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.RingTheory.EssentialFiniteness | {
"line": 83,
"column": 6
} | {
"line": 83,
"column": 36
} | {
"line": 84,
"column": 6
} | [
{
"pp": "case surj\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nσ : Finset S\nhσ : ∀ (s : S), ∃ t ∈ adjoin R ↑σ, IsUnit t ∧ s * t ∈ adjoin R ↑σ\ns : S\n⊢ ∃ x, s * (algebraMap (↥(adjoin R ↑σ)) S) ↑x.2 = (algebraMap (↥(adjoin R ↑σ)) S) x.1",
"ppTerm": "?surj",
... | [
"case surj\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nσ : Finset S\nhσ : ∀ (s : S), ∃ t ∈ adjoin R ↑σ, IsUnit t ∧ s * t ∈ adjoin R ↑σ\ns t : S\nht : t ∈ adjoin R ↑σ\nht' : IsUnit t\nh : s * t ∈ adjoin R ↑σ\n⊢ ∃ x, s * (algebraMap (↥(adjoin R ↑σ)) S) ↑x.2 = (algebraMa... | obtain ⟨t, ht, ht', h⟩ := hσ s | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.EssentialFiniteness | {
"line": 121,
"column": 4
} | {
"line": 123,
"column": 77
} | {
"line": 124,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : CommRing T\ninst✝³ : Algebra R S\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nσ : Subalgebra R S\nhσ : ∀ (s : S), ∃ t ∈ σ, IsUnit t ∧ s * t ∈ σ\nτ : Set T\nt : T\nht ... | [] | intro t ht
exact ⟨1, Subalgebra.one_mem _, isUnit_one,
(one_smul S t).symm ▸ Algebra.mem_sup_right (Algebra.subset_adjoin ht)⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.EssentialFiniteness | {
"line": 121,
"column": 4
} | {
"line": 123,
"column": 77
} | {
"line": 124,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : CommRing T\ninst✝³ : Algebra R S\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nσ : Subalgebra R S\nhσ : ∀ (s : S), ∃ t ∈ σ, IsUnit t ∧ s * t ∈ σ\nτ : Set T\nt : T\nht ... | [] | intro t ht
exact ⟨1, Subalgebra.one_mem _, isUnit_one,
(one_smul S t).symm ▸ Algebra.mem_sup_right (Algebra.subset_adjoin ht)⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Localization.AtPrime.Basic | {
"line": 283,
"column": 45
} | {
"line": 283,
"column": 49
} | {
"line": 283,
"column": 50
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommSemiring R\nS✝ : Type u_2\ninst✝³ : CommSemiring S✝\ninst✝² : Algebra R S✝\nP : Type u_3\ninst✝¹ : CommSemiring P\nI : Ideal R\nhI : I.IsPrime\nS : Type u_4\ninst✝ : CommSemiring S\nJ : Ideal S\nhJ : J.IsPrime\nK : Ideal P\nhK : K.IsPrime\nf : R →+* S\nhIJ : I = Ideal.comap f... | [
"R : Type u_1\ninst✝⁴ : CommSemiring R\nS✝ : Type u_2\ninst✝³ : CommSemiring S✝\ninst✝² : Algebra R S✝\nP : Type u_3\ninst✝¹ : CommSemiring P\nI : Ideal R\nhI : I.IsPrime\nS : Type u_4\ninst✝ : CommSemiring S\nJ : Ideal S\nhJ : J.IsPrime\nK : Ideal P\nhK : K.IsPrime\nf : R →+* S\nhIJ : I = Ideal.comap f J\ng : S →+... | hJK, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Localization.Submodule | {
"line": 140,
"column": 4
} | {
"line": 140,
"column": 76
} | {
"line": 141,
"column": 4
} | [
{
"pp": "case mp.refine_2\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_3\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... | · exact ⟨0, Submodule.zero_mem _, 1, by rw [mk'_one, map_one, one_smul]⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.EssentialFiniteness | {
"line": 162,
"column": 88
} | {
"line": 166,
"column": 19
} | {
"line": 168,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\n⊢ EssFiniteType R S ↔ ∃ S₀ M, FiniteType R ↥S₀ ∧ IsLocalization M S",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"Algebra.EssFiniteType.subalgebra",
... | [] | by
refine ⟨fun h ↦ ⟨subalgebra R S, submonoid R S, inferInstance, inferInstance⟩, ?_⟩
rintro ⟨S₀, M, _, _⟩
letI := of_isLocalization S M
exact comp R S₀ S | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Ring.Regular | {
"line": 68,
"column": 8
} | {
"line": 68,
"column": 16
} | {
"line": 68,
"column": 17
} | [
{
"pp": "α : Type u_1\ninst✝ : Ring α\nh : ∀ (x y z : α), x * y = 1 → x * z = 1 → y = z\nx y : α\neq : x * y = 1\n⊢ x * (1 - y * x + y) = 1",
"ppTerm": "?m.74",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"HMul.hMul",
"AddGroupWithOne.toAddGroup... | [
"α : Type u_1\ninst✝ : Ring α\nh : ∀ (x y z : α), x * y = 1 → x * z = 1 → y = z\nx y : α\neq : x * y = 1\n⊢ x * (1 - y * x) + x * y = 1"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.Adjugate | {
"line": 134,
"column": 29
} | {
"line": 134,
"column": 95
} | {
"line": 136,
"column": 0
} | [
{
"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",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
"Matrix.update... | [] | rw [cramer_apply, det_eq_elem_of_subsingleton _ i, updateCol_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Matrix.Adjugate | {
"line": 134,
"column": 29
} | {
"line": 134,
"column": 95
} | {
"line": 136,
"column": 0
} | [
{
"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",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
"Matrix.update... | [] | rw [cramer_apply, det_eq_elem_of_subsingleton _ i, updateCol_self] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.Adjugate | {
"line": 134,
"column": 29
} | {
"line": 134,
"column": 95
} | {
"line": 136,
"column": 0
} | [
{
"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",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
"Matrix.update... | [] | rw [cramer_apply, det_eq_elem_of_subsingleton _ i, updateCol_self] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Laurent | {
"line": 288,
"column": 18
} | {
"line": 288,
"column": 26
} | {
"line": 288,
"column": 27
} | [
{
"pp": "case add\nR : Type u_1\ninst✝ : Semiring R\nr : R\np✝ q✝ : R[T;T⁻¹]\nhp : r • p✝ = C r * p✝\nhq : r • q✝ = C r * q✝\n⊢ r • p✝ + r • q✝ = C r * (p✝ + q✝)",
"ppTerm": "?add",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"NonAssocSemiring.toAddCo... | [
"case add\nR : Type u_1\ninst✝ : Semiring R\nr : R\np✝ q✝ : R[T;T⁻¹]\nhp : r • p✝ = C r * p✝\nhq : r • q✝ = C r * q✝\n⊢ r • p✝ + r • q✝ = C r * p✝ + C r * q✝"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.Transvection | {
"line": 101,
"column": 4
} | {
"line": 103,
"column": 27
} | {
"line": 104,
"column": 2
} | [
{
"pp": "case neg\nn : Type u_1\nR : Type u₂\ninst✝² : DecidableEq n\ninst✝¹ : CommRing R\ni j : n\ninst✝ : Finite n\nc : R\nval✝ : Fintype n\na b : n\nha : i = a\nhb : ¬j = b\n⊢ updateRow 1 i (1 i + c • 1 j) a b = transvection i j c a b",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
... | [] | · simp only [ha, updateRow_self, Pi.add_apply, one_apply, Pi.smul_apply, hb, ↓reduceIte,
smul_eq_mul, mul_zero, add_zero, transvection, add_apply, and_false, not_false_eq_true,
single_apply_of_ne] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Matrix.Adjugate | {
"line": 248,
"column": 28
} | {
"line": 248,
"column": 40
} | {
"line": 248,
"column": 40
} | [
{
"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\n⊢ A.cramer b = Aᵀ.adjugateᵀ *ᵥ b",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
"NonUnitalCommRing.toNonUnitalNon... | [
"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\n⊢ A.cramer b = (of fun i ↦ Aᵀᵀ.cramer (Pi.single i 1))ᵀ *ᵥ b"
] | adjugate_def | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Dimension.Localization | {
"line": 213,
"column": 4
} | {
"line": 221,
"column": 86
} | {
"line": 224,
"column": 0
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : IsDomain R\nr : R\ns : ↥R⁰\nh : ∀ (r' : R) (s' : ↥R⁰), ↑s' * r ≠ r' * ↑s\nn : ℕ\nIH : ∀ (g : ℕ → R) (x : ℕ), ∑ i ∈ Finset.range n, g i • (r * ↑s ^ (i + x)) = 0 → ∀ i < n, g i = 0\ng : ℕ → R\nx : ℕ\nhg : ∑ i ∈ Finset.range (n + 1), g i • (r * ↑s ^ (i + x... | [] | rw [Finset.sum_range_succ'] at hg
by_cases hg0 : g 0 = 0
· simp only [hg0, zero_smul, add_zero, add_assoc] at hg
cases i; exacts [hg0, IH _ _ hg _ (Nat.succ_lt_succ_iff.mp hin)]
