module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.WittVector.Complete | {
"line": 124,
"column": 4
} | {
"line": 124,
"column": 15
} | {
"line": 124,
"column": 16
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : CommRing k\ninst✝¹ : CharP k p\ninst✝ : PerfectRing k p\nx✝ : 𝕎 k\nn : ℕ\nh : ∀ (n : ℕ), x✝ ≡ 0 [SMOD Ideal.span {↑p} ^ n]\nthis : ∀ m < n + 1, x✝.coeff m = 0\n⊢ x✝.coeff n = coeff 0 n",
"ppTerm": "?m.47",
"assigned": true,
"usedConsta... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : CommRing k\ninst✝¹ : CharP k p\ninst✝ : PerfectRing k p\nx✝ : 𝕎 k\nn : ℕ\nh : ∀ (n : ℕ), x✝ ≡ 0 [SMOD Ideal.span {↑p} ^ n]\nthis : ∀ m < n + 1, x✝.coeff m = 0\n⊢ x✝.coeff n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Perfectoid.FontaineTheta | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 13
} | {
"line": 125,
"column": 14
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\np : ℕ\ninst✝² : Fact (Nat.Prime p)\ninst✝¹ : Fact ¬IsUnit ↑p\ninst✝ : IsAdicComplete 𝔭 R\nn : ℕ\nx : R♭\n⊢ (ghostComponentModPPow n) ((teichmuller p) ((PreTilt.coeff n) x)) = (Ideal.Quotient.mk (𝔭 ^ (n + 1))) (untilt x)",
"ppTerm": "?m.55",
"assigned": false,
... | [
"R : Type u\ninst✝³ : CommRing R\np : ℕ\ninst✝² : Fact (Nat.Prime p)\ninst✝¹ : Fact ¬IsUnit ↑p\ninst✝ : IsAdicComplete 𝔭 R\nn : ℕ\nx : R♭\n⊢ (ghostComponentModPPow n) ((teichmuller p) ((PreTilt.coeff n) x)) = (Ideal.Quotient.mk (𝔭 ^ (n + 1))) (untilt x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.TeichmullerSeries | {
"line": 75,
"column": 2
} | {
"line": 75,
"column": 13
} | {
"line": 75,
"column": 14
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharP R p\nn : ℕ\nx : R\n⊢ ((teichmuller p) x * ↑p ^ n).coeff n = x ^ p ^ n",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharP R p\nn : ℕ\nx : R\n⊢ ((teichmuller p) x * ↑p ^ n).coeff n = x ^ p ^ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Perfectoid.BDeRham | {
"line": 67,
"column": 10
} | {
"line": 67,
"column": 21
} | {
"line": 67,
"column": 22
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\np : ℕ\ninst✝² : Fact (Nat.Prime p)\ninst✝¹ : Fact ¬IsUnit ↑p\ninst✝ : IsAdicComplete (span {↑p}) R\n⊢ IsUnit (((algebraMap R (Localization.Away ↑p)).comp (fontaineTheta R p)) ↑p)",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"R : Type u\ninst✝³ : CommRing R\np : ℕ\ninst✝² : Fact (Nat.Prime p)\ninst✝¹ : Fact ¬IsUnit ↑p\ninst✝ : IsAdicComplete (span {↑p}) R\n⊢ IsUnit ↑p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Eisenstein.Distinguished | {
"line": 47,
"column": 6
} | {
"line": 47,
"column": 39
} | {
"line": 47,
"column": 40
} | [
{
"pp": "case neg.inl\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nI : Ideal R\ndistinguish : f.IsDistinguishedAt I\ni : ℕ\nne : ¬i = f.natDegree\nlt : i < f.natDegree\n⊢ (map (Ideal.Quotient.mk I) f).coeff i = (X ^ f.natDegree).coeff i",
"ppTerm": "?neg.inl✝",
"assigned": true,
"usedConstants": [
... | [
"case neg.inl\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nI : Ideal R\ndistinguish : f.IsDistinguishedAt I\ni : ℕ\nne : ¬i = f.natDegree\nlt : i < f.natDegree\n⊢ f.coeff i ∈ I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Eisenstein.Distinguished | {
"line": 47,
"column": 6
} | {
"line": 47,
"column": 60
} | {
"line": 48,
"column": 4
} | [
{
"pp": "case neg.inl\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nI : Ideal R\ndistinguish : f.IsDistinguishedAt I\ni : ℕ\nne : ¬i = f.natDegree\nlt : i < f.natDegree\n⊢ (map (Ideal.Quotient.mk I) f).coeff i = (X ^ f.natDegree).coeff i",
"ppTerm": "?neg.inl✝",
"assigned": true,
"usedConstants": [
... | [] | simpa [ne, eq_zero_iff_mem] using (distinguish.mem lt) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.Polynomial.Eisenstein.Distinguished | {
"line": 47,
"column": 6
} | {
"line": 47,
"column": 60
} | {
"line": 48,
"column": 4
} | [
{
"pp": "case neg.inl\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nI : Ideal R\ndistinguish : f.IsDistinguishedAt I\ni : ℕ\nne : ¬i = f.natDegree\nlt : i < f.natDegree\n⊢ (map (Ideal.Quotient.mk I) f).coeff i = (X ^ f.natDegree).coeff i",
"ppTerm": "?neg.inl✝",
"assigned": true,
"usedConstants": [
... | [] | simpa [ne, eq_zero_iff_mem] using (distinguish.mem lt) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Eisenstein.Distinguished | {
"line": 47,
"column": 6
} | {
"line": 47,
"column": 60
} | {
"line": 48,
"column": 4
} | [
{
"pp": "case neg.inl\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nI : Ideal R\ndistinguish : f.IsDistinguishedAt I\ni : ℕ\nne : ¬i = f.natDegree\nlt : i < f.natDegree\n⊢ (map (Ideal.Quotient.mk I) f).coeff i = (X ^ f.natDegree).coeff i",
"ppTerm": "?neg.inl✝",
"assigned": true,
"usedConstants": [
... | [] | simpa [ne, eq_zero_iff_mem] using (distinguish.mem lt) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 130,
"column": 2
} | {
"line": 136,
"column": 82
} | {
"line": 138,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝ : Field F\np q : F[X]\nd : ℕ\n⊢ (p + q).hilbertPoly d = p.hilbertPoly d + q.hilbertPoly d",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.recAux",
"instHSMul",
"Semiring.toModule",
"congrArg",
"AddMonoid.toAddZ... | [] | delta hilbertPoly
induction d with
| zero => simp only [add_zero]
| succ d _ =>
simp only
rw [← sum_def _ fun _ r => r • _]
exact sum_add_index _ _ _ (fun _ => zero_smul ..) (fun _ _ _ => add_smul ..) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 130,
"column": 2
} | {
"line": 136,
"column": 82
} | {
"line": 138,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝ : Field F\np q : F[X]\nd : ℕ\n⊢ (p + q).hilbertPoly d = p.hilbertPoly d + q.hilbertPoly d",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.recAux",
"instHSMul",
"Semiring.toModule",
"congrArg",
"AddMonoid.toAddZ... | [] | delta hilbertPoly
induction d with
| zero => simp only [add_zero]
| succ d _ =>
simp only
rw [← sum_def _ fun _ r => r • _]
exact sum_add_index _ _ _ (fun _ => zero_smul ..) (fun _ _ _ => add_smul ..) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Opposites | {
"line": 112,
"column": 6
} | {
"line": 112,
"column": 29
} | {
"line": 112,
"column": 30
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nh : IsLeftCancelMulZero R[X]\na b c : R\neq : (fun x ↦ a + x) b = (fun x ↦ a + x) c\na✝ : Nontrivial R\ntrinomial : R → R[X] := fun r ↦ a • X ^ 2 + r • X + C a\nht : ∀ (r : R), (X + C 1) * trinomial r = a • X ^ 3 + (a + r) • X ^ 2 + (a + r) • X + C a\n⊢ b = c",
"pp... | [
"R : Type u_1\ninst✝ : Semiring R\nh : IsLeftCancelMulZero R[X]\na b c : R\neq : (fun x ↦ a + x) b = (fun x ↦ a + x) c\na✝ : Nontrivial R\ntrinomial : R → R[X] := fun r ↦ a • X ^ 2 + r • X + C a\nht : ∀ (r : R), (X + C 1) * trinomial r = a • X ^ 3 + (a + r) • X ^ 2 + (a + r) • X + C a\n⊢ b = c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Selmer | {
"line": 69,
"column": 98
} | {
"line": 80,
"column": 64
} | {
"line": 82,
"column": 0
} | [
{
"pp": "n : ℕ\nhn1 : n ≠ 1\n⊢ Irreducible (X ^ n - X - 1)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Iff.mpr",
"Polynomial.map_one",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Units.val",
"Eq.mpr",
"Pol... | [] | by
by_cases hn0 : n = 0
· rw [hn0, pow_zero, sub_sub, add_comm, ← sub_sub, sub_self, zero_sub]
exact Associated.irreducible ⟨-1, mul_neg_one X⟩ irreducible_X
have hp : (X ^ n - X - 1 : ℤ[X]) = trinomial 0 1 n (-1) (-1) 1 := by
simp only [trinomial, C_neg, C_1]; ring
have hn : 1 < n := Nat.one_lt_iff_ne_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.SmallDegreeVieta | {
"line": 38,
"column": 2
} | {
"line": 38,
"column": 77
} | {
"line": 39,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\na b c x1 x2 : R\nhroots : (C a * X ^ 2 + C b * X + C c).roots = {x1, x2}\np : R[X] := C a * X ^ 2 + C b * X + C c\nhp_natDegree : p.natDegree = 2\nhp_roots_card : p.roots.card = p.natDegree\n⊢ b = -a * (x1 + x2)",
"ppTerm": "?m.245",
"assig... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\na b c x1 x2 : R\nhroots : (C a * X ^ 2 + C b * X + C c).roots = {x1, x2}\np : R[X] := C a * X ^ 2 + C b * X + C c\nhp_natDegree : p.natDegree = 2\nhp_roots_card : p.roots.card = p.natDegree\n⊢ b = -(a * (x2 + x1))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.SmallDegreeVieta | {
"line": 50,
"column": 2
} | {
"line": 50,
"column": 77
} | {
"line": 51,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\na b c x1 x2 : R\nhroots : (C a * X ^ 2 + C b * X + C c).roots = {x1, x2}\np : R[X] := C a * X ^ 2 + C b * X + C c\nhp_natDegree : p.natDegree = 2\nhp_roots_card : p.roots.card = p.natDegree\n⊢ c = a * x1 * x2",
"ppTerm": "?m.243",
"assigned... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\na b c x1 x2 : R\nhroots : (C a * X ^ 2 + C b * X + C c).roots = {x1, x2}\np : R[X] := C a * X ^ 2 + C b * X + C c\nhp_natDegree : p.natDegree = 2\nhp_roots_card : p.roots.card = p.natDegree\n⊢ c = a * (x1 * x2)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.ShiftedLegendre | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 79
} | {
"line": 114,
"column": 2
} | [
{
"pp": "n : ℕ\nR : Type u_1\ninst✝ : Ring R\nx : R\n⊢ (aeval x) (shiftedLegendre n) = (-1) ^ n * (aeval (1 - x)) (shiftedLegendre n)",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Polynomial.instOne",
"Polynomial.instNeg",
"HMul.hMul",
"congrArg",
"CommSemi... | [
"n : ℕ\nR : Type u_1\ninst✝ : Ring R\nx : R\nthis : (aeval x) ((-1) ^ n * (shiftedLegendre n).comp (1 - X)) = (aeval x) (shiftedLegendre n)\n⊢ (aeval x) (shiftedLegendre n) = (-1) ^ n * (aeval (1 - x)) (shiftedLegendre n)"
] | have := congr(aeval x $(neg_one_pow_mul_shiftedLegendre_comp_one_sub_X_eq n)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Polynomial.ShiftedLegendre | {
"line": 114,
"column": 2
} | {
"line": 114,
"column": 26
} | {
"line": 114,
"column": 27
} | [
{
"pp": "n : ℕ\nR : Type u_1\ninst✝ : Ring R\nx : R\nthis : (aeval x) ((-1) ^ n * (shiftedLegendre n).comp (1 - X)) = (aeval x) (shiftedLegendre n)\n⊢ (aeval x) (shiftedLegendre n) = (-1) ^ n * (aeval (1 - x)) (shiftedLegendre n)",
"ppTerm": "?m.48",
"assigned": false,
"usedConstants": [],
"used... | [
"n : ℕ\nR : Type u_1\ninst✝ : Ring R\nx : R\nthis : (aeval x) ((-1) ^ n * (shiftedLegendre n).comp (1 - X)) = (aeval x) (shiftedLegendre n)\n⊢ (aeval x) (shiftedLegendre n) = (-1) ^ n * (aeval (1 - x)) (shiftedLegendre n)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.CoeffMulMem | {
"line": 58,
"column": 2
} | {
"line": 58,
"column": 13
} | {
"line": 58,
"column": 14
} | [
{
"pp": "A : Type u_1\ninst✝ : Semiring A\nI : Ideal A\nf g : A⟦X⟧\nn : ℕ\nhg : ∀ i ≤ n, (coeff i) g ∈ I\n⊢ ∀ i ≤ n, (coeff i) (f * g) ∈ I",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"A : Type u_1\ninst✝ : Semiring A\nI : Ideal A\nf g : A⟦X⟧\nn : ℕ\nhg : ∀ i ≤ n, (coeff i) g ∈ I\n⊢ ∀ i ≤ n, (coeff i) (f * g) ∈ I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.CoeffMulMem | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 13
} | {
"line": 62,
"column": 14
} | [
{
"pp": "A : Type u_1\ninst✝ : Semiring A\nI : Ideal A\nf g : A⟦X⟧\nhg : ∀ (i : ℕ), (coeff i) g ∈ I\n⊢ ∀ (i : ℕ), (coeff i) (f * g) ∈ I",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"A : Type u_1\ninst✝ : Semiring A\nI : Ideal A\nf g : A⟦X⟧\nhg : ∀ (i : ℕ), (coeff i) g ∈ I\n⊢ ∀ (i : ℕ), (coeff i) (f * g) ∈ I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.CoeffMulMem | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 45
} | {
"line": 69,
"column": 4
} | [
{
"pp": "A : Type u_1\ninst✝¹ : Semiring A\nI : Ideal A\nf g : A⟦X⟧\nn : ℕ\ninst✝ : I.IsTwoSided\nhf : ∀ i ≤ n, (coeff i) f ∈ I\n⊢ ∀ i ≤ n, (coeff i) (f * g) ∈ I",
"ppTerm": "?m.27",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"A : Type u_1\ninst✝¹ : Semiring A\nI : Ideal A\nf g : A⟦X⟧\nn : ℕ\ninst✝ : I.IsTwoSided\nhf : ∀ i ≤ n, (coeff i) f ∈ I\n⊢ ∀ i ≤ n, (coeff i) (f * g) ∈ I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.CoeffMulMem | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 45
} | {
"line": 74,
"column": 4
} | [
{
"pp": "A : Type u_1\ninst✝¹ : Semiring A\nI : Ideal A\nf g : A⟦X⟧\ninst✝ : I.IsTwoSided\nhf : ∀ (i : ℕ), (coeff i) f ∈ I\n⊢ ∀ (i : ℕ), (coeff i) (f * g) ∈ I",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"A : Type u_1\ninst✝¹ : Semiring A\nI : Ideal A\nf g : A⟦X⟧\ninst✝ : I.IsTwoSided\nhf : ∀ (i : ℕ), (coeff i) f ∈ I\n⊢ ∀ (i : ℕ), (coeff i) (f * g) ∈ I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.CoeffMulMem | {
"line": 72,
"column": 60
} | {
"line": 74,
"column": 84
} | {
"line": 76,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝¹ : Semiring A\nI : Ideal A\nf g : A⟦X⟧\ninst✝ : I.IsTwoSided\nhf : ∀ (i : ℕ), (coeff i) f ∈ I\n⊢ ∀ (i : ℕ), (coeff i) (f * g) ∈ I",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Ideal.one_eq_top",
"Submodule.mem_top._simp_1",
"Semiring.toModu... | [] | by
simpa only [Ideal.IsTwoSided.mul_one] using
coeff_mul_mem_ideal_mul_ideal_of_coeff_mem_ideal' (J := 1) (g := g) hf (by simp) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.PowerSeries.Schroder | {
"line": 73,
"column": 4
} | {
"line": 73,
"column": 13
} | {
"line": 74,
"column": 4
} | [
{
"pp": "n : ℕ\nhn : 0 < n\n⊢ ∀ x ∈ range n,\n (coeff x) (X * largeSchroderSeries) * (n - x).largeSchroder =\n if 0 < x then (x - 1).largeSchroder * (n - x).largeSchroder else 0",
"ppTerm": "?m.169",
"assigned": true,
"usedConstants": [
"Finset",
"Membership.mem",
"Finset.r... | [
"n : ℕ\nhn : 0 < n\nx : ℕ\na : x ∈ range n\n⊢ (coeff x) (X * largeSchroderSeries) * (n - x).largeSchroder =\n if 0 < x then (x - 1).largeSchroder * (n - x).largeSchroder else 0"
] | intro x a | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.TensorProduct.DirectLimitFG | {
"line": 221,
"column": 2
} | {
"line": 221,
"column": 29
} | {
"line": 221,
"column": 30
} | [
{
"pp": "R : Type u\nM : Type u_1\nN : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nP : Submodule R M\nhP : P.FG\nt : ↥P ⊗[R] N\nh : (rTensor N P.subtype) t = (rTensor N P.subtype) 0\n⊢ ∃ Q, ∃ (hPQ : P ≤ Q), Q.FG ∧ (rTensor N (in... | [
"R : Type u\nM : Type u_1\nN : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nP : Submodule R M\nhP : P.FG\nt : ↥P ⊗[R] N\nh : (rTensor N P.subtype) t = (rTensor N P.subtype) 0\n⊢ ∃ Q, ∃ (hPQ : P ≤ Q), Q.FG ∧ (rTensor N (inclusion hPQ)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PolynomialLaw.Basic | {
"line": 244,
"column": 2
} | {
"line": 244,
"column": 49
} | {
"line": 246,
"column": 0
} | [
{
"pp": "case e'_3\nR : Type u\ninst✝⁶ : CommSemiring R\nM : Type u_1\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nN : Type u_2\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\nf : M →ₚₗ[R] N\nS : Type u\ninst✝¹ : CommSemiring S\ninst✝ : Algebra R S\nx : M\n⊢ 1 ⊗ₜ[R] x = (rTensor M (Algebra.algHom R R S).toLi... | [] | · rw [rTensor_tmul, toLinearMap_apply, map_one] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Radical.NatInt | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 27
} | {
"line": 62,
"column": 28
} | [
{
"pp": "n : ℕ\n⊢ radical n ≤ 1 ↔ n ≤ 1",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\n⊢ radical n ≤ 1 ↔ n ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Radical.NatInt | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 36
} | {
"line": 72,
"column": 37
} | [
{
"pp": "n : ℕ\n⊢ n < radical n ↔ n = 0",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\n⊢ n < radical n ↔ n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Radical.NatInt | {
"line": 146,
"column": 2
} | {
"line": 146,
"column": 27
} | {
"line": 146,
"column": 28
} | [
{
"pp": "z : ℤ\n⊢ radical z ≤ 1 ↔ z.natAbs ≤ 1",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"z : ℤ\n⊢ radical z ≤ 1 ↔ z.natAbs ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Regular.Flat | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 76
} | {
"line": 73,
"column": 77
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹³ : CommRing R\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : Module R M\ninst✝⁸ : AddCommGroup N\ninst✝⁷ : Module R N\ninst✝⁶ : Module S N\ninst✝⁵ : IsScalarTower R S N\np : Ideal R\ninst✝⁴ : p.IsPrime\nins... | [
"R : Type u_1\nS : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹³ : CommRing R\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : Module R M\ninst✝⁸ : AddCommGroup N\ninst✝⁷ : Module R N\ninst✝⁶ : Module S N\ninst✝⁵ : IsScalarTower R S N\np : Ideal R\ninst✝⁴ : p.IsPrime\ninst✝³ : IsLoca... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 206,
"column": 4
} | {
"line": 206,
"column": 35
} | {
"line": 206,
"column": 36
} | [
{
"pp": "case pos\nA : Type u_1\ninst✝ : CommRing A\ng : A⟦X⟧\nI : Ideal A\ni : ℕ\nh : i < ((map (Ideal.Quotient.mk I)) g).order.toNat\n⊢ (coeff i) g ∈ I",
"ppTerm": "?pos✝",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case pos\nA : Type u_1\ninst✝ : CommRing A\ng : A⟦X⟧\nI : Ideal A\ni : ℕ\nh : i < ((map (Ideal.Quotient.mk I)) g).order.toNat\n⊢ (coeff i) g ∈ I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Regular.Free | {
"line": 65,
"column": 4
} | {
"line": 65,
"column": 20
} | {
"line": 65,
"column": 21
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : FinitePresentation R M\nx : R\nmem : x ∈ ⊥.jacobson\nreg : IsSMulRegular M x\nfree : Free (R ⧸ Ideal.span {x}) (QuotSMulTop x M)\nthis✝¹ : Module.Finite (R ⧸ Ideal.span {x}) (QuotSMulTop x M)\nI : Typ... | [
"R : Type u_1\ninst✝³ : CommRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : FinitePresentation R M\nx : R\nmem : x ∈ ⊥.jacobson\nreg : IsSMulRegular M x\nfree : Free (R ⧸ Ideal.span {x}) (QuotSMulTop x M)\nthis✝¹ : Module.Finite (R ⧸ Ideal.span {x}) (QuotSMulTop x M)\nI : Type u_2 := Fre... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 291,
"column": 2
} | {
"line": 291,
"column": 41
} | {
"line": 291,
"column": 42
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\ng : A⟦X⟧\nI : Ideal A\nH : g.IsWeierstrassDivisorAt I\nf : A⟦X⟧\ninst✝ : IsPrecomplete I A\nk i : ℕ\n⊢ (coeff i) (H.div f - H.seq f k) ∈ I ^ k",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"Semiring.toMo... | [
"A : Type u_1\ninst✝¹ : CommRing A\ng : A⟦X⟧\nI : Ideal A\nH : g.IsWeierstrassDivisorAt I\nf : A⟦X⟧\ninst✝ : IsPrecomplete I A\nk i : ℕ\n⊢ ↑(H.divCoeff f i) - (coeff i) (H.seq f k) ∈ I ^ k"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RegularLocalRing.Defs | {
"line": 85,
"column": 4
} | {
"line": 86,
"column": 57
} | {
"line": 87,
"column": 8
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsLocalRing R\ninst✝¹ : IsDomain R\ninst✝ : IsPrincipalIdealRing R\nisf : ¬IsField R\nx : R\nhx : maximalIdeal R = R ∙ x\n⊢ ↑(Submodule.spanFinrank (maximalIdeal R)) ≤ ringKrullDim R",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants"... | [
"case neg\nR : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsLocalRing R\ninst✝¹ : IsDomain R\ninst✝ : IsPrincipalIdealRing R\nisf : ¬IsField R\nx : R\nhx : maximalIdeal R = R ∙ x\n⊢ (R ∙ x).spanFinrank ≤ {x}.ncard"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RegularLocalRing.Defs | {
"line": 108,
"column": 2
} | {
"line": 112,
"column": 95
} | {
"line": 114,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nR' : Type u_2\ninst✝¹ : CommRing R'\ne : R ≃+* R'\ninst✝ : IsRegularRing R\n⊢ IsRegularRing R'",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"OreLocalization.instAlgebra",
"congrArg",
"CommSemiring.toSemiring",
... | [] | have := isNoetherianRing_of_ringEquiv R e
rw [isRegularRing_iff]
intro p hp
exact IsRegularLocalRing.of_ringEquiv <| IsLocalization.ringEquivOfRingEquiv
(Localization.AtPrime (p.comap e)) (Localization.AtPrime p) e (e.map_primeCompl_comap_eq p) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.RegularLocalRing.Defs | {
"line": 108,
"column": 2
} | {
"line": 112,
"column": 95
} | {
"line": 114,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nR' : Type u_2\ninst✝¹ : CommRing R'\ne : R ≃+* R'\ninst✝ : IsRegularRing R\n⊢ IsRegularRing R'",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"OreLocalization.instAlgebra",
"congrArg",
"CommSemiring.toSemiring",
... | [] | have := isNoetherianRing_of_ringEquiv R e
rw [isRegularRing_iff]
intro p hp
exact IsRegularLocalRing.of_ringEquiv <| IsLocalization.ringEquivOfRingEquiv
(Localization.AtPrime (p.comap e)) (Localization.AtPrime p) e (e.map_primeCompl_comap_eq p) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.RegularLocalRing.Polynomial | {
"line": 40,
"column": 25
} | {
"line": 40,
"column": 47
} | {
"line": 40,
"column": 48
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsRegularLocalRing R\np : Ideal R[X]\ninst✝ : p.IsPrime\nmax : comap C p = maximalIdeal R\nq : Ideal R[X] := Ideal.map C (maximalIdeal R)\n⊢ q ≤ p",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
... | [
"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsRegularLocalRing R\np : Ideal R[X]\ninst✝ : p.IsPrime\nmax : comap C p = maximalIdeal R\nq : Ideal R[X] := Ideal.map C (maximalIdeal R)\n⊢ Ideal.map C (comap C p) ≤ p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RegularLocalRing.Polynomial | {
"line": 60,
"column": 39
} | {
"line": 60,
"column": 56
} | {
"line": 60,
"column": 57
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsRegularLocalRing R\np : Ideal R[X]\ninst✝ : p.IsPrime\nmax : comap C p = maximalIdeal R\nq : Ideal R[X] := Ideal.map C (maximalIdeal R)\nqle : q ≤ p\nreg : ↑(Submodule.spanFinrank (maximalIdeal R)) = ringKrullDim R\nfg' : (maximalIdeal R).FG\nfg : (Submodul... | [
"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsRegularLocalRing R\np : Ideal R[X]\ninst✝ : p.IsPrime\nmax : comap C p = maximalIdeal R\nq : Ideal R[X] := Ideal.map C (maximalIdeal R)\nqle : q ≤ p\nreg : ↑(Submodule.spanFinrank (maximalIdeal R)) = ringKrullDim R\nfg' : (maximalIdeal R).FG\nfg : (Submodule.generators... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Regular.ProjectiveDimension | {
"line": 80,
"column": 6
} | {
"line": 80,
"column": 24
} | {
"line": 80,
"column": 25
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsNoetherianRing R\nn✝ : ℕ\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\nw✝⁴ : Type v\nw✝³ : AddCommGroup w✝⁴\nw✝² : Module R w✝⁴\nw✝¹ : Free R w✝⁴\nw✝ : Module.Finite R w✝⁴\nf : w✝⁴ →ₗ[R] ↑M\nsurjf : Function.Surjective ⇑f\nS : ShortCo... | [
"R : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsNoetherianRing R\nn✝ : ℕ\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\nw✝⁴ : Type v\nw✝³ : AddCommGroup w✝⁴\nw✝² : Module R w✝⁴\nw✝¹ : Free R w✝⁴\nw✝ : Module.Finite R w✝⁴\nf : w✝⁴ →ₗ[R] ↑M\nsurjf : Function.Surjective ⇑f\nS : ShortComplex (Modul... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 388,
"column": 2
} | {
"line": 388,
"column": 13
} | {
"line": 388,
"column": 14
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\ng : A⟦X⟧\nI : Ideal A\nH : g.IsWeierstrassDivisorAt I\ninst✝ : IsAdicComplete I A\n⊢ H.div 0 = 0",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"A : Type u_1\ninst✝¹ : CommRing A\ng : A⟦X⟧\nI : Ideal A\nH : g.IsWeierstrassDivisorAt I\ninst✝ : IsAdicComplete I A\n⊢ H.div 0 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 406,
"column": 2
} | {
"line": 406,
"column": 13
} | {
"line": 406,
"column": 14
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\ng : A⟦X⟧\nI : Ideal A\nH : g.IsWeierstrassDivisorAt I\ninst✝ : IsAdicComplete I A\n⊢ H.mod 0 = 0",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"A : Type u_1\ninst✝¹ : CommRing A\ng : A⟦X⟧\nI : Ideal A\nH : g.IsWeierstrassDivisorAt I\ninst✝ : IsAdicComplete I A\n⊢ H.mod 0 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 434,
"column": 71
} | {
"line": 434,
"column": 82
} | {
"line": 434,
"column": 83
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\ng : A⟦X⟧\nI : Ideal A\nH : g.IsWeierstrassDivisorAt I\ninst✝ : IsAdicComplete I A\nr : A[X]\nhr : r.degree < ↑((map (Ideal.Quotient.mk I)) g).order.toNat\nh1 : (H.mod ↑r).degree < ↑((map (Ideal.Quotient.mk I)) g).order.toNat\nh2 : ↑r = g * H.div ↑r + ↑(H.mod ↑r)\n⊢ g ... | [
"A : Type u_1\ninst✝¹ : CommRing A\ng : A⟦X⟧\nI : Ideal A\nH : g.IsWeierstrassDivisorAt I\ninst✝ : IsAdicComplete I A\nr : A[X]\nhr : r.degree < ↑((map (Ideal.Quotient.mk I)) g).order.toNat\nh1 : (H.mod ↑r).degree < ↑((map (Ideal.Quotient.mk I)) g).order.toNat\nh2 : ↑r = g * H.div ↑r + ↑(H.mod ↑r)\n⊢ g * H.div ↑r +... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 439,
"column": 71
} | {
"line": 439,
"column": 82
} | {
"line": 439,
"column": 83
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\ng : A⟦X⟧\nI : Ideal A\nH : g.IsWeierstrassDivisorAt I\ninst✝ : IsAdicComplete I A\nr : A[X]\nhr : r.degree < ↑((map (Ideal.Quotient.mk I)) g).order.toNat\nh1 : (H.mod ↑r).degree < ↑((map (Ideal.Quotient.mk I)) g).order.toNat\nh2 : ↑r = g * H.div ↑r + ↑(H.mod ↑r)\n⊢ g ... | [
"A : Type u_1\ninst✝¹ : CommRing A\ng : A⟦X⟧\nI : Ideal A\nH : g.IsWeierstrassDivisorAt I\ninst✝ : IsAdicComplete I A\nr : A[X]\nhr : r.degree < ↑((map (Ideal.Quotient.mk I)) g).order.toNat\nh1 : (H.mod ↑r).degree < ↑((map (Ideal.Quotient.mk I)) g).order.toNat\nh2 : ↑r = g * H.div ↑r + ↑(H.mod ↑r)\n⊢ g * H.div ↑r +... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Regular.ProjectiveDimension | {
"line": 134,
"column": 6
} | {
"line": 134,
"column": 24
} | {
"line": 134,
"column": 25
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : Small.{v, u} R\ninst✝² : IsLocalRing R\ninst✝¹ : IsNoetherianRing R\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\nx : R\nreg : IsSMulRegular (↑M) x\nmem : x ∈ maximalIdeal R\nsub : Subsingleton ↑M ↔ Subsingleton (QuotSMulTop x ↑M)\nn✝ n : ℕ\nS : ShortComplex (M... | [
"R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : Small.{v, u} R\ninst✝² : IsLocalRing R\ninst✝¹ : IsNoetherianRing R\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\nx : R\nreg : IsSMulRegular (↑M) x\nmem : x ∈ maximalIdeal R\nsub : Subsingleton ↑M ↔ Subsingleton (QuotSMulTop x ↑M)\nn✝ n : ℕ\nS : ShortComplex (ModuleCat R) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Regular.ProjectiveDimension | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 83
} | {
"line": 138,
"column": 84
} | [
{
"pp": "case bot\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : Small.{v, u} R\ninst✝² : IsLocalRing R\ninst✝¹ : IsNoetherianRing R\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\nx : R\nreg : IsSMulRegular (↑M) x\nmem : x ∈ maximalIdeal R\nsub : Subsingleton ↑M ↔ Subsingleton (QuotSMulTop x ↑M)\naux : ∀ (n : ℕ), pr... | [
"case bot\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : Small.{v, u} R\ninst✝² : IsLocalRing R\ninst✝¹ : IsNoetherianRing R\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\nx : R\nreg : IsSMulRegular (↑M) x\nmem : x ∈ maximalIdeal R\nsub : Subsingleton ↑M ↔ Subsingleton (QuotSMulTop x ↑M)\naux : ∀ (n : ℕ), projectiveDime... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Regular.ProjectiveDimension | {
"line": 142,
"column": 15
} | {
"line": 142,
"column": 26
} | {
"line": 142,
"column": 27
} | [
{
"pp": "case coe.coe\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : Small.{v, u} R\ninst✝² : IsLocalRing R\ninst✝¹ : IsNoetherianRing R\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\nx : R\nreg : IsSMulRegular (↑M) x\nmem : x ∈ maximalIdeal R\nsub : Subsingleton ↑M ↔ Subsingleton (QuotSMulTop x ↑M)\naux : ∀ (n : ℕ)... | [
"case coe.coe\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : Small.{v, u} R\ninst✝² : IsLocalRing R\ninst✝¹ : IsNoetherianRing R\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\nx : R\nreg : IsSMulRegular (↑M) x\nmem : x ∈ maximalIdeal R\nsub : Subsingleton ↑M ↔ Subsingleton (QuotSMulTop x ↑M)\naux : ∀ (n : ℕ), projective... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Regular.ProjectiveDimension | {
"line": 142,
"column": 4
} | {
"line": 142,
"column": 14
} | {
"line": 142,
"column": 15
} | [
{
"pp": "case coe.coe\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : Small.{v, u} R\ninst✝² : IsLocalRing R\ninst✝¹ : IsNoetherianRing R\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\nx : R\nreg : IsSMulRegular (↑M) x\nmem : x ∈ maximalIdeal R\nsub : Subsingleton ↑M ↔ Subsingleton (QuotSMulTop x ↑M)\naux : ∀ (n : ℕ)... | [] | | coe n => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.Regular.ProjectiveDimension | {
"line": 153,
"column": 4
} | {
"line": 153,
"column": 15
} | {
"line": 153,
"column": 16
} | [
{
"pp": "case zero\nR : Type u\ninst✝⁵ : CommRing R\ninst✝⁴ : Small.{v, u} R\ninst✝³ : IsLocalRing R\ninst✝² : IsNoetherianRing R\nM : ModuleCat R\ninst✝¹ : Nontrivial ↑M\ninst✝ : Module.Finite R ↑M\nrs : List R\nreg : IsWeaklyRegular (↑M) rs\nmem : ∀ r ∈ rs, r ∈ maximalIdeal R\nlen : rs.length = 0\n⊢ projectiv... | [
"case zero\nR : Type u\ninst✝⁵ : CommRing R\ninst✝⁴ : Small.{v, u} R\ninst✝³ : IsLocalRing R\ninst✝² : IsNoetherianRing R\nM : ModuleCat R\ninst✝¹ : Nontrivial ↑M\ninst✝ : Module.Finite R ↑M\nrs : List R\nreg : IsWeaklyRegular (↑M) rs\nmem : ∀ r ∈ rs, r ∈ maximalIdeal R\nlen : rs.length = 0\n⊢ projectiveDimension (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Regular.ProjectiveDimension | {
"line": 162,
"column": 6
} | {
"line": 165,
"column": 68
} | {
"line": 167,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝⁵ : CommRing R\ninst✝⁴ : Small.{v, u} R\ninst✝³ : IsLocalRing R\ninst✝² : IsNoetherianRing R\nn : ℕ\nhn :\n ∀ (M : ModuleCat R) [Nontrivial ↑M] [Module.Finite R ↑M] (rs : List R),\n IsWeaklyRegular (↑M) rs →\n (∀ r ∈ rs, r ∈ maximalIdeal R) →\n rs.length = n → projectiv... | [] | rw [Nat.cast_add, Nat.cast_one, projectiveDimension_eq_of_iso
(Submodule.quotOfListConsSMulTopEquivQuotSMulTopInner M x rs').toModuleIso, add_comm _ 1,
← add_assoc, ← projectiveDimension_quotSMulTop_eq_succ_of_isSMulRegular M x reg.1 mem.1,
← hn (ModuleCat.of R (QuotSMulTop x M)) rs' reg.2 mem.2... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Regular.ProjectiveDimension | {
"line": 171,
"column": 4
} | {
"line": 171,
"column": 48
} | {
"line": 172,
"column": 4
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsLocalRing R\ninst✝ : IsNoetherianRing R\nrs : List R\nreg : RingTheory.Sequence.IsRegular R rs\nx : R\nhx : x ∈ rs\n⊢ x ∈ maximalIdeal R",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Submodule",
"i... | [
"R : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsLocalRing R\ninst✝ : IsNoetherianRing R\nrs : List R\nreg : RingTheory.Sequence.IsRegular R rs\nx : R\nhx : x ∈ rs\n⊢ x ∈ Ideal.ofList rs • ⊤"
] | apply IsLocalRing.le_maximalIdeal reg.2.symm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Regular.ProjectiveDimension | {
"line": 172,
"column": 4
} | {
"line": 172,
"column": 15
} | {
"line": 172,
"column": 16
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsLocalRing R\ninst✝ : IsNoetherianRing R\nrs : List R\nreg : RingTheory.Sequence.IsRegular R rs\nx : R\nhx : x ∈ rs\n⊢ x ∈ Ideal.ofList rs • ⊤",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"R : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsLocalRing R\ninst✝ : IsNoetherianRing R\nrs : List R\nreg : RingTheory.Sequence.IsRegular R rs\nx : R\nhx : x ∈ rs\n⊢ x ∈ Ideal.ofList rs"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 709,
"column": 4
} | {
"line": 709,
"column": 19
} | {
"line": 711,
"column": 0
} | [
{
"pp": "case refine_2\nA : Type u_1\ninst✝ : CommRing A\ng : A⟦X⟧\nf : A[X]\nh : A⟦X⟧\nI : Ideal A\nH : g.IsWeierstrassFactorizationAt f h I\na : A\nha : IsUnit a\n⊢ a • g = ↑f * a • h",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Algebra.mul_smul_comm",
"instHSMul",
... | [] | simp [H.eq_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 709,
"column": 4
} | {
"line": 709,
"column": 19
} | {
"line": 711,
"column": 0
} | [
{
"pp": "case refine_2\nA : Type u_1\ninst✝ : CommRing A\ng : A⟦X⟧\nf : A[X]\nh : A⟦X⟧\nI : Ideal A\nH : g.IsWeierstrassFactorizationAt f h I\na : A\nha : IsUnit a\n⊢ a • g = ↑f * a • h",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Algebra.mul_smul_comm",
"instHSMul",
... | [] | simp [H.eq_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 709,
"column": 4
} | {
"line": 709,
"column": 19
} | {
"line": 711,
"column": 0
} | [
{
"pp": "case refine_2\nA : Type u_1\ninst✝ : CommRing A\ng : A⟦X⟧\nf : A[X]\nh : A⟦X⟧\nI : Ideal A\nH : g.IsWeierstrassFactorizationAt f h I\na : A\nha : IsUnit a\n⊢ a • g = ↑f * a • h",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Algebra.mul_smul_comm",
"instHSMul",
... | [] | simp [H.eq_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.SimpleModule.Isotypic | {
"line": 126,
"column": 79
} | {
"line": 130,
"column": 50
} | {
"line": 132,
"column": 0
} | [
{
"pp": "R : Type u_2\nM : Type u\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nN : Submodule R M\n⊢ IsIsotypic R ↥N ↔ ∀ m ≤ N, ∀ [IsSimpleModule R ↥m], IsIsotypicOfType R ↥N ↥m",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"Submo... | [] | by
rw [Subtype.forall', ← (Submodule.MapSubtype.orderIso N).forall_congr_right]
have e := Submodule.equivMapOfInjective _ N.subtype_injective
simp_rw [Submodule.MapSubtype.orderIso, Equiv.coe_fn_mk, ← (e _).isSimpleModule_iff,
← (e _).isIsotypicOfType_iff_type, IsIsotypic] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 801,
"column": 12
} | {
"line": 801,
"column": 23
} | {
"line": 801,
"column": 24
} | [
{
"pp": "A : Type u_1\ninst✝² : CommRing A\ninst✝¹ : IsLocalRing A\ninst✝ : IsHausdorff (IsLocalRing.maximalIdeal A) A\ng : A⟦X⟧\nf f' : A[X]\nh h' : A⟦X⟧\nH : g.IsWeierstrassFactorization f h\nH2 : g.IsWeierstrassFactorization f' h'\nh1 : ↑⋯.unit = ↑⋯.unit\nh2 :\n Polynomial.X ^ ((map (IsLocalRing.residue A))... | [
"A : Type u_1\ninst✝² : CommRing A\ninst✝¹ : IsLocalRing A\ninst✝ : IsHausdorff (IsLocalRing.maximalIdeal A) A\ng : A⟦X⟧\nf f' : A[X]\nh h' : A⟦X⟧\nH : g.IsWeierstrassFactorization f h\nH2 : g.IsWeierstrassFactorization f' h'\nh1 : ↑⋯.unit = ↑⋯.unit\nh2 :\n Polynomial.X ^ ((map (IsLocalRing.residue A)) g).order.to... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 801,
"column": 12
} | {
"line": 801,
"column": 26
} | {
"line": 801,
"column": 26
} | [
{
"pp": "A : Type u_1\ninst✝² : CommRing A\ninst✝¹ : IsLocalRing A\ninst✝ : IsHausdorff (IsLocalRing.maximalIdeal A) A\ng : A⟦X⟧\nf f' : A[X]\nh h' : A⟦X⟧\nH : g.IsWeierstrassFactorization f h\nH2 : g.IsWeierstrassFactorization f' h'\nh1 : ↑⋯.unit = ↑⋯.unit\nh2 :\n Polynomial.X ^ ((map (IsLocalRing.residue A))... | [] | simpa using h2 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 801,
"column": 12
} | {
"line": 801,
"column": 26
} | {
"line": 801,
"column": 26
} | [
{
"pp": "A : Type u_1\ninst✝² : CommRing A\ninst✝¹ : IsLocalRing A\ninst✝ : IsHausdorff (IsLocalRing.maximalIdeal A) A\ng : A⟦X⟧\nf f' : A[X]\nh h' : A⟦X⟧\nH : g.IsWeierstrassFactorization f h\nH2 : g.IsWeierstrassFactorization f' h'\nh1 : ↑⋯.unit = ↑⋯.unit\nh2 :\n Polynomial.X ^ ((map (IsLocalRing.residue A))... | [] | simpa using h2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 801,
"column": 12
} | {
"line": 801,
"column": 26
} | {
"line": 801,
"column": 26
} | [
{
"pp": "A : Type u_1\ninst✝² : CommRing A\ninst✝¹ : IsLocalRing A\ninst✝ : IsHausdorff (IsLocalRing.maximalIdeal A) A\ng : A⟦X⟧\nf f' : A[X]\nh h' : A⟦X⟧\nH : g.IsWeierstrassFactorization f h\nH2 : g.IsWeierstrassFactorization f' h'\nh1 : ↑⋯.unit = ↑⋯.unit\nh2 :\n Polynomial.X ^ ((map (IsLocalRing.residue A))... | [] | simpa using h2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 890,
"column": 29
} | {
"line": 890,
"column": 40
} | {
"line": 890,
"column": 41
} | [
{
"pp": "A : Type u_1\ninst✝² : CommRing A\ninst✝¹ : IsLocalRing A\ninst✝ : IsAdicComplete (IsLocalRing.maximalIdeal A) A\na : A\ng : A⟦X⟧\nhg : (map (IsLocalRing.residue A)) (a • g) ≠ 0\nH : g.IsWeierstrassFactorization (g.weierstrassDistinguished ⋯) (g.weierstrassUnit ⋯)\nH' : (a • g).IsWeierstrassFactorizati... | [
"A : Type u_1\ninst✝² : CommRing A\ninst✝¹ : IsLocalRing A\ninst✝ : IsAdicComplete (IsLocalRing.maximalIdeal A) A\na : A\ng : A⟦X⟧\nhg : (map (IsLocalRing.residue A)) (a • g) ≠ 0\nH : g.IsWeierstrassFactorization (g.weierstrassDistinguished ⋯) (g.weierstrassUnit ⋯)\nH' : (a • g).IsWeierstrassFactorization ((a • g).... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 900,
"column": 29
} | {
"line": 900,
"column": 40
} | {
"line": 900,
"column": 41
} | [
{
"pp": "A : Type u_1\ninst✝² : CommRing A\ninst✝¹ : IsLocalRing A\ninst✝ : IsAdicComplete (IsLocalRing.maximalIdeal A) A\na : A\ng : A⟦X⟧\nhg : (map (IsLocalRing.residue A)) (a • g) ≠ 0\nH : g.IsWeierstrassFactorization (g.weierstrassDistinguished ⋯) (g.weierstrassUnit ⋯)\nH' : (a • g).IsWeierstrassFactorizati... | [
"A : Type u_1\ninst✝² : CommRing A\ninst✝¹ : IsLocalRing A\ninst✝ : IsAdicComplete (IsLocalRing.maximalIdeal A) A\na : A\ng : A⟦X⟧\nhg : (map (IsLocalRing.residue A)) (a • g) ≠ 0\nH : g.IsWeierstrassFactorization (g.weierstrassDistinguished ⋯) (g.weierstrassUnit ⋯)\nH' : (a • g).IsWeierstrassFactorization ((a • g).... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.SimpleRing.DivisionRing | {
"line": 41,
"column": 64
} | {
"line": 43,
"column": 81
} | {
"line": 45,
"column": 0
} | [
{
"pp": "S : Type u_2\ninst✝³ : DivisionRing S\nN : Type u_3\ninst✝² : AddCommGroup N\ninst✝¹ : Module S N\ninst✝ : IsSimpleModule S N\n⊢ Nonempty (N ≃ₗ[S] S)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Submodule",
"Submodule.Quotient.addCommMonoid",
"Semiring.toModu... | [] | by
obtain ⟨I, hI, ⟨e⟩⟩ := isSimpleModule_iff_quot_maximal.mp ‹_›
exact ⟨e ≪≫ₗ I.quotEquivOfEqBot ((eq_bot_or_eq_top I).resolve_right hI.ne_top)⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.SimpleRing.DivisionRing | {
"line": 51,
"column": 24
} | {
"line": 51,
"column": 74
} | {
"line": 51,
"column": 75
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx✝ :\n Nontrivial M ∧\n ∀ (N : Type v) [inst : AddCommGroup N] [inst_1 : Module R N] (f : M →ₗ[R] N), f = 0 ∨ Function.Injective ⇑f\nhM1 : Nontrivial M\nhM2 : ∀ (N : Type v) [inst : AddCommGroup N] [inst_1 : Modul... | [
"R : Type u\nM : Type v\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx✝ :\n Nontrivial M ∧\n ∀ (N : Type v) [inst : AddCommGroup N] [inst_1 : Module R N] (f : M →ₗ[R] N), f = 0 ∨ Function.Injective ⇑f\nhM1 : Nontrivial M\nhM2 : ∀ (N : Type v) [inst : AddCommGroup N] [inst_1 : Module R N] (f : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RegularLocalRing.Polynomial | {
"line": 99,
"column": 4
} | {
"line": 102,
"column": 89
} | {
"line": 103,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsRegularRing R\np : Ideal R[X]\nhp : p.IsPrime\nq : Ideal R := comap C p\nS : Type u_1 := (Localization.AtPrime q)[X]\npc : Submonoid R[X] := Submonoid.map (↑C) q.primeCompl\nthis✝³ : Algebra R[X] S := algebra R (Localization.AtPrime q)\nthis✝² : IsLocalizati... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsRegularRing R\np : Ideal R[X]\nhp : p.IsPrime\nq : Ideal R := comap C p\nS : Type u_1 := (Localization.AtPrime q)[X]\npc : Submonoid R[X] := Submonoid.map (↑C) q.primeCompl\nthis✝³ : Algebra R[X] S := algebra R (Localization.AtPrime q)\nthis✝² : IsLocalization pc S\npS ... | rw [← Polynomial.algebraMap_eq (R := Localization.AtPrime q),
← IsScalarTower.algebraMap_eq R (Localization.AtPrime q) (Localization.AtPrime q)[X],
IsScalarTower.algebraMap_eq R R[X] (Localization.AtPrime q)[X], ← comap_comap,
← Ideal.under_def R[X], IsLocalization.under_map_of_isPrime_disjoint pc _ ‹... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.SimpleModule.Isotypic | {
"line": 315,
"column": 4
} | {
"line": 315,
"column": 47
} | {
"line": 316,
"column": 2
} | [
{
"pp": "case inl\nR : Type u_2\nM : Type u\nN : Type u_3\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\nS : Submodule R M\ninst✝ : IsSimpleModule R ↥S\nf : M →ₗ[R] N\ninj : Function.Injective ⇑(f ∘ₗ S.subtype)\n⊢ (f ∘ₗ S.subtype).range ≤ isotypicCo... | [] | exact le_sSup ⟨.symm <| .ofInjective _ inj⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.SimpleModule.Isotypic | {
"line": 315,
"column": 4
} | {
"line": 315,
"column": 47
} | {
"line": 316,
"column": 2
} | [
{
"pp": "case inl\nR : Type u_2\nM : Type u\nN : Type u_3\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\nS : Submodule R M\ninst✝ : IsSimpleModule R ↥S\nf : M →ₗ[R] N\ninj : Function.Injective ⇑(f ∘ₗ S.subtype)\n⊢ (f ∘ₗ S.subtype).range ≤ isotypicCo... | [] | exact le_sSup ⟨.symm <| .ofInjective _ inj⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.SimpleModule.Isotypic | {
"line": 315,
"column": 4
} | {
"line": 315,
"column": 47
} | {
"line": 316,
"column": 2
} | [
{
"pp": "case inl\nR : Type u_2\nM : Type u\nN : Type u_3\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\nS : Submodule R M\ninst✝ : IsSimpleModule R ↥S\nf : M →ₗ[R] N\ninj : Function.Injective ⇑(f ∘ₗ S.subtype)\n⊢ (f ∘ₗ S.subtype).range ≤ isotypicCo... | [] | exact le_sSup ⟨.symm <| .ofInjective _ inj⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.SimpleModule.Isotypic | {
"line": 369,
"column": 25
} | {
"line": 369,
"column": 51
} | {
"line": 369,
"column": 52
} | [
{
"pp": "R₀ : Type u_1\nR : Type u_2\nM : Type u\nN✝ : Type u_3\nS : Type u_4\ninst✝¹⁰ : CommSemiring R₀\ninst✝⁹ : Ring R\ninst✝⁸ : Algebra R₀ R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup N✝\ninst✝⁵ : AddCommGroup S\ninst✝⁴ : Module R M\ninst✝³ : Module R N✝\ninst✝² : Module R S\ninst✝¹ : IsSimpleModule R ... | [
"R₀ : Type u_1\nR : Type u_2\nM : Type u\nN✝ : Type u_3\nS : Type u_4\ninst✝¹⁰ : CommSemiring R₀\ninst✝⁹ : Ring R\ninst✝⁸ : Algebra R₀ R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup N✝\ninst✝⁵ : AddCommGroup S\ninst✝⁴ : Module R M\ninst✝³ : