module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff | {
"line": 227,
"column": 4
} | {
"line": 227,
"column": 47
} | {
"line": 227,
"column": 48
} | [
{
"pp": "case neg\nR : 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\nthis : Nonempty n\ni : ℕ\nhi : 2 < i\n⊢ M.charpoly.natDegree < i",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
... | [
"case neg\nR : 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\nthis : Nonempty n\ni : ℕ\nhi : 2 < i\n⊢ 2 < i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic | {
"line": 156,
"column": 14
} | {
"line": 156,
"column": 25
} | {
"line": 156,
"column": 26
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nA : Type u_5\nB : Type u_6\ninst✝³ : Ring A\ninst✝² : Ring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : A ≃ₐ[R] B\nx : A\nh : IsIntegral R (f x)\n⊢ IsIntegral R x",
"ppTerm": "?m.28",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"u... | [
"R : Type u_1\ninst✝⁴ : CommRing R\nA : Type u_5\nB : Type u_6\ninst✝³ : Ring A\ninst✝² : Ring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : A ≃ₐ[R] B\nx : A\nh : IsIntegral R (f x)\n⊢ IsIntegral R x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IntegralClosure.Algebra.Basic | {
"line": 185,
"column": 2
} | {
"line": 185,
"column": 35
} | {
"line": 185,
"column": 36
} | [
{
"pp": "R : Type u_1\nS : Type u_4\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nx y : S\nhx : f.IsIntegralElem x\nhy : f.IsIntegralElem y\n⊢ f.IsIntegralElem (x - y)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AddGroupWithOne.toAddGroup",
"congrA... | [
"R : Type u_1\nS : Type u_4\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nx y : S\nhx : f.IsIntegralElem x\nhy : f.IsIntegralElem y\n⊢ f.IsIntegralElem (x + -y)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff | {
"line": 357,
"column": 52
} | {
"line": 357,
"column": 63
} | {
"line": 357,
"column": 64
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\nn : Type v\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\nM : Matrix n n R\nhM : IsNilpotent M\nh✝ : Nonempty n\nthis : IsNilpotent (M.charpolyRev.coeff 1)\n⊢ IsNilpotent M.trace",
"ppTerm": "?m.59",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"R : Type u\ninst✝² : CommRing R\nn : Type v\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\nM : Matrix n n R\nhM : IsNilpotent M\nh✝ : Nonempty n\nthis : IsNilpotent (M.charpolyRev.coeff 1)\n⊢ IsNilpotent M.trace"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.ScaleRoots | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 13
} | {
"line": 56,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\np : R[X]\ns : R\nx✝ : ℕ\n⊢ x✝ ∈ (p.scaleRoots s).support → x✝ ∈ p.support",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"Finset",
"HSub.hSub",
"Membership.mem",
"Polyno... | [
"R : Type u_1\ninst✝ : Semiring R\np : R[X]\ns : R\nx✝ : ℕ\n⊢ ¬p.coeff x✝ * s ^ (p.natDegree - x✝) = 0 → ¬p.coeff x✝ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.ScaleRoots | {
"line": 139,
"column": 63
} | {
"line": 139,
"column": 74
} | {
"line": 139,
"column": 75
} | [
{
"pp": "S : Type u_2\nA : Type u_3\ninst✝¹ : Semiring S\ninst✝ : Semiring A\np : S[X]\nf : S →+* A\na : A\ns : S\nhsa : Commute (f s) a\nhf : ∀ (s₁ s₂ : S), Commute (f s₁) (f s₂)\ni : ℕ\n_hi : i ∈ p.support\nhi' : i ∉ (p.scaleRoots s).support\n⊢ p.coeff i * s ^ (p.natDegree - i) = 0",
"ppTerm": "?m.271",
... | [
"S : Type u_2\nA : Type u_3\ninst✝¹ : Semiring S\ninst✝ : Semiring A\np : S[X]\nf : S →+* A\na : A\ns : S\nhsa : Commute (f s) a\nhf : ∀ (s₁ s₂ : S), Commute (f s₁) (f s₂)\ni : ℕ\n_hi : i ∈ p.support\nhi' : i ∉ (p.scaleRoots s).support\n⊢ p.coeff i * s ^ (p.natDegree - i) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.HasseDeriv | {
"line": 147,
"column": 17
} | {
"line": 147,
"column": 37
} | {
"line": 147,
"column": 37
} | [
{
"pp": "case succ.e_a.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) - n",
"ppTerm": "?succ.e_a.e_a.e_a",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.inst... | [
"case succ.e_a.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 = k + 1"
] | add_tsub_cancel_left | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.HasseDeriv | {
"line": 171,
"column": 6
} | {
"line": 171,
"column": 80
} | {
"line": 171,
"column": 80
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝ : Semiring R\nk l i : ℕ\nhikl : k + l ≤ i\nh1 : l ≤ i\nh2 : k ≤ i - l\nh3 : k ≤ k + l\n⊢ ↑(i - l)! / (↑k ! * ↑(i - l - k)!) * (↑i ! / (↑l ! * ↑(i - l)!)) =\n ↑(k + l)! / (↑k ! * ↑(k + l - k)!) * (↑i ! / (↑(k + l)! * ↑(i - (k + l))!))",
"ppTerm": "?neg✝",
"assign... | [
"case neg\nR : Type u_1\ninst✝ : Semiring R\nk l i : ℕ\nhikl : k + l ≤ i\nh1 : l ≤ i\nh2 : k ≤ i - l\nh3 : k ≤ k + l\n⊢ ↑(i - l)! / (↑k ! * ↑(i - l - k)!) * (↑i ! / (↑l ! * ↑(i - l)!)) =\n ↑(k + l)! / (↑k ! * ↑(k + l - k)!) * (↑i ! / (↑(k + l)! * ↑(i - l - k)!))"
] | show i - (k + l) = i - l - k by rw [add_comm]; apply tsub_add_eq_tsub_tsub | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff | {
"line": 473,
"column": 48
} | {
"line": 473,
"column": 81
} | {
"line": 473,
"column": 81
} | [
{
"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\nhk : k ≤ Fintype.card n\na✝ : Nontrivial R\nhnd : M.charpoly.natDegree = Fintype.card n\nhrev : M.charpoly.coeff (Fintype.card n - k) = M.charpoly.reverse.coeff k\nhcharpolyRev : M.charpolyR... | [
"R : Type u\ninst✝² : CommRing R\nn : Type v\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\nM : Matrix n n R\nk : ℕ\nhk : k ≤ Fintype.card n\na✝ : Nontrivial R\nhnd : M.charpoly.natDegree = Fintype.card n\nhrev : M.charpoly.coeff (Fintype.card n - k) = M.charpoly.reverse.coeff k\nhcharpolyRev : M.charpolyRev = (1 + X ... | (Finset.mem_powersetCard.mp hs).2 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.ScaleRoots | {
"line": 185,
"column": 2
} | {
"line": 185,
"column": 32
} | {
"line": 187,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\np : R[X]\nr s : R\n⊢ p.scaleRoots (r * s) = (p.scaleRoots r).scaleRoots s",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"HMul.hMul",
"Polynomial.ext",
"Monoid.toMulOneClass",
"CommSemiring.toN... | [] | ext; simp [mul_pow, mul_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.ScaleRoots | {
"line": 185,
"column": 2
} | {
"line": 185,
"column": 32
} | {
"line": 187,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\np : R[X]\nr s : R\n⊢ p.scaleRoots (r * s) = (p.scaleRoots r).scaleRoots s",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"HMul.hMul",
"Polynomial.ext",
"Monoid.toMulOneClass",
"CommSemiring.toN... | [] | ext; simp [mul_pow, mul_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Splits | {
"line": 44,
"column": 2
} | {
"line": 44,
"column": 13
} | {
"line": 44,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\n⊢ Splits 0",
"ppTerm": "?m.6",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : Semiring R\n⊢ Splits 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Splits | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 13
} | {
"line": 56,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\n⊢ X.Splits",
"ppTerm": "?m.6",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : Semiring R\n⊢ X.Splits"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Splits | {
"line": 112,
"column": 35
} | {
"line": 112,
"column": 65
} | {
"line": 112,
"column": 66
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\na b : R\nhf : (C a * X + C b).natDegree ≤ 1\nh : Invertible (C a * X + C b).leadingCoeff\nha : a ≠ 0\n⊢ Invertible a",
"ppTerm": "?m.72",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : Semiring R\na b : R\nhf : (C a * X + C b).natDegree ≤ 1\nh : Invertible (C a * X + C b).leadingCoeff\nha : a ≠ 0\n⊢ Invertible a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Splits | {