simp only [zero_add, pow_add _ _ x,
← mul_assoc, pow_succ, ← Finset.sum_mul, smul_eq_mul] at hg
rw [← neg_eq_iff_a... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Dimension.Localization | {
"line": 213,
"column": 4
} | {
"line": 221,
"column": 86
} | {
"line": 224,
"column": 0
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : IsDomain R\nr : R\ns : ↥R⁰\nh : ∀ (r' : R) (s' : ↥R⁰), ↑s' * r ≠ r' * ↑s\nn : ℕ\nIH : ∀ (g : ℕ → R) (x : ℕ), ∑ i ∈ Finset.range n, g i • (r * ↑s ^ (i + x)) = 0 → ∀ i < n, g i = 0\ng : ℕ → R\nx : ℕ\nhg : ∑ i ∈ Finset.range (n + 1), g i • (r * ↑s ^ (i + x... | [] | rw [Finset.sum_range_succ'] at hg
by_cases hg0 : g 0 = 0
· simp only [hg0, zero_smul, add_zero, add_assoc] at hg
cases i; exacts [hg0, IH _ _ hg _ (Nat.succ_lt_succ_iff.mp hin)]
simp only [zero_add, pow_add _ _ x,
← mul_assoc, pow_succ, ← Finset.sum_mul, smul_eq_mul] at hg
rw [← neg_eq_iff_a... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Laurent | {
"line": 433,
"column": 42
} | {
"line": 433,
"column": 53
} | {
"line": 433,
"column": 53
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nf : R[T;T⁻¹]\nh : f = 0\n⊢ degree 0 = ⊥",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AddMonoidAlgebra.semiring",
"WithBot",
"congrArg",
"LaurentPolynomial",
"id",
"Bot.bot",
"Int",
... | [
"R : Type u_1\ninst✝ : Semiring R\nf : R[T;T⁻¹]\nh : f = 0\n⊢ ⊥ = ⊥"
] | degree_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.Adjugate | {
"line": 371,
"column": 30
} | {
"line": 371,
"column": 39
} | {
"line": 371,
"column": 40
} | [
{
"pp": "α : Type w\ninst✝ : CommRing α\nn : ℕ\nA : Matrix (Fin n.succ) (Fin n.succ) α\ni j h : Fin n.succ\nhjk : h ≠ i\n⊢ (-1) ^ (↑j + ↑h) * 0 * (A.submatrix j.succAbove h.succAbove).det = 0",
"ppTerm": "?m.70",
"assigned": true,
"usedConstants": [
"Fin.succAbove",
"Eq.mpr",
"NegZ... | [
"α : Type w\ninst✝ : CommRing α\nn : ℕ\nA : Matrix (Fin n.succ) (Fin n.succ) α\ni j h : Fin n.succ\nhjk : h ≠ i\n⊢ 0 * (A.submatrix j.succAbove h.succAbove).det = 0"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Laurent | {
"line": 684,
"column": 8
} | {
"line": 684,
"column": 16
} | {
"line": 684,
"column": 17
} | [
{
"pp": "case add\nR : Type u_1\nS : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid S\ninst✝¹ : Module R S\ninst✝ : Monoid S\nf : R[T;T⁻¹]\nx : Sˣ\nr : R\np q : R[T;T⁻¹]\nhp : (C r * p).smeval x = r • p.smeval x\nhq : (C r * q).smeval x = r • q.smeval x\n⊢ (C r * (p + q)).smeval x = r • (p + q).smeval x"... | [
"case add\nR : Type u_1\nS : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid S\ninst✝¹ : Module R S\ninst✝ : Monoid S\nf : R[T;T⁻¹]\nx : Sˣ\nr : R\np q : R[T;T⁻¹]\nhp : (C r * p).smeval x = r • p.smeval x\nhq : (C r * q).smeval x = r • q.smeval x\n⊢ (C r * p + C r * q).smeval x = r • (p + q).smeval x"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.Adjugate | {
"line": 417,
"column": 2
} | {
"line": 417,
"column": 28
} | {
"line": 418,
"column": 2
} | [
{
"pp": "case mk\nn : Type v\nα : Type w\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA : Matrix n n α\ni : n\nthis : Nonempty n\nn' : ℕ\nhn' : Fintype.card n = n'.succ\nx✝ : Trunc (n ≃ Fin n'.succ)\ne : n ≃ Fin n'.succ\nA' : Matrix (Fin n'.succ) (Fin n'.succ) α := (reindex e e) A\n⊢ A'.det ... | [
"case mk\nn : Type v\nα : Type w\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA : Matrix n n α\ni : n\nthis : Nonempty n\nn' : ℕ\nhn' : Fintype.card n = n'.succ\nx✝ : Trunc (n ≃ Fin n'.succ)\ne : n ≃ Fin n'.succ\nA' : Matrix (Fin n'.succ) (Fin n'.succ) α := (reindex e e) A\n⊢ ∑ j, (-1) ^ (↑(e i)... | rw [det_succ_row A' (e i)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Matrix.NonsingularInverse | {
"line": 281,
"column": 10
} | {
"line": 281,
"column": 31
} | {
"line": 281,
"column": 32
} | [
{
"pp": "n : 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\nu v : n → α\nhM : u = A *ᵥ v\n⊢ A⁻¹ *ᵥ A *ᵥ v = v",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalN... | [
"n : 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\nu v : n → α\nhM : u = A *ᵥ v\n⊢ (A⁻¹ * A) *ᵥ v = v"
] | Matrix.mulVec_mulVec, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MatrixAlgebra | {
"line": 142,
"column": 8
} | {
"line": 142,
"column": 25
} | {
"line": 142,
"column": 26
} | [
{
"pp": "l : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nR : Type u_5\nS : Type u_6\nA : Type u_7\nB : Type u_8\nM : Type u_9\nN : Type u_10\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Semiring B\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\na₁✝ a₂✝ ... | [
"l : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nR : Type u_5\nS : Type u_6\nA : Type u_7\nB : Type u_8\nM : Type u_9\nN : Type u_10\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Semiring B\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\na₁✝ a₂✝ : A\nb₁✝ b₂✝... | _root_.mul_assoc, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Polynomial.Nilpotent | {
"line": 65,
"column": 2
} | {
"line": 66,
"column": 29
} | {
"line": 68,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nP : R[X]\n⊢ IsNilpotent (P * X) ↔ IsNilpotent P",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.isNilpotent_X_mul_iff",
"HMul.hMul",
"congrArg",
"id",
"Polynomial",
"Monoid.toPow",
... | [] | rw [← commute_X P]
exact isNilpotent_X_mul_iff | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Nilpotent | {
"line": 65,
"column": 2
} | {
"line": 66,
"column": 29
} | {
"line": 68,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nP : R[X]\n⊢ IsNilpotent (P * X) ↔ IsNilpotent P",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.isNilpotent_X_mul_iff",
"HMul.hMul",
"congrArg",
"id",
"Polynomial",
"Monoid.toPow",
... | [] | rw [← commute_X P]
exact isNilpotent_X_mul_iff | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.ENat.Lattice | {
"line": 138,
"column": 2
} | {
"line": 142,
"column": 76
} | {
"line": 143,
"column": 2
} | [
{
"pp": "case inr.inr.coe\nι : Sort u_2\na : ℕ∞\nf : ι → ℕ∞\nhne : a ≠ 0\nhι : Nonempty ι\nd : ℕ\nh : ∀ (i : ι), a * f i ≤ ↑d\n⊢ a * ⨆ i, f i ≤ ↑d",
"ppTerm": "?inr.inr.coe",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENat.self_le_mul_left",
"False",
"Preorder.toLT",
... | [
"case inr.inr.coe\nι : Sort u_2\na : ℕ∞\nf : ι → ℕ∞\nhne : a ≠ 0\nhι : Nonempty ι\nd : ℕ\nh : ∀ (i : ι), a * f i ≤ ↑d\nhlt : ⨆ i, f i < ⊤\n⊢ a * ⨆ i, f i ≤ ↑d"
] | have hlt : ⨆ i, f i < ⊤ := by
rw [lt_top_iff_ne_top]
intro htop
obtain ⟨i, hi : d < f i⟩ := (iSup_eq_top ..).1 htop d (by simp)
exact (((h i).trans_lt hi).trans_le (ENat.self_le_mul_left _ hne)).false | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Polynomial.Nilpotent | {
"line": 141,
"column": 4
} | {
"line": 141,
"column": 33
} | {
"line": 142,
"column": 2
} | [
{
"pp": "case left\nR : Type u_1\ninst✝ : CommRing R\nP : R[X]\nhunit : IsUnit P\nQ : R[X]\nhQ : P * Q = 1\nh : P.coeff 0 * Q.coeff 0 = (P * Q).coeff 0\n⊢ P.coeff 0 * Q.coeff 0 = 1",
"ppTerm": "?left",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOn... | [] | rwa [hQ, coeff_one_zero] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Data.ENat.Lattice | {
"line": 239,
"column": 67
} | {
"line": 239,
"column": 92
} | {
"line": 241,
"column": 0
} | [
{
"pp": "ι : Type u_4\ninst✝¹ : Preorder ι\ninst✝ : IsDirectedOrder ι\nf g : ι → ℕ∞\nhf : Monotone f\nhg : Monotone g\ni j _k : ι\nx✝ : i ≤ _k ∧ j ≤ _k\nhi : i ≤ _k\nhj : j ≤ _k\n⊢ f i + g j ≤ f _k + g _k",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"instCompleteLinearOrderENat",
... | [] | by gcongr <;> apply_rules | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.ENat.Lattice | {
"line": 270,
"column": 66
} | {
"line": 270,
"column": 95
} | {
"line": 272,
"column": 0
} | [
{
"pp": "a : ℕ∞\ns : Set ℕ∞\n⊢ sInf s + a = ⨅ b ∈ s, b + a",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"iInf",
"instCompleteLinearOrderENat",
"congrArg",
"CompletelyDistribLattice.toCompleteLattice",
"Membership.mem",
"instAddENat",
"Conditiona... | [] | simp [sInf_eq_iInf, iInf_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.ENat.Lattice | {
"line": 270,
"column": 66
} | {
"line": 270,
"column": 95
} | {
"line": 272,
"column": 0
} | [
{
"pp": "a : ℕ∞\ns : Set ℕ∞\n⊢ sInf s + a = ⨅ b ∈ s, b + a",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"iInf",
"instCompleteLinearOrderENat",
"congrArg",
"CompletelyDistribLattice.toCompleteLattice",
"Membership.mem",
"instAddENat",
"Conditiona... | [] | simp [sInf_eq_iInf, iInf_add] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.ENat.Lattice | {
"line": 270,
"column": 66
} | {
"line": 270,
"column": 95
} | {
"line": 272,
"column": 0
} | [
{
"pp": "a : ℕ∞\ns : Set ℕ∞\n⊢ sInf s + a = ⨅ b ∈ s, b + a",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"iInf",
"instCompleteLinearOrderENat",
"congrArg",
"CompletelyDistribLattice.toCompleteLattice",
"Membership.mem",
"instAddENat",
"Conditiona... | [] | simp [sInf_eq_iInf, iInf_add] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.ENat.Lattice | {
"line": 285,
"column": 67
} | {
"line": 285,
"column": 92
} | {
"line": 287,
"column": 0
} | [
{
"pp": "ι : Type u_4\ninst✝¹ : Preorder ι\ninst✝ : IsCodirectedOrder ι\nf g : ι → ℕ∞\nhf : Monotone f\nhg : Monotone g\ni j _k : ι\nx✝ : _k ≤ i ∧ _k ≤ j\nhi : _k ≤ i\nhj : _k ≤ j\n⊢ f _k + g _k ≤ f i + g j",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"instCompleteLinearOrderENat"... | [] | by gcongr <;> apply_rules | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic | {
"line": 39,
"column": 2
} | {
"line": 40,
"column": 79
} | {
"line": 42,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_4\ninst✝² : CommRing R\ninst✝¹ : Ring S\ninst✝ : Ring T\nf : R →+* S\nx : S\nhx : f.IsIntegralElem x\ng : S →+* T\n⊢ (g.comp f).IsIntegralElem (g x)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
... | [] | obtain ⟨p, hp, hx⟩ := hx
exact ⟨p, hp, by simp_rw [← hom_eval₂, eval₂_eq_eval_map] at hx ⊢; simp [hx]⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic | {
"line": 39,
"column": 2
} | {
"line": 40,
"column": 79
} | {
"line": 42,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_4\ninst✝² : CommRing R\ninst✝¹ : Ring S\ninst✝ : Ring T\nf : R →+* S\nx : S\nhx : f.IsIntegralElem x\ng : S →+* T\n⊢ (g.comp f).IsIntegralElem (g x)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
... | [] | obtain ⟨p, hp, hx⟩ := hx
exact ⟨p, hp, by simp_rw [← hom_eval₂, eval₂_eq_eval_map] at hx ⊢; simp [hx]⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic | {
"line": 176,
"column": 4
} | {
"line": 176,
"column": 33
} | {
"line": 177,
"column": 2
} | [
{
"pp": "case mp\nR : Type u_5\nS : Type u_6\nT : Type u_7\ninst✝⁴ : CommRing R\ninst✝³ : Ring S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra T S\nφ : R ≃+* T\nh : (algebraMap T S).comp φ.toRingHom = algebraMap R S\na : S\nha : IsIntegral R a\nthis✝ : Algebra R T := φ.toRingHom.toAlgebra\nthis : ... | [] | exact IsIntegral.tower_top ha | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic | {
"line": 183,
"column": 4
} | {
"line": 183,
"column": 33
} | {
"line": 185,
"column": 0
} | [
{
"pp": "case mpr\nR : Type u_5\nS : Type u_6\nT : Type u_7\ninst✝⁴ : CommRing R\ninst✝³ : Ring S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra T S\nφ : R ≃+* T\nh : (algebraMap T S).comp φ.toRingHom = algebraMap R S\na : S\nha : IsIntegral T a\nh' : algebraMap T S = (algebraMap R S).comp φ.symm.t... | [] | exact IsIntegral.tower_top ha | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff | {
"line": 214,
"column": 52
} | {
"line": 227,
"column": 50
} | {
"line": 229,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\nn : Type v\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nM : Matrix n n R\ninst✝ : Nontrivial R\nhn : Fintype.card n = 2\n⊢ M.charpoly = X ^ 2 - C M.trace * X + C M.det",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"one_pow",
"Eq.mpr",... | [] | by
have : Nonempty n := by rw [← Fintype.card_pos_iff]; lia
ext i
by_cases hi : i ∈ Finset.range 3
· fin_cases hi
· simp [det_eq_sign_charpoly_coeff, hn]
· simp [trace_eq_neg_charpoly_coeff, hn]
· simpa [leadingCoeff, charpoly_natDegree_eq_dim, hn, coeff_X] using
M.charpoly_monic.leadingCoef... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic | {
"line": 219,
"column": 52
} | {
"line": 219,
"column": 66
} | {
"line": 219,
"column": 67
} | [
{
"pp": "case empty\nR : Type u_1\nA : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\ns : Set A\nhis : ∀ x ∈ ∅, IsIntegral R x\nx : A\n⊢ x ∈ R ∙ 1 ↔ x ∈ Subalgebra.toSubmodule ⊥",
"ppTerm": "?empty",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiri... | [
"case empty\nR : Type u_1\nA : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\ns : Set A\nhis : ∀ x ∈ ∅, IsIntegral R x\nx : A\n⊢ x ∈ 1 ↔ x ∈ Subalgebra.toSubmodule ⊥"
] | ← one_eq_span, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.IntegralClosure.Algebra.Basic | {
"line": 108,
"column": 7
} | {
"line": 108,
"column": 40
} | {
"line": 108,
"column": 40
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R A\ninst✝⁵ : IsDomain A\nM : Type u_5\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Module A M\ninst✝¹ : IsScalarTower R A M\ninst✝ : Module.IsTorsionFree A M\nN : Submodule R M\nhN : N ≠ ⊥\nhN' : N.FG\nx ... | [] | by intro x y; ext; apply mul_smul | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic | {
"line": 252,
"column": 74
} | {
"line": 252,
"column": 83
} | {
"line": 253,
"column": 4
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : Ring A\ninst✝² : Ring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nx : A × B\np₁ : R[X]\nhp₁Monic : p₁.Monic\nhp₁Eval : (aeval x.1) p₁ = 0\np₂ : R[X]\nhp₂Monic : p₂.Monic\nhp₂Eval : (aeval x.2) p₂ = 0\n⊢ (0, (aeval x.2) p₁ * 0) = ... | [
"R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : Ring A\ninst✝² : Ring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nx : A × B\np₁ : R[X]\nhp₁Monic : p₁.Monic\nhp₁Eval : (aeval x.1) p₁ = 0\np₂ : R[X]\nhp₂Monic : p₂.Monic\nhp₂Eval : (aeval x.2) p₂ = 0\n⊢ (0, 0) = 0"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.IntegralClosure.Algebra.Basic | {
"line": 139,
"column": 44
} | {
"line": 139,
"column": 63
} | {
"line": 139,
"column": 63
} | [
{
"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\nthis : (Subalgebra.toSubmodule (Algebra.adjoin R ({x} ∪ {y}))).FG\n⊢ f.IsIntegralElem z",
... | [
"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\nthis : (Subalgebra.toSubmodule R[x, y]).FG\n⊢ f.IsIntegralElem z"
] | Set.singleton_union | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.IntegralClosure.Algebra.Basic | {
"line": 245,
"column": 2
} | {
"line": 247,
"column": 61
} | {
"line": 249,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Ring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nx : A\ny : B\nh : IsIntegral R y\n⊢ IsIntegral A (x • 1 ⊗ₜ[R] y)",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
... | [] | exact smul _ (h.map_of_comp_eq (algebraMap R A)
(Algebra.TensorProduct.includeRight (R := R) (A := A) (B := B)).toRingHom
Algebra.TensorProduct.includeLeftRingHom_comp_algebraMap) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Polynomial.Lifts | {
"line": 164,
"column": 2
} | {
"line": 165,
"column": 40
} | {
