module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Algebra.Order.ToIntervalMod | {
"line": 588,
"column": 4
} | {
"line": 588,
"column": 31
} | [
{
"pp": "case refine_2\nα : Type u_1\ninst✝² : AddCommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedAddMonoid α\nhα : Archimedean α\np : α\nhp : 0 < p\na b c : α\nz : ℤ\nhz : c = z • p + b\n⊢ toIocMod hp a b = toIocMod hp a c",
"usedConstants": [
"Eq.mpr",
"instHSMul",
"congrArg",
... | rw [hz, toIocMod_zsmul_add] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Order.ToIntervalMod | {
"line": 824,
"column": 61
} | {
"line": 826,
"column": 32
} | [
{
"pp": "α : Type u_1\ninst✝² : AddCommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedAddMonoid α\nhα : Archimedean α\np : α\nhp : 0 < p\nx₁ x₂ x₃ : α\n⊢ toIcoMod hp x₁ x₂ ≤ toIocMod hp x₁ x₃ ↔ toIcoMod hp 0 (x₂ - x₁) + toIcoMod hp 0 (x₁ - x₃) ≤ p",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.... | by
rw [toIcoMod_eq_sub, toIocMod_eq_sub _ x₁, add_le_add_iff_right, ← neg_sub x₁ x₃, toIocMod_neg,
neg_zero, le_sub_iff_add_le] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.ToIntervalMod | {
"line": 930,
"column": 48
} | {
"line": 937,
"column": 65
} | [
{
"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 c : α\nn : ℤ\nhp' : Fact (0 < p)\nx₁ x₂ x₃ : α ⧸ AddSubgroup.zmultiples p\nh₁₂₃ : btw x₁ x₂ x₃\nh₃₂₁ : btw x₃ x₂ x₁\n⊢ x₁ = x₂ ∨ x₂ = x₃ ∨ x₃ = x₁",
"usedConstants... | by
induction x₁ using QuotientAddGroup.induction_on
induction x₂ using QuotientAddGroup.induction_on
induction x₃ using QuotientAddGroup.induction_on
rw [btw_cyclic] at h₃₂₁
simp_rw [btw_coe_iff] at h₁₂₃ h₃₂₁
simp_rw [← modEq_iff_eq_mod_zmultiples]
simpa only [modEq_comm] using... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Order.LeftRightNhds | {
"line": 84,
"column": 2
} | {
"line": 86,
"column": 75
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\nha : ¬IsTop a\n⊢ 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b",
"usedConstants": [
"Eq.mpr",
"False",
"Set.Ioi",
"Preorder.toLT",
"Lattice.toSemilatticeSup",
"eq_fa... | · simp only [ha, false_or]
rw [isTop_iff_isMax, not_isMax_iff] at ha
simp only [(nhdsGT_basis_of_exists_gt ha).eq_bot_iff, covBy_iff_Ioo_eq] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Algebra.Order.Field | {
"line": 70,
"column": 28
} | {
"line": 72,
"column": 17
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"Se... | by
convert! h.comp tendsto_inv_nhdsGT_zero
grind [inv_inv] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Order.Basic | {
"line": 210,
"column": 2
} | {
"line": 211,
"column": 72
} | [
{
"pp": "case refine_2\nα : Type u\nts : TopologicalSpace α\ninst✝ : Preorder α\nc : Set α\nh : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}\nk : Set (Set α)\nk_fin : k.Finite\nhk : k ⊆ {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}\nkl : Set (Set α) := {s | s ∈ k ∧ ∃ a ∈ c, s = Ioi a}\nkr : Set (Set α) := {s |... | refine ⟨range al, by simp [range_subset_iff, alc], range ar,
by simp [range_subset_iff, arc], finite_range _, finite_range _, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Order.Basic | {
"line": 288,
"column": 6
} | {
"line": 288,
"column": 27
} | [
{
"pp": "case pos\nα : Type u\nβ : Type v\ninst✝³ : Preorder α\ninst✝² : Preorder β\ninst✝¹ : TopologicalSpace β\ninst✝ : OrderTopology β\nf : α → β\nhf : ∀ {x y : α}, f x < f y ↔ x < y\nH₁ : ∀ {a : α} {b : β} {x : α}, b < f a → ¬b < f x → ∃ y < a, b ≤ f y\nH₂ : ∀ {a : α} {b : β} {x : α}, f a < b → ¬f x < b → ∃... | rcases h with ⟨x, hx⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Topology.Order.Basic | {
"line": 293,
"column": 6
} | {
"line": 293,
"column": 27
} | [
{
"pp": "case pos\nα : Type u\nβ : Type v\ninst✝³ : Preorder α\ninst✝² : Preorder β\ninst✝¹ : TopologicalSpace β\ninst✝ : OrderTopology β\nf : α → β\nhf : ∀ {x y : α}, f x < f y ↔ x < y\nH₁ : ∀ {a : α} {b : β} {x : α}, b < f a → ¬b < f x → ∃ y < a, b ≤ f y\nH₂ : ∀ {a : α} {b : β} {x : α}, f a < b → ¬f x < b → ∃... | rcases h with ⟨x, hx⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Topology.Algebra.Order.Field | {
"line": 240,
"column": 2
} | {
"line": 248,
"column": 78
} | [
{
"pp": "𝕜 : Type u_1\nα : Type u_2\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nl : Filter α\nf g : α → 𝕜\nC : 𝕜\nhf : ∀ᶠ (x : α) in l, |f x| ≤ C\nhg : Tendsto g l (𝓝 0)\n⊢ Tendsto (fun x ↦ f x * g x) l (𝓝 0)",
"u... | rw [tendsto_zero_iff_abs_tendsto_zero]
have hC : Tendsto (fun x ↦ |C * g x|) l (𝓝 0) := by
convert! (hg.const_mul C).abs
simp_rw [mul_zero, abs_zero]
apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hC
· filter_upwards [hf] with x _ using abs_nonneg _
· filter_upwards [hf] with x hx
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Order.Field | {
"line": 240,
"column": 2
} | {
"line": 248,
"column": 78
} | [
{
"pp": "𝕜 : Type u_1\nα : Type u_2\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nl : Filter α\nf g : α → 𝕜\nC : 𝕜\nhf : ∀ᶠ (x : α) in l, |f x| ≤ C\nhg : Tendsto g l (𝓝 0)\n⊢ Tendsto (fun x ↦ f x * g x) l (𝓝 0)",
"u... | rw [tendsto_zero_iff_abs_tendsto_zero]
have hC : Tendsto (fun x ↦ |C * g x|) l (𝓝 0) := by
convert! (hg.const_mul C).abs
simp_rw [mul_zero, abs_zero]
apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hC
· filter_upwards [hf] with x _ using abs_nonneg _
· filter_upwards [hf] with x hx
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Order.Basic | {
"line": 530,
"column": 6
} | {
"line": 532,
"column": 98
} | [
{
"pp": "case refine_2.inl\nα : Type u\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\ns : Set α\nhs : Dense s\na : α\n⊢ IsOpen[generateFrom (Ioi '' s ∪ Iio '' s)] (Ioi a)",
"usedConstants": [
"Eq.mpr",
"Set.Ioi",
"congrArg",
... | rw [hs.Ioi_eq_biUnion]
let _ := generateFrom (Ioi '' s ∪ Iio '' s)
exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inl <| mem_image_of_mem _ h.1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order.Basic | {
"line": 530,
"column": 6
} | {
"line": 532,
"column": 98
} | [
{
"pp": "case refine_2.inl\nα : Type u\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\ns : Set α\nhs : Dense s\na : α\n⊢ IsOpen[generateFrom (Ioi '' s ∪ Iio '' s)] (Ioi a)",
"usedConstants": [
"Eq.mpr",
"Set.Ioi",
"congrArg",
... | rw [hs.Ioi_eq_biUnion]
let _ := generateFrom (Ioi '' s ∪ Iio '' s)
exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inl <| mem_image_of_mem _ h.1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Instances.AddCircle.Defs | {
"line": 363,
"column": 2
} | {
"line": 367,
"column": 9
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : AddCommGroup 𝕜\np : 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsOrderedAddMonoid 𝕜\nhp : Fact (0 < p)\na : 𝕜\ninst✝ : Archimedean 𝕜\nx y : 𝕜\nhx : x ∈ Ioc a (a + p)\nhy : y ∈ Ioc a (a + p)\n⊢ ↑x = ↑y ↔ x = y",
"usedConstants": [
"Eq.mpr",
"Set.Ioc",
"Equiv... | refine ⟨fun h => ?_, by tauto⟩
suffices (⟨x, hx⟩ : Ioc a (a + p)) = ⟨y, hy⟩ by exact Subtype.mk.inj this
apply_fun equivIoc p a at h
rw [← (equivIoc p a).right_inv ⟨x, hx⟩, ← (equivIoc p a).right_inv ⟨y, hy⟩]
exact h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Instances.AddCircle.Defs | {
"line": 363,
"column": 2
} | {
"line": 367,
"column": 9
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : AddCommGroup 𝕜\np : 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsOrderedAddMonoid 𝕜\nhp : Fact (0 < p)\na : 𝕜\ninst✝ : Archimedean 𝕜\nx y : 𝕜\nhx : x ∈ Ioc a (a + p)\nhy : y ∈ Ioc a (a + p)\n⊢ ↑x = ↑y ↔ x = y",
"usedConstants": [
"Eq.mpr",
"Set.Ioc",
"Equiv... | refine ⟨fun h => ?_, by tauto⟩
suffices (⟨x, hx⟩ : Ioc a (a + p)) = ⟨y, hy⟩ by exact Subtype.mk.inj this
apply_fun equivIoc p a at h
rw [← (equivIoc p a).right_inv ⟨x, hx⟩, ← (equivIoc p a).right_inv ⟨y, hy⟩]
exact h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Instances.AddCircle.Defs | {
"line": 564,
"column": 78
} | {
"line": 565,
"column": 49
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : FloorRing 𝕜\nx : 𝕜\n⊢ ↑(Int.fract x) = ↑x",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast",
"zsmul_eq_mul",
"instHSMul",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMu... | by
simp [← Int.self_sub_floor, mem_zmultiples_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Instances.AddCircle.Defs | {
"line": 615,
"column": 2
} | {
"line": 618,
"column": 41
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\ninst✝² : Field 𝕜\np : 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\nhp : Fact (0 < p)\nu : AddCircle p\nn : ℕ\nh : 0 < n\n⊢ (∃ m < n, ↑(↑m / ↑n * p) = u) → n • u = 0",
"usedConstants": [
"Nat.gcd",
"Eq.mpr",
"instHSMul",
"Dvd.dvd... | · rw [← addOrderOf_dvd_iff_nsmul_eq_zero]
rintro ⟨m, -, rfl⟩
constructor; rw [mul_comm, eq_comm]
exact gcd_mul_addOrderOf_div_eq p m h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Monoidal.Grp | {
"line": 303,
"column": 59
} | {
"line": 303,
"column": 77
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : CartesianMonoidalCategory C\nA : C\ninst✝ : GrpObj A\ns : PullbackCone μ μ\n⊢ lift (toUnit s.pt ≫ η) (s.fst ≫ fst A A) ≫ μ = s.fst ≫ fst A A",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
... | lift_comp_one_left | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Monoidal.Grp | {
"line": 321,
"column": 6
} | {
"line": 321,
"column": 34
} | [
{
"pp": "case refine_3\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : CartesianMonoidalCategory C\nA : C\ninst✝ : GrpObj A\ns : PullbackCone μ μ\nm : s.pt ⟶ (A ⊗ A) ⊗ A\nhm₁ : m ≫ μ ▷ A = s.fst\nhm₂ : m ≫ (α_ A A A).hom ≫ A ◁ μ = s.snd\n⊢ m ≫ snd (A ⊗ A) A =\n (fun s ↦ lift (lift (s.snd ≫ fst A A) (lif... | · simpa using hm₁ =≫ snd _ _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Homology.ExactSequence | {
"line": 297,
"column": 6
} | {
"line": 298,
"column": 34
} | [
{
"pp": "case mp.right.hS\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nn : ℕ\nS : ComposableArrows C (n + 2)\nh : S.Exact\n⊢ (mk₂ (S.map' n (n + 1) ⋯ ⋯) (S.map' (n + 1) (n + 2) ⋯ ⋯)).IsComplex",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.ComposableArrows.exact_iff... | · rw [isComplex₂_iff]
exact h.toIsComplex.zero n | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Limits.Types.Pullbacks | {
"line": 222,
"column": 4
} | {
"line": 234,
"column": 41
} | [
{
"pp": "case mpr\nX₁ X₂ X₃ X₄ : Type u\nt : X₁ ⟶ X₂\nr : X₂ ⟶ X₄\nl : X₁ ⟶ X₃\nb : X₃ ⟶ X₄\n⊢ (t ≫ r = l ≫ b ∧\n (∀ (x₁ y₁ : (fun X ↦ X) X₁), (hom t) x₁ = (hom t) y₁ ∧ (hom l) x₁ = (hom l) y₁ → x₁ = y₁) ∧\n ∀ (x₂ : (fun X ↦ X) X₂) (x₃ : (fun X ↦ X) X₃),\n (hom r) x₂ = (hom b) x₃ → ∃ x₁, (h... | rintro ⟨w, h₁, h₂⟩
let φ : X₁ ⟶ PullbackObj r b := ↾fun x₁ ↦ ⟨⟨t x₁, l x₁⟩, congr_hom w x₁⟩
have hφ : IsIso φ := by
rw [isIso_iff_bijective]
constructor
· intro _ _ h
simp [φ] at h
grind
· intro x
obtain ⟨a, ha⟩ := h₂ x.1.1 x.1.2 (by grind)
cat_disch
e... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Types.Pullbacks | {
"line": 222,
"column": 4
} | {
"line": 234,
"column": 41
} | [
{
"pp": "case mpr\nX₁ X₂ X₃ X₄ : Type u\nt : X₁ ⟶ X₂\nr : X₂ ⟶ X₄\nl : X₁ ⟶ X₃\nb : X₃ ⟶ X₄\n⊢ (t ≫ r = l ≫ b ∧\n (∀ (x₁ y₁ : (fun X ↦ X) X₁), (hom t) x₁ = (hom t) y₁ ∧ (hom l) x₁ = (hom l) y₁ → x₁ = y₁) ∧\n ∀ (x₂ : (fun X ↦ X) X₂) (x₃ : (fun X ↦ X) X₃),\n (hom r) x₂ = (hom b) x₃ → ∃ x₁, (h... | rintro ⟨w, h₁, h₂⟩
let φ : X₁ ⟶ PullbackObj r b := ↾fun x₁ ↦ ⟨⟨t x₁, l x₁⟩, congr_hom w x₁⟩
have hφ : IsIso φ := by
rw [isIso_iff_bijective]
constructor
· intro _ _ h
simp [φ] at h
grind
· intro x
obtain ⟨a, ha⟩ := h₂ x.1.1 x.1.2 (by grind)
cat_disch
e... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.SnakeLemma | {
"line": 451,
"column": 15
} | {
"line": 451,
"column": 54
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nS S₁ S₂ S₃ : SnakeInput C\nf : S₁.Hom S₂\ng : S₂.Hom S₃\n⊢ (f.f₀ ≫ g.f₀) ≫ S₃.v₀₁ = S₁.v₀₁ ≫ f.f₁ ≫ g.f₁",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"CategoryTheory.Category.assoc",
"CategoryTheory.C... | simp only [assoc, comm₀₁, comm₀₁_assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.ShortComplex.SnakeLemma | {
"line": 451,
"column": 15
} | {
"line": 451,
"column": 54
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nS S₁ S₂ S₃ : SnakeInput C\nf : S₁.Hom S₂\ng : S₂.Hom S₃\n⊢ (f.f₀ ≫ g.f₀) ≫ S₃.v₀₁ = S₁.v₀₁ ≫ f.f₁ ≫ g.f₁",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"CategoryTheory.Category.assoc",
"CategoryTheory.C... | simp only [assoc, comm₀₁, comm₀₁_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.SnakeLemma | {
"line": 451,
"column": 15
} | {
"line": 451,
"column": 54
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nS S₁ S₂ S₃ : SnakeInput C\nf : S₁.Hom S₂\ng : S₂.Hom S₃\n⊢ (f.f₀ ≫ g.f₀) ≫ S₃.v₀₁ = S₁.v₀₁ ≫ f.f₁ ≫ g.f₁",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"CategoryTheory.Category.assoc",
"CategoryTheory.C... | simp only [assoc, comm₀₁, comm₀₁_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.Reflexive | {
"line": 351,
"column": 4
} | {
"line": 353,
"column": 47
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nA B : C\nf g : A ⟶ B\ns : B ⟶ A\nsl : s ≫ f = 𝟙 B\nsr : s ≫ g = 𝟙 B\n⊢ ∀ {X Y Z : WalkingReflexivePair} (f_1 : X ⟶ Y) (g_1 : Y ⟶ Z),\n (match X, Z, f_1 ≫ g_1 with\n | x, .(x), Hom.id .(x) =>\n 𝟙\n (match x with\n | zero => B\n ... | rintro _ _ _ ⟨⟩ g <;> cases g <;>
simp only [Category.id_comp, Category.comp_id, Category.assoc, sl, sr,
reassoc_of% sl, reassoc_of% sr] <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.Limits.Shapes.Reflexive | {
"line": 351,
"column": 4
} | {
"line": 353,
"column": 47
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nA B : C\nf g : A ⟶ B\ns : B ⟶ A\nsl : s ≫ f = 𝟙 B\nsr : s ≫ g = 𝟙 B\n⊢ ∀ {X Y Z : WalkingReflexivePair} (f_1 : X ⟶ Y) (g_1 : Y ⟶ Z),\n (match X, Z, f_1 ≫ g_1 with\n | x, .(x), Hom.id .(x) =>\n 𝟙\n (match x with\n | zero => B\n ... | rintro _ _ _ ⟨⟩ g <;> cases g <;>
simp only [Category.id_comp, Category.comp_id, Category.assoc, sl, sr,
reassoc_of% sl, reassoc_of% sr] <;> rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.Reflexive | {
"line": 351,
"column": 4
} | {
"line": 353,
"column": 47
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nA B : C\nf g : A ⟶ B\ns : B ⟶ A\nsl : s ≫ f = 𝟙 B\nsr : s ≫ g = 𝟙 B\n⊢ ∀ {X Y Z : WalkingReflexivePair} (f_1 : X ⟶ Y) (g_1 : Y ⟶ Z),\n (match X, Z, f_1 ≫ g_1 with\n | x, .(x), Hom.id .(x) =>\n 𝟙\n (match x with\n | zero => B\n ... | rintro _ _ _ ⟨⟩ g <;> cases g <;>
simp only [Category.id_comp, Category.comp_id, Category.assoc, sl, sr,
reassoc_of% sl, reassoc_of% sr] <;> rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monad.Equalizer | {
"line": 104,
"column": 6
} | {
"line": 104,
"column": 99
} | [
{
"pp": "case refine_2.h\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nT : Comonad C\nX : T.Coalgebra\ns : Fork (CofreeEqualizer.topMap X) (CofreeEqualizer.bottomMap X)\nh₁ : s.ι.f ≫ T.map X.a = s.ι.f ≫ T.δ.app X.A\nh₂ : s.pt.a ≫ T.map s.ι.f = s.ι.f ≫ T.δ.app X.A\n⊢ ({ f := s.ι.f ≫ T.ε.app X.A, h := ⋯ } ≫ (beckCoa... | simpa [← T.ε.naturality_assoc, T.left_counit_assoc] using h₁ =≫ T.ε.app ((T : C ⥤ C).obj X.A) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.IsPrimary | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 49
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nS : Submodule R M\nhI : S.IsPrimary\nx y : R\nx✝ : x * y ∈ (S.colon Set.univ).radical\nn : ℕ\nhn : (x * y) ^ n ∈ S.colon Set.univ\nh : ∀ {r : R} {x : M}, r • x ∈ S → x ∈ S ∨ r ∈ (S.colon ↑⊤).radical\n⊢ x ... | refine or_iff_not_imp_left.mpr fun hx ↦ ⟨n, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Ideal.IsPrimary | {
"line": 55,
"column": 4
} | {
"line": 56,
"column": 41
} | [
{
"pp": "case left\nR : Type u_1\ninst✝ : CommSemiring R\nI : Ideal R\nhi : I.radical.IsMaximal\n⊢ I ≠ ⊤",
"usedConstants": [
"False",
"Semiring.toModule",
"CommSemiring.toSemiring",
"Eq.rec",
"Submodule.instTop",
"Ideal",
"NonUnitalNonAssocSemiring.toAddCommMonoid"... | rintro rfl
exact (radical_top R ▸ hi).ne_top rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.IsPrimary | {
"line": 55,
"column": 4
} | {
"line": 56,
"column": 41
} | [
{
"pp": "case left\nR : Type u_1\ninst✝ : CommSemiring R\nI : Ideal R\nhi : I.radical.IsMaximal\n⊢ I ≠ ⊤",
"usedConstants": [
"False",
"Semiring.toModule",
"CommSemiring.toSemiring",
"Eq.rec",
"Submodule.instTop",
"Ideal",
"NonUnitalNonAssocSemiring.toAddCommMonoid"... | rintro rfl
exact (radical_top R ▸ hi).ne_top rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Colon | {
"line": 183,
"column": 6
} | {
"line": 183,
"column": 37
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Ring R\nI : Ideal R\ninst✝ : I.IsTwoSided\n⊢ Module.annihilator R (R ⧸ I) = I",
"usedConstants": [
"Eq.mpr",
"Submodule",
"Submodule.annihilator_quotient",
"Submodule.Quotient.addCommMonoid",
"Submodule.colon",
"Semiring.toModule",
"M... | Submodule.annihilator_quotient, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.LocalizedModule.Submodule | {
"line": 59,
"column": 4
} | {
"line": 59,
"column": 56
} | [
{
"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... | have ⟨y, t, hyt⟩ := IsLocalization.exists_mk'_eq p r | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Ideal.MinimalPrime.Basic | {
"line": 175,
"column": 4
} | {
"line": 176,
"column": 33
} | [
{
"pp": "case mpr\nR : Type u_1\ninst✝ : CommSemiring R\nI : Ideal R\n⊢ I = ⊤ → I.minimalPrimes = ∅",
"usedConstants": [
"Semiring.toModule",
"Ideal.minimalPrimes_top",
"Ideal.minimalPrimes",
"CommSemiring.toSemiring",
"Submodule.instTop",
"Ideal",
"NonUnitalNonAsso... | rintro rfl
exact Ideal.minimalPrimes_top | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.MinimalPrime.Basic | {
"line": 175,
"column": 4
} | {
"line": 176,
"column": 33
} | [
{
"pp": "case mpr\nR : Type u_1\ninst✝ : CommSemiring R\nI : Ideal R\n⊢ I = ⊤ → I.minimalPrimes = ∅",
"usedConstants": [
"Semiring.toModule",
"Ideal.minimalPrimes_top",
"Ideal.minimalPrimes",
"CommSemiring.toSemiring",
"Submodule.instTop",
"Ideal",
"NonUnitalNonAsso... | rintro rfl
exact Ideal.minimalPrimes_top | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Localization.Ideal | {
"line": 70,
"column": 4
} | {
"line": 70,
"column": 70
} | [
{
"pp": "case h\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nc x : S\nx' : ↥I × ↥M\nhx : x * (algebraMap R S) ↑x'.2 = (algebraMap R S) ↑x'.1\nc' : R × ↥M\nhc : c * (algebraMap R S) ↑c'.2 = (algebraM... | simp only [Z, ← hx, ← hc, smul_eq_mul, Submonoid.coe_mul, map_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Localization.Ideal | {
"line": 206,
"column": 6
} | {
"line": 206,
"column": 28
} | [
{
"pp": "case mp.refine_2\nR : 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\nJ : Ideal S\nh : J.IsPrime\nx y : R\nhxy : (algebraMap R S) x * (algebraMap R S) y ∈ J\n⊢ x ∈ Ideal.under R J ∨ y ∈ Ideal.under R J",
"u... | exact h.mem_or_mem hxy | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Module.LocalizedModule.Submodule | {
"line": 296,
"column": 4
} | {
"line": 297,
"column": 98
} | [
{
"pp": "R✝ : Type u_1\nS✝ : Type u_2\nM✝ : Type u_3\nN✝ : Type u_4\ninst✝²¹ : CommSemiring R✝\ninst✝²⁰ : CommSemiring S✝\ninst✝¹⁹ : AddCommMonoid M✝\ninst✝¹⁸ : AddCommMonoid N✝\ninst✝¹⁷ : Module R✝ M✝\ninst✝¹⁶ : Module R✝ N✝\ninst✝¹⁵ : Algebra R✝ S✝\ninst✝¹⁴ : Module S✝ N✝\ninst✝¹³ : IsScalarTower R✝ S✝ N✝\np✝... | exact ⟨⟨Submodule.Quotient.mk y, s⟩,
by simp only [Function.uncurry_apply_pair, toLocalizedQuotient'_mk, ← mk_smul, mk'_cancel']⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Ideal.Quotient.Nilpotent | {
"line": 43,
"column": 2
} | {
"line": 44,
"column": 23
} | [
{
"pp": "case neg.succ.zero\nP : ⦃S : Type u_1⦄ → [inst : CommRing S] → Ideal S → Prop\nh₁ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I : Ideal S), I ^ 2 = ⊥ → P I\nh₂ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I J : Ideal S), I ≤ J → P I → P (map (Quotient.mk I) J) → P J\nS : Type u_1\ninst✝ : CommRing S\nI : Ideal ... | · rw [pow_one] at hI
exact (hI' hI).elim | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Ideal.Quotient.Nilpotent | {
"line": 46,
"column": 2
} | {
"line": 49,
"column": 31
} | [
{
"pp": "case neg.succ.succ.a\nP : ⦃S : Type u_1⦄ → [inst : CommRing S] → Ideal S → Prop\nh₁ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I : Ideal S), I ^ 2 = ⊥ → P I\nh₂ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I J : Ideal S), I ≤ J → P I → P (map (Quotient.mk I) J) → P J\nS : Type u_1\ninst✝ : CommRing S\nI : Idea... | · apply H n.succ _ (I ^ 2)
· rw [← pow_mul, eq_bot_iff, ← hI, Nat.succ_eq_add_one]
apply Ideal.pow_le_pow_right (by lia)
· exact n.succ.lt_succ_self | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Ideal.Quotient.Nilpotent | {
"line": 80,
"column": 77
} | {
"line": 85,
"column": 26
} | [
{
"pp": "S : Type u_1\ninst✝ : CommRing S\nI : Ideal S\nx : S\nn : ℕ\nhn : n ≠ 0\n⊢ IsUnit ((mk (I ^ n)) x) ↔ IsUnit ((mk I) x)",
"usedConstants": [
"Ideal.Quotient.commSemiring",
"Eq.mpr",
"Submodule",
"RingHom.instRingHomClass",
"Semiring.toModule",
"IsScalarTower.right... | by
rw [← IsNilpotent.isUnit_quotient_mk_iff (I := Ideal.map (Ideal.Quotient.mk (I ^ n)) I)]
· rw [← isUnit_map_iff (DoubleQuot.quotQuotEquivQuotOfLE (Ideal.pow_le_self hn))]
rfl
· use n
simp [← Ideal.map_pow] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.Adjugate | {
"line": 101,
"column": 6
} | {
"line": 101,
"column": 19
} | [
{
"pp": "n : Type v\nα : Type w\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA : Matrix n n α\nb : n → α\ni : n\n⊢ Aᵀ.cramer b i = (A.updateRow i b).det",
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring... | cramer_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.Adjugate | {
"line": 105,
"column": 6
} | {
"line": 105,
"column": 19
} | [
{
"pp": "case h\nn : Type v\nα : Type w\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA : Matrix n n α\ni j : n\n⊢ Aᵀ.cramer (A i) j = Pi.single i A.det j",
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semirin... | cramer_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.Adjugate | {
"line": 134,
"column": 33
} | {
"line": 134,
"column": 46
} | [
{
"pp": "n : Type v\nα : Type w\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\ninst✝¹ : CommRing α\ninst✝ : Subsingleton n\nA : Matrix n n α\nb : n → α\ni : n\n⊢ A.cramer b i = b i",
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Se... | cramer_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.Adjugate | {
"line": 161,
"column": 32
} | {
"line": 161,
"column": 45
} | [
{
"pp": "case h\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix m m α\ne : n ≃ m\nb : n → α\ni : n\n⊢ (A.submatrix ⇑e ⇑e).cramer b i = A.cramer (b ∘ ⇑e.symm) (e i)",
"usedConstants": [
"Pi.Funct... | cramer_apply, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.Matrix.Adjugate | {
"line": 196,
"column": 30
} | {
"line": 196,
"column": 43
} | [
{
"pp": "n : Type v\nα : Type w\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA : Matrix n n α\ni j : n\n⊢ Aᵀ.cramer (Pi.single i 1) j = (A.updateRow j (Pi.single i 1)).det",
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
"NonUnitalCommRing.toNonUnitalNonAssocCommRin... | cramer_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Laurent | {
"line": 574,
"column": 4
} | {
"line": 575,
"column": 83
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝¹ : CommSemiring R\nS : Type u_3\ninst✝ : CommSemiring S\nf : R →+* S\nx : Sˣ\nn : ℤ\nhn : n < 0\n⊢ (eval₂ f x) (T n) = ↑(x ^ n)",
"usedConstants": [
"LaurentPolynomial.T",
"zpow_natCast",
"Units.val",
"Eq.mpr",
"Preorder.toLT",
"Divi... | obtain ⟨m, rfl⟩ := Int.exists_eq_neg_ofNat hn.le
rw [eval₂_T_neg_n, zpow_neg, zpow_natCast, ← inv_pow, Units.val_pow_eq_pow_val] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Laurent | {
"line": 574,
"column": 4
} | {
"line": 575,
"column": 83
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝¹ : CommSemiring R\nS : Type u_3\ninst✝ : CommSemiring S\nf : R →+* S\nx : Sˣ\nn : ℤ\nhn : n < 0\n⊢ (eval₂ f x) (T n) = ↑(x ^ n)",
"usedConstants": [
"LaurentPolynomial.T",
"zpow_natCast",
"Units.val",
"Eq.mpr",
"Preorder.toLT",
"Divi... | obtain ⟨m, rfl⟩ := Int.exists_eq_neg_ofNat hn.le
rw [eval₂_T_neg_n, zpow_neg, zpow_natCast, ← inv_pow, Units.val_pow_eq_pow_val] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Dimension.Localization | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 71
} | [
{
"pp": "case inr\nR : Type uR\nM : Type uM\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\nT : Type uT\ninst✝⁷ : CommRing T\ninst✝⁶ : NoZeroDivisors T\ninst✝⁵ : Algebra R T\ninst✝⁴ : FaithfulSMul R T\nP : Type uP\ninst✝³ : AddCommGroup P\ninst✝² : Module R P\ninst✝¹ : Module T P\ninst✝ : I... | let _ : Algebra FT (FR ⊗[R] FT) := Algebra.TensorProduct.rightAlgebra | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.LinearAlgebra.Matrix.Adjugate | {
"line": 440,
"column": 4
} | {
"line": 441,
"column": 62
} | [
{
"pp": "case right\nn : Type v\nα : Type w\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA : Matrix n n α\nhA : IsLeftRegular A.det\nB C : Matrix n n α\nh : B * A = C * A\n⊢ A.det • B = A.det • C",
"usedConstants": [
"Eq.mpr",
"Matrix.smul",
"NonAssocSemiring.toAddCommM... | rw [← Matrix.mul_one B, ← Matrix.mul_one C, ← Matrix.mul_smul, ← Matrix.mul_smul, ←
mul_adjugate, ← Matrix.mul_assoc, ← Matrix.mul_assoc, h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Matrix.Adjugate | {
"line": 466,
"column": 2
} | {
"line": 468,
"column": 32
} | [
{
"pp": "n : Type v\nα : Type w\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA B : Matrix n n α\ng : Matrix n n α → Matrix n n α[X] := fun M ↦ M.map ⇑C + X • 1\nf' : Matrix n n α[X] →+* Matrix n n α := (evalRingHom 0).mapMatrix\nf'_inv : ∀ (M : Matrix n n α), f' (g M) = M\nf'_adj : ∀ (M : Ma... | have f'_g_mul : ∀ M N : Matrix n n α, f' (g M * g N) = M * N := by
intro M N
rw [map_mul, f'_inv, f'_inv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Data.Matrix.Invertible | {
"line": 154,
"column": 6
} | {
"line": 154,
"column": 65
