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.LinearAlgebra.Matrix.Charpoly.Coeff | {
"line": 241,
"column": 29
} | {
"line": 241,
"column": 37
} | [
{
"pp": "case pos\nn : Type v\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nK : Type u_1\nk : ℕ\ninst✝ : CommRing K\nM : Matrix n n K\nm : ℕ\ni j : n\nhij : ¬i = j\nm0 : m = 0\nmp : m = k\n⊢ -(M ^ k) i j = (1 - M ^ k) i j",
"usedConstants": [
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoi... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff | {
"line": 241,
"column": 29
} | {
"line": 241,
"column": 37
} | [
{
"pp": "case neg\nn : Type v\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nK : Type u_1\nk : ℕ\ninst✝ : CommRing K\nM : Matrix n n K\nm : ℕ\ni j : n\nhij : ¬i = j\nm0 : m = 0\nmp : ¬m = k\n⊢ -(M ^ k) i j = (0 - M ^ k) i j",
"usedConstants": [
"NegZeroClass.toNeg",
"AddGroupWithOne.toAddGroup",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff | {
"line": 241,
"column": 29
} | {
"line": 241,
"column": 37
} | [
{
"pp": "case pos\nn : Type v\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nK : Type u_1\nk : ℕ\ninst✝ : CommRing K\nM : Matrix n n K\nm : ℕ\ni j : n\nhij : ¬i = j\nm0 : ¬m = 0\nh✝ : m = k\n⊢ -0 = (1 - 0) i j",
"usedConstants": [
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff | {
"line": 241,
"column": 29
} | {
"line": 241,
"column": 37
} | [
{
"pp": "case neg\nn : Type v\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nK : Type u_1\nk : ℕ\ninst✝ : CommRing K\nM : Matrix n n K\nm : ℕ\ni j : n\nhij : ¬i = j\nm0 : ¬m = 0\nh✝ : ¬m = k\n⊢ -0 = (0 - 0) i j",
"usedConstants": [
"NegZeroClass.toNeg",
"sub_self",
"AddGroupWithOne.toAddGrou... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Polynomial.Lifts | {
"line": 73,
"column": 6
} | {
"line": 73,
"column": 31
} | [
{
"pp": "R : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\np : S[X]\n⊢ p ∈ lifts f ↔ ∀ (n : ℕ), p.coeff n ∈ Set.range ⇑f",
"usedConstants": [
"Eq.mpr",
"Subsemiring.instSetLike",
"congrArg",
"RingHom",
"Membership.mem",
"id",
"Polynomial.... | lifts_iff_ringHom_rangeS, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Lifts | {
"line": 127,
"column": 6
} | {
"line": 127,
"column": 31
} | [
{
"pp": "R : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\np : S[X]\nn : ℕ\nh : p ∈ lifts f\n⊢ erase n p ∈ lifts f",
"usedConstants": [
"Subsemiring.instSetLike",
"congrArg",
"Membership.mem",
"Eq.mp",
"Polynomial.lifts_iff_ringHom_rangeS",
"Su... | lifts_iff_ringHom_rangeS, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.HasseDeriv | {
"line": 69,
"column": 35
} | {
"line": 69,
"column": 43
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nk : ℕ\nf : R[X]\nn : ℕ\n⊢ (f.sum fun a b ↦ ((monomial (a - k)) (↑(a.choose k) * b)).coeff n) = ↑((n + k).choose k) * f.coeff (n + k)",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Nat.choose",
"Semiring.toModule",... | sum_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.IntegralNormalization | {
"line": 111,
"column": 6
} | {
"line": 111,
"column": 48
} | [
{
"pp": "case pos\nR : Type u\ninst✝ : Semiring R\np : R[X]\ni : ℕ\nhp : 1 ≤ p.natDegree\nh : ¬p.degree = ↑i\nh' : ↑i < p.degree\n⊢ i ≤ p.natDegree - 1",
"usedConstants": [
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"Nat.instOrderedSub",
"Nat.instIsOrderedAddMonoid",
"AddLeftCa... | rw [le_tsub_iff_right hp, Nat.succ_le_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Polynomial.ScaleRoots | {
"line": 237,
"column": 4
} | {
"line": 237,
"column": 54
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : IsReduced R\np : R[X]\nr : R\nn : ℕ\nhp : p = 0\n⊢ (p ^ n).scaleRoots r = p.scaleRoots r ^ n",
"usedConstants": [
"MulOne.toOne",
"Polynomial.instOne",
"congrArg",
"CommSemiring.toSemiring",
"Polynomial.one_scale... | simp [hp, zero_pow_eq, apply_ite (scaleRoots · r)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Polynomial.ScaleRoots | {
"line": 237,
"column": 4
} | {
"line": 237,
"column": 54
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : IsReduced R\np : R[X]\nr : R\nn : ℕ\nhp : p = 0\n⊢ (p ^ n).scaleRoots r = p.scaleRoots r ^ n",
"usedConstants": [
"MulOne.toOne",
"Polynomial.instOne",
"congrArg",
"CommSemiring.toSemiring",
"Polynomial.one_scale... | simp [hp, zero_pow_eq, apply_ite (scaleRoots · r)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.ScaleRoots | {
"line": 237,
"column": 4
} | {
"line": 237,
"column": 54
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : IsReduced R\np : R[X]\nr : R\nn : ℕ\nhp : p = 0\n⊢ (p ^ n).scaleRoots r = p.scaleRoots r ^ n",
"usedConstants": [
"MulOne.toOne",
"Polynomial.instOne",
"congrArg",
"CommSemiring.toSemiring",
"Polynomial.one_scale... | simp [hp, zero_pow_eq, apply_ite (scaleRoots · r)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.ScaleRoots | {
"line": 240,
"column": 34
} | {
"line": 240,
"column": 42
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : IsReduced R\np : R[X]\nr : R\nn : ℕ\nhp : ¬p = 0\nhn : ¬n = 0\n⊢ p.leadingCoeff ^ n ≠ 0",
"usedConstants": [
"False",
"eq_false",
"congrArg",
"CommSemiring.toSemiring",
"Polynomial.leadingCoeff",
"instOfNatNat",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Polynomial.ScaleRoots | {
"line": 240,
"column": 34
} | {
"line": 240,
"column": 42
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : IsReduced R\np : R[X]\nr : R\nn : ℕ\nhp : ¬p = 0\nhn : ¬n = 0\n⊢ p.leadingCoeff ^ n ≠ 0",
"usedConstants": [
"False",
"eq_false",
"congrArg",
"CommSemiring.toSemiring",
"Polynomial.leadingCoeff",
"instOfNatNat",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.ScaleRoots | {
