true,
"usedConst... | [
"R : 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\n⊢ ∀ (X : Free R C), ((single (𝟙 X) 1).sum fun f' r ↦ r • F.map f') = 𝟙 (F.obj X)"
] | dsimp +instances [CategoryTheory.categoryFree] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.CategoryTheory.Abelian.Refinements | {
"line": 223,
"column": 14
} | {
"line": 223,
"column": 16
} | {
"line": 223,
"column": 17
} | [
{
"pp": "case mp\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nh : ∀ ⦃A : C⦄ (y : A ⟶ S₂.homology), ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ y = x ≫ homologyMap φ\nA : C\n⊢ ∀ (y₂ : A ⟶ S₂.X₂),\n y₂ ≫ S₂.g = 0 → ∃ A' π, ∃ (_ : Epi π), ∃ x₂, ∃ (_ : x₂ ≫ S₁.g = 0)... | [
"case mp\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nh : ∀ ⦃A : C⦄ (y : A ⟶ S₂.homology), ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ y = x ≫ homologyMap φ\nA : C\ny₂ : A ⟶ S₂.X₂\n⊢ y₂ ≫ S₂.g = 0 → ∃ A' π, ∃ (_ : Epi π), ∃ x₂, ∃ (_ : x₂ ≫ S₁.g = 0), ∃ y₁, π ≫ y₂ = x₂ ≫... | y₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Abelian.Refinements | {
"line": 238,
"column": 42
} | {
"line": 238,
"column": 72
} | {
"line": 238,
"column": 72
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nh :\n ∀ ⦃A : C⦄ (y₂ : A ⟶ S₂.X₂),\n y₂ ≫ S₂.g = 0 → ∃ A' π, ∃ (_ : Epi π), ∃ x₂, ∃ (_ : x₂ ≫ S₁.g = 0), ∃ y₁, π ≫ y₂ = x₂ ≫ φ.τ₂ + y₁ ≫ S₂.f\nA : C\nγ : A ⟶ S₂.homology\nA₁ : C\nπ₁ : A₁ ⟶ A\nhπ₁ : ... | [] | simpa only [id_comp] using hy₁ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.CategoryTheory.Abelian.Refinements | {
"line": 238,
"column": 42
} | {
"line": 238,
"column": 72
} | {
"line": 238,
"column": 72
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nh :\n ∀ ⦃A : C⦄ (y₂ : A ⟶ S₂.X₂),\n y₂ ≫ S₂.g = 0 → ∃ A' π, ∃ (_ : Epi π), ∃ x₂, ∃ (_ : x₂ ≫ S₁.g = 0), ∃ y₁, π ≫ y₂ = x₂ ≫ φ.τ₂ + y₁ ≫ S₂.f\nA : C\nγ : A ⟶ S₂.homology\nA₁ : C\nπ₁ : A₁ ⟶ A\nhπ₁ : ... | [] | simpa only [id_comp] using hy₁ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Abelian.Refinements | {
"line": 238,
"column": 42
} | {
"line": 238,
"column": 72
} | {
"line": 238,
"column": 72
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nh :\n ∀ ⦃A : C⦄ (y₂ : A ⟶ S₂.X₂),\n y₂ ≫ S₂.g = 0 → ∃ A' π, ∃ (_ : Epi π), ∃ x₂, ∃ (_ : x₂ ≫ S₁.g = 0), ∃ y₁, π ≫ y₂ = x₂ ≫ φ.τ₂ + y₁ ≫ S₂.f\nA : C\nγ : A ⟶ S₂.homology\nA₁ : C\nπ₁ : A₁ ⟶ A\nhπ₁ : ... | [] | simpa only [id_comp] using hy₁ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.SnakeLemma | {
"line": 207,
"column": 4
} | {
"line": 207,
"column": 18
} | {
"line": 208,
"column": 2
} | [
{
"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.L₂.g ≫ S.v₂₃.τ₃)",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"CategoryTheory.ShortComplex.SnakeInput.L₃",
... | [] | apply epi_comp | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer | {
"line": 1310,
"column": 6
} | {
"line": 1310,
"column": 25
} | {
"line": 1311,
"column": 4
} | [
{
"pp": "case inr.inl\nC : Type u\ninst✝¹ : Category.{v, u} C\nι : Type w\ninst✝ : LinearOrder ι\nI : MultispanIndex (MultispanShape.prod ι) C\nc : Multicofork I.toLinearOrder\nh : I.SymmStruct\nx : ι\n⊢ I.fst (x, x) ≫ c.π ((MultispanShape.prod ι).fst (x, x)) = I.snd (x, x) ≫ c.π ((MultispanShape.prod ι).snd (x... | [] | simp [h.fst_eq_snd] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer | {
"line": 1310,
"column": 6
} | {
"line": 1310,
"column": 25
} | {
"line": 1311,
"column": 4
} | [
{
"pp": "case inr.inl\nC : Type u\ninst✝¹ : Category.{v, u} C\nι : Type w\ninst✝ : LinearOrder ι\nI : MultispanIndex (MultispanShape.prod ι) C\nc : Multicofork I.toLinearOrder\nh : I.SymmStruct\nx : ι\n⊢ I.fst (x, x) ≫ c.π ((MultispanShape.prod ι).fst (x, x)) = I.snd (x, x) ≫ c.π ((MultispanShape.prod ι).snd (x... | [] | simp [h.fst_eq_snd] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer | {
"line": 1310,
"column": 6
} | {
"line": 1310,
"column": 25
} | {
"line": 1311,
"column": 4
} | [
{
"pp": "case inr.inl\nC : Type u\ninst✝¹ : Category.{v, u} C\nι : Type w\ninst✝ : LinearOrder ι\nI : MultispanIndex (MultispanShape.prod ι) C\nc : Multicofork I.toLinearOrder\nh : I.SymmStruct\nx : ι\n⊢ I.fst (x, x) ≫ c.π ((MultispanShape.prod ι).fst (x, x)) = I.snd (x, x) ≫ c.π ((MultispanShape.prod ι).snd (x... | [] | simp [h.fst_eq_snd] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monad.Adjunction | {
"line": 335,
"column": 8
} | {
"line": 335,
"column": 54
} | {
"line": 336,
"column": 8
} | [
{
"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⊢ X.a ≫ (reflectorAdjunction R).unit.app X.A = 𝟙 ((reflector R ⋙ R).obj X.A)",
"ppTerm": "?m.62",
"assigned": true,
... | [
"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⊢ (reflectorAdjunction R).unit.app ((reflector R ⋙ R).obj X.A) ≫ R.map ((reflector R).map X.a) =\n 𝟙 ((reflector R ⋙ R).obj X.A)"
] | rw [← (reflectorAdjunction R).unit_naturality] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Monad.Limits | {
"line": 71,
"column": 48
} | {
"line": 71,
"column": 71
} | {
"line": 71,
"column": 72
} | [
{
"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\nj : J\n⊢ T.η.app c.pt ≫ T.map (c.π.app j) ≫ (D.obj j).a = 𝟙 c.pt ≫ c.π.app j",
"ppTerm": "?m.96",
"assigned": true,
"usedConstants": [
... | [
"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\nj : J\n⊢ (𝟭 C).map (c.π.app j) ≫ T.η.app (D.obj j).A ≫ (D.obj j).a = 𝟙 c.pt ≫ c.π.app j"
] | ← T.η.naturality_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Monad.Comonadicity | {
"line": 348,
"column": 2
} | {
"line": 350,
"column": 48
} | {
"line": 351,
"column": 2
} | [
{
"pp": "C : Type u₁\nD : Type u₂\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : Category.{v₁, u₂} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\ninst✝² : F.ReflectsIsomorphisms\ninst✝¹ : HasEqualizerOfIsCosplitPair F\ninst✝ : PreservesLimitOfIsCosplitPair F\n⊢ ComonadicLeftAdjoint F",