"line": 156,
"column": 46
} | {
"line": 156,
"column": 69
} | {
"line": 156,
"column": 70
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\np : R[X]\nhp : p.Splits\nr i : R\n⊢ (X + C r + C i).Splits",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne",
"AddMonoid.toAddSemigroup",
"congrArg... | [
"R : Type u_1\ninst✝ : CommSemiring R\np : R[X]\nhp : p.Splits\nr i : R\n⊢ (X + (C r + C i)).Splits"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.TensorProduct.MvPolynomial | {
"line": 80,
"column": 2
} | {
"line": 80,
"column": 17
} | {
"line": 82,
"column": 0
} | [
{
"pp": "R : Type u\nN : Type v\ninst✝⁵ : CommSemiring R\nσ : Type u_1\nS : Type u_3\ninst✝⁴ : CommSemiring S\ninst✝³ : Algebra R S\ninst✝² : CommSemiring N\ninst✝¹ : Algebra R N\ninst✝ : DecidableEq σ\ne : σ →₀ ℕ\ns : S\nn : N\nd : σ →₀ ℕ\n⊢ coeff d (rTensorAlgEquiv (s ⊗ₜ[R] (monomial e) n)) = if e = d then s ... | [] | simp [tmul_ite] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.TensorProduct.MvPolynomial | {
"line": 80,
"column": 2
} | {
"line": 80,
"column": 17
} | {
"line": 82,
"column": 0
} | [
{
"pp": "R : Type u\nN : Type v\ninst✝⁵ : CommSemiring R\nσ : Type u_1\nS : Type u_3\ninst✝⁴ : CommSemiring S\ninst✝³ : Algebra R S\ninst✝² : CommSemiring N\ninst✝¹ : Algebra R N\ninst✝ : DecidableEq σ\ne : σ →₀ ℕ\ns : S\nn : N\nd : σ →₀ ℕ\n⊢ coeff d (rTensorAlgEquiv (s ⊗ₜ[R] (monomial e) n)) = if e = d then s ... | [] | simp [tmul_ite] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.TensorProduct.MvPolynomial | {
"line": 80,
"column": 2
} | {
"line": 80,
"column": 17
} | {
"line": 82,
"column": 0
} | [
{
"pp": "R : Type u\nN : Type v\ninst✝⁵ : CommSemiring R\nσ : Type u_1\nS : Type u_3\ninst✝⁴ : CommSemiring S\ninst✝³ : Algebra R S\ninst✝² : CommSemiring N\ninst✝¹ : Algebra R N\ninst✝ : DecidableEq σ\ne : σ →₀ ℕ\ns : S\nn : N\nd : σ →₀ ℕ\n⊢ coeff d (rTensorAlgEquiv (s ⊗ₜ[R] (monomial e) n)) = if e = d then s ... | [] | simp [tmul_ite] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Subring | {
"line": 59,
"column": 2
} | {
"line": 62,
"column": 85
} | {
"line": 64,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\np : R[X]\nT : Subring R\nhp : ↑p.coeffs ⊆ ↑T\n⊢ (p.toSubring T hp).support = p.support",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subring.instSetLike",
"Ring.toNonAssocRing",
"congrArg",
"Finset",
"A... | [] | ext i
simp only [mem_support_iff, not_iff_not, Ne]
conv_rhs => rw [← coeff_toSubring p T hp]
exact ⟨fun H => by rw [H, ZeroMemClass.coe_zero], fun H => Subtype.coe_injective H⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Subring | {
"line": 59,
"column": 2
} | {
"line": 62,
"column": 85
} | {
"line": 64,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\np : R[X]\nT : Subring R\nhp : ↑p.coeffs ⊆ ↑T\n⊢ (p.toSubring T hp).support = p.support",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subring.instSetLike",
"Ring.toNonAssocRing",
"congrArg",
"Finset",
"A... | [] | ext i
simp only [mem_support_iff, not_iff_not, Ne]
conv_rhs => rw [← coeff_toSubring p T hp]
exact ⟨fun H => by rw [H, ZeroMemClass.coe_zero], fun H => Subtype.coe_injective H⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Splits | {
"line": 190,
"column": 23
} | {
"line": 190,
"column": 93
} | {
"line": 190,
"column": 94
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nf : R[X]\nhf : f.Splits\na✝ : Nontrivial R\na : R\nm : Multiset R\nhm : f = C f.leadingCoeff * ((X + C a) * (Multiset.map (fun x ↦ X + C x) m).prod)\nha : a ∈ a ::ₘ m\nh : (Irreducible (X + C a) ∧ ∀ m_1 ∈ (Multiset.map (fun x ↦ X + C x) m).toList, IsUnit m_1) ∧ IsU... | [
"R : Type u_1\ninst✝ : CommSemiring R\nf : R[X]\nhf : f.Splits\na✝ : Nontrivial R\na : R\nm : Multiset R\nhm : f = C f.leadingCoeff * ((X + C a) * (Multiset.map (fun x ↦ X + C x) m).prod)\nha : a ∈ a ::ₘ m\nh : (Irreducible (X + C a) ∧ ∀ m_1 ∈ (Multiset.map (fun x ↦ X + C x) m).toList, IsUnit m_1) ∧ IsUnit (C f.lea... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Splits | {
"line": 180,
"column": 2
} | {
"line": 192,
"column": 8
} | {
"line": 194,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nf : R[X]\nhf : f.Splits\nh : Irreducible f\n⊢ f.natDegree ≤ 1",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Nontrivial",
"Eq.mpr",
"Polynomial.C",
"MulOne.toOne",
"le_refl",
"False",
"Polynomial.nat... | [] | nontriviality R
obtain ⟨m, hm⟩ := splits_iff_exists_multiset'.mp hf
rcases m.empty_or_exists_mem with rfl | ⟨a, ha⟩
· rw [hm]
simp
· obtain ⟨m, rfl⟩ := Multiset.exists_cons_of_mem ha
rw [Multiset.map_cons, Multiset.prod_cons] at hm
rw [hm] at h
simp only [irreducible_mul_iff, IsUnit.mul_iff, not... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Splits | {
"line": 180,
"column": 2
} | {
"line": 192,
"column": 8
} | {
"line": 194,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nf : R[X]\nhf : f.Splits\nh : Irreducible f\n⊢ f.natDegree ≤ 1",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Nontrivial",
"Eq.mpr",
"Polynomial.C",
"MulOne.toOne",
"le_refl",
"False",
"Polynomial.nat... | [] | nontriviality R
obtain ⟨m, hm⟩ := splits_iff_exists_multiset'.mp hf
rcases m.empty_or_exists_mem with rfl | ⟨a, ha⟩
· rw [hm]
simp
· obtain ⟨m, rfl⟩ := Multiset.exists_cons_of_mem ha
rw [Multiset.map_cons, Multiset.prod_cons] at hm
rw [hm] at h
simp only [irreducible_mul_iff, IsUnit.mul_iff, not... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Splits | {
"line": 382,
"column": 2
} | {
"line": 382,
"column": 42
} | {
"line": 382,
"column": 43
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nf : R[X]\ninst✝ : IsDomain R\nhf : f.Splits\nhf0 : f.natDegree ≠ 0\n⊢ f.roots ≠ 0",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Polynomial.roots",
"Multiset",
"id",
"Ne",
"Zero.toOfNat0",
"OfNat.ofNat",
... | [
"R : Type u_1\ninst✝¹ : CommRing R\nf : R[X]\ninst✝ : IsDomain R\nhf : f.Splits\nhf0 : f.natDegree ≠ 0\n⊢ ¬f.roots = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.Integral | {
"line": 41,
"column": 85
} | {
"line": 45,
"column": 46
} | {
"line": 47,
"column": 0
} | [
{
"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]\n⊢ ∃ b ∈ M, ∃ q, Polynomial.map (algebraMap R S) q = b • p ∧ q.support ⊆ p.support",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
... | [] | by
obtain ⟨⟨_, b, hb, rfl⟩, h⟩ := exists_integer_multiple (Submonoid.map C M) p
rw [Subtype.coe_mk, C_eq_algebraMap, algebraMap_smul] at h
obtain ⟨q', h₁, h₂⟩ := exists_support_eq_of_mem_lifts h
exact ⟨b, hb, q', h₁, h₂ ▸ support_smul b p⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Splits | {
"line": 668,
"column": 2
} | {
"line": 668,
"column": 13
} | {
"line": 668,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝ : Field R\nF : Subfield R\nf : (↥F)[X]\nhf : f.Splits\nhf0 : f ≠ 0\nx : R\nhx : (map F.subtype f).IsRoot x\n⊢ x ∈ F",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : Field R\nF : Subfield R\nf : (↥F)[X]\nhf : f.Splits\nhf0 : f ≠ 0\nx : R\nhx : (map F.subtype f).IsRoot x\n⊢ x ∈ F"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Splits | {
"line": 674,
"column": 4
} | {
"line": 674,
"column": 72
} | {
"line": 675,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : Field R\nf : R[X]\nx : R\nh₁ : f.natDegree = 2\nh₂ : eval x f = 0\n⊢ (f /ₘ (X - C x)).natDegree = 1",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Polynomial.monic_X_sub_C",
"Eq.mpr",
"Polynomial.C",
"congrArg",
"Polynomial.nat... | [] | rw [natDegree_divByMonic f (monic_X_sub_C x), h₁, natDegree_X_sub_C] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Ideal.GoingUp | {
"line": 61,
"column": 4
} | {
"line": 61,
"column": 26
} | {
"line": 62,
"column": 4
} | [
{
"pp": "case refine_3\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]\np_nonzero : p ≠ 0\nih : p ≠ 0 → eval₂ f r p = 0 → ∃ i, p.coeff i ≠ 0 ∧ p.coeff i ∈ comap f I\na✝ : p * X ≠ 0\n... | [