"line": 240,
"column": 34
} | {
"line": 240,
"column": 42
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : IsReduced R\np : R[X]\nr : R\nn : ℕ\nhp : ¬p = 0\nhn : ¬n = 0\n⊢ p.leadingCoeff ^ n ≠ 0",
"usedConstants": [
"False",
"eq_false",
"congrArg",
"CommSemiring.toSemiring",
"Polynomial.leadingCoeff",
"instOfNatNat",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Subring | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 28
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\nT : Subring R\np : (↥T)[X]\ni : R\nhi : ∃ x, ¬(ofSubring T p).coeff x = 0 ∧ (ofSubring T p).coeff x = i\n⊢ i ∈ ↑T",
"usedConstants": []
}
] | rcases hi with ⟨n, _, h'n⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.RingTheory.Polynomial.ScaleRoots | {
"line": 282,
"column": 11
} | {
"line": 282,
"column": 24
} | [
{
"pp": "case H\nR : Type u_1\ninst✝ : CommSemiring R\np q : R[X]\nr : R\nhr : IsUnit r\na b : R[X]\ne : a * p + b * q = 1\ns : R := ↑hr.unit⁻¹\n⊢ natDegree 1 = 0",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Polynomial.instOne",
"congrArg",
"CommSe... | natDegree_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Splits | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 53
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nf : R[X]\nhf : f.natDegree = 0\n⊢ f.Splits",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Exists.choose_spec",
"Polynomial.Splits.C._simp_1",
"congrArg",
"RingHom",
"Exists",
"id",
"instOfNatNat",
"Polyno... | rw [← (natDegree_eq_zero.mp hf).choose_spec]; aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Splits | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 53
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nf : R[X]\nhf : f.natDegree = 0\n⊢ f.Splits",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Exists.choose_spec",
"Polynomial.Splits.C._simp_1",
"congrArg",
"RingHom",
"Exists",
"id",
"instOfNatNat",
"Polyno... | rw [← (natDegree_eq_zero.mp hf).choose_spec]; aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.ScaleRoots | {
"line": 336,
"column": 40
} | {
"line": 336,
"column": 48
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\np : R[X]\nr a : R\nhr : IsLeftRegular r\nh✝ : Nontrivial R\nq : R[X]\ne : p = (X - C a) ^ rootMultiplicity a p * q\nhq : ¬X - C a ∣ q\nhp : q = 0\n⊢ p = 0",
"usedConstants": [
"Polynomial.C",
"False",
"Dvd.dvd",
"CommRing.toNonUnitalCommRing... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Polynomial.Splits | {
"line": 192,
"column": 2
} | {
"line": 192,
"column": 28
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝ : CommSemiring R\nf g : R[X]\nhf : f.Splits\nhg✝ : g.natDegree ≤ 1\nh : Invertible g.leadingCoeff\nhg : g.natDegree = 1\nm : Multiset R\nhm : f = C f.leadingCoeff * (Multiset.map (fun x ↦ X + C x) m).prod\na : R\n⊢ Invertible ((X + C a).comp g).leadingCoeff",
"usedCons... | rw [leadingCoeff, hg] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.TensorProduct.MvPolynomial | {
"line": 114,
"column": 54
} | {
"line": 114,
"column": 63
} | [
{
"pp": "R : Type u\nN : Type v\ninst✝³ : CommSemiring R\nσ : Type u_1\ninst✝² : DecidableEq σ\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\ne : σ →₀ ℕ\nr : R\nn : N\nd : σ →₀ ℕ\n⊢ (if e = d then r else 0) • n = if e = d then r • n else 0",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
... | ite_smul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.TensorProduct.MvPolynomial | {
"line": 118,
"column": 48
} | {
"line": 118,
"column": 57
} | [
{
"pp": "R : Type u\nN : Type v\ninst✝³ : CommSemiring R\nσ : Type u_1\ninst✝² : DecidableEq σ\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\ns : σ\nn : N\nd : σ →₀ ℕ\n⊢ (if Finsupp.single s 1 = d then 1 else 0) • n = if Finsupp.single s 1 = d then n else 0",
"usedConstants": [
"Eq.mpr",
"NonAss... | ite_smul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Splits | {
"line": 446,
"column": 29
} | {
"line": 446,
"column": 37
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nf : R[X]\ninst✝ : IsDomain R\ng : R[X]\nhg : (f * g).Splits\nhg₀ : f * g ≠ 0\n⊢ f ≠ 0",
"usedConstants": [
"False",
"Semigroup.toMul",
"IsDomain.to_noZeroDivisors",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"eq_false",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Polynomial.Splits | {
"line": 446,
"column": 29
} | {
"line": 446,
"column": 37
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nf : R[X]\ninst✝ : IsDomain R\ng : R[X]\nhg : (f * g).Splits\nhg₀ : f * g ≠ 0\n⊢ f ≠ 0",
"usedConstants": [
"False",
"Semigroup.toMul",
"IsDomain.to_noZeroDivisors",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"eq_false",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Splits | {
"line": 446,
"column": 29
} | {
"line": 446,
"column": 37
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nf : R[X]\ninst✝ : IsDomain R\ng : R[X]\nhg : (f * g).Splits\nhg₀ : f * g ≠ 0\n⊢ f ≠ 0",
"usedConstants": [
"False",
"Semigroup.toMul",
"IsDomain.to_noZeroDivisors",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"eq_false",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Splits | {
"line": 446,
"column": 43
} | {
"line": 446,
"column": 51
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nf : R[X]\ninst✝ : IsDomain R\ng : R[X]\nhg : (f * g).Splits\nhg₀ : f * g ≠ 0\n⊢ g ≠ 0",
"usedConstants": [
"False",
"Semigroup.toMul",
"IsDomain.to_noZeroDivisors",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"eq_false",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Polynomial.Splits | {
"line": 446,
"column": 43
} | {
"line": 446,
"column": 51