"ppTerm": "?m.25",
"assigned": tr... | [
"C : Type u₁\nD : Type u₂\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : Category.{v₁, u₂} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\ninst✝² : F.ReflectsIsomorphisms\ninst✝¹ : HasEqualizerOfIsCosplitPair F\ninst✝ : PreservesLimitOfIsCosplitPair F\nthis : ReflectsLimitOfIsCosplitPair F\n⊢ ComonadicLeftAdjoint F"
] | have : ReflectsLimitOfIsCosplitPair F := ⟨fun f g _ => by
have := HasEqualizerOfIsCosplitPair.out F f g
apply reflectsLimit_of_reflectsIsomorphisms⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.IsPrimary | {
"line": 70,
"column": 26
} | {
"line": 70,
"column": 34
} | {
"line": 70,
"column": 35
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nS T : Submodule R M\nh : (S.colon Set.univ).radical = (T.colon Set.univ).radical\nleft✝¹ : S ≠ ⊤\nleft✝ : T ≠ ⊤\nr✝ : R\nx✝¹ : M\nx✝ : r✝ • x✝¹ ∈ S ⊓ T\nhS' : r✝ • x✝¹ ∈ ↑S\nhT' : r✝ • x✝¹ ∈ ↑T\nhT : ∀ {r... | [
"R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nS T : Submodule R M\nh : (S.colon Set.univ).radical = (T.colon Set.univ).radical\nleft✝¹ : S ≠ ⊤\nleft✝ : T ≠ ⊤\nr✝ : R\nx✝¹ : M\nx✝ : r✝ • x✝¹ ∈ S ⊓ T\nhS' : r✝ • x✝¹ ∈ ↑S\nhT' : r✝ • x✝¹ ∈ ↑T\nhT : ∀ {r : R} {x : M... | mem_inf, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Module.LocalizedModule.Submodule | {
"line": 94,
"column": 4
} | {
"line": 95,
"column": 46
} | {
"line": 96,
"column": 4
} | [
{
"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 ... | rw [mem_comap, restrictScalars_mem, ← IsLocalizedModule.mk'_cancel' _ _ s,
Submonoid.smul_def, ← algebraMap_smul S] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Ideal.MinimalPrime.Basic | {
"line": 160,
"column": 69
} | {
"line": 164,
"column": 45
} | {
"line": 166,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\n⊢ ⊤.minimalPrimes = ∅",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"False",
"Semiring.toModule",
"iff_false",
"Ideal.minimalPrimes",
"congrArg",
"CommSemiring.toSemiring",
... | [] | by
ext p
simp only [Set.notMem_empty, iff_false]
intro h
exact h.isPrime.ne_top (top_le_iff.mp h.le) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Nilpotent.Lemmas | {
"line": 131,
"column": 2
} | {
"line": 133,
"column": 57
} | {
"line": 135,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf : M →ₗ[R] M\np : Submodule R M\nhf : MapsTo ⇑f ↑p ↑p\nhnil : IsNilpotent f\n⊢ IsNilpotent (f.restrict hf)",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Module.End.pow_apply_me... | [] | obtain ⟨n, hn⟩ := hnil
exact ⟨n, LinearMap.ext fun m ↦ by simp only [Module.End.pow_restrict n, hn,
LinearMap.restrict_apply, LinearMap.zero_apply]; rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Nilpotent.Lemmas | {
"line": 131,
"column": 2
} | {
"line": 133,
"column": 57
} | {
"line": 135,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf : M →ₗ[R] M\np : Submodule R M\nhf : MapsTo ⇑f ↑p ↑p\nhnil : IsNilpotent f\n⊢ IsNilpotent (f.restrict hf)",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Module.End.pow_apply_me... | [] | obtain ⟨n, hn⟩ := hnil
exact ⟨n, LinearMap.ext fun m ↦ by simp only [Module.End.pow_restrict n, hn,
LinearMap.restrict_apply, LinearMap.zero_apply]; rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Localization.Ideal | {
"line": 178,
"column": 4
} | {
"line": 181,
"column": 41
} | {
"line": 183,
"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\n⊢ ∀ {a b : Ideal S},\n { toFun := fun J ↦ Ideal.under R J, inj' := ⋯ } a ≤ { toFun := fun J ↦ Ideal.under R J, inj' := ⋯ } b ↔ a ≤ b",
"ppTerm": "?m.31... | [] | rintro J₁ J₂
constructor
· exact fun hJ => (map_under M S) J₁ ▸ (map_under M S) J₂ ▸ Ideal.map_mono hJ
· exact fun hJ => Ideal.comap_mono hJ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Localization.Ideal | {
"line": 178,
"column": 4
} | {
"line": 181,
"column": 41
} | {
"line": 183,
"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\n⊢ ∀ {a b : Ideal S},\n { toFun := fun J ↦ Ideal.under R J, inj' := ⋯ } a ≤ { toFun := fun J ↦ Ideal.under R J, inj' := ⋯ } b ↔ a ≤ b",
"ppTerm": "?m.31... | [] | rintro J₁ J₂
constructor
· exact fun hJ => (map_under M S) J₁ ▸ (map_under M S) J₂ ▸ Ideal.map_mono hJ
· exact fun hJ => Ideal.comap_mono hJ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Localization.LocalizationLocalization | {
"line": 273,
"column": 4
} | {
"line": 273,
"column": 12
} | {
"line": 274,
"column": 4
} | [
{
"pp": "case h\nR : Type u_1\ninst✝⁸ : CommRing R\nM : Submonoid R\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsFractionRing R T\nhM : M ≤ nonZeroDivi... | [
"case h\nR : Type u_1\ninst✝⁸ : CommRing R\nM : Submonoid R\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsFractionRing R T\nhM : M ≤ nonZeroDivisors R\nthis... | apply hx | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.TensorProduct.Quotient | {
"line": 57,
"column": 6
} | {
"line": 57,
"column": 60
} | {
"line": 58,
"column": 6
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nI : Ideal A\nu v : B\n⊢ (quotIdealMapEquivTensorQuotAux B I)\n ((Ideal.Quotient.mk (Ideal.map (algebraMap A B) I)) u * (Ideal.Quotient.mk (Ideal.map (algebraMap A B) I)) v) =\n (quotIdealMapEquivTensorQuo... | [
"A : Type u_1\nB : Type u_2\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nI : Ideal A\nu v : B\n⊢ (u * v) ⊗ₜ[A] 1 = u ⊗ₜ[A] 1 * v ⊗ₜ[A] 1"
] | simp_rw [← map_mul, quotIdealMapEquivTensorQuotAux_mk] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.RingTheory.Localization.Submodule | {
"line": 152,
"column": 6
} | {
"line": 152,
"column": 60
} | {
"line": 153,
"column": 6