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nα : Type u_3\ninst✝⁷ : Fintype n\ninst✝⁶ : DecidableEq n\ninst✝⁵ : Fintype m\ninst✝⁴ : DecidableEq m\ninst✝³ : Ring α\nA : Matrix n n α\nU : Matrix n m α\nC : Matrix m m α\nV : Matrix m n α\ninst✝² : Invertible A\ninst✝¹ : Invertible C\ninst✝ : Invertible (⅟C + V * ⅟A * U)\n... | simp_rw [add_sub_assoc, add_mul, mul_sub, Matrix.mul_assoc] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Data.Matrix.Invertible | {
"line": 154,
"column": 6
} | {
"line": 154,
"column": 65
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nα : Type u_3\ninst✝⁷ : Fintype n\ninst✝⁶ : DecidableEq n\ninst✝⁵ : Fintype m\ninst✝⁴ : DecidableEq m\ninst✝³ : Ring α\nA : Matrix n n α\nU : Matrix n m α\nC : Matrix m m α\nV : Matrix m n α\ninst✝² : Invertible A\ninst✝¹ : Invertible C\ninst✝ : Invertible (⅟C + V * ⅟A * U)\n... | simp_rw [add_sub_assoc, add_mul, mul_sub, Matrix.mul_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Matrix.Invertible | {
"line": 154,
"column": 6
} | {
"line": 154,
"column": 65
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nα : Type u_3\ninst✝⁷ : Fintype n\ninst✝⁶ : DecidableEq n\ninst✝⁵ : Fintype m\ninst✝⁴ : DecidableEq m\ninst✝³ : Ring α\nA : Matrix n n α\nU : Matrix n m α\nC : Matrix m m α\nV : Matrix m n α\ninst✝² : Invertible A\ninst✝¹ : Invertible C\ninst✝ : Invertible (⅟C + V * ⅟A * U)\n... | simp_rw [add_sub_assoc, add_mul, mul_sub, Matrix.mul_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Matrix.Invertible | {
"line": 163,
"column": 13
} | {
"line": 165,
"column": 10
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nα : Type u_3\ninst✝⁷ : Fintype n\ninst✝⁶ : DecidableEq n\ninst✝⁵ : Fintype m\ninst✝⁴ : DecidableEq m\ninst✝³ : Ring α\nA : Matrix n n α\nU : Matrix n m α\nC : Matrix m m α\nV : Matrix m n α\ninst✝² : Invertible A\ninst✝¹ : Invertible C\ninst✝ : Invertible (⅟C + V * ⅟A * U)\n... | by
rw [Matrix.mul_invOf_cancel_right]
abel | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.Transvection | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 60
} | [
{
"pp": "n : Type u_1\nR : Type u₂\ninst✝² : DecidableEq n\ninst✝¹ : CommRing R\ninst✝ : Fintype n\ni✝ j✝ : n\nt_hij : i✝ ≠ j✝\nc✝ : R\n⊢ { i := i✝, j := j✝, hij := t_hij, c := c✝ }.inv.toMatrix * { i := i✝, j := j✝, hij := t_hij, c := c✝ }.toMatrix = 1",
"usedConstants": [
"NegZeroClass.toNeg",
... | simp [toMatrix, transvection_mul_transvection_same, t_hij] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Matrix.Transvection | {
"line": 195,
"column": 2
} | {
"line": 195,
"column": 60
} | [
{
"pp": "n : Type u_1\nR : Type u₂\ninst✝² : DecidableEq n\ninst✝¹ : CommRing R\ninst✝ : Fintype n\ni✝ j✝ : n\nt_hij : i✝ ≠ j✝\nc✝ : R\n⊢ { i := i✝, j := j✝, hij := t_hij, c := c✝ }.toMatrix * { i := i✝, j := j✝, hij := t_hij, c := c✝ }.inv.toMatrix = 1",
"usedConstants": [
"add_neg_cancel",
"Ne... | simp [toMatrix, transvection_mul_transvection_same, t_hij] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Matrix.Block | {
"line": 402,
"column": 2
} | {
"line": 403,
"column": 39
} | [
{
"pp": "α : Type u_1\nm : Type u_3\nR : Type v\nM : Matrix m m R\nb : m → α\ninst✝⁴ : CommRing R\ninst✝³ : DecidableEq m\ninst✝² : Fintype m\ninst✝¹ : LinearOrder α\ninst✝ : Invertible M\nhM : M.BlockTriangular b\nk : α\np : m → Prop := fun i ↦ b i < k\nq : m → Prop := fun i ↦ ¬b i < k\nh_sum : M⁻¹.toBlock q p... | rw [this, ← Matrix.zero_mul (M.toBlock p p)⁻¹, ← h_mul_eq_zero,
mul_inv_cancel_right_of_invertible] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.MatrixPolynomialAlgebra | {
"line": 121,
"column": 2
} | {
"line": 122,
"column": 92
} | [
{
"pp": "case a.a\nR : Type u_1\ninst✝² : CommSemiring R\nn : Type w\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\np : R[X]\nm : ℕ\ni j : n\n⊢ (matPolyEquiv (p • 1)).coeff m i j = (Polynomial.map (algebraMap R (Matrix n n R)) p).coeff m i j",
"usedConstants": [
"Eq.mpr",
"Matrix.smul",
"Poly... | simp only [matPolyEquiv_coeff_apply, smul_apply, one_apply, smul_eq_mul, mul_ite, mul_one,
mul_zero, coeff_map, algebraMap_matrix_apply, Algebra.algebraMap_self, RingHom.id_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Matrix.SchurComplement | {
"line": 307,
"column": 2
} | {
"line": 308,
"column": 36
} | [
{
"pp": "l : Type u_1\nm : Type u_2\nn : Type u_3\nα : Type u_4\ninst✝⁸ : Fintype l\ninst✝⁷ : Fintype m\ninst✝⁶ : Fintype n\ninst✝⁵ : DecidableEq l\ninst✝⁴ : DecidableEq m\ninst✝³ : DecidableEq n\ninst✝² : CommRing α\nA : Matrix m m α\nB : Matrix m n α\nC : Matrix n m α\nD : Matrix n n α\ninst✝¹ : Invertible D\... | letI iBD : Invertible (fromBlocks 1 (B * ⅟D) 0 1 : Matrix (m ⊕ n) (m ⊕ n) α) :=
fromBlocksZero₂₁Invertible _ _ _ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLetI___1 | Lean.Parser.Tactic.tacticLetI__ |
Mathlib.LinearAlgebra.Matrix.SchurComplement | {
"line": 320,
"column": 2
} | {
"line": 321,
"column": 89
} | [
{
"pp": "l : Type u_1\nm : Type u_2\nn : Type u_3\nα : Type u_4\ninst✝⁸ : Fintype l\ninst✝⁷ : Fintype m\ninst✝⁶ : Fintype n\ninst✝⁵ : DecidableEq l\ninst✝⁴ : DecidableEq m\ninst✝³ : DecidableEq n\ninst✝² : CommRing α\nA : Matrix m m α\nB : Matrix m n α\nC : Matrix n m α\nD : Matrix n n α\ninst✝¹ : Invertible A\... | letI iABCD' :=
submatrixEquivInvertible (fromBlocks A B C D) (Equiv.sumComm _ _) (Equiv.sumComm _ _) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLetI___1 | Lean.Parser.Tactic.tacticLetI__ |
Mathlib.RingTheory.Polynomial.Nilpotent | {
"line": 79,
"column": 4
} | {
"line": 79,
"column": 88
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝ : CommRing R\nP : R[X]\nh : ∀ (i : ℕ), IsNilpotent (P.coeff i)\n⊢ IsNilpotent (P.sum fun n a ↦ (monomial n) a)",
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"CommSemiring.toSemiring",
"Finset",
"LinearMap.instFunLike",
"Me... | exact isNilpotent_sum (fun i _ ↦ by simpa only [isNilpotent_monomial_iff] using h i) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Module.SpanRank | {
"line": 90,
"column": 6
} | {
"line": 90,
"column": 36
} | [
{
"pp": "R : Type u_1\nM : Type u\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np : Submodule R M\n⊢ toENat p.spanRank = ⨅ s, ↑(↑s).card",
"usedConstants": [
"Eq.mpr",
"Submodule",
"Set.encard",
"iInf",
"instCompleteLinearOrderENat",
"Submodule.spanR... | spanRank_toENat_eq_iInf_encard | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.ENat.Lattice | {
"line": 258,
"column": 2
} | {
"line": 258,
"column": 37
} | [
{