"case refine_3.refine_1\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]\np_nonzero : p ≠ 0\nih : p ≠ 0 → eval₂ f r p = 0 → ∃ i, p.coeff i ≠ 0 ∧ p.coeff i ∈ comap f I\na✝ : p * X ≠ 0\nhp ... | refine ⟨i + 1, ?_, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Ideal.GoingUp | {
"line": 63,
"column": 6
} | {
"line": 63,
"column": 22
} | {
"line": 63,
"column": 23
} | [
{
"pp": "case refine_3.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]\np_nonzero : p ≠ 0\nih : p ≠ 0 → eval₂ f r p = 0 → ∃ i, p.coeff i ≠ 0 ∧ p.coeff i ∈ comap f I\na✝ : p ... | [
"case refine_3.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]\np_nonzero : p ≠ 0\nih : p ≠ 0 → eval₂ f r p = 0 → ∃ i, p.coeff i ≠ 0 ∧ p.coeff i ∈ comap f I\na✝ : p * X ≠ 0\nhp ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.FaithfullyFlat.Basic | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 15
} | {
"line": 69,
"column": 16
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nm : Ideal R\nh : m.IsMaximal\nr : m • ⊤ = ⊤\n⊢ 1 ∈ m",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\nM : Type v\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nm : Ideal R\nh : m.IsMaximal\nr : m • ⊤ = ⊤\n⊢ 1 ∈ m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.FaithfullyFlat.Basic | {
"line": 79,
"column": 83
} | {
"line": 79,
"column": 91
} | {
"line": 79,
"column": 91
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx✝ : Flat R M ∧ ∀ ⦃m : Ideal R⦄, m.IsMaximal → m • ⊤ ≠ ⊤\nflat : Flat R M\nh : ∀ ⦃m : Ideal R⦄, m.IsMaximal → m • ⊤ ≠ ⊤\nI : Ideal R\nhI : I ≠ ⊤\nr : I • ⊤ = ⊤\nm : Ideal R\nhm : m.IsMaximal\nle : I ≤ m\n⊢ ⊤ ≤ I •... | [] | simp [r] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Flat.FaithfullyFlat.Basic | {
"line": 79,
"column": 83
} | {
"line": 79,
"column": 91
} | {
"line": 79,
"column": 91
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx✝ : Flat R M ∧ ∀ ⦃m : Ideal R⦄, m.IsMaximal → m • ⊤ ≠ ⊤\nflat : Flat R M\nh : ∀ ⦃m : Ideal R⦄, m.IsMaximal → m • ⊤ ≠ ⊤\nI : Ideal R\nhI : I ≠ ⊤\nr : I • ⊤ = ⊤\nm : Ideal R\nhm : m.IsMaximal\nle : I ≤ m\n⊢ ⊤ ≤ I •... | [] | simp [r] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Flat.FaithfullyFlat.Basic | {
"line": 79,
"column": 83
} | {
"line": 79,
"column": 91
} | {
"line": 79,
"column": 91
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx✝ : Flat R M ∧ ∀ ⦃m : Ideal R⦄, m.IsMaximal → m • ⊤ ≠ ⊤\nflat : Flat R M\nh : ∀ ⦃m : Ideal R⦄, m.IsMaximal → m • ⊤ ≠ ⊤\nI : Ideal R\nhI : I ≠ ⊤\nr : I • ⊤ = ⊤\nm : Ideal R\nhm : m.IsMaximal\nle : I ≤ m\n⊢ ⊤ ≤ I •... | [] | simp [r] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.GoingUp | {
"line": 148,
"column": 4
} | {
"line": 148,
"column": 15
} | {
"line": 148,
"column": 16
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝² : CommRing R\nS : Type u_2\ninst✝¹ : CommRing S\nf : R →+* S\nI J : Ideal S\ninst✝ : I.IsPrime\nhIJ : I ≤ J\nr : S\np : R[X]\np_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p ≠ 0\nhpI : eval₂ f r p ∈ I\nhrJ : r ∈ ↑J\nhrI : r ∉ ↑I\nrbar_ne_zero : (Quotient.mk I)... | [
"case refine_1\nR : Type u_1\ninst✝² : CommRing R\nS : Type u_2\ninst✝¹ : CommRing S\nf : R →+* S\nI J : Ideal S\ninst✝ : I.IsPrime\nhIJ : I ≤ J\nr : S\np : R[X]\np_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p ≠ 0\nhpI : eval₂ f r p ∈ I\nhrJ : r ∈ ↑J\nhrI : r ∉ ↑I\nrbar_ne_zero : (Quotient.mk I) r ≠ 0\nrbar... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.FaithfullyFlat.Basic | {
"line": 100,
"column": 4
} | {
"line": 101,
"column": 56
} | {
"line": 101,
"column": 57
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nfl : FaithfullyFlat R M\nN : Type u_1\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\ninst✝ : Nontrivial N\nn : N\nhn : n ≠ 0\nI : Ideal R := (R ∙ n).annihilator\nI_ne_top : ¬I = ⊤\nr : R\nhr : r ∈ I\n⊢ r ∈ ((Line... | [
"R : Type u\nM : Type v\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nfl : FaithfullyFlat R M\nN : Type u_1\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\ninst✝ : Nontrivial N\nn : N\nhn : n ≠ 0\nI : Ideal R := (R ∙ n).annihilator\nI_ne_top : ¬I = ⊤\nr : R\nhr : r ∈ I\n⊢ r • n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.FaithfullyFlat.Basic | {
"line": 105,
"column": 4
} | {
"line": 107,
"column": 64
} | {
"line": 107,
"column": 65
} | [
{
"pp": "case h\nR : Type u\nM : Type v\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nfl : FaithfullyFlat R M\nN : Type u_1\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\ninst✝ : Nontrivial N\nn : N\nhn : n ≠ 0\nI : Ideal R := (R ∙ n).annihilator\nI_ne_top : ¬I = ⊤\ninc : R ⧸ I →ₗ[R] N := ... | [
"case h\nR : Type u\nM : Type v\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nfl : FaithfullyFlat R M\nN : Type u_1\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\ninst✝ : Nontrivial N\nn : N\nhn : n ≠ 0\nI : Ideal R := (R ∙ n).annihilator\nI_ne_top : ¬I = ⊤\ninc : R ⧸ I →ₗ[R] N := Submodule.li... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.Integral | {
"line": 301,
"column": 51
} | {
"line": 301,
"column": 76
} | {
"line": 301,
"column": 77
} | [
{
"pp": "R : Type u_5\nS : Type u_6\nSₘ : Type u_7\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : CommRing Sₘ\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra S Sₘ\ninst✝² : Algebra R Sₘ\ninst✝¹ : IsScalarTower R S Sₘ\nr : S\nhr : IsIntegral R r\ninst✝ : Away r Sₘ\nx : S\na✝ : Nontrivial S\np : R[X]\nhpm : p.Monic\... | [
"R : Type u_5\nS : Type u_6\nSₘ : Type u_7\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : CommRing Sₘ\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra S Sₘ\ninst✝² : Algebra R Sₘ\ninst✝¹ : IsScalarTower R S Sₘ\nr : S\nhr : IsIntegral R r\ninst✝ : Away r Sₘ\nx : S\na✝ : Nontrivial S\np : R[X]\nhpm : p.Monic\nm : ℕ\nhm :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.FaithfullyFlat.Basic | {
"line": 196,
"column": 2
} | {
"line": 196,
"column": 28
} | {
"line": 197,
"column": 2
} | [
{
"pp": "R : Type u\nM✝ : Type v\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M✝\ninst✝⁴ : Module R M✝\nι : Type u_1\ninst✝³ : Nonempty ι\nM : ι → Type u_2\ninst✝² : (i : ι) → AddCommGroup (M i)\ninst✝¹ : (i : ι) → Module R (M i)\ninst✝ : ∀ (i : ι), FaithfullyFlat R (M i)\nN : Type (max u u_1 u_2)\nx✝¹ : AddComm... | [
"R : Type u\nM✝ : Type v\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M✝\ninst✝⁴ : Module R M✝\nι : Type u_1\ninst✝³ : Nonempty ι\nM : ι → Type u_2\ninst✝² : (i : ι) → AddCommGroup (M i)\ninst✝¹ : (i : ι) → Module R (M i)\ninst✝ : ∀ (i : ι), FaithfullyFlat R (M i)\nN : Type (max u u_1 u_2)\nx✝¹ : AddCommGroup N\nx✝ ... | obtain ⟨i⟩ := ‹Nonempty ι› | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Ideal.GoingUp | {
"line": 327,
"column": 4
} | {
"line": 327,
"column": 44
} | {
"line": 327,
"column": 45
} | [
{
"pp": "case refine_2\nR : 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\nP : Ideal R\ninst✝¹ : P.IsPrime\nI : Ideal S\ninst✝ : I.IsPrime\nhIP : comap (algebraMap R S) I ≤ P\nQ' : Ideal (S ⧸ I)\nQ'_prime : Q'.IsPrime\nhQ' : comap (algebr... | [
"case refine_2\nR : 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\nP : Ideal R\ninst✝¹ : P.IsPrime\nI : Ideal S\ninst✝ : I.IsPrime\nhIP : comap (algebraMap R S) I ≤ P\nQ' : Ideal (S ⧸ I)\nQ'_prime : Q'.IsPrime\nhQ' : comap (algebraMap (R ⧸ co... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.FaithfullyFlat.Basic | {
"line": 297,
"column": 42
} | {
"line": 297,
"column": 64
} | {
"line": 297,
"column": 64
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nN1 : Type u_1\ninst✝⁵ : AddCommGroup N1\ninst✝⁴ : Module R N1\nN2 : Type u_2\ninst✝³ : AddCommGroup N2\ninst✝² : Module R N2\nN3 : Type u_3\ninst✝¹ : AddCommGroup N3\ninst✝ : Module R N3\nl12 : N1 →ₗ[R] N2\nl23 :... | [