"line": 167,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\np : S[X]\nhlifts : p ∈ lifts f\n⊢ ∃ q, map f q = p ∧ q.degree = p.degree",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"WithBot",
"congrArg",
"Finset",
"Exists",
"Polynomi... | [] | obtain ⟨q, hq, hq'⟩ := exists_support_eq_of_mem_lifts hlifts
exact ⟨q, hq, congrArg Finset.max hq'⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Lifts | {
"line": 164,
"column": 2
} | {
"line": 165,
"column": 40
} | {
"line": 167,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\np : S[X]\nhlifts : p ∈ lifts f\n⊢ ∃ q, map f q = p ∧ q.degree = p.degree",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"WithBot",
"congrArg",
"Finset",
"Exists",
"Polynomi... | [] | obtain ⟨q, hq, hq'⟩ := exists_support_eq_of_mem_lifts hlifts
exact ⟨q, hq, congrArg Finset.max hq'⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff | {
"line": 436,
"column": 10
} | {
"line": 436,
"column": 27
} | {
"line": 437,
"column": 8
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\nn : Type v\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\nM : Matrix n n R\nk : ℕ\nD : (n → R[X]) [⋀^n]→ₗ[R[X]] R[X] := detRowAlternating\nh_map : ∀ (s : Finset n), (s.piecewise (fun i ↦ M.map (⇑C) i) fun i ↦ 1 i) = map (s.piecewise M 1) ⇑C\nh_det : ∀ (s : Finset n), D (s.... | [] | split_ifs <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.Polynomial.HasseDeriv | {
"line": 144,
"column": 2
} | {
"line": 144,
"column": 67
} | {
"line": 145,
"column": 2
} | [
{
"pp": "case succ.e_a.e_a\nR : Type u_1\ninst✝ : Semiring R\nk✝ k : ℕ\nih : ⇑(k ! • hasseDeriv k) = (⇑derivative)^[k]\nf : R[X]\nn : ℕ\nthis : n + k + 1 = n + (k + 1)\n⊢ (k + 1) * (n + k + 1).choose (k + 1) = (n + k + 1).choose (n + 1) * (n + 1)",
"ppTerm": "?succ.e_a.e_a",
"assigned": true,
"usedC... | [
"case succ.e_a.e_a\nR : Type u_1\ninst✝ : Semiring R\nk✝ k : ℕ\nih : ⇑(k ! • hasseDeriv k) = (⇑derivative)^[k]\nf : R[X]\nn : ℕ\nthis : n + k + 1 = n + (k + 1)\n⊢ (n + k + 1).choose n * (k + 1) = (n + k + 1).choose n * (n + k + 1 - n)"
] | rw [← choose_symm_of_eq_add this, choose_succ_right_eq, mul_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.HasseDeriv | {
"line": 179,
"column": 25
} | {
"line": 179,
"column": 42
} | {
"line": 179,
"column": 42
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\np : R[X]\nn : ℕ\n⊢ max (Finset.fold max 0 (fun x ↦ 0) ({x ∈ p.support | ↑(x.choose n) * p.coeff x = 0}))\n (Finset.fold max 0 (fun x ↦ x - n) ({i ∈ p.support | ¬↑(i.choose n) * p.coeff i = 0})) ≤\n p.natDegree - n",
"ppTerm": "?m.55",
"assigned": true,
... | [
"R : Type u_1\ninst✝ : Semiring R\np : R[X]\nn : ℕ\n⊢ max (if {x ∈ p.support | ↑(x.choose n) * p.coeff x = 0} = ∅ then 0 else max 0 0)\n (Finset.fold max 0 (fun x ↦ x - n) ({i ∈ p.support | ¬↑(i.choose n) * p.coeff i = 0})) ≤\n p.natDegree - n",
"case h\nR : Type u_1\ninst✝ : Semiring R\np : R[X]\nn : ℕ\n... | Finset.fold_const | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.HasseDeriv | {
"line": 225,
"column": 8
} | {
"line": 225,
"column": 24
} | {
"line": 225,
"column": 24
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nk : ℕ\nD : ℕ → R[X] →+ R[X] := fun k ↦ (hasseDeriv k).toAddMonoidHom\nΦ : R[X] →+ R[X] →+ R[X] := mul\nm : ℕ\nr : R\nn : ℕ\ns : R\nx : ℕ × ℕ\nhx : x ∈ antidiagonal k\n⊢ (monomial (m - x.1 + (n - x.2))) (↑(m.choose x.1) * r * (↑(n.choose x.2) * s)) =\n (monomial (m +... | [
"R : Type u_1\ninst✝ : Semiring R\nk : ℕ\nD : ℕ → R[X] →+ R[X] := fun k ↦ (hasseDeriv k).toAddMonoidHom\nΦ : R[X] →+ R[X] →+ R[X] := mul\nm : ℕ\nr : R\nn : ℕ\ns : R\nx : ℕ × ℕ\nhx : x.1 + x.2 = k\n⊢ (monomial (m - x.1 + (n - x.2))) (↑(m.choose x.1) * r * (↑(n.choose x.2) * s)) =\n (monomial (m + n - k)) (↑(m.cho... | mem_antidiagonal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.HasseDeriv | {
"line": 206,
"column": 91
} | {
"line": 238,
"column": 35
} | {
"line": 240,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nk : ℕ\nf g : R[X]\n⊢ (hasseDeriv k) (f * g) = ∑ ij ∈ antidiagonal k, (hasseDeriv ij.1) f * (hasseDeriv ij.2) g",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"LinearMap.toAddMonoidHom",
"Nat.cast_mul._simp_1",
"Eq.mpr",
"N... | [] | by
let D k := (@hasseDeriv R _ k).toAddMonoidHom
let Φ := @AddMonoidHom.mul R[X] _
change
(compHom (D k)).comp Φ f g =
∑ ij ∈ antidiagonal k, ((compHom.comp ((compHom Φ) (D ij.1))).flip (D ij.2) f) g
simp only [← finsetSum_apply]
congr 2
clear f g
ext m r n s : 4
simp only [Φ, D, finsetSum_app... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Splits | {
"line": 648,
"column": 2
} | {
"line": 648,
"column": 30
} | {
"line": 650,
"column": 0
} | [
{
"pp": "case ha\nR : Type u_1\ninst✝ : Field R\nf : R[X]\nhf : f.Splits\nx : R\nhx : eval x f ≠ 0\nz : R\nhz : z ∈ f.roots\n⊢ x - z ≠ 0",
"ppTerm": "?ha",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
"False",
"Polynomial.roots",
"AddGroupWithOne.to... | [] | aesop (add simp sub_eq_zero) | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Ideal.GoingUp | {
"line": 56,
"column": 4
} | {
"line": 56,
"column": 58
} | {
"line": 57,
"column": 4
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝¹ : CommRing R\nS : Type u_2\ninst✝ : CommRing S\nf : R →+* S\nI : Ideal S\nr : S\nr_non_zero_divisor : ∀ {x : S}, x * r = 0 → x = 0\nhr : r ∈ I\np✝ p : R[X]\na : R\ncoeff_eq_zero : p.coeff 0 = 0\na_ne_zero : a ≠ 0\na✝¹ : p ≠ 0 → eval₂ f r p = 0 → ∃ i, p.coeff i ≠ 0 ∧ ... | [
"case refine_2\nR : Type u_1\ninst✝¹ : CommRing R\nS : Type u_2\ninst✝ : CommRing S\nf : R →+* S\nI : Ideal S\nr : S\nr_non_zero_divisor : ∀ {x : S}, x * r = 0 → x = 0\nhr : r ∈ I\np✝ p : R[X]\na : R\ncoeff_eq_zero : p.coeff 0 = 0\na_ne_zero : a ≠ 0\na✝¹ : p ≠ 0 → eval₂ f r p = 0 → ∃ i, p.coeff i ≠ 0 ∧ p.coeff i ∈ ... | refine ⟨0, ?_, coeff_zero_mem_comap_of_root_mem hr hp⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Localization.Integral | {
"line": 244,
"column": 6
} | {
"line": 244,
"column": 23
} | {
"line": 244,
"column": 24
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝³ : CommRing R\nM : Submonoid R\nRₘ : Type u_3\ninst✝² : CommRing Rₘ\ninst✝¹ : Algebra R Rₘ\ninst✝ : IsLocalization M Rₘ\np : Rₘ[X]\nhp : p.leadingCoeff ∈ (algebraMap R Rₘ).range\nn : ℕ\nh₁ : n ∈ p.support\nh₂ : ¬n = p.natDegree\nthis : n + 1 ≤ p.natDegree\n⊢ (algebraMap R ... | [
"case neg\nR : Type u_1\ninst✝³ : CommRing R\nM : Submonoid R\nRₘ : Type u_3\ninst✝² : CommRing Rₘ\ninst✝¹ : Algebra R Rₘ\ninst✝ : IsLocalization M Rₘ\np : Rₘ[X]\nhp : p.leadingCoeff ∈ (algebraMap R Rₘ).range\nn : ℕ\nh₁ : n ∈ p.support\nh₂ : ¬n = p.natDegree\nthis : n + 1 ≤ p.natDegree\n⊢ (algebraMap R Rₘ) ↑(common... | _root_.mul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Algebraic.Integral | {
"line": 70,
"column": 2
} | {
"line": 70,
"column": 21
} | {
"line": 71,
"column": 2
} | [
{
"pp": "K : Type u\nA : Type v\ninst✝² : Field K\ninst✝¹ : Ring A\ninst✝ : Algebra K A\nx : A\n⊢ IsAlgebraic K x → IsIntegral K x",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"IsAlgebraic",
"CommSemiring.toSemiring",
"AlgHom",
"AlgHom.funLike",
"Polynomial... | [
"K : Type u\nA : Type v\ninst✝² : Field K\ninst✝¹ : Ring A\ninst✝ : Algebra K A\nx : A\np : K[X]\nhp : p ≠ 0\nhpx : (aeval x) p = 0\n⊢ IsIntegral K x"
] | rintro ⟨p, hp, hpx⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.RingTheory.Algebraic.Integral | {
"line": 266,
"column": 2
} | {
"line": 272,
"column": 63
} | {
"line": 273,
"column": 2
} | [
{
"pp": "case pos\nR : Type u_1\nS : Type u_2\nA : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Ring A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra S A\ninst✝² : IsScalarTower R S A\ninst✝¹ : NoZeroDivisors S\ninst✝ : Algebra.IsAlgebraic R S\na : A\nh : IsAlgebraic S a\np✝ : S[X... | [
"case pos\nR : Type u_1\nS : Type u_2\nA : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Ring A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra S A\ninst✝² : IsScalarTower R S A\ninst✝¹ : NoZeroDivisors S\ninst✝ : Algebra.IsAlgebraic R S\na : A\nh : IsAlgebraic S a\np✝ : S[X]\nhp : p✝ ≠... | have : IsAlgebraic (integralClosure R S) a := by
refine ⟨p, ?_, ?_⟩
· simpa only [← Polynomial.map_ne_zero_iff (f := Subring.subtype _) (p := p)
Subtype.val_injective, p, map_toSubring, smul_ne_zero_iff] using And.intro hr hp
rw [← eval_map_algebraMap, Subalgebra.algebraMap_eq, ← map_map, ← Subalgeb... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Algebraic.Integral | {
"line": 435,
"column": 2
} | {
"line": 435,
"column": 56
} | {
"line": 437,
"column": 0
} | [
{
"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)\nt : S\nht : t ∈ R[y]\nr : R\nhr : r ≠ 0\neq : y * t = (algebraMap R S) r\nthis : IsAlgebrai... | [] | exact this.of_smul (mem_nonZeroDivisors_of_ne_zero hr) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Localization.Integral | {
"line": 353,
"column": 15
} | {
"line": 353,
"column": 32
} | {
"line": 353,
"column": 33
} | [
{
"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\nφ : R[X] →ₐ[R] S := ⋯\nq : R[X][X]\nhqm : q.Monic\nhqt : eval₂ φ.toRingHom t q = 0\nN : ℕ := ⋯\nhN : ∀ (i : ℕ), (q.coeff i).natDegree ≤ N\nq... | [
"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\nφ : R[X] →ₐ[R] S := aeval s\nq : R[X][X]\nhqm : q.Monic\nhqt : eval₂ φ.toRingHom t q = 0\nN : ℕ := q.support.sup fun x ↦ (q.coeff x).natDegree\nhN : ∀ (... | mul_comm (t ^ _), | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Flat.FaithfullyFlat.Basic | {
"line": 576,
"column": 2
} | {
"line": 581,
"column": 52
} | {
"line": 583,
"column": 0
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\nS : Type u_1\nN : Type u_2\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : FaithfullyFlat R S\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : Module S N\ninst✝ : IsScalarTower R S N\nf : M →ₗ[R] N\nh... | [] | simpa only [← Submodule.Quotient.subsingleton_iff.not] using not_congr <|
(tensorQuotEquivQuotSMul N (I.map (algebraMap R S))).symm ≪≫ₗ TensorProduct.comm S N _ ≪≫ₗ
hf.tensorEquiv _ ≪≫ₗ AlgebraTensorModule.congr (I.qoutMapEquivTensorQout S) (.refl R M) ≪≫ₗ
AlgebraTensorModule.assoc R R S S _ M ≪≫ₗ (Te... | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.Flat.FaithfullyFlat.Basic | {
"line": 576,
"column": 2
} | {
"line": 581,
"column": 52
} | {
"line": 583,
"column": 0
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\nS : Type u_1\nN : Type u_2\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : FaithfullyFlat R S\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : Module S N\ninst✝ : IsScalarTower R S N\nf : M →ₗ[R] N\nh... | [] | simpa only [← Submodule.Quotient.subsingleton_iff.not] using not_congr <|
(tensorQuotEquivQuotSMul N (I.map (algebraMap R S))).symm ≪≫ₗ TensorProduct.comm S N _ ≪≫ₗ
hf.tensorEquiv _ ≪≫ₗ AlgebraTensorModule.congr (I.qoutMapEquivTensorQout S) (.refl R M) ≪≫ₗ
AlgebraTensorModule.assoc R R S S _ M ≪≫ₗ (Te... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Flat.FaithfullyFlat.Basic | {
"line": 576,
"column": 2
} | {
"line": 581,
"column": 52
} | {
"line": 583,
"column": 0
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\nS : Type u_1\nN : Type u_2\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : FaithfullyFlat R S\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : Module S N\ninst✝ : IsScalarTower R S N\nf : M →ₗ[R] N\nh... | [] | simpa only [← Submodule.Quotient.subsingleton_iff.not] using not_congr <|
(tensorQuotEquivQuotSMul N (I.map (algebraMap R S))).symm ≪≫ₗ TensorProduct.comm S N _ ≪≫ₗ
hf.tensorEquiv _ ≪≫ₗ AlgebraTensorModule.congr (I.qoutMapEquivTensorQout S) (.refl R M) ≪≫ₗ
AlgebraTensorModule.assoc R R S S _ M ≪≫ₗ (Te... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.SurjectiveOnStalks | {
"line": 142,
"column": 13
} | {
"line": 142,
"column": 22
} | {
"line": 142,
"column": 23
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nS : Type u_2\ninst✝³ : CommRing S\nT : Type u_3\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\ninst✝ : Algebra R S\nhf₂ : (algebraMap R T).SurjectiveOnStalks\nJ : Ideal T\nhJ : J.IsPrime\n⊢ 1 ⊗ₜ[R] (1 • 1) * 0 = 0 ⊗ₜ[R] 1",
"ppTerm": "?m.101",
"assigned": true,
... | [
"R : Type u_1\ninst✝⁴ : CommRing R\nS : Type u_2\ninst✝³ : CommRing S\nT : Type u_3\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\ninst✝ : Algebra R S\nhf₂ : (algebraMap R T).SurjectiveOnStalks\nJ : Ideal T\nhJ : J.IsPrime\n⊢ 0 = 0 ⊗ₜ[R] 1"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.SurjectiveOnStalks | {
"line": 155,
"column": 72
} | {
"line": 155,
"column": 80
} | {
"line": 155,
"column": 81
} | [
{
"pp": "case add\nR : Type u_1\ninst✝⁴ : CommRing R\nS : Type u_2\ninst✝³ : CommRing S\nT : Type u_3\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\ninst✝ : Algebra R S\nhf₂ : (algebraMap R T).SurjectiveOnStalks\nJ : Ideal T\nhJ : J.IsPrime\nx₁ x₂ : S ⊗[R] T\nt₁ : T\nr₁ : R\na₁ : S\nhr₁ : r₁ • t₁ ∉ J\ne₁ : 1 ⊗ₜ[R]... | [
"case add\nR : Type u_1\ninst✝⁴ : CommRing R\nS : Type u_2\ninst✝³ : CommRing S\nT : Type u_3\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\ninst✝ : Algebra R S\nhf₂ : (algebraMap R T).SurjectiveOnStalks\nJ : Ideal T\nhJ : J.IsPrime\nx₁ x₂ : S ⊗[R] T\nt₁ : T\nr₁ : R\na₁ : S\nhr₁ : r₁ • t₁ ∉ J\ne₁ : 1 ⊗ₜ[R] (r₁ • t₁) *... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Ring.Idempotent | {
"line": 60,
"column": 39
} | {
"line": 60,
"column": 48
} | {
"line": 60,
"column": 49
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\na✝ b✝ a b : { a // a.1 * a.2 = 0 ∧ a.1 + a.2 = 1 }\n⊢ (↑a).1 * ((↑a).2 * (↑b).2) + (↑a).2 * (↑a).2 * 0 = 0",
"ppTerm": "?m.134",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
... | [
"R : Type u_1\ninst✝ : CommSemiring R\na✝ b✝ a b : { a // a.1 * a.2 = 0 ∧ a.1 + a.2 = 1 }\n⊢ (↑a).1 * ((↑a).2 * (↑b).2) + 0 = 0"
] | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Order.Ring.Idempotent | {
"line": 63,
"column": 22
} | {
"line": 63,
"column": 30
} | {
"line": 63,
"column": 31
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\na✝ b✝ a b : { a // a.1 * a.2 = 0 ∧ a.1 + a.2 = 1 }\n⊢ (↑a).1 * ((↑a).1 + (↑a).2 * (↑b).1) = (↑a).1",
"ppTerm": "?m.146",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOn... | [
"R : Type u_1\ninst✝ : CommSemiring R\na✝ b✝ a b : { a // a.1 * a.2 = 0 ∧ a.1 + a.2 = 1 }\n⊢ (↑a).1 * (↑a).1 + (↑a).1 * ((↑a).2 * (↑b).1) = (↑a).1"
] | mul_add, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Order.Ring.Idempotent | {
"line": 66,
"column": 22
} | {
"line": 66,
"column": 30
} | {
"line": 66,
"column": 31
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\na✝ b✝ a b : { a // a.1 * a.2 = 0 ∧ a.1 + a.2 = 1 }\n⊢ (↑b).1 * ((↑a).1 + (↑a).2 * (↑b).1) = (↑b).1",
"ppTerm": "?m.153",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOn... | [
"R : Type u_1\ninst✝ : CommSemiring R\na✝ b✝ a b : { a // a.1 * a.2 = 0 ∧ a.1 + a.2 = 1 }\n⊢ (↑b).1 * (↑a).1 + (↑b).1 * ((↑a).2 * (↑b).1) = (↑b).1"
] | mul_add, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Order.Ring.Idempotent | {
"line": 79,
"column": 77
} | {
"line": 79,
"column": 85
} | {
"line": 80,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\na✝ b✝ a b c : { a // a.1 * a.2 = 0 ∧ a.1 + a.2 = 1 }\n⊢ (↑a).2 * (↑b).2 + ((↑a).1 * ((↑a).2 * (↑c).2) + (↑a).2 * (↑b).1 * ((↑a).2 * (↑c).2)) =\n (↑a).2 * ((↑b).2 + (↑b).1 * (↑c).2)",
"ppTerm": "?m.206",
"assigned": true,
"usedConstants": [
"Dis... | [
"R : Type u_1\ninst✝ : CommSemiring R\na✝ b✝ a b c : { a // a.1 * a.2 = 0 ∧ a.1 + a.2 = 1 }\n⊢ (↑a).2 * (↑b).2 + ((↑a).1 * ((↑a).2 * (↑c).2) + (↑a).2 * (↑b).1 * ((↑a).2 * (↑c).2)) =\n (↑a).2 * (↑b).2 + (↑a).2 * ((↑b).1 * (↑c).2)"
] | mul_add, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Order.Ring.Idempotent | {
"line": 130,
"column": 35
} | {
"line": 130,
"column": 43
} | {
"line": 130,
"column": 44
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\na b c : { x // IsIdempotentElem x }\n⊢ ↑a * (↑b + ↑c) - ↑a * (↑b * ↑c) = ↑a * ↑b + ↑a * ↑c - ↑a * ↑b * (↑a * ↑c)",
"ppTerm": "?m.120",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"NonUnitalCommRing.toNonUnita... | [
"R : Type u_1\ninst✝ : CommRing R\na b c : { x // IsIdempotentElem x }\n⊢ ↑a * ↑b + ↑a * ↑c - ↑a * (↑b * ↑c) = ↑a * ↑b + ↑a * ↑c - ↑a * ↑b * (↑a * ↑c)"
] | mul_add, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Ideal.MinimalPrime.Localization | {
"line": 75,
"column": 2
} | {
"line": 75,
"column": 84
} | {
"line": 76,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nI p : Ideal R\nhp : p ∈ I.minimalPrimes\nx : R\nhx : x ∈ p\n⊢ ∃ y ∉ I, x * y ∈ I",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"HMul.hMul",
"Ideal.minimalPrimes",
"CommSemiring.toSemiring",
... | [
"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\n⊢ ∃ y ∉ I, x * y ∈ I"
] | obtain ⟨y, hy, n, hx⟩ := Ideal.iUnion_minimalPrimes.subset (Set.mem_biUnion hp hx) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Order.RelSeries | {
"line": 196,
"column": 6
} | {
"line": 196,
"column": 23
} | {
"line": 196,
"column": 24
} | [
{
"pp": "α : Type u_1\nr : SetRel α α\ns : RelSeries r\nx : α\n⊢ x ∈ s.toList ↔ x ∈ s",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"List.ofFn",
"Membership.mem",
"id",
"RelSeries.length",
"RelSeries.toList",
"instOfNa... | [
"α : Type u_1\nr : SetRel α α\ns : RelSeries r\nx : α\n⊢ x ∈ List.ofFn s.toFun ↔ x ∈ s"
] | RelSeries.toList, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.KrullDimension | {
"line": 381,
"column": 32
} | {
"line": 381,
"column": 45
} | {
"line": 381,
"column": 45
} | [
{
"pp": "case coe\nα : Type u_1\ninst✝ : Preorder α\na : α\nn : ℕ\nhne : Nonempty { p // RelSeries.last p = a }\nm : ℕ\nh : n ≤ m\nha : ⨆ p, ⨆ (_ : RelSeries.last p = a), ↑p.length = ↑m\n⊢ ∃ p, RelSeries.last p = a ∧ p.length = n",
"ppTerm": "?coe",
"assigned": true,
"usedConstants": [
"Preord... | [
"case coe\nα : Type u_1\ninst✝ : Preorder α\na : α\nn : ℕ\nhne : Nonempty { p // RelSeries.last p = a }\nm : ℕ\nh : n ≤ m\nha : ⨆ x, ↑(↑x).length = ↑m\n⊢ ∃ p, RelSeries.last p = a ∧ p.length = n"
] | iSup_subtype' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.KrullDimension | {
"line": 426,
"column": 82
} | {
"line": 431,
"column": 13
} | {
"line": 433,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : Preorder α\nx : α\nn : ℕ\n⊢ height x ≤ ↑n ↔ ∀ y < x, height y < ↑n",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENat.coe_ne_top._simp_1",
"False",
"Preorder.toLT",
"instCompleteLinearOrderENat",
"instCharZeroE... | [] | by
conv_lhs => rw [height_eq_iSup_lt_height, iSup₂_le_iff]
congr! 2 with y _
cases height y
· simp
· norm_cast | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Compactness.Bases | {
"line": 62,
"column": 6
} | {
"line": 62,
"column": 49
} | {
"line": 63,
"column": 4
} | [
{
"pp": "case mpr.left\nX : Type u_1\nι : Type u_2\ninst✝ : TopologicalSpace X\nb : ι → Set X\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\ns : Set ι\nhs : s.Finite\n⊢ IsCompact (⋃ i ∈ s, b i)",
"ppTerm": "?mpr.left",
"assigned": true,
"usedConstants": [
"Membership.mem... | [] | exact hs.isCompact_biUnion fun i _ => hb' i | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Compactness.Bases | {
"line": 62,
"column": 6
} | {
"line": 62,
"column": 49
} | {
"line": 63,
"column": 4
} | [
{
"pp": "case mpr.left\nX : Type u_1\nι : Type u_2\ninst✝ : TopologicalSpace X\nb : ι → Set X\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\ns : Set ι\nhs : s.Finite\n⊢ IsCompact (⋃ i ∈ s, b i)",
"ppTerm": "?mpr.left",
"assigned": true,
"usedConstants": [
"Membership.mem... | [] | exact hs.isCompact_biUnion fun i _ => hb' i | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Compactness.Bases | {
"line": 62,
"column": 6
} | {
"line": 62,
"column": 49
} | {
"line": 63,
"column": 4
} | [
{
"pp": "case mpr.left\nX : Type u_1\nι : Type u_2\ninst✝ : TopologicalSpace X\nb : ι → Set X\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\ns : Set ι\nhs : s.Finite\n⊢ IsCompact (⋃ i ∈ s, b i)",
"ppTerm": "?mpr.left",
"assigned": true,
"usedConstants": [
"Membership.mem... | [] | exact hs.isCompact_biUnion fun i _ => hb' i | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.KrullDimension | {
"line": 840,
"column": 8
} | {
"line": 840,
"column": 75
} | {
"line": 841,
"column": 8
} | [
{
"pp": "case coe.coe\nα✝ : Type u_1\ninst✝² : Preorder α✝\nα : Type u_1\ninst✝¹ : Preorder α\ninst✝ : Nonempty α\nhnottop : krullDim α < ⊤\na : α\nthis✝ : height a < ⊤\nthis : coheight a < ⊤\nn : ℕ\nhh : height a = ↑n\nm : ℕ\nhch : coheight a = ↑m\np₁ : LTSeries α\nhlast : RelSeries.last p₁ = a\nhlen₁ : p₁.len... | [
"case coe.coe\nα✝ : Type u_1\ninst✝² : Preorder α✝\nα : Type u_1\ninst✝¹ : Preorder α\ninst✝ : Nonempty α\nhnottop : krullDim α < ⊤\na : α\nthis✝ : height a < ⊤\nthis : coheight a < ⊤\nn : ℕ\nhh : height a = ↑n\nm : ℕ\nhch : coheight a = ↑m\np₁ : LTSeries α\nhlast : RelSeries.last p₁ = a\nhlen₁ : p₁.length = n\np₂ ... | obtain ⟨p₂, hhead, hlen₂⟩ := exists_series_of_coheight_eq_coe a hch | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Order.KrullDimension | {
"line": 1053,
"column": 23
} | {
"line": 1053,
"column": 88
} | {
"line": 1053,
"column": 89
} | [
{
"pp": "α : Type u_1\ninst✝ : Preorder α\nx : α\np : LTSeries (WithTop α)\nhlast : RelSeries.last p = ↑x\ni : Fin p.length\n⊢ ((p.toFun i.castSucc).untop ⋯, (p.toFun i.succ).untop ⋯) ∈ {(a, b) | a < b}",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"lt_of_le_of_lt",... | [] | by simpa [WithTop.untop_lt_iff, WithTop.coe_untop] using p.step i | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.KrullDimension | {
"line": 1072,
"column": 21
} | {
"line": 1072,
"column": 34
} | {
"line": 1072,
"column": 34
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : Nonempty α\n⊢ ⨆ y, ⨆ (_ : y ≠ ⊤), height y + 1 = ⨆ i, height i + 1",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instCompleteLinearOrderENat",
"instAddMonoidWithOneENat",
"WithTop.instPreorder",
... | [
"α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : Nonempty α\n⊢ ⨆ x, height ↑x + 1 = ⨆ i, height i + 1"
] | iSup_subtype' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.RelSeries | {
"line": 990,
"column": 6
} | {
"line": 991,
"column": 83
} | {
"line": 992,
"column": 6
} | [
{
"pp": "case succ.refine_1\nα : Type u_3\ninst✝² : PartialOrder α\ninst✝¹ : WellFoundedLT α\ninst✝ : WellFoundedGT α\nn : ℕ\nIH :\n ∀ (s : Fin (n + 1) → α) (h : ∀ (i : Fin n), (s i.castSucc, s i.succ) ∈ {(a, b) | a < b}),\n ∃ t i,\n t.toFun ∘ ⇑i = { length := n, toFun := s, step := h }.toFun ∧\n ... | [
"case succ.refine_1.refine_1\nα : Type u_3\ninst✝² : PartialOrder α\ninst✝¹ : WellFoundedLT α\ninst✝ : WellFoundedGT α\nn : ℕ\nIH :\n ∀ (s : Fin (n + 1) → α) (h : ∀ (i : Fin n), (s i.castSucc, s i.succ) ∈ {(a, b) | a < b}),\n ∃ t i,\n t.toFun ∘ ⇑i = { length := n, toFun := s, step := h }.toFun ∧\n i... | refine Fin.lastCases (Fin.lastCases (fun _ ↦ rfl) fun j eq ↦ ?_) fun j ↦ Fin.lastCases
(fun eq ↦ ?_) fun k eq ↦ Fin.ext (congr_arg Fin.val (by simpa using! eq) :) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Order.KrullDimension | {
"line": 1085,
"column": 2
} | {
"line": 1085,
"column": 64
} | {
"line": 1087,
"column": 0
} | [
{
"pp": "⊢ krullDim (WithTop ℕ) = ⊤",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Order.krullDim_of_noMaxOrder",
"WithBot.some",
"WithBot",
"instAddMonoidWithOneENat",
"instTopENat",
"WithTop.instPreorder",
"congrArg",
"PartialOrder.toPreo... | [] | simp [← WithBot.coe_top, ← WithBot.coe_one, ← WithBot.coe_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Order.Ideal | {
"line": 376,
"column": 8
} | {
"line": 376,
"column": 43
} | {
"line": 377,
"column": 8
} | [
{
"pp": "P : Type u_1\ninst✝¹ : SemilatticeSup P\ninst✝ : IsCodirectedOrder P\nx : P\nI✝ J✝ s t I J : Ideal P\n⊢ {x | ∃ i ∈ I, ∃ j ∈ J, x ≤ i ⊔ j}.Nonempty",
"ppTerm": "?m.317",
"assigned": true,
"usedConstants": [
"PartialOrder.toPreorder",
"setOf",
"Preorder.toLE",
"Members... | [
"P : Type u_1\ninst✝¹ : SemilatticeSup P\ninst✝ : IsCodirectedOrder P\nx : P\nI✝ J✝ s t I J : Ideal P\nw : P\nh : w ∈ ↑I ∩ ↑J\n⊢ {x | ∃ i ∈ I, ∃ j ∈ J, x ≤ i ⊔ j}.Nonempty"
] | obtain ⟨w, h⟩ := inter_nonempty I J | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.QuasiSeparated | {
"line": 165,
"column": 4
} | {
"line": 165,
"column": 76
} | {
"line": 167,
"column": 0
} | [
{
"pp": "case insert.inr\nα✝ : Type u_1\ninst✝² : TopologicalSpace α✝\nα : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : QuasiSeparatedSpace α\ns✝ : Set (Set α)\na✝ : Set α\ns : Set (Set α)\nha : a✝ ∉ s\nhs : s.Finite\nih :\n (∀ t ∈ s, IsOpen[inst✝¹] t ∨ IsClosed[inst✝¹] t) →\n (∀ t ∈ s, IsCompact t) → (∀ ... | [] | · grind [IsCompact.inter_of_isOpen, hs.isOpen_sInter, Set.sInter_insert] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Sets.Compacts | {
"line": 624,
"column": 2
} | {
"line": 626,
"column": 53
} | {
"line": 628,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : WeaklyLocallyCompactSpace α\ninst✝ : Nonempty α\n⊢ Nonempty (PositiveCompacts α)",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Filter.i... | [] | inhabit α
rcases exists_compact_mem_nhds (default : α) with ⟨K, hKc, hK⟩
exact ⟨⟨K, hKc⟩, _, mem_interior_iff_mem_nhds.2 hK⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Sets.Compacts | {
"line": 624,
"column": 2
} | {
"line": 626,
"column": 53
} | {
"line": 628,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : WeaklyLocallyCompactSpace α\ninst✝ : Nonempty α\n⊢ Nonempty (PositiveCompacts α)",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Filter.i... | [] | inhabit α
rcases exists_compact_mem_nhds (default : α) with ⟨K, hKc, hK⟩
exact ⟨⟨K, hKc⟩, _, mem_interior_iff_mem_nhds.2 hK⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Constructible | {
"line": 278,
"column": 2
} | {
"line": 293,
"column": 84
} | {
"line": 295,
"column": 0
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\ns : Set X\nhf : IsClosedEmbedding f\nhfcomp : IsRetrocompact (range f)ᶜ\nhs : IsConstructible s\n⊢ IsConstructible (f '' s)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Set.in... | [] | induction hs using IsConstructible.empty_union_induction with
| open_retrocompact U hUopen hUcomp =>
have hfU : IsOpen (f '' U ∪ (range f)ᶜ) := by
simpa [← range_sdiff_image hf.injective, sdiff_eq, compl_inter, union_comm]
using (hf.isClosedMap _ hUopen.isClosed_compl).isOpen_compl
suffices h : ... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Topology.Constructible | {
"line": 278,
"column": 2
} | {
"line": 293,
"column": 84
} | {
"line": 295,
"column": 0
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\ns : Set X\nhf : IsClosedEmbedding f\nhfcomp : IsRetrocompact (range f)ᶜ\nhs : IsConstructible s\n⊢ IsConstructible (f '' s)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Set.in... | [] | induction hs using IsConstructible.empty_union_induction with
| open_retrocompact U hUopen hUcomp =>
have hfU : IsOpen (f '' U ∪ (range f)ᶜ) := by
simpa [← range_sdiff_image hf.injective, sdiff_eq, compl_inter, union_comm]
using (hf.isClosedMap _ hUopen.isClosed_compl).isOpen_compl
suffices h : ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Constructible | {
"line": 278,
"column": 2
} | {
"line": 293,
"column": 84
} | {
"line": 295,
"column": 0
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\ns : Set X\nhf : IsClosedEmbedding f\nhfcomp : IsRetrocompact (range f)ᶜ\nhs : IsConstructible s\n⊢ IsConstructible (f '' s)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Set.in... | [] | induction hs using IsConstructible.empty_union_induction with
| open_retrocompact U hUopen hUcomp =>
have hfU : IsOpen (f '' U ∪ (range f)ᶜ) := by
simpa [← range_sdiff_image hf.injective, sdiff_eq, compl_inter, union_comm]
using (hf.isClosedMap _ hUopen.isClosed_compl).isOpen_compl
suffices h : ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Constructible | {
"line": 484,
"column": 4
} | {
"line": 485,
"column": 37
} | {
"line": 486,
"column": 2
} | [
{
"pp": "case a\nX : Type u_2\ninst✝² : TopologicalSpace X\ns t : Set X\ninst✝¹ : PrespectralSpace X\ninst✝ : QuasiSeparatedSpace X\nhs : IsLocallyConstructible s\nhst : s ⊆ t\nht : IsCompact t\nU : X → Set X\nhU : ∀ (x : X), IsOpen[inst✝²] (U x)\nhU' : ∀ (x : X), IsCompact (U x)\nhxU : ∀ (x : X), x ∈ U x\nhUs ... | [] | rw [← Set.iUnion₂_inter, Set.subset_inter_iff]
exact ⟨hst.trans htσ, subset_rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Constructible | {
"line": 484,
"column": 4
} | {
"line": 485,
"column": 37
} | {
"line": 486,
"column": 2
} | [
{
"pp": "case a\nX : Type u_2\ninst✝² : TopologicalSpace X\ns t : Set X\ninst✝¹ : PrespectralSpace X\ninst✝ : QuasiSeparatedSpace X\nhs : IsLocallyConstructible s\nhst : s ⊆ t\nht : IsCompact t\nU : X → Set X\nhU : ∀ (x : X), IsOpen[inst✝²] (U x)\nhU' : ∀ (x : X), IsCompact (U x)\nhxU : ∀ (x : X), x ∈ U x\nhUs ... | [] | rw [← Set.iUnion₂_inter, Set.subset_inter_iff]
exact ⟨hst.trans htσ, subset_rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 621,
"column": 2
} | {
"line": 621,
"column": 44
} | {
"line": 623,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nf : R\n⊢ IsCompact (Set.range (comap (algebraMap R (Localization (Submonoid.powers f)))))",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Algebra.algebraMap",
"OreLocalization.instAlgebra",
"CommSemiring.toSemiring",
"Al... | [] | exact isCompact_range (continuous_comap _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 928,
"column": 2
} | {
"line": 928,
"column": 88
} | {
"line": 929,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : R →+* S\nh₁ : Function.Surjective (comap f)\nh₂ : GeneralizingMap (comap f)\ns : Set (PrimeSpectrum R)\nhsc : IsClosed (comap f ⁻¹' s)\n⊢ StableUnderGeneralization sᶜ",
"ppTerm": "?m.69",
"assigned": true,
"use... | [
"R : Type u_1\nS : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : R →+* S\nh₁ : Function.Surjective (comap f)\nh₂ : GeneralizingMap (comap f)\ns : Set (PrimeSpectrum R)\nhsc : IsClosed (comap f ⁻¹' s)\n⊢ sᶜ = comap f '' (comap f ⁻¹' s)ᶜ"
] | convert! h₂.stableUnderGeneralization_image hsc.isOpen_compl.stableUnderGeneralization | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.GroupTheory.Exponent | {
"line": 168,
"column": 2
} | {
"line": 172,
"column": 23
} | {
"line": 174,
"column": 0
} | [
{
"pp": "G : Type u\ninst✝ : Monoid G\nn : ℕ\nhpos : 0 < n\nhG : ∀ (g : G), g ^ n = 1\n⊢ exponent G ≤ n",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"Monoid.ExponentExists",
"Monoid.exponen... | [] | classical
rw [exponent, dif_pos]