"pp": "case inr\nι : Sort u_2\nf : ι → ℕ∞\na : ℕ∞\ninst✝ : Nonempty ι\nha : a ≠ ⊤\nh : ∀ (x : ι), f x ≤ a\ni : ι\n⊢ f i ≤ a - ⨅ i, a - f i",
"usedConstants": [
"Eq.mpr",
"iInf",
"instCompleteLinearOrderENat",
"instSubENat",
"congrArg",
"CompletelyDistribLattice.toComple... | rw [← ENat.sub_sub_cancel ha (h _)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.ENat.Lattice | {
"line": 309,
"column": 2
} | {
"line": 309,
"column": 46
} | [
{
"pp": "ι : Sort u_2\na : ℕ∞\nκ : Sort u_4\nq₁ : ι → Sort u_5\nq₂ : κ → Sort u_6\nf : (i : ι) → q₁ i → ℕ∞\ng : (k : κ) → q₂ k → ℕ∞\nh : ∀ (i : ι) (pi : q₁ i) (k : κ) (qk : q₂ k), a ≤ f i pi + g k qk\n⊢ a ≤ ⨅ i, ⨅ j, ⨅ i_1, ⨅ j_1, f i j + g i_1 j_1",
"usedConstants": [
"iInf",
"instCompleteLinea... | exact le_iInf₂ fun i hi => le_iInf₂ (h i hi) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Module.SpanRank | {
"line": 188,
"column": 2
} | {
"line": 188,
"column": 46
} | [
{
"pp": "R : Type u_1\nM : Type u\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np : Submodule R M\nh : p.FG\ns : Set M\nhs₁ : #↑s = p.spanRank\nhs₂ : span R s = p\n⊢ s.encard = ↑p.spanFinrank",
"usedConstants": [
"Submodule.fg_iff_spanRank_eq_spanFinrank",
"Iff.mpr",
... | have := fg_iff_spanRank_eq_spanFinrank.mpr h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap | {
"line": 60,
"column": 32
} | {
"line": 60,
"column": 71
} | [
{
"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\nf : Module.End R M\ni : ι\n⊢ f ((Fintype.linearCombination R b) (Pi.single i 1)) = f (b i)",
"usedConstants": [
"Eq.mpr",
"P... | Fintype.linearCombination_apply_single, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap | {
"line": 104,
"column": 73
} | {
"line": 110,
"column": 5
} | [
{
"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\nA A' : Matrix ι ι R\nf f' : Module.End R M\nh : Represents b A f\nh' : Represents b A' f'\n⊢ Represents b (A * A') (f * f')",
"usedConst... | by
delta Matrix.Represents PiToModule.fromMatrix
rw [LinearMap.comp_apply, AlgEquiv.toLinearMap_apply, map_mul]
ext
dsimp [PiToModule.fromEnd]
rw [← h'.congr_fun, ← h.congr_fun]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap | {
"line": 114,
"column": 2
} | {
"line": 114,
"column": 64
} | [
{
"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\n⊢ ((LinearMap.llcomp R (ι → R) (ι → R) M) (Fintype.linearCombination R b) ∘ₗ algEquivMatrix'.symm.toLinearMap) 1 =\n (PiToModule.fromEnd ... | rw [LinearMap.comp_apply, AlgEquiv.toLinearMap_apply, map_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.IntegralClosure.Algebra.Basic | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 85
} | [
{
"pp": "R : Type u_1\nS : Type u_4\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nx y z : S\nhx : f.IsIntegralElem x\nhy : f.IsIntegralElem y\nhz : z ∈ Subring.closure {x, y}\nthis : Algebra R S := f.toAlgebra\n⊢ f.IsIntegralElem z",
"usedConstants": [
"Submodule",
"HMul.hMul",
"I... | have := (IsIntegral.fg_adjoin_singleton hx).mul (IsIntegral.fg_adjoin_singleton hy) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.IntegralClosure.Algebra.Basic | {
"line": 137,
"column": 2
} | {
"line": 142,
"column": 82
} | [
{
"pp": "R : Type u_1\nS : Type u_4\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nx y z : S\nhx : f.IsIntegralElem x\nhy : f.IsIntegralElem y\nhz : z ∈ Subring.closure {x, y}\n⊢ f.IsIntegralElem z",
"usedConstants": [
"Subalgebra.instSetLike",
"Iff.mpr",
"Submodule",
"HMul.h... | letI : Algebra R S := f.toAlgebra
have := (IsIntegral.fg_adjoin_singleton hx).mul (IsIntegral.fg_adjoin_singleton hy)
rw [← Algebra.adjoin_union_coe_submodule, Set.singleton_union] at this
exact
IsIntegral.of_mem_of_fg (Algebra.adjoin R {x, y}) this z
(Algebra.mem_adjoin_iff.2 <| Subring.closure_mono Se... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.IntegralClosure.Algebra.Basic | {
"line": 137,
"column": 2
} | {
"line": 142,
"column": 82
} | [
{
"pp": "R : Type u_1\nS : Type u_4\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nx y z : S\nhx : f.IsIntegralElem x\nhy : f.IsIntegralElem y\nhz : z ∈ Subring.closure {x, y}\n⊢ f.IsIntegralElem z",
"usedConstants": [
"Subalgebra.instSetLike",
"Iff.mpr",
"Submodule",
"HMul.h... | letI : Algebra R S := f.toAlgebra
have := (IsIntegral.fg_adjoin_singleton hx).mul (IsIntegral.fg_adjoin_singleton hy)
rw [← Algebra.adjoin_union_coe_submodule, Set.singleton_union] at this
exact
IsIntegral.of_mem_of_fg (Algebra.adjoin R {x, y}) this z
(Algebra.mem_adjoin_iff.2 <| Subring.closure_mono Se... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Taylor | {
"line": 193,
"column": 17
} | {
"line": 193,
"column": 45
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nr✝ : R\nf : R[X]\nr : R\nP : R[X]\n⊢ (taylorAlgHom (-r)) ((↑↑(taylorAlgHom r).toRingHom).toFun P) = P",
"usedConstants": [
"AddHom.mk.congr_simp",
"Polynomial.C",
"Polynomial.comp_assoc",
"NegZeroClass.toNeg",
"RingHom.instRingHomClass... | by simp [taylor, comp_assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Taylor | {
"line": 194,
"column": 17
} | {
"line": 194,
"column": 45
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nr✝ : R\nf : R[X]\nr : R\nP : R[X]\n⊢ (↑↑(taylorAlgHom r).toRingHom).toFun ((taylorAlgHom (-r)) P) = P",
"usedConstants": [
"AddHom.mk.congr_simp",
"Polynomial.C",
"Polynomial.comp_assoc",
"NegZeroClass.toNeg",
"RingHom.instRingHomClass... | by simp [taylor, comp_assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.ScaleRoots | {
"line": 195,
"column": 4
} | {
"line": 195,
"column": 20
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\np q : R[X]\nr : R\nn : ℕ\n⊢ r ^ (p.natDegree + q.natDegree - (p * q).natDegree) * ((p * q).coeff n * r ^ ((p * q).natDegree - n)) =\n (∑ x ∈ Finset.antidiagonal n, p.coeff x.1 * q.coeff x.2) * r ^ (p.natDegree + q.natDegree - n)",
"usedConstants": [
"E... | rw [← coeff_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Polynomial.ScaleRoots | {
"line": 209,
"column": 6
} | {
"line": 209,
"column": 17
} | [
{
"pp": "case inr.inr\nR : Type u_1\ninst✝ : CommSemiring R\np q : R[X]\nr : R\nn a b : ℕ\ne : a + b = n\nha : a ≤ p.natDegree\nhb : b ≤ q.natDegree\n⊢ p.coeff a * q.coeff b * r ^ (p.natDegree + q.natDegree - n) =\n p.coeff a * r ^ (p.natDegree - a) * (q.coeff b * r ^ (q.natDegree - b))",
"usedConstants"... | | inr hb => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.RingTheory.Polynomial.ScaleRoots | {
"line": 215,
"column": 24
} | {
"line": 215,
"column": 41