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nf : R[X]\ninst✝ : IsDomain R\ng : R[X]\nhg : (f * g).Splits\nhg₀ : f * g ≠ 0\n⊢ g ≠ 0",
"usedConstants": [
"False",
"Semigroup.toMul",
"IsDomain.to_noZeroDivisors",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"eq_false",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Splits | {
"line": 446,
"column": 43
} | {
"line": 446,
"column": 51
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nf : R[X]\ninst✝ : IsDomain R\ng : R[X]\nhg : (f * g).Splits\nhg₀ : f * g ≠ 0\n⊢ g ≠ 0",
"usedConstants": [
"False",
"Semigroup.toMul",
"IsDomain.to_noZeroDivisors",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"eq_false",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Splits | {
"line": 489,
"column": 4
} | {
"line": 489,
"column": 12
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : DivisionSemiring R\na b : R\nhf : (C a * X + C b).natDegree ≤ 1\nha : a = 0\n⊢ (C a * X + C b).Splits",
"usedConstants": [
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClass",
"Polynomial.Splits.C._simp_1",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Polynomial.Splits | {
"line": 489,
"column": 4
} | {
"line": 489,
"column": 12
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : DivisionSemiring R\na b : R\nhf : (C a * X + C b).natDegree ≤ 1\nha : a = 0\n⊢ (C a * X + C b).Splits",
"usedConstants": [
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClass",
"Polynomial.Splits.C._simp_1",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Splits | {
"line": 489,
"column": 4
} | {
"line": 489,
"column": 12
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : DivisionSemiring R\na b : R\nhf : (C a * X + C b).natDegree ≤ 1\nha : a = 0\n⊢ (C a * X + C b).Splits",
"usedConstants": [
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClass",
"Polynomial.Splits.C._simp_1",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Localization.Integral | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 50
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\np : S[X]\ni : ℕ\nh : i ∉ p.support\n⊢ coeffIntegerNormalization M p i = 0",
"usedConstants": [
"congrArg",
"CommSemiring.toSemiring",
"Finset",
... | simp only [mem_support_iff, ne_eq, not_not] at h | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Localization.Integral | {
"line": 244,
"column": 9
} | {
"line": 244,
"column": 12
} | [
{
"pp": "case pos\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\n⊢ p.coeff n * (algebraMap R Rₘ) ↑(commonDenom... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.GoingUp | {
"line": 347,
"column": 25
} | {
"line": 347,
"column": 41
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\nS : Type u_2\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : Algebra.IsIntegral R S\ninst✝¹ : FaithfulSMul R S\nP : Ideal R\ninst✝ : P.IsMaximal\n⊢ ∃ Q, Q.IsMaximal ∧ P = under R Q",
"usedConstants": [
"Eq.mpr",
"congrArg",
"CommSemiring.toS... | eq_comm (a := P) | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Localization.Integral | {
"line": 334,
"column": 6
} | {
"line": 334,
"column": 14
} | [
{
"pp": "case neg\nR : 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 ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Localization.Integral | {
"line": 334,
"column": 6
} | {
"line": 334,
"column": 14
} | [
{
"pp": "case neg\nR : 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 ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Localization.Integral | {
"line": 334,
"column": 6
} | {
"line": 334,
"column": 14
} | [
{
"pp": "case neg\nR : 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 ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Spectrum.Prime.Basic | {
"line": 284,
"column": 2
} | {
"line": 291,
"column": 11
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nI : Ideal R\n⊢ zeroLocus ↑I = ∅ ↔ I = ⊤",
"usedConstants": [
"Eq.mpr",
"PrimeSpectrum.mk",
"Semiring.toModule",
"PrimeSpectrum.zeroLocus",
"CommSemiring.toSemiring",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Memb... | constructor
· contrapose!
intro h
rcases Ideal.exists_le_maximal I h with ⟨M, hM, hIM⟩
exact ⟨⟨M, hM.isPrime⟩, hIM⟩
· rintro rfl
apply zeroLocus_empty_of_one_mem
trivial | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Spectrum.Prime.Basic | {
"line": 284,
"column": 2
} | {
"line": 291,
"column": 11
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nI : Ideal R\n⊢ zeroLocus ↑I = ∅ ↔ I = ⊤",
"usedConstants": [
"Eq.mpr",
"PrimeSpectrum.mk",
"Semiring.toModule",
"PrimeSpectrum.zeroLocus",
"CommSemiring.toSemiring",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Memb... | constructor
· contrapose!
intro h
rcases Ideal.exists_le_maximal I h with ⟨M, hM, hIM⟩
exact ⟨⟨M, hM.isPrime⟩, hIM⟩
· rintro rfl
apply zeroLocus_empty_of_one_mem
trivial | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Localization.Integral | {
"line": 341,
"column": 4
} | {
"line": 353,
"column": 24
} | [
{
"pp": "case refine_1\nR : 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.c... | refine monic_of_natDegree_le_of_coeff_eq_one (q.natDegree + N) ?_ ?_
· refine natDegree_sum_le_of_forall_le _ _ fun i hi ↦ ?_
grw [natDegree_mul_le, natDegree_pow_le, natDegree_X_le, natDegree_reflect_le]
simp [max_eq_left (hN _), le_natDegree_of_mem_supp _ hi]
· simp only [sum, finsetSum_coeff, coe... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Localization.Integral | {
"line": 341,
"column": 4
} | {
"line": 353,
"column": 24
} | [
{
"pp": "case refine_1\nR : 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.c... | refine monic_of_natDegree_le_of_coeff_eq_one (q.natDegree + N) ?_ ?_