· apply Nat.find_min'
exact ⟨hpos, hG⟩
· exact ⟨n, hpos, hG⟩ | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.GroupTheory.Exponent | {
"line": 168,
"column": 2
} | {
"line": 172,
"column": 23
} | {
"line": 174,
"column": 0
} | [
{
"pp": "G : Type u\ninst✝ : Monoid G\nn : ℕ\nhpos : 0 < n\nhG : ∀ (g : G), g ^ n = 1\n⊢ exponent G ≤ n",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"Monoid.ExponentExists",
"Monoid.exponen... | [] | classical
rw [exponent, dif_pos]
· apply Nat.find_min'
exact ⟨hpos, hG⟩
· exact ⟨n, hpos, hG⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Exponent | {
"line": 168,
"column": 2
} | {
"line": 172,
"column": 23
} | {
"line": 174,
"column": 0
} | [
{
"pp": "G : Type u\ninst✝ : Monoid G\nn : ℕ\nhpos : 0 < n\nhG : ∀ (g : G), g ^ n = 1\n⊢ exponent G ≤ n",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"Monoid.ExponentExists",
"Monoid.exponen... | [] | classical
rw [exponent, dif_pos]
· apply Nat.find_min'
exact ⟨hpos, hG⟩
· exact ⟨n, hpos, hG⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.SpecificGroups.Cyclic.Basic | {
"line": 334,
"column": 19
} | {
"line": 334,
"column": 28
} | {
"line": 334,
"column": 29
} | [
{
"pp": "α : Type u_1\ninst✝³ : Group α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : IsCyclic α\nn : ℕ\nhn0 : 0 < n\ng : α\nhg : ∀ (x : α), x ∈ zpowers g\nm : ℕ\nhm : Fintype.card α = n.gcd (Fintype.card α) * 0\nhm0 : m = 0\n⊢ False",
"ppTerm": "?m.224",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_1\ninst✝³ : Group α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : IsCyclic α\nn : ℕ\nhn0 : 0 < n\ng : α\nhg : ∀ (x : α), x ∈ zpowers g\nm : ℕ\nhm : Fintype.card α = 0\nhm0 : m = 0\n⊢ False"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.PGroup | {
"line": 153,
"column": 91
} | {
"line": 158,
"column": 33
} | {
"line": 160,
"column": 0
} | [
{
"pp": "p : ℕ\nG : Type u_1\ninst✝² : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\nα : Type u_2\ninst✝¹ : MulAction G α\na : α\ninst✝ : Finite ↑(orbit G a)\n⊢ ∃ n, Nat.card ↑(orbit G a) = p ^ n",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finite.of_equiv"... | [] | by
let ϕ := orbitEquivQuotientStabilizer G a
haveI := Finite.of_equiv (orbit G a) ϕ
haveI := (stabilizer G a).finiteIndex_of_finite_quotient
rw [Nat.card_congr ϕ]
exact hG.index (stabilizer G a) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 1279,
"column": 2
} | {
"line": 1279,
"column": 87
} | {
"line": 1280,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\n⊢ zeroLocus ∘ SetLike.coe '' minimalPrimes R = irreducibleComponents (PrimeSpectrum R)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"irreducibleComponents",
"Semiring.toModule",
"PrimeSpectrum.zeroLocus",
... | [
"R : Type u\ninst✝ : CommSemiring R\n⊢ Set.EqOn ((zeroLocus ∘ SetLike.coe) ∘ vanishingIdeal) id (irreducibleComponents (PrimeSpectrum R))"
] | rw [← vanishingIdeal_irreducibleComponents, ← Set.image_comp, Set.EqOn.image_eq_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 1359,
"column": 67
} | {
"line": 1361,
"column": 16
} | {
"line": 1363,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : CommSemiring R\ninst✝ : IsLocalRing R\n⊢ IsClosed {closedPoint R}",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PrimeSpectrum.mk",
"IsLocalRing.closedPoint._proof_1",
"congrArg",
"CommSemiring.toSemiring",
"IsLoc... | [] | by
rw [PrimeSpectrum.isClosed_singleton_iff_isMaximal, closedPoint]
infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Rank | {
"line": 54,
"column": 2
} | {
"line": 54,
"column": 35
} | {
"line": 55,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : FG G\nh : rank G = 0\ns : Finset G\nhs : s.card = rank G\nhs' : Subgroup.closure ↑s = ⊤\n⊢ Subsingleton G",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Subgroup.closure",
"congrArg",
"Finset",
"Eq.mp",
"Subgr... | [
"G : Type u_1\ninst✝¹ : Group G\ninst✝ : FG G\nh : rank G = 0\ns : Finset G\nhs : s = ∅\nhs' : Subgroup.closure ↑s = ⊤\n⊢ Subsingleton G"
] | rw [h, Finset.card_eq_zero] at hs | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 195,
"column": 68
} | {
"line": 195,
"column": 87
} | {
"line": 195,
"column": 87
} | [
{
"pp": "G : Type u_2\nG' : Type u_3\ninst✝² : Group G\ninst✝¹ : Group G'\ninst✝ : IsCyclic G'\nf : G →* G'\nhf : f.ker ≤ center G\na b : G\nx : G'\ny : G\nhxy : f y = x\nhx : ∀ (a : ↥f.range), a ∈ zpowers ⟨x, ⋯⟩\nm : ℤ\nhm✝ : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) m = ⟨f a, ⋯⟩\nn : ℤ\nhn✝ : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) n = ⟨f b... | [
"G : Type u_2\nG' : Type u_3\ninst✝² : Group G\ninst✝¹ : Group G'\ninst✝ : IsCyclic G'\nf : G →* G'\nhf : f.ker ≤ center G\na b : G\nx : G'\ny : G\nhxy : f y = x\nhx : ∀ (a : ↥f.range), a ∈ zpowers ⟨x, ⋯⟩\nm : ℤ\nhm✝ : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) m = ⟨f a, ⋯⟩\nn : ℤ\nhn✝ : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) n = ⟨f b, ⋯⟩\nhm : x... | mem_center_iff.1 ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 603,
"column": 2
} | {
"line": 603,
"column": 63
} | {
"line": 604,
"column": 2
} | [
{
"pp": "G : Type u_2\ninst✝¹ : Group G\ninst✝ : Finite G\nh : IsCyclic G\nthis : NeZero (Nat.card G)\n⊢ Nat.card (MulAut G) = (Nat.card G).totient",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ZMod.commRing",
"Monoid.toMulOneClass",
"congrArg",
"... | [
"G : Type u_2\ninst✝¹ : Group G\ninst✝ : Finite G\nh : IsCyclic G\nthis : NeZero (Nat.card G)\n⊢ Nat.card (MulAut G) = Nat.card (ZMod (Nat.card G))ˣ"
] | rw [← ZMod.card_units_eq_totient, ← Nat.card_eq_fintype_card] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.CompactlyGenerated.Intervals | {
"line": 25,
"column": 2
} | {
"line": 25,
"column": 9
} | {
"line": 25,
"column": 10
} | [
{
"pp": "α : Type u_2\ninst✝ : CompleteLattice α\na : α\nb : ↑(Iic a)\nh : ∀ (ι : Type u_2) (s : ι → α), ↑b ≤ iSup s → ∃ t, ↑b ≤ ⨆ a ∈ t, s a\n⊢ ∀ (ι : Type u_2) (s : ι → ↑(Iic a)), b ≤ iSup s → ∃ t, b ≤ ⨆ a_2 ∈ t, s a_2",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [],
"usedFVars": []... | [
"α : Type u_2\ninst✝ : CompleteLattice α\na : α\nb : ↑(Iic a)\nh : ∀ (ι : Type u_2) (s : ι → α), ↑b ≤ iSup s → ∃ t, ↑b ≤ ⨆ a ∈ t, s a\nι : Type u_2\n⊢ ∀ (s : ι → ↑(Iic a)), b ≤ iSup s → ∃ t, b ≤ ⨆ a_2 ∈ t, s a_2"
] | intro ι | Lean.Elab.Tactic.evalIntro | null |
Mathlib.GroupTheory.Sylow | {
"line": 750,
"column": 37
} | {
"line": 750,
"column": 48
} | {
"line": 750,
"column": 49
} | [
{
"pp": "G : Type u\ninst✝¹ : Group G\ninst✝ : Fintype G\np k : ℕ\nhp : Nat.Prime p\nh : p ^ k ∣ Nat.card G\nthis✝ : Fact (Nat.Prime p)\nH : Subgroup G\nhH : Nat.card ↥H = p ^ k\nthis : ∀ g ∈ (↑H \\ {1}).toFinset, p ∣ orderOf g\n⊢ p ^ k ∣ ∏ i ∈ (↑H \\ {1}).toFinset, p",
"ppTerm": "?m.183",
"assigned": t... | [
"G : Type u\ninst✝¹ : Group G\ninst✝ : Fintype G\np k : ℕ\nhp : Nat.Prime p\nh : p ^ k ∣ Nat.card G\nthis✝ : Fact (Nat.Prime p)\nH : Subgroup G\nhH : Nat.card ↥H = p ^ k\nthis : ∀ g ∈ (↑H \\ {1}).toFinset, p ∣ orderOf g\n⊢ p ^ k ∣ p ^ #(↑H \\ {1}).toFinset"
] | prod_const, | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.GroupTheory.Sylow | {
"line": 774,
"column": 41
} | {
"line": 776,
"column": 40
} | {
"line": 778,
"column": 0
} | [
{
"pp": "G : Type u\ninst✝² : Group G\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nh : (↑P).Normal\n⊢ (↑P).Characteristic",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Sylow",
"Unique",
"Sylow.characteristic_of_subsingleton",
"Un... | [] | by
have _ := unique_of_normal P h
exact characteristic_of_subsingleton _ | [anonymous] | Lean.Parser.Term.byTactic |
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