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\np q : R[X]\nr : R\nh : p.leadingCoeff * q.leadingCoeff ≠ 0\n⊢ (p * q).scaleRoots r = r ^ (p.natDegree + q.natDegree - (p * q).natDegree) • (p * q).scaleRoots r",
"usedConstants": [
"Eq.mpr",
"instHSMul",
"HMul.hMul",
"congrArg",
"C... | natDegree_mul' h, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Splits | {
"line": 330,
"column": 6
} | {
"line": 330,
"column": 32
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nf : R[X]\nS : Type u_4\ninst✝¹ : CommRing S\ninst✝ : IsDomain S\ni : R →+* S\nhi : Function.Injective ⇑i\nhf : (map i f).Splits\nj : (a : S) → a ∈ (map i f).roots → R\nhj : ∀ (a : S) (a_1 : a ∈ (map i f).roots), i (j a a_1) = a\n⊢ f.Splits",
"usedConstants": [
... | splits_iff_exists_multiset | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Localization.Integral | {
"line": 323,
"column": 2
} | {
"line": 323,
"column": 27
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nS : Type u_2\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nt s : S\nhst : s * t = 1\nht : IsIntegral (↥R[s]) t\na✝ : Nontrivial S\n⊢ IsIntegral R t",
"usedConstants": [
"CommSemiring.toSemiring",
"AlgHom",
"Polynomial.algebraOfAlgebra",
"Algeb... | let φ := aeval (R := R) s | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.Flat.FaithfullyFlat.Basic | {
"line": 492,
"column": 17
} | {
"line": 492,
"column": 28
} | [
{
"pp": "case mp.a.h.h\nR : Type u\nM : Type v\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nA : Type u_1\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : FaithfullyFlat R A\nm : M\nf : R →ₗ[R] M := (LinearMap.lsmul R M).flip m\na : A\nh : a • 1 ⊗ₜ[R] m = 0\n⊢ ((AlgebraTensorModule.curry (L... | smul_tmul', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Flat.FaithfullyFlat.Basic | {
"line": 560,
"column": 42
} | {
"line": 566,
"column": 56
} | [
{
"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\n⊢ FaithfullyFlat S (S ⊗[R] M)",
"usedConstants": [
"Eq.mpr",
"TensorProduct.AlgebraTensorModule.cancelBaseChang... | by
rw [Module.FaithfullyFlat.iff_flat_and_rTensor_reflects_triviality]
refine ⟨inferInstance, fun N _ _ hN ↦ ?_⟩
let _ : Module R N := Module.compHom N (algebraMap R S)
have : IsScalarTower R S N := IsScalarTower.of_algebraMap_smul fun r ↦ congrFun rfl
have := (AlgebraTensorModule.cancelBaseChange R S S N M).... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Algebraic.Integral | {
"line": 431,
"column": 84
} | {
"line": 435,
"column": 56
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : NoZeroDivisors R\ny z : S\nhy : y ∈ nonZeroDivisors S\nalg_y : IsAlgebraic R y\nalg_yz : IsAlgebraic R (y * z)\n⊢ IsAlgebraic R z",
"usedConstants": [
"Subalgebra.instSetLike",
"Iff.mpr",... | by
have ⟨t, ht, r, hr, eq⟩ := alg_y.exists_nonzero_eq_adjoin_mul hy
have := alg_yz.mul (Algebra.isAlgebraic_adjoin_singleton_iff.mpr alg_y _ ht)
rw [mul_right_comm, eq, ← Algebra.smul_def] at this
exact this.of_smul (mem_nonZeroDivisors_of_ne_zero hr) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Ring.Idempotent | {
"line": 84,
"column": 69
} | {
"line": 86,
"column": 70
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\na✝ b a : { a // a.1 * a.2 = 0 ∧ a.1 + a.2 = 1 }\n⊢ (↑(a ⊓ aᶜ)).2 = (↑⟨(0, 1), ⋯⟩).2",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"IsIdempotentElem.of_mul_add",
"instBooleanAlgebraSubtype... | by
simp_rw +instances [(· ⊔ ·), (· ⊓ ·), (·ᶜ), SemilatticeSup.sup,
(IsIdempotentElem.of_mul_add a.2.1 a.2.2).1.eq, add_comm, a.2.2] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Ideal.MinimalPrime.Localization | {
"line": 77,
"column": 2
} | {
"line": 78,
"column": 78
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nI p : Ideal R\nhp : p ∈ I.minimalPrimes\nx : R\nhx✝ : x ∈ p\ny : R\nhy : y ∉ I.radical\nn : ℕ\nhx : (x * y) ^ n ∈ I\nH : ∃ m, x ^ m * y ^ n ∈ I\n⊢ ∃ y ∉ I, x * y ∈ I",
"usedConstants": [
"MulOne.toOne",
"Semiring.toModule",
"HMul.hMul",
... | have : Nat.find H ≠ 0 :=
fun h ↦ hy ⟨n, by simpa only [h, pow_zero, one_mul] using Nat.find_spec H⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Spectrum.Prime.RingHom | {
"line": 316,
"column": 2
} | {
"line": 320,
"column": 47
} | [
{
"pp": "case refine_2\nR : Type u\ninst✝² : CommRing R\nS : Type u_1\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : PrimeSpectrum R\nk : Type u := p.asIdeal.ResidueField\nh : p ∈ Set.range (comap (algebraMap R S))\n⊢ Nontrivial (p.asIdeal.ResidueField ⊗[R] S)",
"usedConstants": [
"Nontrivial",
... | · obtain ⟨q, rfl⟩ := h
let f : k ⊗[R] S →ₐ[R] q.asIdeal.ResidueField :=
Algebra.TensorProduct.lift (Ideal.ResidueField.mapₐ _ _ (Algebra.ofId _ _) rfl)
(IsScalarTower.toAlgHom _ _ _) (fun _ _ ↦ Commute.all ..)
exact RingHom.domain_nontrivial f.toRingHom | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Ideal.MinimalPrime.Localization | {
"line": 207,
"column": 98
} | {
"line": 210,
"column": 84
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommSemiring R\nS : Submonoid R\nA : Type u_2\ninst✝² : CommSemiring A\ninst✝¹ : Algebra R A\ninst✝ : IsLocalization S A\nJ : Ideal A\n⊢ (Ideal.comap (algebraMap R A) J).minimalPrimes = Ideal.comap (algebraMap R A) '' J.minimalPrimes",
"usedConstants": [
"Iff.mpr",
... | by
conv_rhs => rw [← map_under S A J, minimalPrimes_map S]
refine (Set.image_preimage_eq_iff.mpr ?_).symm
exact subset_trans (Ideal.minimalPrimes_comap_subset (algebraMap R A) J) (by simp) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.RelSeries | {
"line": 318,
"column": 39
} | {
"line": 318,
"column": 57
} | [
{
"pp": "case h.e'_5.h.e'_4.h.e'_4.h.e'_2\nα : Type u_1\nr : SetRel α α\nβ : Type u_2\ns : SetRel β β\np q : RelSeries r\nconnect : (p.last, q.head) ∈ r\ni : Fin (p.length + q.length + 1)\nhi : Fin.castLE (@RelSeries.append._proof_3 α r p q) (Fin.last p.length) < i\nx : Fin (p.length + 1 + (q.length + 1)) := Fi... | Nat.sub_add_cancel | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Spectrum.Maximal.Localization | {
"line": 203,
"column": 2
} | {
"line": 213,
"column": 95
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nsurj : Function.Surjective ⇑(toPiLocalization R)\nI : PrimeSpectrum R\n⊢ I.asIdeal.IsMaximal",
"usedConstants": [
"PrimeSpectrum.mk",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClass",
"PrimeSpectrum.toPiLocalization"... | classical
have ⟨J, max, le⟩ := I.1.exists_le_maximal I.2.ne_top
obtain ⟨r, hr⟩ := surj (Function.update 0 ⟨J, max.isPrime⟩ 1)
by_contra h
have hJ : algebraMap _ _ r = _ := (congr_fun hr _).trans (Function.update_self ..)
have hI : algebraMap _ _ r = _ := congr_fun hr I
rw [← IsLocalization.lift_eq (M := J.p... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.RingTheory.Spectrum.Maximal.Localization | {
"line": 203,
"column": 2
} | {
"line": 213,
"column": 95
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nsurj : Function.Surjective ⇑(toPiLocalization R)\nI : PrimeSpectrum R\n⊢ I.asIdeal.IsMaximal",
"usedConstants": [
"PrimeSpectrum.mk",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClass",
"PrimeSpectrum.toPiLocalization"... | classical
have ⟨J, max, le⟩ := I.1.exists_le_maximal I.2.ne_top
obtain ⟨r, hr⟩ := surj (Function.update 0 ⟨J, max.isPrime⟩ 1)
by_contra h
have hJ : algebraMap _ _ r = _ := (congr_fun hr _).trans (Function.update_self ..)
have hI : algebraMap _ _ r = _ := congr_fun hr I
rw [← IsLocalization.lift_eq (M := J.p... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Spectrum.Maximal.Localization | {
"line": 203,
"column": 2
} | {
"line": 213,
"column": 95
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nsurj : Function.Surjective ⇑(toPiLocalization R)\nI : PrimeSpectrum R\n⊢ I.asIdeal.IsMaximal",
"usedConstants": [
"PrimeSpectrum.mk",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClass",
"PrimeSpectrum.toPiLocalization"... | classical
have ⟨J, max, le⟩ := I.1.exists_le_maximal I.2.ne_top
obtain ⟨r, hr⟩ := surj (Function.update 0 ⟨J, max.isPrime⟩ 1)
by_contra h
have hJ : algebraMap _ _ r = _ := (congr_fun hr _).trans (Function.update_self ..)
have hI : algebraMap _ _ r = _ := congr_fun hr I
rw [← IsLocalization.lift_eq (M := J.p... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Compactness.Bases | {
"line": 29,
"column": 83
} | {
"line": 29,
"column": 92
} | [
{
"pp": "X : Type u_1\nι : Type u_2\ninst✝ : TopologicalSpace X\nb : ι → Set X\nhb : IsTopologicalBasis (range b)\nU : Set X\nhUc : IsCompact U\nhUo : IsOpen U\nY : Type u_1\nf' : Y → ι\ne : U = ⋃ i, (b ∘ f') i\nhf' : ∀ (i : Y), b (f' i) = (b ∘ f') i\n⊢ U ⊆ ⋃ i, (b ∘ f') i",
"usedConstants": [
"Eq.mpr... | by rw [e] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.KrullDimension | {
"line": 275,
"column": 2
} | {
"line": 275,
"column": 88
} | [
{
"pp": "α : Type u_1\ninst✝ : Preorder α\na : α\nn : ℕ∞\n⊢ height a + n = ⨆ p, ⨆ (_ : RelSeries.last p = a), ↑p.length + n",
"usedConstants": [
"Preorder.toLT",
"RelSeries.last",
"setOf",
"FiniteDimensionalOrder.match_1",
"Subtype",
"Nonempty.intro",
"Subtype.mk",
... | have hne : Nonempty { p : LTSeries α // p.last = a } := ⟨RelSeries.singleton _ a, rfl⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Order.KrullDimension | {
"line": 368,
"column": 2
} | {
"line": 368,
"column": 88
} | [
{
"pp": "α : Type u_1\ninst✝ : Preorder α\na : α\nn : ℕ\nh : ↑n ≤ height a\n⊢ ∃ p, RelSeries.last p = a ∧ p.length = n",
"usedConstants": [
"Preorder.toLT",
"RelSeries.last",
"setOf",
"FiniteDimensionalOrder.match_1",
"Subtype",
"Nonempty.intro",
"Subtype.mk",
... | have hne : Nonempty { p : LTSeries α // p.last = a } := ⟨RelSeries.singleton _ a, rfl⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Order.RelSeries | {
"line": 496,
"column": 2
} | {
"line": 496,
"column": 26
} | [
{
"pp": "α : Type u_1\nr : SetRel α α\np : RelSeries r\nx : α\nhx : (x, p.head) ∈ r\n⊢ (p.cons x hx).toList = x :: p.toList",
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"RelSeries.toList",
"RelSeries",
"List.cons",
"instHAppendOfAppend",
"List",
"Rel... | rw [cons, toList_append] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.RelSeries | {
"line": 698,
"column": 4
} | {
"line": 698,
"column": 22
} | [
{
"pp": "α : Type u_1\nr : SetRel α α\nβ : Type u_2\ns : SetRel β β\np q : RelSeries r\nconnect : p.last = q.head\n⊢ ∀ (i : Fin (p.length + q.length)),\n (Fin.addCases (p.toFun ∘ Fin.castSucc) q.toFun i.castSucc, Fin.addCases (p.toFun ∘ Fin.castSucc) q.toFun i.succ) ∈ r",
"usedConstants": [
"SetRel... | apply Fin.addCases | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.Sets.Opens | {
"line": 335,
"column": 4
} | {
"line": 335,
"column": 13
} | [
{
"pp": "case hb'\nα : Type u_2\ninst✝ : TopologicalSpace α\nι : Type u_5\nb : ι → Opens α\nhb : IsBasis (range b)\nhb' : ∀ (i : ι), IsCompact ↑(b i)\nU : Set α\n⊢ ∀ (i : ι), IsCompact (b i).carrier",
"usedConstants": []
}
] | exact hb' | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Sets.Opens | {
"line": 335,
"column": 4
} | {
"line": 335,
"column": 13
} | [
{
"pp": "case hb'\nα : Type u_2\ninst✝ : TopologicalSpace α\nι : Type u_5\nb : ι → Opens α\nhb : IsBasis (range b)\nhb' : ∀ (i : ι), IsCompact ↑(b i)\nU : Set α\n⊢ ∀ (i : ι), IsCompact (b i).carrier",
"usedConstants": []
}
] | exact hb' | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Sets.Opens | {
"line": 335,
"column": 4
} | {
"line": 335,
"column": 13
} | [
{
"pp": "case hb'\nα : Type u_2\ninst✝ : TopologicalSpace α\nι : Type u_5\nb : ι → Opens α\nhb : IsBasis (range b)\nhb' : ∀ (i : ι), IsCompact ↑(b i)\nU : Set α\n⊢ ∀ (i : ι), IsCompact (b i).carrier",
"usedConstants": []
}
] | exact hb' | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.KrullDimension | {
"line": 824,
"column": 4
} | {
"line": 824,
"column": 64
} | [
{
"pp": "case a\nα : Type u_1\ninst✝¹ : Preorder α\ninst✝ : Nonempty α\n⊢ krullDim α ≤ ↑(⨆ a, height a + coheight a)",
"usedConstants": [
"Eq.mpr",
"WithBot.some",
"WithBot",
"instCompleteLinearOrderENat",
"congrArg",
"iSup",
"PartialOrder.toPreorder",
"Preord... | rw [krullDim_eq_iSup_height_of_nonempty, WithBot.coe_le_coe] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.KrullDimension | {
"line": 826,
"column": 4
} | {
"line": 826,
"column": 33
} | [
{
"pp": "case a\nα : Type u_1\ninst✝¹ : Preorder α\ninst✝ : Nonempty α\n⊢ ↑(⨆ a, height a + coheight a) ≤ krullDim α",
"usedConstants": [
"Nonempty",
"Preorder"
]
}
] | wlog hnottop : krullDim α < ⊤ | Mathlib.Tactic._aux_Mathlib_Tactic_WLOG___elabRules_Mathlib_Tactic_wlog_1 | Mathlib.Tactic.wlog |
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