· refine natDegree_sum_le_of_forall_le _ _ fun i hi ↦ ?_
grw [natDegree_mul_le, natDegree_pow_le, natDegree_X_le, natDegree_reflect_le]
simp [max_eq_left (hN _), le_natDegree_of_mem_supp _ hi]
· simp only [sum, finsetSum_coeff, coe... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Spectrum.Prime.Basic | {
"line": 474,
"column": 4
} | {
"line": 474,
"column": 57
} | [
{
"pp": "case right\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nhA_nont : Nontrivial A\nhgt : ∀ J > ⊤, J ≠ ⊥ → ∃ Z, (Multiset.map asIdeal Z).prod ≤ J ∧ (Multiset.map asIdeal Z).prod ≠ ⊥\nh_nzI : ⊤ ≠ ⊥\np_id : Ideal A\nh_nzp : p_id ≠ ⊥\nh_pp : p_id.IsPrim... | rwa [Multiset.map_singleton, Multiset.prod_singleton] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.RingTheory.Spectrum.Prime.Basic | {
"line": 479,
"column": 2
} | {
"line": 479,
"column": 85
} | [
{
"pp": "case neg\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nM : Ideal A\nhgt : ∀ J > M, J ≠ ⊥ → ∃ Z, (Multiset.map asIdeal Z).prod ≤ J ∧ (Multiset.map asIdeal Z).prod ≠ ⊥\nh_nzI : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : ¬M.IsPrime\n⊢ ∃ ... | obtain ⟨x, hx, y, hy, h_xy⟩ := (Ideal.not_isPrime_iff.mp h_prM).resolve_left h_topM | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Localization.Integral | {
"line": 451,
"column": 4
} | {
"line": 451,
"column": 26
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝¹⁵ : CommRing R\nS : Type u_2\ninst✝¹⁴ : CommRing S\ninst✝¹³ : Algebra R S\nRf : Type u_5\nSf : Type u_6\ninst✝¹² : CommRing Rf\ninst✝¹¹ : CommRing Sf\ninst✝¹⁰ : Algebra R Rf\ninst✝⁹ : Algebra S Sf\ninst✝⁸ : Algebra Rf Sf\ninst✝⁷ : Algebra R Sf\ninst✝⁶ : IsScalarTower ... | rintro ⟨_, f, hf, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.RingTheory.Spectrum.Prime.Basic | {
"line": 466,
"column": 2
} | {
"line": 495,
"column": 38
} | [
{
"pp": "A : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI : Ideal A\nh_nzI : I ≠ ⊥\n⊢ ∃ Z, (Multiset.map asIdeal Z).prod ≤ I ∧ (Multiset.map asIdeal Z).prod ≠ ⊥",
"usedConstants": [
"Nontrivial",
"add_mul",
"Distrib.leftDistribClass",
... | induction I using IsNoetherian.induction with | hgt M hgt =>
change Ideal A at M
have hA_nont : Nontrivial A := IsDomain.toNontrivial
by_cases h_topM : M = ⊤
· rcases h_topM with rfl
obtain ⟨p_id, h_nzp, h_pp⟩ : ∃ p : Ideal A, p ≠ ⊥ ∧ p.IsPrime := by
apply Ring.not_isField_iff_exists_prime.mp h_fA
... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.RingTheory.Spectrum.Prime.Basic | {
"line": 466,
"column": 2
} | {
"line": 495,
"column": 38
} | [
{
"pp": "A : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI : Ideal A\nh_nzI : I ≠ ⊥\n⊢ ∃ Z, (Multiset.map asIdeal Z).prod ≤ I ∧ (Multiset.map asIdeal Z).prod ≠ ⊥",
"usedConstants": [
"Nontrivial",
"add_mul",
"Distrib.leftDistribClass",
... | induction I using IsNoetherian.induction with | hgt M hgt =>
change Ideal A at M
have hA_nont : Nontrivial A := IsDomain.toNontrivial
by_cases h_topM : M = ⊤
· rcases h_topM with rfl
obtain ⟨p_id, h_nzp, h_pp⟩ : ∃ p : Ideal A, p ≠ ⊥ ∧ p.IsPrime := by
apply Ring.not_isField_iff_exists_prime.mp h_fA
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Spectrum.Prime.Basic | {
"line": 466,
"column": 2
} | {
"line": 495,
"column": 38
} | [
{
"pp": "A : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI : Ideal A\nh_nzI : I ≠ ⊥\n⊢ ∃ Z, (Multiset.map asIdeal Z).prod ≤ I ∧ (Multiset.map asIdeal Z).prod ≠ ⊥",
"usedConstants": [
"Nontrivial",
"add_mul",
"Distrib.leftDistribClass",
... | induction I using IsNoetherian.induction with | hgt M hgt =>
change Ideal A at M
have hA_nont : Nontrivial A := IsDomain.toNontrivial
by_cases h_topM : M = ⊤
· rcases h_topM with rfl
obtain ⟨p_id, h_nzp, h_pp⟩ : ∃ p : Ideal A, p ≠ ⊥ ∧ p.IsPrime := by
apply Ring.not_isField_iff_exists_prime.mp h_fA
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.SurjectiveOnStalks | {
"line": 61,
"column": 6
} | {
"line": 62,
"column": 80
} | [
{
"pp": "case mpr.refine_2\nR : Type u_1\ninst✝² : CommRing R\nS : Type u_2\ninst✝¹ : CommRing S\nf : R →+* S\nP : Ideal S\ninst✝ : P.IsPrime\nH : ∀ (s : S), ∃ x r, ∃ c ∉ P, f r ∉ P ∧ c * f r * s = c * f x\ny✝ : Localization.AtPrime P\nx✝ : S × ↥P.primeCompl\ny t : S\nh : t ∈ P.primeCompl\nyx ys : R\nyc : S\nhy... | simp only [Localization.mk_eq_mk', Localization.localRingHom_mk', map_mul f,
IsLocalization.mk'_eq_iff_eq, IsLocalization.eq_iff_exists P.primeCompl] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Flat.FaithfullyFlat.Basic | {
"line": 489,
"column": 2
} | {
"line": 489,
"column": 54
} | [
{
"pp": "case mp\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\nh : 1 ⊗ₜ[R] m = 0\nf : R →ₗ[R] M := (LinearMap.lsmul R M).flip m\n⊢ f = 0",
"usedConstants": [
"Eq.mpr",
... | rw [Module.FaithfullyFlat.zero_iff_lTensor_zero R A] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Algebraic.Integral | {
"line": 236,
"column": 55
} | {
"line": 249,
"column": 41
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Ring A\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\ninst✝ : NoZeroDivisors S\nint : Algebra.IsIntegral R S\na : A\nh : IsAlgebraic S a\n⊢ IsAlgebraic R a",
... | by
by_cases hRS : Function.Injective (algebraMap R S)
on_goal 2 => exact (Algebra.isAlgebraic_of_not_injective
fun h ↦ hRS <| .of_comp (IsScalarTower.algebraMap_eq R S A ▸ h)).1 _
have := hRS.noZeroDivisors _ (map_zero _) (map_mul _)
have ⟨s, hs, int_s⟩ := h.exists_integral_multiple
cases subsingleton_or_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Flat.FaithfullyFlat.Algebra | {
"line": 76,
"column": 33
} | {
"line": 80,
"column": 49