"R : Type u\nM : Type v\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nN1 : Type u_1\ninst✝⁵ : AddCommGroup N1\ninst✝⁴ : Module R N1\nN2 : Type u_2\ninst✝³ : AddCommGroup N2\ninst✝² : Module R N2\nN3 : Type u_3\ninst✝¹ : AddCommGroup N3\ninst✝ : Module R N3\nl12 : N1 →ₗ[R] N2\nl23 : N2 →ₗ[R] N3... | Submodule.mem_span_set | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Localization.Integral | {
"line": 348,
"column": 8
} | {
"line": 348,
"column": 80
} | {
"line": 349,
"column": 10
} | [
{
"pp": "case refine_1.refine_2.h₀\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 ... | [
"case refine_1.refine_2.h₀\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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Algebraic.Integral | {
"line": 152,
"column": 21
} | {
"line": 152,
"column": 69
} | {
"line": 152,
"column": 70
} | [
{
"pp": "R : Type u_1\nA : Type u_3\ninst✝² : CommRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nz : A\nhz : IsAlgebraic R z\ninj : ∃ a, (algebraMap R A) a = 0 ∧ a ≠ 0\nr : R\neq : (algebraMap R A) r = 0\nne : r ≠ 0\n⊢ IsIntegral R (r • z)",
"ppTerm": "?m.88",
"assigned": true,
"usedConstants": [
... | [
"R : Type u_1\nA : Type u_3\ninst✝² : CommRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nz : A\nhz : IsAlgebraic R z\ninj : ∃ a, (algebraMap R A) a = 0 ∧ a ≠ 0\nr : R\neq : (algebraMap R A) r = 0\nne : r ≠ 0\n⊢ IsIntegral R 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.Basic | {
"line": 195,
"column": 2
} | {
"line": 195,
"column": 49
} | {
"line": 195,
"column": 50
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nideal_gc : GaloisConnection Ideal.span SetLike.coe\n⊢ GaloisConnection (fun s ↦ zeroLocus s) fun t ↦ ↑(vanishingIdeal t)",
"ppTerm": "?m.27",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝ : CommSemiring R\nideal_gc : GaloisConnection Ideal.span SetLike.coe\n⊢ GaloisConnection (fun s ↦ zeroLocus s) fun t ↦ ↑(vanishingIdeal t)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Algebraic.Integral | {
"line": 185,
"column": 49
} | {
"line": 185,
"column": 91
} | {
"line": 185,
"column": 92
} | [
{
"pp": "R : Type u_1\nA : Type u_3\ninst✝² : CommRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nz : A\ny : R\nhy : y ∈ nonZeroDivisors R\nh : IsAlgebraic R (y • z)\np : R[X]\nhp : p ≠ 0\neval0 : (aeval (y • z)) p = 0\n⊢ (aeval z) (p.comp (C y * X)) = 0",
"ppTerm": "?m.46",
"assigned": true,
"usedCon... | [
"R : Type u_1\nA : Type u_3\ninst✝² : CommRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nz : A\ny : R\nhy : y ∈ nonZeroDivisors R\nh : IsAlgebraic R (y • z)\np : R[X]\nhp : p ≠ 0\neval0 : (aeval (y • z)) p = 0\n⊢ (aeval ((algebraMap R A) y * z)) p = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Algebraic.Integral | {
"line": 214,
"column": 21
} | {
"line": 214,
"column": 32
} | {
"line": 214,
"column": 33
} | [
{
"pp": "S : Type u_2\ninst✝³ : CommRing S\nK : Type u_4\ninst✝² : CommRing K\ninst✝¹ : Algebra S K\ninst✝ : IsIntegralClosure S ℤ K\nx : K\nhx : IsAlgebraic ℤ x\ns : S\nn : ℕ\nha : -↑n ≠ 0\nh : -↑n • x = (algebraMap S K) s\n⊢ n ≠ 0",
"ppTerm": "?m.127",
"assigned": true,
"usedConstants": [
"i... | [
"S : Type u_2\ninst✝³ : CommRing S\nK : Type u_4\ninst✝² : CommRing K\ninst✝¹ : Algebra S K\ninst✝ : IsIntegralClosure S ℤ K\nx : K\nhx : IsAlgebraic ℤ x\ns : S\nn : ℕ\nha : -↑n ≠ 0\nh : -↑n • x = (algebraMap S K) s\n⊢ ¬n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.Basic | {
"line": 350,
"column": 22
} | {
"line": 350,
"column": 33
} | {
"line": 350,
"column": 34
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nf g : R\nx : PrimeSpectrum R\n⊢ x ∈ zeroLocus {f * g} ↔ x ∈ zeroLocus {f} ∪ zeroLocus {g}",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"PrimeSpectrum.mem_zeroLocus._simp_1",
"Semiri... | [
"R : Type u\ninst✝ : CommSemiring R\nf g : R\nx : PrimeSpectrum R\n⊢ f * g ∈ x.asIdeal ↔ f ∈ x.asIdeal ∨ g ∈ x.asIdeal"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.Basic | {
"line": 360,
"column": 22
} | {
"line": 360,
"column": 33
} | {
"line": 360,
"column": 34
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nf : R\nn : ℕ\nhn : 0 < n\nx : PrimeSpectrum R\n⊢ x ∈ zeroLocus {f ^ n} ↔ x ∈ zeroLocus {f}",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"PrimeSpectrum.mem_zeroLocus._simp_1",
"Semir... | [
"R : Type u\ninst✝ : CommSemiring R\nf : R\nn : ℕ\nhn : 0 < n\nx : PrimeSpectrum R\n⊢ f ^ n ∈ x.asIdeal ↔ f ∈ x.asIdeal"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.Integral | {
"line": 393,
"column": 42
} | {
"line": 393,
"column": 53
} | {
"line": 393,
"column": 54
} | [
{
"pp": "A : Type u_3\ninst✝⁹ : CommRing A\nL : Type u_5\ninst✝⁸ : Field L\ninst✝⁷ : Algebra A L\nC : Type u_6\ninst✝⁶ : CommRing C\ninst✝⁵ : IsDomain C\ninst✝⁴ : Algebra C L\ninst✝³ : IsIntegralClosure C A L\ninst✝² : Algebra A C\ninst✝¹ : IsScalarTower A C L\ninst✝ : Algebra.IsAlgebraic A L\ninj : ∀ (x : A), ... | [
"A : Type u_3\ninst✝⁹ : CommRing A\nL : Type u_5\ninst✝⁸ : Field L\ninst✝⁷ : Algebra A L\nC : Type u_6\ninst✝⁶ : CommRing C\ninst✝⁵ : IsDomain C\ninst✝⁴ : Algebra C L\ninst✝³ : IsIntegralClosure C A L\ninst✝² : Algebra A C\ninst✝¹ : IsScalarTower A C L\ninst✝ : Algebra.IsAlgebraic A L\ninj : ∀ (x : A), (algebraMap ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Algebraic.Integral | {
"line": 265,
"column": 6
} | {
"line": 266,
"column": 72
} | {
"line": 266,
"column": 73
} | [
{
"pp": "case refine_1\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✝ ... | [
"case refine_1\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 :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Algebraic.Integral | {
"line": 259,
"column": 2
} | {
"line": 270,
"column": 44
} | {
"line": 272,
"column": 0
} | [
{
"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]... | [] | have ⟨r, hr, int⟩ := Algebra.IsAlgebraic.exists_integral_multiples R (p.support.image (coeff p))
let p := (r • p).toSubring (integralClosure R S).toSubring fun s hs ↦ by
obtain ⟨n, hn, rfl⟩ := mem_coeffs_iff.mp hs
exact int _ (Finset.mem_image_of_mem _ <| support_smul _ _ hn)
have : IsAlgebraic (integralClo... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Algebraic.Integral | {
"line": 259,
"column": 2
} | {
"line": 270,
"column": 44
} | {
"line": 272,
"column": 0
} | [
{
"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]... | [] | have ⟨r, hr, int⟩ := Algebra.IsAlgebraic.exists_integral_multiples R (p.support.image (coeff p))
let p := (r • p).toSubring (integralClosure R S).toSubring fun s hs ↦ by
obtain ⟨n, hn, rfl⟩ := mem_coeffs_iff.mp hs
exact int _ (Finset.mem_image_of_mem _ <| support_smul _ _ hn)
have : IsAlgebraic (integralClo... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Spectrum.Prime.Basic | {
"line": 415,
"column": 4
} | {
"line": 415,
"column": 15
} | {
"line": 415,
"column": 16
} | [
{
"pp": "case refine_1\nR : Type u\ninst✝¹ : CommSemiring R\nm : Ideal R\ninst✝ : m.IsMaximal\nI : PrimeSpectrum R\nh : m ≤ I.asIdeal\n⊢ I ∈ {{ asIdeal := m, isPrime := ⋯ }}",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PrimeSpectrum.mk",
"CommSemiring.to... | [
"case refine_1\nR : Type u\ninst✝¹ : CommSemiring R\nm : Ideal R\ninst✝ : m.IsMaximal\nI : PrimeSpectrum R\nh : m ≤ I.asIdeal\n⊢ I = { asIdeal := m, isPrime := ⋯ }"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Algebraic.Integral | {
"line": 555,
"column": 2
} | {
"line": 555,
"column": 13
} | {
"line": 555,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝¹⁵ : CommRing R\nR' : Type u_4\nS : Type u\ninst✝¹⁴ : CommRing R'\ninst✝¹³ : CommRing S\ninst✝¹² : Algebra R S\ninst✝¹¹ : Algebra R R'\ninst✝¹⁰ : IsFractionRing R R'\ninst✝⁹ : FaithfulSMul R S\ninst✝⁸ : Algebra.IsAlgebraic R S\ninst✝⁷ : NoZeroDivisors S\nS' : Type u\ninst✝⁶ : CommRin... | [