} | [
{
"pp": "case mp.refine_4\nR : Type u_1\ninst✝⁷ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝⁶ : CommSemiring S\ninst✝⁵ : Algebra R S\ninst✝⁴ : IsLocalization M S\nN : Type u_3\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : Module S N\ninst✝ : IsScalarTower R S N\nx : N\na : Set N\nh : x ∈ Su... | [
"case mp.refine_4\nR : Type u_1\ninst✝⁷ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝⁶ : CommSemiring S\ninst✝⁵ : Algebra R S\ninst✝⁴ : IsLocalization M S\nN : Type u_3\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : Module S N\ninst✝ : IsScalarTower R S N\nx : N\na : Set N\nh : x ∈ Submodule.span... | refine ⟨y' • y, Submodule.smul_mem _ _ hy, z' * z, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Localization.Submodule | {
"line": 160,
"column": 2
} | {
"line": 168,
"column": 66
} | {
"line": 170,
"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\nx : S\na : Set R\n⊢ x ∈ Ideal.span (⇑(algebraMap R S) '' a) ↔ ∃ y ∈ Ideal.span a, ∃ z, x = mk' S y z",
"ppTerm": "?m.35",
"assigned": true,
"usedC... | [] | refine (mem_span_iff M).trans ?_
constructor
· rw [← coeSubmodule_span]
rintro ⟨_, ⟨y, hy, rfl⟩, z, hz⟩
refine ⟨y, hy, z, ?_⟩
rw [hz, Algebra.linearMap_apply, smul_eq_mul, mul_comm, mul_mk'_eq_mk'_of_mul, mul_one]
· rintro ⟨y, hy, z, hz⟩
refine ⟨algebraMap R S y, Submodule.map_mem_span_algebraMap_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Localization.Submodule | {
"line": 160,
"column": 2
} | {
"line": 168,
"column": 66
} | {
"line": 170,
"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\nx : S\na : Set R\n⊢ x ∈ Ideal.span (⇑(algebraMap R S) '' a) ↔ ∃ y ∈ Ideal.span a, ∃ z, x = mk' S y z",
"ppTerm": "?m.35",
"assigned": true,
"usedC... | [] | refine (mem_span_iff M).trans ?_
constructor
· rw [← coeSubmodule_span]
rintro ⟨_, ⟨y, hy, rfl⟩, z, hz⟩
refine ⟨y, hy, z, ?_⟩
rw [hz, Algebra.linearMap_apply, smul_eq_mul, mul_comm, mul_mk'_eq_mk'_of_mul, mul_one]
· rintro ⟨y, hy, z, hz⟩
refine ⟨algebraMap R S y, Submodule.map_mem_span_algebraMap_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.LocalProperties.Basic | {
"line": 393,
"column": 2
} | {
"line": 393,
"column": 17
} | {
"line": 394,
"column": 2
} | [
{
"pp": "P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nhP : OfLocalizationSpan fun {R S} [CommRing R] [CommRing S] ↦ P\nhPi : RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ P\nR S : Type u\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\ns : Set R\nhs : Ideal.span... | [
"P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nhP : OfLocalizationSpan fun {R S} [CommRing R] [CommRing S] ↦ P\nhPi : RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ P\nR S : Type u\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\ns : Set R\nhs : Ideal.span s = ⊤\nhT :... | apply hP _ s hs | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.LocalProperties.Basic | {
"line": 431,
"column": 2
} | {
"line": 431,
"column": 17
} | {
"line": 432,
"column": 2
} | [
{
"pp": "P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nhP : OfLocalizationSpanTarget fun {R S} [CommRing R] [CommRing S] ↦ P\nhP' : RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ P\nR S : Type u\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\ns : Set S\nhs : Idea... | [
"P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nhP : OfLocalizationSpanTarget fun {R S} [CommRing R] [CommRing S] ↦ P\nhP' : RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ P\nR S : Type u\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\ns : Set S\nhs : Ideal.span s = ⊤... | apply hP _ s hs | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.LinearAlgebra.Matrix.Symmetric | {
"line": 146,
"column": 2
} | {
"line": 146,
"column": 36
} | {
"line": 147,
"column": 2
} | [
{
"pp": "α : Type u_1\nn : Type u_3\nm : Type u_4\nA : Matrix n n α\nf : n ≃ m\n⊢ ((reindex f f) A).IsSymm ↔ A.IsSymm",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Equiv.instEquivLike",
"Matrix",
"Matrix.IsSymm.reindex",
"Equiv",
"Iff.intro",
"Matrix.... | [
"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"
] | refine ⟨fun h ↦ ?_, (·.reindex f)⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.Matrix.Polynomial | {
"line": 51,
"column": 6
} | {
"line": 53,
"column": 56
} | {
"line": 54,
"column": 4
} | [
{
"pp": "n : Type u_1\nα : Type u_2\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA B : Matrix n n α\ng : Equiv.Perm n\n⊢ (sign g • ∏ i, (X • A.map ⇑C + B.map ⇑C) (g i) i).natDegree ≤ (∏ i, (X • A.map ⇑C + B.map ⇑C) (g i) i).natDegree",
"ppTerm": "?m.153",
"assigned": true,
"usedC... | [] | rcases Int.units_eq_one_or (sign g) with sg | sg
· rw [sg, one_smul]
· rw [sg, Units.neg_smul, one_smul, natDegree_neg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.Polynomial | {
"line": 51,
"column": 6
} | {
"line": 53,
"column": 56
} | {
"line": 54,
"column": 4
} | [
{
"pp": "n : Type u_1\nα : Type u_2\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA B : Matrix n n α\ng : Equiv.Perm n\n⊢ (sign g • ∏ i, (X • A.map ⇑C + B.map ⇑C) (g i) i).natDegree ≤ (∏ i, (X • A.map ⇑C + B.map ⇑C) (g i) i).natDegree",
"ppTerm": "?m.153",
"assigned": true,
"usedC... | [] | rcases Int.units_eq_one_or (sign g) with sg | sg
· rw [sg, one_smul]
· rw [sg, Units.neg_smul, one_smul, natDegree_neg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Laurent | {
"line": 581,
"column": 81
} | {
"line": 582,
"column": 70
} | {
"line": 584,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommSemiring R\nS : Type u_3\ninst✝ : CommSemiring S\nf : R →+* S\nx : Sˣ\nr : R\nn : ℕ\n⊢ (eval₂ f x) (C r * T ↑n) = f r * ↑x ^ n",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"LaurentPolynomial.T",
"Units.val",
"Eq.mpr",
"Semiring... | [] | by
rw [← Polynomial.toLaurent_C_mul_T, eval₂_toLaurent, eval₂_monomial] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.Trace | {
"line": 287,
"column": 2
} | {
"line": 287,
"column": 55
} | {
"line": 289,
"column": 0
} | [
{
"pp": "m : Type u_2\nn : Type u_3\nR : Type u_6\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : NonAssocSemiring R\nA B : Matrix m n R\nh : ∀ (x : Matrix n m R), (A * x).trace = (B * x).trace\ni : m\nj : n\n⊢ A i j = B i j",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"NonAssocS... | [] | simpa [trace_mul_single] using h (single j i (1 : R)) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.Matrix.Trace | {
"line": 287,
"column": 2
} | {
"line": 287,
"column": 55
} | {
"line": 289,
"column": 0
} | [
{
"pp": "m : Type u_2\nn : Type u_3\nR : Type u_6\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : NonAssocSemiring R\nA B : Matrix m n R\nh : ∀ (x : Matrix n m R), (A * x).trace = (B * x).trace\ni : m\nj : n\n⊢ A i j = B i j",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"NonAssocS... | [] | simpa [trace_mul_single] using h (single j i (1 : R)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.Trace | {
"line": 287,
"column": 2
} | {
"line": 287,
"column": 55
} | {
"line": 289,
"column": 0
} | [
{
"pp": "m : Type u_2\nn : Type u_3\nR : Type u_6\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : NonAssocSemiring R\nA B : Matrix m n R\nh : ∀ (x : Matrix n m R), (A * x).trace = (B * x).trace\ni : m\nj : n\n⊢ A i j = B i j",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"NonAssocS... | [] | simpa [trace_mul_single] using h (single j i (1 : R)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.Adjugate | {
"line": 451,
"column": 4
} | {
"line": 451,
"column": 13
} | {
"line": 451,
"column": 14
} | [
{
"pp": "n : Type v\nα : Type w\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA B : Matrix n n α\nhA : IsLeftRegular A.det\nhB : IsLeftRegular B.det\nhAB : IsLeftRegular (A * B).det\n⊢ (A * B).det • 1 = A * (B.det • 1 * A.adjugate)",
"ppTerm": "?m.115",
"assigned": true,
"usedCons... | [
"n : Type v\nα : Type w\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA B : Matrix n n α\nhA : IsLeftRegular A.det\nhB : IsLeftRegular B.det\nhAB : IsLeftRegular (A * B).det\n⊢ (A * B).det • 1 = A * B.det • (1 * A.adjugate)"
] | smul_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.NonsingularInverse | {
"line": 277,
"column": 12
} | {
"line": 277,
"column": 57
} | {
"line": 277,
"column": 57
} | [
{
"pp": "m : Type u\nn : Type u'\nα : Type v\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : CommRing α\nA : Matrix n n α\ninst✝ : Invertible A\nB C : Matrix m n α\nh : B = C * A\n⊢ B * A⁻¹ = C",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"... | [] | by rw [h, mul_inv_cancel_right_of_invertible] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.NonsingularInverse | {
"line": 328,
"column": 6
} | {
"line": 328,
"column": 26
} | {
"line": 328,
"column": 27
} | [
{
"pp": "case intro\nm : Type u\nn : Type u'\nR : Type u_2\ninst✝³ : Semiring R\ninst✝² : DecidableEq n\ninst✝¹ : Fintype m\ninst✝ : Finite n\nA : Matrix m n R\nval✝ : Fintype n\nh : Function.Surjective fun v ↦ v ᵥ* A\nrows : n → m → R\nhrows : ∀ (x : n), (fun v ↦ v ᵥ* A) (rows x) = Pi.single x 1\ni j : n\n⊢ (o... | [
"case intro\nm : Type u\nn : Type u'\nR : Type u_2\ninst✝³ : Semiring R\ninst✝² : DecidableEq n\ninst✝¹ : Fintype m\ninst✝ : Finite n\nA : Matrix m n R\nval✝ : Fintype n\nh : Function.Surjective fun v ↦ v ᵥ* A\nrows : n → m → R\nhrows : ∀ (x : n), (fun v ↦ v ᵥ* A) (rows x) = Pi.single x 1\ni j : n\n⊢ (of rows i ᵥ* ... | mul_apply_eq_vecMul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.Adjugate | {
"line": 498,
"column": 2
} | {
"line": 498,
"column": 25
} | {
"line": 499,
"column": 2
} | [
{
"pp": "case succ.zero\nn : Type v\nα : Type w\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA : Matrix n n α\nh : Fintype.card n ≠ 1\nh_card : Fintype.card n = 0 + 1\n⊢ A.adjugate.adjugate = A.det ^ (0 + 1 - 2) • A",
"ppTerm": "?succ.zero",
"assigned": true,
"usedConstants": [
... | [
"case succ.succ\nn : Type v\nα : Type w\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA : Matrix n n α\nh : Fintype.card n ≠ 1\nn✝ : ℕ\nh_card : Fintype.card n = n✝ + 1 + 1\n⊢ A.adjugate.adjugate = A.det ^ (n✝ + 1 + 1 - 2) • A"
] | · exact (h h_card).elim | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Matrix.NonsingularInverse | {
"line": 356,
"column": 8
} | {
"line": 356,
"column": 38
} | {
"line": 356,
"column": 38
} | [
{
"pp": "case refine_1\nm : Type u\ninst✝² : DecidableEq m\nK : Type u_3\ninst✝¹ : Field K\ninst✝ : Fintype m\nA : Matrix m m K\nh : Function.Injective fun v ↦ v ᵥ* A\n⊢ IsUnit A",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAsso... | [
"case refine_1\nm : Type u\ninst✝² : DecidableEq m\nK : Type u_3\ninst✝¹ : Field K\ninst✝ : Fintype m\nA : Matrix m m K\nh : Function.Injective fun v ↦ v ᵥ* A\n⊢ Function.Surjective fun v ↦ v ᵥ* A"
] | ← vecMul_surjective_iff_isUnit | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.NonsingularInverse | {
"line": 362,
"column": 6
} | {
"line": 362,
"column": 25
} | {
"line": 362,
"column": 26
} | [
{
"pp": "m : Type u\ninst✝² : DecidableEq m\nK : Type u_3\ninst✝¹ : Field K\ninst✝ : Fintype m\nA : Matrix m m K\n⊢ Function.Injective A.mulVec ↔ IsUnit A",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.... | [
"m : Type u\ninst✝² : DecidableEq m\nK : Type u_3\ninst✝¹ : Field K\ninst✝ : Fintype m\nA : Matrix m m K\n⊢ Function.Injective A.mulVec ↔ IsUnit Aᵀ"
] | ← isUnit_transpose, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.NonsingularInverse | {
"line": 516,
"column": 6
} | {
"line": 516,
"column": 15
} | {
"line": 516,
"column": 16
} | [
{
"pp": "n : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA : Matrix n n α\nh : IsUnit A.det\n⊢ h.unit⁻¹ • A * A.adjugate = 1",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matrix.smul",
"instHSMul",
"Semiring.toMo... | [
"n : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA : Matrix n n α\nh : IsUnit A.det\n⊢ h.unit⁻¹ • (A * A.adjugate) = 1"
] | smul_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap | {
"line": 178,
"column": 2
} | {
"line": 191,
"column": 36
} | {
"line": 193,
"column": 0
} | [
{
"pp": "ι : Type u_1\ninst✝⁴ : Fintype ι\nM : Type u_2\ninst✝³ : AddCommGroup M\nR : Type u_3\ninst✝² : CommRing R\ninst✝¹ : Module R M\nb : ι → M\ninst✝ : DecidableEq ι\nhb : Submodule.span R (Set.range b) = ⊤\nf : Module.End R M\nI : Ideal R\nhI : LinearMap.range f ≤ I • ⊤\n⊢ ∃ M_1, (toEnd R b hb) M_1 = f ∧ ... | [] | have : ∀ x, f x ∈ LinearMap.range (Ideal.finsuppTotal ι M I b) := by
rw [Ideal.range_finsuppTotal, hb]
exact fun x => hI (LinearMap.mem_range_self f x)
choose bM' hbM' using this
let A : Matrix ι ι R := fun i j => bM' (b j) i
have : A.Represents b f := by
rw [Matrix.represents_iff']
dsimp [A]
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap | {
"line": 178,
"column": 2
} | {
"line": 191,
"column": 36
} | {
"line": 193,
"column": 0
} | [
{
"pp": "ι : Type u_1\ninst✝⁴ : Fintype ι\nM : Type u_2\ninst✝³ : AddCommGroup M\nR : Type u_3\ninst✝² : CommRing R\ninst✝¹ : Module R M\nb : ι → M\ninst✝ : DecidableEq ι\nhb : Submodule.span R (Set.range b) = ⊤\nf : Module.End R M\nI : Ideal R\nhI : LinearMap.range f ≤ I • ⊤\n⊢ ∃ M_1, (toEnd R b hb) M_1 = f ∧ ... | [] | have : ∀ x, f x ∈ LinearMap.range (Ideal.finsuppTotal ι M I b) := by
rw [Ideal.range_finsuppTotal, hb]
exact fun x => hI (LinearMap.mem_range_self f x)
choose bM' hbM' using this
let A : Matrix ι ι R := fun i j => bM' (b j) i
have : A.Represents b f := by
rw [Matrix.represents_iff']
dsimp [A]
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Eval.Subring | {
"line": 42,
"column": 4
} | {
"line": 42,
"column": 27
} | {
"line": 43,
"column": 4
} | [
{
"pp": "case mpr\nR : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nf : R →+* S\np : S[X]\nh : ∀ (n : ℕ), p.coeff n ∈ f.rangeS\ni : ℕ\n_hi : i ∈ Finset.range (p.natDegree + 1)\n⊢ C (p.coeff i) * X ^ i ∈ (mapRingHom f).rangeS",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
... | [
"case mpr\nR : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nf : R →+* S\np : S[X]\nh : ∀ (n : ℕ), p.coeff n ∈ f.rangeS\ni : ℕ\n_hi : i ∈ Finset.range (p.natDegree + 1)\nc : R\nhc : f c = p.coeff i\n⊢ C (p.coeff i) * X ^ i ∈ (mapRingHom f).rangeS"
] | rcases h i with ⟨c, hc⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff | {
"line": 189,
"column": 4
} | {
"line": 191,
"column": 83
} | {
"line": 193,
"column": 0
} | [
{
"pp": "case e'_3\nR : Type u\ninst✝² : CommRing R\nn : Type v\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\nM : Matrix n n R\ne : Matrix n n R ≃ₗ[R] Matrix (Fin (Fintype.card n)) (Fin (Fintype.card n)) R :=\n reindexLinearEquiv R R (Fintype.equivFin n) (Fintype.equivFin n)\n⊢ M.trace = (e M).trace",
"ppTer... | [] | delta trace
rw [← (Fintype.equivFin n).symm.sum_comp]
simp_rw [e, coe_reindexLinearEquiv, reindex_apply, diag_apply, submatrix_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff | {
"line": 189,
"column": 4
} | {
"line": 191,
"column": 83
} | {
"line": 193,
"column": 0
} | [
{
"pp": "case e'_3\nR : Type u\ninst✝² : CommRing R\nn : Type v\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\nM : Matrix n n R\ne : Matrix n n R ≃ₗ[R] Matrix (Fin (Fintype.card n)) (Fin (Fintype.card n)) R :=\n reindexLinearEquiv R R (Fintype.equivFin n) (Fintype.equivFin n)\n⊢ M.trace = (e M).trace",
"ppTer... | [] | delta trace
rw [← (Fintype.equivFin n).symm.sum_comp]
simp_rw [e, coe_reindexLinearEquiv, reindex_apply, diag_apply, submatrix_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Lifts | {
"line": 195,
"column": 4
} | {
"line": 195,
"column": 47
} | {
"line": 196,
"column": 4
} | [
{
"pp": "case inl\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\np : S[X]\nhlifts : p ∈ lifts f\nhp : p.Monic\nhR : Subsingleton S\n⊢ ∃ q, map f q = p ∧ q.natDegree = p.natDegree ∧ q.Monic",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Polynomial.inst... | [
"case inl\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nhR : Subsingleton S\nhlifts : 1 ∈ lifts f\nhp : Monic 1\n⊢ ∃ q, map f q = 1 ∧ q.natDegree = natDegree 1 ∧ q.Monic"
] | obtain rfl : p = 1 := Subsingleton.elim _ _ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Polynomial.IntegralNormalization | {
"line": 117,
"column": 2
} | {
"line": 126,
"column": 76
} | {
"line": 128,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\n⊢ p.integralNormalization * C p.leadingCoeff = p.scaleRoots p.leadingCoeff",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Polynomial.integralNormalization",
"WithBot.instPreorder",
"Eq.mpr",
"Polynomial.C",
... | [] | ext i
rw [coeff_mul_C, integralNormalization_coeff]
split_ifs with h
· simp [natDegree_eq_of_degree_eq_some h, leadingCoeff]
· simp only [coeff_scaleRoots]
by_cases h' : i < p.degree
· rw [mul_assoc, ← pow_succ, tsub_right_comm, tsub_add_cancel_of_le]
rw [le_tsub_iff_left (coe_lt_degree.mp h').le,... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.IntegralNormalization | {
"line": 117,
"column": 2
} | {
"line": 126,
"column": 76
} | {
"line": 128,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\n⊢ p.integralNormalization * C p.leadingCoeff = p.scaleRoots p.leadingCoeff",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Polynomial.integralNormalization",