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝⁶ : CommRing A\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra A B\ninst✝³ : IsDomain B\ninst✝² : Flat A B\ninst✝¹ : Algebra.IsIntegral A B\ninst✝ : FaithfulSMul A B\n⊢ FaithfullyFlat A B",
"usedConstants": [
"Iff.mpr",
"PrimeSpectrum.mk",
"RingHom.instRing... | by
refine Module.FaithfullyFlat.of_comap_surjective fun P ↦ ?_
obtain ⟨P, hP₁, hP₂⟩ := Ideal.exists_ideal_over_prime_of_isIntegral_of_isDomain P.1 (S := B)
(by simp [(RingHom.injective_iff_ker_eq_bot _).mp (FaithfulSMul.algebraMap_injective A B)])
exact ⟨⟨P, hP₁⟩, PrimeSpectrum.ext_iff.mpr hP₂⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.LocalRing.ResidueField.Ideal | {
"line": 54,
"column": 2
} | {
"line": 56,
"column": 50
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\nf : R →+* S\nH : f.SurjectiveOnStalks\nI : Ideal R\ninst✝¹ : I.IsPrime\nJ : Ideal S\ninst✝ : J.IsPrime\nhf : I = Ideal.comap f J\n⊢ Function.Bijective ⇑(Ideal.ResidueField.map I J f hf)",
"usedConstants": [
"RingHom.instRin... | subst hf
exact ⟨RingHom.injective _, Ideal.Quotient.lift_surjective_of_surjective _ _
(Ideal.Quotient.mk_surjective.comp (H J ‹_›))⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.LocalRing.ResidueField.Ideal | {
"line": 54,
"column": 2
} | {
"line": 56,
"column": 50
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\nf : R →+* S\nH : f.SurjectiveOnStalks\nI : Ideal R\ninst✝¹ : I.IsPrime\nJ : Ideal S\ninst✝ : J.IsPrime\nhf : I = Ideal.comap f J\n⊢ Function.Bijective ⇑(Ideal.ResidueField.map I J f hf)",
"usedConstants": [
"RingHom.instRin... | subst hf
exact ⟨RingHom.injective _, Ideal.Quotient.lift_surjective_of_surjective _ _
(Ideal.Quotient.mk_surjective.comp (H J ‹_›))⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.KrullDimension.Basic | {
"line": 91,
"column": 2
} | {
"line": 92,
"column": 53
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : FiniteRingKrullDim R\n⊢ Nontrivial R",
"usedConstants": [
"Nontrivial",
"Eq.mpr",
"LTSeries.nonempty_of_finiteDimensionalOrder",
"congrArg",
"PartialOrder.toPreorder",
"id",
"PrimeSpectrum.instPartialOrder",
... | rw [← PrimeSpectrum.nonempty_iff_nontrivial]
exact LTSeries.nonempty_of_finiteDimensionalOrder _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.KrullDimension.Basic | {
"line": 91,
"column": 2
} | {
"line": 92,
"column": 53
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : FiniteRingKrullDim R\n⊢ Nontrivial R",
"usedConstants": [
"Nontrivial",
"Eq.mpr",
"LTSeries.nonempty_of_finiteDimensionalOrder",
"congrArg",
"PartialOrder.toPreorder",
"id",
"PrimeSpectrum.instPartialOrder",
... | rw [← PrimeSpectrum.nonempty_iff_nontrivial]
exact LTSeries.nonempty_of_finiteDimensionalOrder _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.RelSeries | {
"line": 75,
"column": 2
} | {
"line": 75,
"column": 10
} | [
{
"pp": "α : Type u_1\nr : SetRel α α\ny : RelSeries r\nfx : Fin (y.length + 1) → α\nstep✝ : ∀ (i : Fin y.length), (fx i.castSucc, fx i.succ) ∈ r\ntoFun_eq : { length := y.length, toFun := fx, step := step✝ }.toFun = y.toFun ∘ Fin.cast ⋯\n⊢ { length := y.length, toFun := fx, step := step✝ } = y",
"usedConst... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Order.KrullDimension | {
"line": 145,
"column": 4
} | {
"line": 145,
"column": 12
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝ : Preorder α\na : α\nn : ℕ∞\nh : ∀ (p : LTSeries α), RelSeries.last p = a → ↑p.length ≤ n\np : LTSeries α\nhlast : RelSeries.last p ≤ a\nthis :\n ∀ {α : Type u_1} [inst : Preorder α] {a : α} {n : ℕ∞},\n (∀ (p : LTSeries α), RelSeries.last p = a → ↑p.length ≤ n) →\n ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Order.KrullDimension | {
"line": 145,
"column": 4
} | {
"line": 145,
"column": 12
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝ : Preorder α\na : α\nn : ℕ∞\nh : ∀ (p : LTSeries α), RelSeries.last p = a → ↑p.length ≤ n\np : LTSeries α\nhlast : RelSeries.last p ≤ a\nthis :\n ∀ {α : Type u_1} [inst : Preorder α] {a : α} {n : ℕ∞},\n (∀ (p : LTSeries α), RelSeries.last p = a → ↑p.length ≤ n) →\n ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.KrullDimension | {
"line": 145,
"column": 4
} | {
"line": 145,
"column": 12
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝ : Preorder α\na : α\nn : ℕ∞\nh : ∀ (p : LTSeries α), RelSeries.last p = a → ↑p.length ≤ n\np : LTSeries α\nhlast : RelSeries.last p ≤ a\nthis :\n ∀ {α : Type u_1} [inst : Preorder α] {a : α} {n : ℕ∞},\n (∀ (p : LTSeries α), RelSeries.last p = a → ↑p.length ≤ n) →\n ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.KrullDimension | {
"line": 147,
"column": 78
} | {
"line": 147,
"column": 86
} | [
{
"pp": "α✝ : Type u_1\ninst✝¹ : Preorder α✝\nα : Type u_1\ninst✝ : Preorder α\na : α\nn : ℕ∞\nh : ∀ (p : LTSeries α), RelSeries.last p = a → ↑p.length ≤ n\np : LTSeries α\nhlast : RelSeries.last p ≤ a\nhlenpos : p.length ≠ 0\n⊢ p.length ≠ 0",
"usedConstants": [
"False",
"Preorder.toLT",
"... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Order.KrullDimension | {
"line": 147,
"column": 78
} | {
"line": 147,
"column": 86
} | [
{
"pp": "α✝ : Type u_1\ninst✝¹ : Preorder α✝\nα : Type u_1\ninst✝ : Preorder α\na : α\nn : ℕ∞\nh : ∀ (p : LTSeries α), RelSeries.last p = a → ↑p.length ≤ n\np : LTSeries α\nhlast : RelSeries.last p ≤ a\nhlenpos : p.length ≠ 0\n⊢ p.length ≠ 0",
"usedConstants": [
"False",
"Preorder.toLT",
"... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.KrullDimension | {
"line": 147,
"column": 78
} | {
"line": 147,
"column": 86
} | [
{
"pp": "α✝ : Type u_1\ninst✝¹ : Preorder α✝\nα : Type u_1\ninst✝ : Preorder α\na : α\nn : ℕ∞\nh : ∀ (p : LTSeries α), RelSeries.last p = a → ↑p.length ≤ n\np : LTSeries α\nhlast : RelSeries.last p ≤ a\nhlenpos : p.length ≠ 0\n⊢ p.length ≠ 0",