"R : Type u_1\ninst✝¹⁵ : CommRing R\nR' : Type u_4\nS : Type u\ninst✝¹⁴ : CommRing R'\ninst✝¹³ : CommRing S\ninst✝¹² : Algebra R S\ninst✝¹¹ : Algebra R R'\ninst✝¹⁰ : IsFractionRing R R'\ninst✝⁹ : FaithfulSMul R S\ninst✝⁸ : Algebra.IsAlgebraic R S\ninst✝⁷ : NoZeroDivisors S\nS' : Type u\ninst✝⁶ : CommRing S'\ninst✝⁵... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.FaithfullyFlat.Basic | {
"line": 421,
"column": 6
} | {
"line": 421,
"column": 27
} | {
"line": 421,
"column": 28
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\niff_exact :\n ∀ {N1 : Type (max u v)} [inst : AddCommGroup N1] [inst_1 : Module R N1] {N2 : Type (max u v)}\n [inst_2 : AddCommGroup N2] [inst_3 : Module R N2] {N3 : Type (max u v)} [inst_4 : AddCommGroup N3]\... | [
"R : Type u\nM : Type v\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\niff_exact :\n ∀ {N1 : Type (max u v)} [inst : AddCommGroup N1] [inst_1 : Module R N1] {N2 : Type (max u v)}\n [inst_2 : AddCommGroup N2] [inst_3 : Module R N2] {N3 : Type (max u v)} [inst_4 : AddCommGroup N3]\n [inst_5... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.FaithfullyFlat.Basic | {
"line": 422,
"column": 8
} | {
"line": 422,
"column": 19
} | {
"line": 422,
"column": 20
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\niff_exact :\n ∀ {N1 : Type (max u v)} [inst : AddCommGroup N1] [inst_1 : Module R N1] {N2 : Type (max u v)}\n [inst_2 : AddCommGroup N2] [inst_3 : Module R N2] {N3 : Type (max u v)} [inst_4 : AddCommGroup N3]\... | [
"R : Type u\nM : Type v\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\niff_exact :\n ∀ {N1 : Type (max u v)} [inst : AddCommGroup N1] [inst_1 : Module R N1] {N2 : Type (max u v)}\n [inst_2 : AddCommGroup N2] [inst_3 : Module R N2] {N3 : Type (max u v)} [inst_4 : AddCommGroup N3]\n [inst_5... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.FaithfullyFlat.Basic | {
"line": 463,
"column": 11
} | {
"line": 463,
"column": 22
} | {
"line": 463,
"column": 23
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nh : FaithfullyFlat R M\nN : Type u_1\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nN' : Type u_2\ninst✝¹ : AddCommGroup N'\ninst✝ : Module R N'\nf : N →ₗ[R] N'\nhf : LinearMap.lTensor M f = 0\nthis : Function.Ex... | [
"R : Type u\nM : Type v\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nh : FaithfullyFlat R M\nN : Type u_1\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nN' : Type u_2\ninst✝¹ : AddCommGroup N'\ninst✝ : Module R N'\nf : N →ₗ[R] N'\nhf : LinearMap.lTensor M f = 0\nthis : Function.Exact ⇑f ⇑Line... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.FaithfullyFlat.Basic | {
"line": 476,
"column": 8
} | {
"line": 476,
"column": 19
} | {
"line": 476,
"column": 20
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nh✝ : FaithfullyFlat R M\nN : Type u_1\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nN' : Type u_2\ninst✝¹ : AddCommGroup N'\ninst✝ : Module R N'\nf : N →ₗ[R] N'\nh : LinearMap.lTensor M f = 0\nn : N\nm : M\n⊢ (T... | [
"R : Type u\nM : Type v\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nh✝ : FaithfullyFlat R M\nN : Type u_1\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nN' : Type u_2\ninst✝¹ : AddCommGroup N'\ninst✝ : Module R N'\nf : N →ₗ[R] N'\nh : LinearMap.lTensor M f = 0\nn : N\nm : M\n⊢ m ⊗ₜ[R] f n = ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.FaithfullyFlat.Basic | {
"line": 477,
"column": 64
} | {
"line": 477,
"column": 75
} | {
"line": 477,
"column": 76
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nh✝ : FaithfullyFlat R M\nN : Type u_1\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nN' : Type u_2\ninst✝¹ : AddCommGroup N'\ninst✝ : Module R N'\nf : N →ₗ[R] N'\nh : LinearMap.rTensor M f = 0\nm : M\nn : N\n⊢ (T... | [
"R : Type u\nM : Type v\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nh✝ : FaithfullyFlat R M\nN : Type u_1\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nN' : Type u_2\ninst✝¹ : AddCommGroup N'\ninst✝ : Module R N'\nf : N →ₗ[R] N'\nh : LinearMap.rTensor M f = 0\nm : M\nn : N\n⊢ f n ⊗ₜ[R] m = ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.FaithfullyFlat.Basic | {
"line": 488,
"column": 20
} | {
"line": 488,
"column": 35
} | {
"line": 488,
"column": 36
} | [
{
"pp": "R : 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\nthis : f = 0\n⊢ m = 0",
"ppTerm": "?m.88",
"assign... | [
"R : 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\nthis : f = 0\n⊢ m = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.FaithfullyFlat.Basic | {
"line": 554,
"column": 2
} | {
"line": 554,
"column": 13
} | {
"line": 554,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝⁸ : CommRing R\nS : Type u_2\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module S M\ninst✝² : IsScalarTower R S M\ninst✝¹ : FaithfullyFlat R S\ninst✝ : FaithfullyFlat S M\nN : Type (max u_1 u_3)\nx✝³ : AddCommGroup ... | [
"R : Type u_1\ninst✝⁸ : CommRing R\nS : Type u_2\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module S M\ninst✝² : IsScalarTower R S M\ninst✝¹ : FaithfullyFlat R S\ninst✝ : FaithfullyFlat S M\nN : Type (max u_1 u_3)\nx✝³ : AddCommGroup N\nx✝² : Mod... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.FaithfullyFlat.Basic | {
"line": 576,
"column": 2
} | {
"line": 576,
"column": 62
} | {
"line": 576,
"column": 63
} | [
{
"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... | [
"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\nhf : IsBaseCh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.FaithfullyFlat.Basic | {
"line": 599,
"column": 2
} | {
"line": 599,
"column": 20
} | {
"line": 599,
"column": 21
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\nS : Type u_1\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\ninst✝⁵ : FaithfullyFlat R S\ninst✝⁴ : Flat S (S ⊗[R] M)\nN P : Type (max u v)\ninst✝³ : AddCommGroup N\ninst✝² : AddCommGroup P\ninst✝¹ : Module R N\ninst... | [
"R : Type u\nM : Type v\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\nS : Type u_1\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\ninst✝⁵ : FaithfullyFlat R S\ninst✝⁴ : Flat S (S ⊗[R] M)\nN P : Type (max u v)\ninst✝³ : AddCommGroup N\ninst✝² : AddCommGroup P\ninst✝¹ : Module R N\ninst✝ : Module R... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.FaithfullyFlat.Algebra | {
"line": 137,
"column": 17
} | {
"line": 138,
"column": 9
} | {
"line": 138,
"column": 10
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : Module.FaithfullyFlat A B\nx✝¹ x✝ : Ideal A\nh : map (algebraMap A B) x✝¹ = map (algebraMap A B) x✝\n⊢ x✝¹ = x✝",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [... | [
"A : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : Module.FaithfullyFlat A B\nx✝¹ x✝ : Ideal A\nh : map (algebraMap A B) x✝¹ = map (algebraMap A B) x✝\n⊢ x✝¹ = x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Ring.Idempotent | {
"line": 136,
"column": 45
} | {
"line": 138,
"column": 37
} | {
"line": 139,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\na : { a // IsIdempotentElem a }\n⊢ ↑(a ⊔ ⟨1 - ↑a, ⋯⟩) = ↑⊤",
"ppTerm": "?m.134",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsIdempotentElem.one_sub",
"Lattice.toSemilatticeSup",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | [] | by
simp_rw [(· ⊔ ·), SemilatticeSup.sup, add_sub_cancel, mul_sub, mul_one]
rw [a.2, sub_self, sub_zero]; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.KrullDimension.Basic | {
"line": 57,
"column": 8
} | {
"line": 57,
"column": 19
} | {
"line": 57,
"column": 20
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : R →+* S\nhf : Function.Surjective ⇑f\nx✝¹ x✝ : PrimeSpectrum S\nh : { asIdeal := Ideal.comap f x✝¹.asIdeal, isPrime := ⋯ } = { asIdeal := Ideal.comap f x✝.asIdeal, isPrime := ⋯ }\n⊢ Ideal.comap f x✝¹.asIdeal = Ideal.comap ... | [