"WithBot.instPreorder",
"Eq.mpr",
"Polynomial.C",
... | [] | ext i
rw [coeff_mul_C, integralNormalization_coeff]
split_ifs with h
· simp [natDegree_eq_of_degree_eq_some h, leadingCoeff]
· simp only [coeff_scaleRoots]
by_cases h' : i < p.degree
· rw [mul_assoc, ← pow_succ, tsub_right_comm, tsub_add_cancel_of_le]
rw [le_tsub_iff_left (coe_lt_degree.mp h').le,... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.TensorProduct.MvPolynomial | {
"line": 182,
"column": 8
} | {
"line": 182,
"column": 35
} | {
"line": 182,
"column": 35
} | [
{
"pp": "case map_one\nR : Type u\nN : Type v\ninst✝⁵ : CommSemiring R\nσ : Type u_1\nι : Type u_2\nS : Type u_3\ninst✝⁴ : CommSemiring S\ninst✝³ : Algebra R S\ninst✝² : CommSemiring N\ninst✝¹ : Algebra R N\ninst✝ : DecidableEq σ\n⊢ 1 = rTensor (1 ⊗ₜ[R] 1)",
"ppTerm": "?map_one",
"assigned": true,
"... | [
"case map_one\nR : Type u\nN : Type v\ninst✝⁵ : CommSemiring R\nσ : Type u_1\nι : Type u_2\nS : Type u_3\ninst✝⁴ : CommSemiring S\ninst✝³ : Algebra R S\ninst✝² : CommSemiring N\ninst✝¹ : Algebra R N\ninst✝ : DecidableEq σ\n⊢ rTensor.symm 1 = 1 ⊗ₜ[R] 1"
] | ← LinearEquiv.symm_apply_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.TensorProduct.MvPolynomial | {
"line": 232,
"column": 2
} | {
"line": 239,
"column": 44
} | {
"line": 241,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝² : CommSemiring R\nσ : Type u_1\nA : Type u_4\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nm : σ →₀ ℕ\na : A\n⊢ (algebraTensorAlgEquiv R A).symm ((monomial m) a) = a ⊗ₜ[R] (monomial m) 1",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"one_pow",
"Fi... | [] | apply @Finsupp.induction σ ℕ _ _ m
· simp [algebraTensorAlgEquiv]
· intro i n f _ _ hfa
simp only [algebraTensorAlgEquiv, AlgEquiv.ofAlgHom_symm_apply] at hfa ⊢
simp only [add_comm, monomial_add_single, map_mul, map_pow, aeval_X,
Algebra.TensorProduct.tmul_pow, one_pow, hfa]
nth_rw 2 [← mul_one a]... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.TensorProduct.MvPolynomial | {
"line": 232,
"column": 2
} | {
"line": 239,
"column": 44
} | {
"line": 241,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝² : CommSemiring R\nσ : Type u_1\nA : Type u_4\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nm : σ →₀ ℕ\na : A\n⊢ (algebraTensorAlgEquiv R A).symm ((monomial m) a) = a ⊗ₜ[R] (monomial m) 1",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"one_pow",
"Fi... | [] | apply @Finsupp.induction σ ℕ _ _ m
· simp [algebraTensorAlgEquiv]
· intro i n f _ _ hfa
simp only [algebraTensorAlgEquiv, AlgEquiv.ofAlgHom_symm_apply] at hfa ⊢
simp only [add_comm, monomial_add_single, map_mul, map_pow, aeval_X,
Algebra.TensorProduct.tmul_pow, one_pow, hfa]
nth_rw 2 [← mul_one a]... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Basic | {
"line": 302,
"column": 27
} | {
"line": 302,
"column": 50
} | {
"line": 303,
"column": 6
} | [
{
"pp": "R : Type u_1\nS : Type u_4\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\np : R[X]\nx : S\nh : f (p.coeff 0) = 0\nh' : p.natDegree = 0\nhp : ¬Polynomial.map f p = 0\nthis : Polynomial.map f p = 0\n⊢ f.IsIntegralElem (f p.leadingCoeff * x)",
"ppTerm": "?m.262",
"assigned": true,
"use... | [] | by exact (hp this).elim | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Splits | {
"line": 558,
"column": 19
} | {
"line": 561,
"column": 15
} | {
"line": 563,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : Field R\ninst✝¹ : CommRing S\ninst✝ : IsDomain S\nf : R[X]\nhf : f.Splits\nhf0 : f ≠ 0\ni : R →+* S\nx : S\nhx : (map i f).IsRoot x\n⊢ x ∈ i.range",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Polynomial.roots",
"Subring.instSetL... | [] | by
rw [← mem_roots (map_ne_zero hf0), hf.roots_map, Multiset.mem_map] at hx
obtain ⟨x, -, hx⟩ := hx
exact ⟨x, hx⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Splits | {
"line": 645,
"column": 2
} | {
"line": 648,
"column": 30
} | {
"line": 650,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Field R\nf : R[X]\nhf : f.Splits\nx : R\nhx : eval x f ≠ 0\n⊢ eval x (derivative f) = eval x f * (Multiset.map (fun z ↦ 1 / (x - z)) f.roots).sum",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Multiset.sum",
"Polynomial.derivative",
"Eq.mp... | [] | simp only [hf.eval_derivative, hf.eval_eq_prod_roots, ← Multiset.sum_map_mul_left, mul_assoc]
refine congr_arg Multiset.sum (Multiset.map_congr rfl fun z hz ↦ ?_)
rw [← Multiset.prod_map_erase hz, mul_one_div, mul_div_cancel_left₀]
aesop (add simp sub_eq_zero) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Splits | {
"line": 645,
"column": 2
} | {
"line": 648,
"column": 30
} | {
"line": 650,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Field R\nf : R[X]\nhf : f.Splits\nx : R\nhx : eval x f ≠ 0\n⊢ eval x (derivative f) = eval x f * (Multiset.map (fun z ↦ 1 / (x - z)) f.roots).sum",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Multiset.sum",
"Polynomial.derivative",
"Eq.mp... | [] | simp only [hf.eval_derivative, hf.eval_eq_prod_roots, ← Multiset.sum_map_mul_left, mul_assoc]
refine congr_arg Multiset.sum (Multiset.map_congr rfl fun z hz ↦ ?_)
rw [← Multiset.prod_map_erase hz, mul_one_div, mul_div_cancel_left₀]
aesop (add simp sub_eq_zero) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Localization.Integral | {
"line": 214,
"column": 54
} | {
"line": 214,
"column": 75
} | {
"line": 214,
"column": 75
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝⁶ : CommRing Rₘ\ninst✝⁵ : CommRing Sₘ\ninst✝⁴ : Algebra R Rₘ\ninst✝³ : IsLocalization M Rₘ\ninst✝² : Algebra S Sₘ\ninst✝¹ : IsLocalization (Alge... | [
"case refine_1\nR : Type u_1\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝⁶ : CommRing Rₘ\ninst✝⁵ : CommRing Sₘ\ninst✝⁴ : Algebra R Rₘ\ninst✝³ : IsLocalization M Rₘ\ninst✝² : Algebra S Sₘ\ninst✝¹ : IsLocalization (Algebra.algebraM... | IsLocalization.map_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Flat.FaithfullyFlat.Basic | {
"line": 76,
"column": 6
} | {
"line": 76,
"column": 24
} | {
"line": 76,
"column": 24
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\n⊢ FaithfullyFlat R M ↔ Flat R M ∧ ∀ (I : Ideal R), I ≠ ⊤ → I • ⊤ ≠ ⊤",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"instHSMul",
"Semiring.toMo... | [
"R : Type u\nM : Type v\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\n⊢ (Flat R M ∧ ∀ ⦃m : Ideal R⦄, m.IsMaximal → m • ⊤ ≠ ⊤) ↔ Flat R M ∧ ∀ (I : Ideal R), I ≠ ⊤ → I • ⊤ ≠ ⊤"
] | faithfullyFlat_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.GoingUp | {
"line": 238,
"column": 2
} | {
"line": 240,
"column": 99
} | {
"line": 242,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nA : Type u_3\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : Algebra.IsIntegral R A\nI : Ideal A\ninst✝ : I.IsPrime\n⊢ I ∈ (map (algebraMap R A) (under R I)).minimalPrimes",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"LE.le.lt_of_no... | [] | refine ⟨⟨inferInstance, map_comap_le⟩, fun r ⟨hr, hpr⟩ hrq ↦ ?_⟩
contrapose! hpr
exact mt map_le_iff_le_comap.mp (not_le_of_gt (IsIntegral.comap_lt_comap (hrq.lt_of_not_ge hpr))) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.GoingUp | {
"line": 238,
"column": 2
} | {
"line": 240,
"column": 99
} | {
"line": 242,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nA : Type u_3\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : Algebra.IsIntegral R A\nI : Ideal A\ninst✝ : I.IsPrime\n⊢ I ∈ (map (algebraMap R A) (under R I)).minimalPrimes",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"LE.le.lt_of_no... | [] | refine ⟨⟨inferInstance, map_comap_le⟩, fun r ⟨hr, hpr⟩ hrq ↦ ?_⟩
contrapose! hpr
exact mt map_le_iff_le_comap.mp (not_le_of_gt (IsIntegral.comap_lt_comap (hrq.lt_of_not_ge hpr))) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Spectrum.Prime.Basic | {
"line": 360,
"column": 19
} | {
"line": 360,
"column": 58
} | {
"line": 362,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nf : R\nn : ℕ\nhn : 0 < n\nx : PrimeSpectrum R\n⊢ x ∈ zeroLocus {f ^ n} ↔ x ∈ zeroLocus {f}",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"PrimeSpectrum.mem_zeroLocus._simp_1",
"Semir... | [] | by simpa using x.2.pow_mem_iff_mem n hn | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Localization.Integral | {
"line": 508,
"column": 4
} | {
"line": 508,
"column": 14
} | {
"line": 509,
"column": 4
} | [
{
"pp": "case mpr\nR : Type u_1\ninst✝⁹ : CommRing R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nK : Type u_4\ninst✝⁶ : Field K\ninst✝⁵ : IsDomain R\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : Module.IsTorsionFree R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : ... | [
"case right\nR : Type u_1\ninst✝⁹ : CommRing R\nS : Type u_2\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nK : Type u_4\ninst✝⁶ : Field K\ninst✝⁵ : IsDomain R\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : Module.IsTorsionFree R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : K), IsAlge... | use f, hf₁ | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.RingTheory.LocalRing.ResidueField.Basic | {
"line": 228,
"column": 17
} | {
"line": 230,
"column": 58
} | {
"line": 232,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : IsLocalRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : IsLocalRing S\ninst✝⁴ : CommRing T\ninst✝³ : IsLocalRing T\ninst✝² : Algebra R S\ninst✝¹ : Algebra R T\ne : S →ₐ[R] T\ninst✝ : IsLocalHom e\nx : R\n⊢ (↑↑(map ↑e)).toFun ((algebraMap R (R... | [] | by
simp [IsScalarTower.algebraMap_apply R S (ResidueField S),
IsScalarTower.algebraMap_apply R T (ResidueField T)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Flat.FaithfullyFlat.Basic | {
"line": 563,
"column": 2
} | {
"line": 563,
"column": 57
} | {
"line": 564,
"column": 2
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nS : Type u_1\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : FaithfullyFlat R M\nN : Type (max u_1 v)\nx✝¹ : AddCommGroup N\nx✝ : Module S N\nhN : Subsingleton (N ⊗[S] (S ⊗[R] M))\n⊢ Subsingleton N",
"ppT... | [
"R : Type u\nM : Type v\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nS : Type u_1\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : FaithfullyFlat R M\nN : Type (max u_1 v)\nx✝² : AddCommGroup N\nx✝¹ : Module S N\nhN : Subsingleton (N ⊗[S] (S ⊗[R] M))\nx✝ : Module R N := compHom N (algebraM... | let _ : Module R N := Module.compHom N (algebraMap R S) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.Flat.FaithfullyFlat.Algebra | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 60
} | {
"line": 67,
"column": 2
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝⁶ : CommRing A\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra A B\ninst✝³ : IsLocalRing A\ninst✝² : IsLocalRing B\ninst✝¹ : Flat A B\ninst✝ : IsLocalHom (algebraMap A B)\nm : Ideal A\nhm : m.IsMaximal\n⊢ m • ⊤ ≠ ⊤",
"ppTerm": "?m.29",
"assigned": true,
"usedConstant... | [
"A : Type u_1\nB : Type u_2\ninst✝⁶ : CommRing A\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra A B\ninst✝³ : IsLocalRing A\ninst✝² : IsLocalRing B\ninst✝¹ : Flat A B\ninst✝ : IsLocalHom (algebraMap A B)\nm : Ideal A\nhm : m.IsMaximal\n⊢ Submodule.restrictScalars A (Ideal.map (algebraMap A B) (IsLocalRing.maximalIdeal A)) ... | rw [Ideal.smul_top_eq_map, IsLocalRing.eq_maximalIdeal hm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Spectrum.Prime.RingHom | {
"line": 336,
"column": 69
} | {
"line": 338,
"column": 83
} | {
"line": 340,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : PrimeSpectrum R\n⊢ Set.range (comap (algebraMap R I.asIdeal.ResidueField)) = {I}",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Inhabited.default",
"PrimeSpectrum.ext",
"Algebra.algebraMap",
"OreLocali... | [] | by
rw [Set.range_unique, Set.singleton_eq_singleton_iff]
exact PrimeSpectrum.ext (Ideal.ext fun x ↦ Ideal.algebraMap_residueField_eq_zero) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.RelSeries | {
"line": 106,
"column": 65
} | {
"line": 106,
"column": 92
} | {
"line": 108,
"column": 0
} | [
{
"pp": "α : Type u_1\nr : SetRel α α\nx : α\n⊢ (singleton r x).toList = [x]",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"instNeZeroNatHAdd_1",
"Fin.succ",
"congrArg",
"AddMonoid.toAddZeroClass",
"List.ofFn",
"Nat.instAddMonoid",
"Fin.instOfNat"... | [] | by simp [toList, singleton] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.KrullDimension | {
"line": 443,
"column": 4
} | {
"line": 444,
"column": 12
} | {
"line": 445,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : Preorder α\nx : α\nh : height x = ⊤\nn : ℕ\n⊢ ∃ p, RelSeries.last p = x ∧ p.length = n",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Order.exists_series_of_le_height",
"ENat.instNatCast",
"instTopENat",
"congrArg",
"le_top._si... | [] | apply exists_series_of_le_height x (n := n)
simp [h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.KrullDimension | {
"line": 443,
"column": 4
} | {
"line": 444,
"column": 12
} | {
"line": 445,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : Preorder α\nx : α\nh : height x = ⊤\nn : ℕ\n⊢ ∃ p, RelSeries.last p = x ∧ p.length = n",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Order.exists_series_of_le_height",
"ENat.instNatCast",
"instTopENat",
"congrArg",
"le_top._si... | [] | apply exists_series_of_le_height x (n := n)
simp [h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.KrullDimension | {
"line": 616,
"column": 25
} | {
"line": 616,
"column": 44
} | {
"line": 616,
"column": 44
} | [
{
"pp": "α : Type u_1\ninst✝ : Preorder α\n⊢ krullDim α ≠ ⊥ ↔ Nonempty α",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"WithBot",
"instCompleteLinearOrderENat",
"congrArg",
"WithBot.instOrderBot",
"OrderBot.toBot",
"PartialOrder.toPreor... | [
"α : Type u_1\ninst✝ : Preorder α\n⊢ Nonempty α ↔ Nonempty α"
] | krullDim_ne_bot_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.RelSeries | {
"line": 741,
"column": 2
} | {
"line": 741,
"column": 26
} | {
"line": 742,
"column": 2
} | [
{
"pp": "case succ\nα : Type u_1\nr : SetRel α α\nq : RelSeries r\nn✝ : ℕ\ntoFun✝ : Fin (n✝ + 1 + 1) → α\nstep✝ : ∀ (i : Fin (n✝ + 1)), (toFun✝ i.castSucc, toFun✝ i.succ) ∈ r\nh : { length := n✝ + 1, toFun := toFun✝, step := step✝ }.last = q.head\n⊢ ({ length := n✝ + 1, toFun := toFun✝, step := step✝ }.smash q ... | [
"case succ\nα : Type u_1\nr : SetRel α α\nq : RelSeries r\nn✝ : ℕ\ntoFun✝ : Fin (n✝ + 1 + 1) → α\nstep✝ : ∀ (i : Fin (n✝ + 1)), (toFun✝ i.castSucc, toFun✝ i.succ) ∈ r\nh : { length := n✝ + 1, toFun := toFun✝, step := step✝ }.last = q.head\n⊢ Fin.addCases (toFun✝ ∘ Fin.castSucc) q.toFun 0 = toFun✝ 0"
] | dsimp only [smash, head] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.Order.KrullDimension | {
"line": 801,
"column": 8
} | {
"line": 801,
"column": 40
} | {
"line": 801,
"column": 40
} | [
{
"pp": "case a\nα : Type u_1\ninst✝¹ : Preorder α\ninst✝ : Nonempty α\n⊢ ↑(⨆ a, height a) ≤ krullDim α",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"WithBot.instSupSet",
"Eq.mpr",
"WithBot.some",
"WithBot",
"instCompleteLinearOrderENat",
"congrArg",
... | [
"case a\nα : Type u_1\ninst✝¹ : Preorder α\ninst✝ : Nonempty α\n⊢ ⨆ i, ↑(height i) ≤ krullDim α"
] | WithBot.coe_iSup (by bddDefault) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.RelSeries | {
"line": 795,
"column": 6
} | {
"line": 795,
"column": 26
} | {
"line": 796,
"column": 6
} | [
{
"pp": "case mp.succ\nα : Type u_1\nr : SetRel α α\ninst✝ : Nonempty α\nH : ∀ (x : RelSeries r), ∃ y, x.length < y.length\nn : ℕ\nIH : ∃ x, x.length = n\n⊢ ∃ x, x.length = n + 1",
"ppTerm": "?mp.succ",
"assigned": true,
"usedConstants": [
"Exists",
"RelSeries.length",
"instOfNatNa... | [
"case mp.succ\nα : Type u_1\nr : SetRel α α\ninst✝ : Nonempty α\nH : ∀ (x : RelSeries r), ∃ y, x.length < y.length\nn : ℕ\nl : RelSeries r\nhl : l.length = n\n⊢ ∃ x, x.length = n + 1"
] | obtain ⟨l, hl⟩ := IH | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.LocalAtTarget | {
"line": 181,
"column": 4
} | {
"line": 181,
"column": 38
} | {
"line": 183,
"column": 0
} | [
{
"pp": "case h₂\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\nι : Type u_3\nU : ι → Opens β\nhU : IsOpenCover U\nh : Continuous[inst✝¹, inst✝] f\n⊢ IsClosed[inst✝] (range f) ↔ ∀ (x : ι), IsClosed[instTopologicalSpaceSubtype] (Subtype.val ⁻¹' range f)",
"pp... | [] | exact hU.isClosed_iff_coe_preimage | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.LocalAtTarget | {
"line": 223,
"column": 4
} | {
"line": 223,
"column": 30
} | {
"line": 223,
"column": 31
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\nι : Type u_3\nU : ι → Opens α\nhU : IsOpenCover U\nhf : ∀ (i : ι), IsOpenMap (f ∘ Subtype.val)\nV : Set α\nhV : IsOpen[inst✝¹] V\n⊢ f '' V = f '' ⋃ i, V ∩ range Subtype.val",
"ppTerm": "?m.115",
"ass... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\nι : Type u_3\nU : ι → Opens α\nhU : IsOpenCover U\nhf : ∀ (i : ι), IsOpenMap (f ∘ Subtype.val)\nV : Set α\nhV : IsOpen[inst✝¹] V\n⊢ f '' V = f '' ⋃ i, V ∩ {x | x ∈ U i}"
] | Subtype.range_coe_subtype, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.