"usedConstants": [
"False",
"Preorder.toLT",
"... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.KrullDimension | {
"line": 212,
"column": 4
} | {
"line": 212,
"column": 12
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝ : Preorder α\np : LTSeries α\nx : α\nhlast : RelSeries.last p ≤ x\nhlen0 : ¬p.length ≠ 0\n⊢ ↑p.length ≤ height x",
"usedConstants": [
"Preorder.toLT",
"instAddMonoidWithOneENat",
"ENat.instNatCast",
"congrArg",
"instIsBotZeroClass",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Order.KrullDimension | {
"line": 212,
"column": 4
} | {
"line": 212,
"column": 12
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝ : Preorder α\np : LTSeries α\nx : α\nhlast : RelSeries.last p ≤ x\nhlen0 : ¬p.length ≠ 0\n⊢ ↑p.length ≤ height x",
"usedConstants": [
"Preorder.toLT",
"instAddMonoidWithOneENat",
"ENat.instNatCast",
"congrArg",
"instIsBotZeroClass",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.KrullDimension | {
"line": 212,
"column": 4
} | {
"line": 212,
"column": 12
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝ : Preorder α\np : LTSeries α\nx : α\nhlast : RelSeries.last p ≤ x\nhlen0 : ¬p.length ≠ 0\n⊢ ↑p.length ≤ height x",
"usedConstants": [
"Preorder.toLT",
"instAddMonoidWithOneENat",
"ENat.instNatCast",
"congrArg",
"instIsBotZeroClass",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.RelSeries | {
"line": 419,
"column": 14
} | {
"line": 419,
"column": 49
} | [
{
"pp": "case h.e'_5.h.e'_3\nα : Type u_1\nr : SetRel α α\nβ : Type u_2\ns : SetRel β β\np : RelSeries r\ni : Fin p.length\na : α\nprev_connect : (p.toFun i.castSucc, a) ∈ r\nconnect_next : (a, p.toFun i.succ) ∈ r\nm : Fin (p.length + 1)\nx : α := i.succ.castSucc.insertNth a p.toFun m.castSucc\ny : α := i.succ.... | Fin.insertNth_apply_above (h := hm) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.RelSeries | {
"line": 615,
"column": 8
} | {
"line": 615,
"column": 17
} | [
{
"pp": "case h.e'_1.length_eq\nα : Type u_1\nr : SetRel α α\nβ : Type u_2\ns : SetRel β β\nmotive : RelSeries r → Sort u_3\nsingleton : (x : α) → motive (RelSeries.singleton r x)\ncons : (p : RelSeries r) → (x : α) → (hx : (x, p.head) ∈ r) → motive p → motive (p.cons x hx)\np : RelSeries r\nheq : p.length = 0\... | exact heq | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.RelSeries | {
"line": 615,
"column": 8
} | {
"line": 615,
"column": 17
} | [
{
"pp": "case h.e'_1.length_eq\nα : Type u_1\nr : SetRel α α\nβ : Type u_2\ns : SetRel β β\nmotive : RelSeries r → Sort u_3\nsingleton : (x : α) → motive (RelSeries.singleton r x)\ncons : (p : RelSeries r) → (x : α) → (hx : (x, p.head) ∈ r) → motive p → motive (p.cons x hx)\np : RelSeries r\nheq : p.length = 0\... | exact heq | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.RelSeries | {
"line": 615,
"column": 8
} | {
"line": 615,
"column": 17
} | [
{
"pp": "case h.e'_1.length_eq\nα : Type u_1\nr : SetRel α α\nβ : Type u_2\ns : SetRel β β\nmotive : RelSeries r → Sort u_3\nsingleton : (x : α) → motive (RelSeries.singleton r x)\ncons : (p : RelSeries r) → (x : α) → (hx : (x, p.head) ∈ r) → motive p → motive (p.cons x hx)\np : RelSeries r\nheq : p.length = 0\... | exact heq | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.RelSeries | {
"line": 624,
"column": 2
} | {
"line": 624,
"column": 16
} | [
{
"pp": "α : Type u_1\nr : SetRel α α\nβ : Type u_2\ns : SetRel β β\nmotive : RelSeries r → Sort u_3\nsingleton : (x : α) → motive (RelSeries.singleton r x)\ncons : (p : RelSeries r) → (x : α) → (hx : (x, p.head) ∈ r) → motive p → motive (p.cons x hx)\np : RelSeries r\nthis : {n : ℕ} → p.length = n → motive p :... | exact this rfl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.RelSeries | {
"line": 680,
"column": 8
} | {
"line": 680,
"column": 17
} | [
{
"pp": "case h.e'_1.length_eq\nα : Type u_1\nr : SetRel α α\nβ : Type u_2\ns : SetRel β β\nmotive : RelSeries r → Sort u_3\nsingleton : (x : α) → motive (RelSeries.singleton r x)\nsnoc : (p : RelSeries r) → (x : α) → (hx : (p.last, x) ∈ r) → motive p → motive (p.snoc x hx)\np : RelSeries r\nheq : p.length = 0\... | exact heq | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.RelSeries | {
"line": 680,
"column": 8
} | {
"line": 680,
"column": 17
} | [
{
"pp": "case h.e'_1.length_eq\nα : Type u_1\nr : SetRel α α\nβ : Type u_2\ns : SetRel β β\nmotive : RelSeries r → Sort u_3\nsingleton : (x : α) → motive (RelSeries.singleton r x)\nsnoc : (p : RelSeries r) → (x : α) → (hx : (p.last, x) ∈ r) → motive p → motive (p.snoc x hx)\np : RelSeries r\nheq : p.length = 0\... | exact heq | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.RelSeries | {
"line": 680,
"column": 8
} | {
"line": 680,
"column": 17
} | [
{
"pp": "case h.e'_1.length_eq\nα : Type u_1\nr : SetRel α α\nβ : Type u_2\ns : SetRel β β\nmotive : RelSeries r → Sort u_3\nsingleton : (x : α) → motive (RelSeries.singleton r x)\nsnoc : (p : RelSeries r) → (x : α) → (hx : (p.last, x) ∈ r) → motive p → motive (p.snoc x hx)\np : RelSeries r\nheq : p.length = 0\... | exact heq | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.KrullDimension | {
"line": 521,
"column": 8
} | {
"line": 521,
"column": 24
} | [
{
"pp": "α✝ : Type u_1\ninst✝¹ : Preorder α✝\nα : Type u_1\ninst✝ : Preorder α\nx : α\nn : ℕ\nhfin : height x < ⊤\n⊢ height x = ↑n + 1 ↔ ↑n < height x ∧ height x ≤ ↑n + 1",
"usedConstants": [
"Eq.mpr",
"instCompleteLinearOrderENat",
"instAddMonoidWithOneENat",
"ENat.instNatCast",
... | le_antisymm_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.RelSeries | {