"R : Type u_1\nS : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : R →+* S\nhf : Function.Surjective ⇑f\nx✝¹ x✝ : PrimeSpectrum S\nh : { asIdeal := Ideal.comap f x✝¹.asIdeal, isPrime := ⋯ } = { asIdeal := Ideal.comap f x✝.asIdeal, isPrime := ⋯ }\n⊢ Ideal.comap f x✝¹.asIdeal = Ideal.comap f x✝.asIdeal... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.MinimalPrime.Localization | {
"line": 67,
"column": 6
} | {
"line": 67,
"column": 46
} | {
"line": 67,
"column": 47
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nI : Ideal R\nx y : R\nhy : y ∉ I.radical\nhx : x * y ∈ I.radical\n⊢ ∃ p ∈ I.minimalPrimes, y ∉ p",
"ppTerm": "?m.414",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"Ideal.minimalPrimes",
"CommSemiring.toSemiring",
"M... | [
"R : Type u_1\ninst✝ : CommSemiring R\nI : Ideal R\nx y : R\nhy : y ∉ I.radical\nhx : x * y ∈ I.radical\n⊢ ∃ p, I.IsMinimalPrime p ∧ y ∉ p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.MinimalPrime.Localization | {
"line": 78,
"column": 22
} | {
"line": 78,
"column": 61
} | {
"line": 78,
"column": 62
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nI p : Ideal R\nhp : p ∈ I.minimalPrimes\nx : R\nhx✝ : x ∈ p\ny : R\nhy : y ∉ I.radical\nn : ℕ\nhx : (x * y) ^ n ∈ I\nH : ∃ m, x ^ m * y ^ n ∈ I\nh : Nat.find H = 0\n⊢ y ^ n ∈ I",
"ppTerm": "?m.111",
"assigned": false,
"usedConstants": [],
"usedFVars... | [
"R : Type u_1\ninst✝ : CommSemiring R\nI p : Ideal R\nhp : p ∈ I.minimalPrimes\nx : R\nhx✝ : x ∈ p\ny : R\nhy : y ∉ I.radical\nn : ℕ\nhx : (x * y) ^ n ∈ I\nH : ∃ m, x ^ m * y ^ n ∈ I\nh : Nat.find H = 0\n⊢ y ^ n ∈ I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.RingHom | {
"line": 141,
"column": 4
} | {
"line": 141,
"column": 15
} | {
"line": 141,
"column": 16
} | [
{
"pp": "case inl\nι : Type u_3\nR : ι → Type u_2\ninst✝ : (i : ι) → CommSemiring (R i)\nx✝¹ x✝ : (i : ι) × PrimeSpectrum (R i)\ni : ι\np q : PrimeSpectrum (R i)\neq : sigmaToPi R ⟨i, p⟩ = sigmaToPi R ⟨i, q⟩\nx : R i\n⊢ x ∈ p.asIdeal ↔ x ∈ q.asIdeal",
"ppTerm": "?inl",
"assigned": false,
"usedConsta... | [
"case inl\nι : Type u_3\nR : ι → Type u_2\ninst✝ : (i : ι) → CommSemiring (R i)\nx✝¹ x✝ : (i : ι) × PrimeSpectrum (R i)\ni : ι\np q : PrimeSpectrum (R i)\neq : sigmaToPi R ⟨i, p⟩ = sigmaToPi R ⟨i, q⟩\nx : R i\n⊢ x ∈ p.asIdeal ↔ x ∈ q.asIdeal"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.RingHom | {
"line": 167,
"column": 4
} | {
"line": 167,
"column": 79
} | {
"line": 167,
"column": 80
} | [
{
"pp": "ι : Type u_3\nR : ι → Type u_2\ninst✝² : (i : ι) → CommSemiring (R i)\ninst✝¹ : Infinite ι\ninst✝ : ∀ (i : ι), Nontrivial (R i)\nJ : Ideal ((i : ι) → R i) :=\n let __spread.0 := AddMonoidHom.mrange DFinsupp.coeFnAddMonoidHom;\n { toAddSubmonoid := __spread.0, smul_mem' := ⋯ }\nx : Π₀ (i : ι), R i\nhx... | [
"ι : Type u_3\nR : ι → Type u_2\ninst✝² : (i : ι) → CommSemiring (R i)\ninst✝¹ : Infinite ι\ninst✝ : ∀ (i : ι), Nontrivial (R i)\nJ : Ideal ((i : ι) → R i) :=\n let __spread.0 := AddMonoidHom.mrange DFinsupp.coeFnAddMonoidHom;\n { toAddSubmonoid := __spread.0, smul_mem' := ⋯ }\nx : Π₀ (i : ι), R i\nhx : DFinsupp.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.KrullDimension.Basic | {
"line": 200,
"column": 75
} | {
"line": 200,
"column": 86
} | {
"line": 200,
"column": 87
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : NoZeroDivisors R\ninst✝ : Ring.KrullDimLE 1 R\na : R\nha : Prime a\n⊢ Ideal.span {a} ≠ ⊥",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Submodule.span_eq_bot._simp_1",
"Eq.mpr",
"Submodule",
"Semiring.toModu... | [
"R : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : NoZeroDivisors R\ninst✝ : Ring.KrullDimLE 1 R\na : R\nha : Prime a\n⊢ ¬a = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.RingHom | {
"line": 171,
"column": 4
} | {
"line": 171,
"column": 15
} | {
"line": 171,
"column": 16
} | [
{
"pp": "ι : Type u_3\nR : ι → Type u_2\ninst✝² : (i : ι) → CommSemiring (R i)\ninst✝¹ : Infinite ι\ninst✝ : ∀ (i : ι), Nontrivial (R i)\nJ : Ideal ((i : ι) → R i) :=\n let __spread.0 := AddMonoidHom.mrange DFinsupp.coeFnAddMonoidHom;\n { toAddSubmonoid := __spread.0, smul_mem' := ⋯ }\nI : Ideal ((i : ι) → R ... | [
"ι : Type u_3\nR : ι → Type u_2\ninst✝² : (i : ι) → CommSemiring (R i)\ninst✝¹ : Infinite ι\ninst✝ : ∀ (i : ι), Nontrivial (R i)\nJ : Ideal ((i : ι) → R i) :=\n let __spread.0 := AddMonoidHom.mrange DFinsupp.coeFnAddMonoidHom;\n { toAddSubmonoid := __spread.0, smul_mem' := ⋯ }\nI : Ideal ((i : ι) → R i)\nmax : I.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.RingHom | {
"line": 191,
"column": 4
} | {
"line": 191,
"column": 49
} | {
"line": 191,
"column": 50
} | [
{
"pp": "ι : Type u_3\nR : ι → Type u_4\ninst✝¹ : (i : ι) → CommRing (R i)\ninst✝ : _root_.Finite ι\np : PrimeSpectrum ((i : ι) → R i)\nval✝ : Fintype ι\ne : ι → (i : ι) → R i := fun i ↦ Function.update 1 i 0\nH : ∏ i, e i = 0\n⊢ ∃ i, e i ∈ p.asIdeal",
"ppTerm": "?m.104",
"assigned": false,
"usedCon... | [
"ι : Type u_3\nR : ι → Type u_4\ninst✝¹ : (i : ι) → CommRing (R i)\ninst✝ : _root_.Finite ι\np : PrimeSpectrum ((i : ι) → R i)\nval✝ : Fintype ι\ne : ι → (i : ι) → R i := fun i ↦ Function.update 1 i 0\nH : ∏ i, e i = 0\n⊢ ∃ i, e i ∈ p.asIdeal"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalRing.ResidueField.Ideal | {
"line": 155,
"column": 30
} | {
"line": 155,
"column": 61
} | {
"line": 155,
"column": 62
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing A\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\nI : Ideal R\ninst✝² : I.IsPrime\nk : Type u_5\ninst✝¹ : Field k\np : Ideal k\ninst✝ : p.IsPrime\n⊢ p.IsMaximal",
"p... | [
"R : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing A\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\nI : Ideal R\ninst✝² : I.IsPrime\nk : Type u_5\ninst✝¹ : Field k\np : Ideal k\ninst✝ : p.IsPrime\n⊢ ⊥.IsMaximal"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.Bases | {
"line": 35,
"column": 4
} | {
"line": 35,
"column": 15
} | {
"line": 35,
"column": 16
} | [
{
"pp": "case refine_1\nX : Type u_1\nι : Type u_2\ninst✝ : TopologicalSpace X\nb : ι → Set X\nhb : IsTopologicalBasis (range b)\nU : Set X\nhUc : IsCompact U\nhUo : IsOpen U\nY : Type u_1\nf' : Y → ι\ne : U = ⋃ i, (b ∘ f') i\nhf' : ∀ (i : Y), b (f' i) = (b ∘ f') i\nt : Finset Y\nht : U ⊆ ⋃ i ∈ t, (b ∘ f') i\ni... | [
"case refine_1\nX : Type u_1\nι : Type u_2\ninst✝ : TopologicalSpace X\nb : ι → Set X\nhb : IsTopologicalBasis (range b)\nU : Set X\nhUc : IsCompact U\nhUo : IsOpen U\nY : Type u_1\nf' : Y → ι\ne : U = ⋃ i, (b ∘ f') i\nhf' : ∀ (i : Y), b (f' i) = (b ∘ f') i\nt : Finset Y\nht : U ⊆ ⋃ i ∈ t, (b ∘ f') i\ni : Y\nhi : i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Maximal.Localization | {
"line": 43,
"column": 40
} | {
"line": 43,
"column": 59
} | {
"line": 43,
"column": 60
} | [
{
"pp": "R : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : IsDomain R\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nx : K\nhrange : x ∉ range ⇑(algebraMap R K)\nhlocal : ∀ (i : MaximalSpectrum R), x ∈ Localization.subalgebra.ofField K i.asIdeal.primeCompl ⋯\ndenom : Ideal R := Sub... | [
"R : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : IsDomain R\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nx : K\nhrange : x ∉ range ⇑(algebraMap R K)\nhlocal : ∀ (i : MaximalSpectrum R), x ∈ Localization.subalgebra.ofField K i.asIdeal.primeCompl ⋯\ndenom : Ideal R := Submodule.comap... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Maximal.Localization | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 37
} | {
"line": 63,
"column": 38
} | [
{
"pp": "R : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : IsDomain R\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nx✝ : K\n⊢ x✝ ∈ ⨅ v, Localization.subalgebra.ofField K v.asIdeal.primeCompl ⋯ →\n x✝ ∈ ⨅ v, Localization.subalgebra.ofField K v.asIdeal.primeCompl ⋯",
"ppTerm"... | [