"line": 687,
"column": 2
} | {
"line": 687,
"column": 16
} | [
{
"pp": "α : Type u_1\nr : SetRel α α\nβ : Type u_2\ns : SetRel β β\nmotive : RelSeries r → Sort u_3\nsingleton : (x : α) → motive (RelSeries.singleton r x)\nsnoc : (p : RelSeries r) → (x : α) → (hx : (p.last, x) ∈ r) → motive p → motive (p.snoc x hx)\np : RelSeries r\nthis : {n : ℕ} → p.length = n → motive p :... | exact this rfl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.RelSeries | {
"line": 698,
"column": 4
} | {
"line": 698,
"column": 34
} | [
{
"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 <;> intro i | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Order.KrullDimension | {
"line": 536,
"column": 4
} | {
"line": 536,
"column": 12
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝ : Preorder α\nx : α\nn : ℕ\nthis :\n ∀ {α : Type u_1} [inst : Preorder α] {x : α} {n : ℕ},\n height x < ⊤ → (height x = ↑n ↔ height x < ⊤ ∧ (n = 0 ∨ ∃ y < x, height y = ↑n - 1) ∧ ∀ y < x, height y < ↑n)\nhfin : ¬height x < ⊤\n⊢ height x = ↑n ↔ height x < ⊤ ∧ (n = 0 ∨ ∃... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Order.KrullDimension | {
"line": 536,
"column": 4
} | {
"line": 536,
"column": 12
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝ : Preorder α\nx : α\nn : ℕ\nthis :\n ∀ {α : Type u_1} [inst : Preorder α] {x : α} {n : ℕ},\n height x < ⊤ → (height x = ↑n ↔ height x < ⊤ ∧ (n = 0 ∨ ∃ y < x, height y = ↑n - 1) ∧ ∀ y < x, height y < ↑n)\nhfin : ¬height x < ⊤\n⊢ height x = ↑n ↔ height x < ⊤ ∧ (n = 0 ∨ ∃... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.KrullDimension | {
"line": 536,
"column": 4
} | {
"line": 536,
"column": 12
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝ : Preorder α\nx : α\nn : ℕ\nthis :\n ∀ {α : Type u_1} [inst : Preorder α] {x : α} {n : ℕ},\n height x < ⊤ → (height x = ↑n ↔ height x < ⊤ ∧ (n = 0 ∨ ∃ y < x, height y = ↑n - 1) ∧ ∀ y < x, height y < ↑n)\nhfin : ¬height x < ⊤\n⊢ height x = ↑n ↔ height x < ⊤ ∧ (n = 0 ∨ ∃... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Sets.Opens | {
"line": 304,
"column": 4
} | {
"line": 304,
"column": 53
} | [
{
"pp": "case mpr\nα : Type u_2\ninst✝ : TopologicalSpace α\nB : Set (Opens α)\nh : ∀ {U : Opens α} {x : α}, x ∈ U → ∃ U' ∈ B, x ∈ U' ∧ U' ≤ U\n⊢ IsBasis B",
"usedConstants": [
"TopologicalSpace.isTopologicalBasis_of_isOpen_of_nhds",
"TopologicalSpace.Opens",
"TopologicalSpace.Opens.instSe... | refine isTopologicalBasis_of_isOpen_of_nhds ?_ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Order.KrullDimension | {
"line": 827,
"column": 6
} | {
"line": 827,
"column": 14
} | [
{
"pp": "case a.inr\nα : Type u_1\ninst✝¹ : Preorder α\ninst✝ : Nonempty α\nthis : ∀ {α : Type u_1} [inst : Preorder α] [Nonempty α], krullDim α < ⊤ → ↑(⨆ a, height a + coheight a) ≤ krullDim α\nhnottop : ¬krullDim α < ⊤\n⊢ ↑(⨆ a, height a + coheight a) ≤ krullDim α",
"usedConstants": [
"WithBot.instP... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Order.KrullDimension | {
"line": 827,
"column": 6
} | {
"line": 827,
"column": 14
} | [
{
"pp": "case a.inr\nα : Type u_1\ninst✝¹ : Preorder α\ninst✝ : Nonempty α\nthis : ∀ {α : Type u_1} [inst : Preorder α] [Nonempty α], krullDim α < ⊤ → ↑(⨆ a, height a + coheight a) ≤ krullDim α\nhnottop : ¬krullDim α < ⊤\n⊢ ↑(⨆ a, height a + coheight a) ≤ krullDim α",
"usedConstants": [
"WithBot.instP... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.KrullDimension | {
"line": 827,
"column": 6
} | {
"line": 827,
"column": 14
} | [
{
"pp": "case a.inr\nα : Type u_1\ninst✝¹ : Preorder α\ninst✝ : Nonempty α\nthis : ∀ {α : Type u_1} [inst : Preorder α] [Nonempty α], krullDim α < ⊤ → ↑(⨆ a, height a + coheight a) ≤ krullDim α\nhnottop : ¬krullDim α < ⊤\n⊢ ↑(⨆ a, height a + coheight a) ≤ krullDim α",
"usedConstants": [
"WithBot.instP... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.KrullDimension | {
"line": 834,
"column": 13
} | {
"line": 834,
"column": 21
} | [
{
"pp": "case h.top\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 < ⊤\nhh : height a = ⊤\n⊢ ⊤ + coheight a ≤ ⨆ p, ↑p.length",
"usedConstants": [
"False",
"Preorder.toLT",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Order.KrullDimension | {
"line": 834,
"column": 13
} | {
"line": 834,
"column": 21
} | [
{
"pp": "case h.top\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 < ⊤\nhh : height a = ⊤\n⊢ ⊤ + coheight a ≤ ⨆ p, ↑p.length",
"usedConstants": [
"False",
"Preorder.toLT",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.KrullDimension | {
"line": 834,
"column": 13
} | {
"line": 834,
"column": 21
} | [
{
"pp": "case h.top\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 < ⊤\nhh : height a = ⊤\n⊢ ⊤ + coheight a ≤ ⨆ p, ↑p.length",
"usedConstants": [
"False",
"Preorder.toLT",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.KrullDimension | {
"line": 837,
"column": 15
} | {
"line": 837,
"column": 23
} | [
{
"pp": "case h.coe.top\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\nhch : coheight a = ⊤\n⊢ ↑n + ⊤ ≤ ⨆ p, ↑p.length",
"usedConstants": [
"False",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Order.KrullDimension | {
"line": 837,
"column": 15
} | {
"line": 837,
"column": 23
} | [
{
"pp": "case h.coe.top\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\nhch : coheight a = ⊤\n⊢ ↑n + ⊤ ≤ ⨆ p, ↑p.length",
"usedConstants": [
"False",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.KrullDimension | {
"line": 837,
"column": 15
} | {
"line": 837,
"column": 23
} | [
{
"pp": "case h.coe.top\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\nhch : coheight a = ⊤\n⊢ ↑n + ⊤ ≤ ⨆ p, ↑p.length",
"usedConstants": [
"False",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.KrullDimension | {
"line": 947,
"column": 6
} | {
"line": 947,
"column": 22
} | [
{
"pp": "α : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : BoundedOrder α\n⊢ krullDim α = 1 ↔ IsSimpleOrder α",
"usedConstants": [
"Eq.mpr",
"WithBot",
"instCompleteLinearOrderENat",
"instAddMonoidWithOneENat",
"congrArg",
"PartialOrder.toPreorder",
"IsSimpleOrder",
... | le_antisymm_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.KrullDimension | {
"line": 1015,
"column": 6
} | {
"line": 1015,
"column": 14
} | [
{
"pp": "case a.h.inr\nα : Type u_1\ninst✝ : Preorder α\nx : α\np : LTSeries (WithBot α)\nhlast : RelSeries.last p = ↑x\nthis :\n ∀ {α : Type u_1} [inst : Preorder α] (x : α) (p : LTSeries (WithBot α)),\n RelSeries.last p = ↑x → p.length ≠ 0 → ↑p.length ≤ height x + 1\nhlenpos : ¬p.length ≠ 0\n⊢ ↑p.length ≤... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Order.KrullDimension | {
"line": 1015,
"column": 6
} | {
"line": 1015,
"column": 14
} | [
{
"pp": "case a.h.inr\nα : Type u_1\ninst✝ : Preorder α\nx : α\np : LTSeries (WithBot α)\nhlast : RelSeries.last p = ↑x\nthis :\n ∀ {α : Type u_1} [inst : Preorder α] (x : α) (p : LTSeries (WithBot α)),\n RelSeries.last p = ↑x → p.length ≠ 0 → ↑p.length ≤ height x + 1\nhlenpos : ¬p.length ≠ 0\n⊢ ↑p.length ≤... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.KrullDimension | {
"line": 1015,
"column": 6
} | {
"line": 1015,
"column": 14
} | [
{
"pp": "case a.h.inr\nα : Type u_1\ninst✝ : Preorder α\nx : α\np : LTSeries (WithBot α)\nhlast : RelSeries.last p = ↑x\nthis :\n ∀ {α : Type u_1} [inst : Preorder α] (x : α) (p : LTSeries (WithBot α)),\n RelSeries.last p = ↑x → p.length ≠ 0 → ↑p.length ≤ height x + 1\nhlenpos : ¬p.length ≠ 0\n⊢ ↑p.length ≤... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.BooleanSubalgebra | {
"line": 264,
"column": 53
} | {
"line": 264,
"column": 74
} | [
{
"pp": "ι : Sort u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : BooleanAlgebra α\ninst✝¹ : BooleanAlgebra β\ninst✝ : BooleanAlgebra γ\nL✝ M : BooleanSubalgebra α\nf✝ : BoundedLatticeHom α β\ns t : Set α\na✝ b : α\nf : BoundedLatticeHom α β\nL : BooleanSubalgebra α\na : α\nha : a ∈ ↑L\n⊢ aᶜ ∈ ↑L ∧ f aᶜ... | by simpa [map_compl'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.KrullDimension | {
"line": 1089,
"column": 2
} | {
"line": 1091,
"column": 62
} | [
{
"pp": "n : ℕ∞\n⊢ height n = n",
"usedConstants": [
"WithBot.some",
"WithBot",
"Order.height_nat",
"ENat.instNatCast",
"instTopENat",
"WithTop.instPreorder",
"congrArg",
"ENat.recTopCoe",
"Order.height_coe_withTop",
"Preorder.toLE",
"instPre... | cases n with
| top => simp only [← WithBot.coe_eq_coe, height_top_eq_krullDim, krullDim_enat, WithBot.coe_top]
| coe n => exact (height_coe_withTop _).trans (height_nat _) | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.Order.KrullDimension | {
"line": 1089,
"column": 2
} | {
"line": 1091,
"column": 62
} | [
{
"pp": "n : ℕ∞\n⊢ height n = n",
"usedConstants": [
"WithBot.some",
"WithBot",
"Order.height_nat",
"ENat.instNatCast",
"instTopENat",
"WithTop.instPreorder",
"congrArg",
"ENat.recTopCoe",
"Order.height_coe_withTop",
"Preorder.toLE",
"instPre... | cases n with
| top => simp only [← WithBot.coe_eq_coe, height_top_eq_krullDim, krullDim_enat, WithBot.coe_top]
| coe n => exact (height_coe_withTop _).trans (height_nat _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.KrullDimension | {
"line": 1089,
"column": 2
} | {
"line": 1091,
"column": 62
} | [
{
"pp": "n : ℕ∞\n⊢ height n = n",
"usedConstants": [
"WithBot.some",
"WithBot",
"Order.height_nat",
"ENat.instNatCast",
"instTopENat",
"WithTop.instPreorder",
"congrArg",
"ENat.recTopCoe",
"Order.height_coe_withTop",
"Preorder.toLE",
"instPre... | cases n with
| top => simp only [← WithBot.coe_eq_coe, height_top_eq_krullDim, krullDim_enat, WithBot.coe_top]
| coe n => exact (height_coe_withTop _).trans (height_nat _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.LocalAtTarget | {
"line": 169,
"column": 2
} | {
"line": 173,
"column": 36
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\nι : Type u_3\nU : ι → Opens β\nhU : IsOpenCover U\nh : Continuous[inst✝¹, inst✝] f\n⊢ IsOpenEmbedding f ↔ ∀ (i : ι), IsOpenEmbedding ((U i).carrier.restrictPreimage f)",
"usedConstants": [
"Set.res... | simp_rw [isOpenEmbedding_iff, forall_and]
apply and_congr
· exact hU.isEmbedding_iff_restrictPreimage h
· simp_rw [range_restrictPreimage]
exact hU.isOpen_iff_coe_preimage | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.LocalAtTarget | {
"line": 169,
"column": 2
} | {
"line": 173,
"column": 36
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\nι : Type u_3\nU : ι → Opens β\nhU : IsOpenCover U\nh : Continuous[inst✝¹, inst✝] f\n⊢ IsOpenEmbedding f ↔ ∀ (i : ι), IsOpenEmbedding ((U i).carrier.restrictPreimage f)",
"usedConstants": [
"Set.res... | simp_rw [isOpenEmbedding_iff, forall_and]
apply and_congr
· exact hU.isEmbedding_iff_restrictPreimage h
· simp_rw [range_restrictPreimage]
exact hU.isOpen_iff_coe_preimage | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Spectral.Prespectral | {
"line": 44,
"column": 35
} | {
"line": 44,
"column": 43
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nι : Type u_3\nb : ι → Set X\nbasis : IsTopologicalBasis (Set.range b)\nisCompact_basis : ∀ (i : ι), IsCompact (b i)\n⊢ ∀ U ∈ Set.range b, IsCompact U",
"usedConstants": [
"Membership.mem",
"Exists",
"Set.mem_range._simp_1",
"forall_e... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
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