"R : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : IsDomain R\nK : Type u_5\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nx✝ : K\n⊢ (∀ (i : PrimeSpectrum R), x✝ ∈ Localization.subalgebra.ofField K i.asIdeal.primeCompl ⋯) →\n ∀ (i : MaximalSpectrum R), x✝ ∈ Localization.subalgebra.ofField K i.asI... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Maximal.Localization | {
"line": 180,
"column": 2
} | {
"line": 181,
"column": 93
} | {
"line": 183,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\n⊢ Function.Surjective ⇑(piLocalizationToMaximal R)",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"MaximalSpectrum.asIdeal",
"OreLocalization.instAlgebra",
"PrimeSpectrum.isPrime",
"CommSemiring.toSemiring",
"AlgH... | [] | classical
exact fun r ↦ ⟨fun I ↦ if h : I.1.IsMaximal then r ⟨_, h⟩ else 0, funext fun _ ↦ dif_pos _⟩ | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.RingTheory.Spectrum.Maximal.Localization | {
"line": 180,
"column": 2
} | {
"line": 181,
"column": 93
} | {
"line": 183,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\n⊢ Function.Surjective ⇑(piLocalizationToMaximal R)",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"MaximalSpectrum.asIdeal",
"OreLocalization.instAlgebra",
"PrimeSpectrum.isPrime",
"CommSemiring.toSemiring",
"AlgH... | [] | classical
exact fun r ↦ ⟨fun I ↦ if h : I.1.IsMaximal then r ⟨_, h⟩ else 0, funext fun _ ↦ dif_pos _⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Spectrum.Maximal.Localization | {
"line": 180,
"column": 2
} | {
"line": 181,
"column": 93
} | {
"line": 183,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\n⊢ Function.Surjective ⇑(piLocalizationToMaximal R)",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"MaximalSpectrum.asIdeal",
"OreLocalization.instAlgebra",
"PrimeSpectrum.isPrime",
"CommSemiring.toSemiring",
"AlgH... | [] | classical
exact fun r ↦ ⟨fun I ↦ if h : I.1.IsMaximal then r ⟨_, h⟩ else 0, funext fun _ ↦ dif_pos _⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Sets.OpenCover | {
"line": 42,
"column": 2
} | {
"line": 42,
"column": 36
} | {
"line": 42,
"column": 37
} | [
{
"pp": "ι : Type u_1\nX : Type u_3\ninst✝ : TopologicalSpace X\nu : ι → Opens X\nhu : IsOpenCover u\n⊢ ⋃ i, ↑(u i) = univ",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_1\nX : Type u_3\ninst✝ : TopologicalSpace X\nu : ι → Opens X\nhu : IsOpenCover u\n⊢ ⋃ i, ↑(u i) = univ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sets.OpenCover | {
"line": 50,
"column": 2
} | {
"line": 50,
"column": 37
} | {
"line": 50,
"column": 38
} | [
{
"pp": "ι : Type u_1\nX : Type u_3\ninst✝ : TopologicalSpace X\nu : ι → Opens X\nhu : IsOpenCover u\na : X\n⊢ ∃ i, a ∈ u i",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_1\nX : Type u_3\ninst✝ : TopologicalSpace X\nu : ι → Opens X\nhu : IsOpenCover u\na : X\n⊢ ∃ i, a ∈ u i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sets.OpenCover | {
"line": 68,
"column": 8
} | {
"line": 68,
"column": 19
} | {
"line": 68,
"column": 20
} | [
{
"pp": "ι : Type u_1\nX : Type u_3\ninst✝¹ : TopologicalSpace X\nu : ι → Opens X\nhu : ↑⊤ ⊆ ↑(iSup u)\ninst✝ : CompactSpace X\n⊢ univ ⊆ ⋃ i, (u i).carrier",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.univ",
"TopologicalSpace.Opens",
"id",
"L... | [
"ι : Type u_1\nX : Type u_3\ninst✝¹ : TopologicalSpace X\nu : ι → Opens X\nhu : ↑⊤ ⊆ ↑(iSup u)\ninst✝ : CompactSpace X\n⊢ ⋃ i, ↑(u i) = univ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sets.OpenCover | {
"line": 70,
"column": 2
} | {
"line": 70,
"column": 85
} | {
"line": 70,
"column": 86
} | [
{
"pp": "case h\nι : Type u_1\nX : Type u_3\ninst✝¹ : TopologicalSpace X\nu : ι → Opens X\nhu : ↑⊤ ⊆ ↑(iSup u)\ninst✝ : CompactSpace X\ns : Finset ι\nhs : univ ⊆ ⋃ i ∈ s, (u i).carrier\n⊢ IsOpenCover fun i ↦ u ↑i",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.t... | [
"case h\nι : Type u_1\nX : Type u_3\ninst✝¹ : TopologicalSpace X\nu : ι → Opens X\nhu : ↑⊤ ⊆ ↑(iSup u)\ninst✝ : CompactSpace X\ns : Finset ι\nhs : univ ⊆ ⋃ i ∈ s, (u i).carrier\n⊢ ⋃ x ∈ s, ↑(u x) = univ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sets.OpenCover | {
"line": 118,
"column": 48
} | {
"line": 118,
"column": 59
} | {
"line": 118,
"column": 60
} | [
{
"pp": "X : Type u_1\nι : Type u_2\ninst✝ : TopologicalSpace X\nU : ι → Opens X\nhn : Pairwise ((fun x1 x2 ↦ ¬Disjoint x1 x2) on U)\nh : ∀ (i : ι), IsPreirreducible ↑(U i)\ns : Set X\nhs : IsOpen[inst✝] s\nhsU : s ⊆ ⋃ i, ↑(U i)\nx : X\nhx : x ∈ s\ni : ι\nhi : x ∈ ↑(U i)\nu : Set X\nhu : u ∈ irreducibleComponen... | [
"X : Type u_1\nι : Type u_2\ninst✝ : TopologicalSpace X\nU : ι → Opens X\nhn : Pairwise ((fun x1 x2 ↦ ¬Disjoint x1 x2) on U)\nh : ∀ (i : ι), IsPreirreducible ↑(U i)\ns : Set X\nhs : IsOpen[inst✝] s\nhsU : s ⊆ ⋃ i, ↑(U i)\nx : X\nhx : x ∈ s\ni : ι\nhi : x ∈ ↑(U i)\nu : Set X\nhu : u ∈ irreducibleComponents X\nhUu : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sets.OpenCover | {
"line": 128,
"column": 59
} | {
"line": 128,
"column": 70
} | {
"line": 128,
"column": 71
} | [
{
"pp": "X : Type u_1\nι : Type u_2\ninst✝ : TopologicalSpace X\nU : ι → Opens X\nhn : Pairwise ((fun x1 x2 ↦ ¬Disjoint x1 x2) on U)\nhU : IsOpenCover U\nh : ∀ (i : ι), PreirreducibleSpace ↥(U i)\nh' : ∀ (i : ι), IsPreirreducible (U i).carrier\n⊢ univ ⊆ ⋃ i, ↑(U i)",
"ppTerm": "?m.37",
"assigned": true,... | [
"X : Type u_1\nι : Type u_2\ninst✝ : TopologicalSpace X\nU : ι → Opens X\nhn : Pairwise ((fun x1 x2 ↦ ¬Disjoint x1 x2) on U)\nhU : IsOpenCover U\nh : ∀ (i : ι), PreirreducibleSpace ↥(U i)\nh' : ∀ (i : ι), IsPreirreducible (U i).carrier\n⊢ ⋃ i, ↑(U i) = univ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.LocalAtTarget | {
"line": 113,
"column": 4
} | {
"line": 113,
"column": 53
} | {
"line": 113,
"column": 54
} | [
{
"pp": "case mpr\nβ : Type u_2\ninst✝ : TopologicalSpace β\nι : Type u_3\nU : ι → Opens β\ns : Set β\nhU : IsOpenCover U\nH : ∀ (i : ι), IsOpen[inst✝] (s ∩ ↑(U i))\n⊢ IsOpen[inst✝] s",
"ppTerm": "?mpr",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case mpr\nβ : Type u_2\ninst✝ : TopologicalSpace β\nι : Type u_3\nU : ι → Opens β\ns : Set β\nhU : IsOpenCover U\nH : ∀ (i : ι), IsOpen[inst✝] (s ∩ ↑(U i))\n⊢ IsOpen[inst✝] s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.LocalAtTarget | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 13
} | {
"line": 122,
"column": 14
} | [
{
"pp": "β : Type u_2\ninst✝ : TopologicalSpace β\nι : Type u_3\nU : ι → Opens β\nhU : IsOpenCover U\ns : Set β\n⊢ IsClosed[inst✝] s ↔ ∀ (i : ι), IsClosed[instTopologicalSpaceSubtype] (Subtype.val ⁻¹' s)",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals... | [
"β : Type u_2\ninst✝ : TopologicalSpace β\nι : Type u_3\nU : ι → Opens β\nhU : IsOpenCover U\ns : Set β\n⊢ IsClosed[inst✝] s ↔ ∀ (i : ι), IsClosed[instTopologicalSpaceSubtype] (Subtype.val ⁻¹' s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.LocalAtTarget | {
"line": 159,
"column": 4
} | {
"line": 160,
"column": 34
} | {
"line": 160,
"column": 35
} | [
{
"pp": "case mpr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\nι : Type u_3\nU : ι → Opens β\nhU : IsOpenCover U\nh : Continuous[inst✝¹, inst✝] f\nH : ∀ (i : ι) (x : ↑(f ⁻¹' (U i).carrier)), 𝓝 x = Filter.comap Subtype.val (Filter.comap f (𝓝 ((f ∘ Subtype.val... | [
"case mpr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\nι : Type u_3\nU : ι → Opens β\nhU : IsOpenCover U\nh : Continuous[inst✝¹, inst✝] f\nH : ∀ (i : ι) (x : ↑(f ⁻¹' (U i).carrier)), 𝓝 x = Filter.comap Subtype.val (Filter.comap f (𝓝 ((f ∘ Subtype.val) x)))\nx : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.LocalAtTarget | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 43
} | {
"line": 164,
"column": 44
} | [
{
"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⊢ IsEmbedding f ↔ ∀ (i : ι), IsEmbedding ((U i).carrier.restrictPreimage f)",
"ppTerm": "?m.23",
"assigned": true,
... | [
"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⊢ IsInducing f ∧ Function.Injective f ↔\n (∀ (x : ι), IsInducing ((↑(U x)).restrictPreimage f)) ∧ ∀ (x : ι), Function.Injective ((↑(... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.BooleanSubalgebra | {
"line": 48,
"column": 61
} | {
"line": 48,
"column": 72
} | {
"line": 48,
"column": 73
} | [
{
"pp": "α : Type u_2\ninst✝ : BooleanAlgebra α\nL : BooleanSubalgebra α\na : α\nha : aᶜ ∈ L\n⊢ a ∈ L",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_2\ninst✝ : BooleanAlgebra α\nL : BooleanSubalgebra α\na : α\nha : aᶜ ∈ L\n⊢ a ∈ L"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.BooleanSubalgebra | {
"line": 50,
"column": 36
} | {
"line": 50,
"column": 47
} | {
"line": 50,
"column": 48
} | [
{
"pp": "α : Type u_2\ninst✝ : BooleanAlgebra α\nL : BooleanSubalgebra α\n⊢ ⊤ ∈ L",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_2\ninst✝ : BooleanAlgebra α\nL : BooleanSubalgebra α\n⊢ ⊤ ∈ L"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.LocallyClosed | {
"line": 185,
"column": 12
} | {
"line": 185,
"column": 23
} | {
"line": 185,
"column": 24
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\ntfae_1_to_2 : IsLocallyClosed s → IsOpen[inst✝] (coborder s)\ntfae_2_to_3 : IsOpen[inst✝] (coborder s) → ∀ x ∈ s, ∃ U ∈ 𝓝 x, IsClosed[instTopologicalSpaceSubtype] (U ↓∩ s)\ntfae_3_to_4 :\n (∀ x ∈ s, ∃ U ∈ 𝓝 x, IsClosed[instTopologicalSpaceSubtype]... | [
"X : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\ntfae_1_to_2 : IsLocallyClosed s → IsOpen[inst✝] (coborder s)\ntfae_2_to_3 : IsOpen[inst✝] (coborder s) → ∀ x ∈ s, ∃ U ∈ 𝓝 x, IsClosed[instTopologicalSpaceSubtype] (U ↓∩ s)\ntfae_3_to_4 :\n (∀ x ∈ s, ∃ U ∈ 𝓝 x, IsClosed[instTopologicalSpaceSubtype] (U ↓∩ s)) →... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.LocallyClosed | {
"line": 186,
"column": 4
} | {
"line": 186,
"column": 15
} | {
"line": 186,
"column": 16
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\ntfae_1_to_2 : IsLocallyClosed s → IsOpen[inst✝] (coborder s)\ntfae_2_to_3 : IsOpen[inst✝] (coborder s) → ∀ x ∈ s, ∃ U ∈ 𝓝 x, IsClosed[instTopologicalSpaceSubtype] (U ↓∩ s)\ntfae_3_to_4 :\n (∀ x ∈ s, ∃ U ∈ 𝓝 x, IsClosed[instTopologicalSpaceSubtype]... | [
"X : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\ntfae_1_to_2 : IsLocallyClosed s → IsOpen[inst✝] (coborder s)\ntfae_2_to_3 : IsOpen[inst✝] (coborder s) → ∀ x ∈ s, ∃ U ∈ 𝓝 x, IsClosed[instTopologicalSpaceSubtype] (U ↓∩ s)\ntfae_3_to_4 :\n (∀ x ∈ s, ∃ U ∈ 𝓝 x, IsClosed[instTopologicalSpaceSubtype] (U ↓∩ s)) →... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.BooleanSubalgebra | {
"line": 373,
"column": 4
} | {
"line": 373,
"column": 15
} | {
"line": 373,
"column": 16
} | [
{
"pp": "α : Type u_2\ninst✝ : BooleanAlgebra α\ns : Set α\np : (g : α) → g ∈ closure s → Prop\nmem : ∀ (x : α) (hx : x ∈ s), p x ⋯\nbot : p ⊥ ⋯\nsup : ∀ (x : α) (hx : x ∈ closure s) (y : α) (hy : y ∈ closure s), p x hx → p y hy → p (x ⊔ y) ⋯\ncompl : ∀ (x : α) (hx : x ∈ closure s), p x hx → p xᶜ ⋯\nx✝ : α\nhx✝... | [
"α : Type u_2\ninst✝ : BooleanAlgebra α\ns : Set α\np : (g : α) → g ∈ closure s → Prop\nmem : ∀ (x : α) (hx : x ∈ s), p x ⋯\nbot : p ⊥ ⋯\nsup : ∀ (x : α) (hx : x ∈ closure s) (y : α) (hy : y ∈ closure s), p x hx → p y hy → p (x ⊔ y) ⋯\ncompl : ∀ (x : α) (hx : x ∈ closure s), p x hx → p xᶜ ⋯\nx✝ : α\nhx✝ : x✝ ∈ clos... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sets.Opens | {
"line": 428,
"column": 51
} | {
"line": 428,
"column": 62
} | {
"line": 428,
"column": 63
} | [
{
"pp": "α : Type u_2\ninst✝ : TopologicalSpace α\ns : Opens α\nH : ∀ {ι : Type u_2} (U : ι → Set α), (∀ (i : ι), IsOpen[inst✝] (U i)) → ↑s ⊆ ⋃ i, U i → ∃ t, ↑s ⊆ ⋃ i ∈ t, U i\nι : Type u_2\nU : ι → Opens α\nhU : s ≤ iSup U\n⊢ ↑s ⊆ ⋃ i, ↑(U i)",
"ppTerm": "?m.76",
"assigned": false,
"usedConstants":... | [
"α : Type u_2\ninst✝ : TopologicalSpace α\ns : Opens α\nH : ∀ {ι : Type u_2} (U : ι → Set α), (∀ (i : ι), IsOpen[inst✝] (U i)) → ↑s ⊆ ⋃ i, U i → ∃ t, ↑s ⊆ ⋃ i ∈ t, U i\nι : Type u_2\nU : ι → Opens α\nhU : s ≤ iSup U\n⊢ ↑s ⊆ ⋃ i, ↑(U i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.LocalAtTarget | {
"line": 259,
"column": 6
} | {
"line": 259,
"column": 56
} | {
"line": 259,
"column": 57
} | [
{
"pp": "X : Type u_6\nY : Type u_7\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nf : X → Y\nh : Continuous[inst✝², inst✝¹] f\nι : Type u_4\nU : ι → Opens Y\nhU : range f ⊆ ↑(iSup U)\nV : ι → Type u_5\ninst✝ : (i : ι) → TopologicalSpace (V i)\niV : (i : ι) → V i → X\nhiV : ∀ (i : ι), Continuous[ins... | [
"X : Type u_6\nY : Type u_7\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nf : X → Y\nh : Continuous[inst✝², inst✝¹] f\nι : Type u_4\nU : ι → Opens Y\nhU : range f ⊆ ↑(iSup U)\nV : ι → Type u_5\ninst✝ : (i : ι) → TopologicalSpace (V i)\niV : (i : ι) → V i → X\nhiV : ∀ (i : ι), Continuous[inst✝ i, inst✝²... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.BooleanSubalgebra | {
"line": 413,
"column": 25
} | {
"line": 413,
"column": 36
} | {
"line": 413,
"column": 37
} | [
{
"pp": "α : Type u_2\ninst✝ : BooleanAlgebra α\ns : Set α\nisSublattice : IsSublattice s\nbot_mem : ⊥ ∈ s\ntop_mem : ⊤ ∈ s\np : (g : α) → g ∈ closure s → Prop\nsdiff : ∀ (x : α) (hx : x ∈ s) (y : α) (hy : y ∈ s), p (x \\ y) ⋯\nsup : ∀ (x : α) (hx : x ∈ closure s) (y : α) (hy : y ∈ closure s), p x hx → p y hy →... | [
"α : Type u_2\ninst✝ : BooleanAlgebra α\ns : Set α\nisSublattice : IsSublattice s\nbot_mem : ⊥ ∈ s\ntop_mem : ⊤ ∈ s\np : (g : α) → g ∈ closure s → Prop\nsdiff : ∀ (x : α) (hx : x ∈ s) (y : α) (hy : y ∈ s), p (x \\ y) ⋯\nsup : ∀ (x : α) (hx : x ∈ closure s) (y : α) (hy : y ∈ closure s), p x hx → p y hy → p (x ⊔ y) ⋯... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sets.Closeds | {
"line": 542,
"column": 4
} | {
"line": 543,
"column": 34
} | {
"line": 543,
"column": 35
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : β → α\nh : IsOpenEmbedding f\na b : IrreducibleCloseds β\nhle :\n { toFun := fun T ↦ ⟨map f ⋯ T, ⋯⟩, invFun := fun V ↦ { carrier := f ⁻¹' ↑↑V, isIrreducible' := ⋯, isClosed' := ⋯ },\n left_inv... | [
"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : β → α\nh : IsOpenEmbedding f\na b : IrreducibleCloseds β\nhle :\n { toFun := fun T ↦ ⟨map f ⋯ T, ⋯⟩, invFun := fun V ↦ { carrier := f ⁻¹' ↑↑V, isIrreducible' := ⋯, isClosed' := ⋯ },\n left_inv := ⋯, right... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.LocalAtTarget | {
"line": 271,
"column": 4
} | {
"line": 271,
"column": 64
} | {
"line": 271,
"column": 65
} | [
{
"pp": "X : Type u_6\nY : Type u_7\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nf : X → Y\nh : Continuous[inst✝², inst✝¹] f\nι : Type u_4\nU : ι → Opens Y\nhU : range f ⊆ ↑(iSup U)\nV : ι → Type u_5\ninst✝ : (i : ι) → TopologicalSpace (V i)\niV : (i : ι) → V i → X\nhiV : ∀ (i : ι), Continuous[ins... | [
"X : Type u_6\nY : Type u_7\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nf : X → Y\nh : Continuous[inst✝², inst✝¹] f\nι : Type u_4\nU : ι → Opens Y\nhU : range f ⊆ ↑(iSup U)\nV : ι → Type u_5\ninst✝ : (i : ι) → TopologicalSpace (V i)\niV : (i : ι) → V i → X\nhiV : ∀ (i : ι), Continuous[inst✝ i, inst✝²... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.NoetherianSpace | {
"line": 189,
"column": 2
} | {
"line": 190,
"column": 9
} | {
"line": 190,
"column": 10
} | [
{
"pp": "α : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : NoetherianSpace α\ns : Closeds α\n⊢ ∃ S, (∀ (k : ↥S), IsIrreducible ↑↑k) ∧ s = S.sup id",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"congrArg",
"Finset",
"IsI... | [
"α : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : NoetherianSpace α\ns : Closeds α\n⊢ ∃ S, (∀ a ∈ S, IsIrreducible ↑a) ∧ s = sSup ↑S"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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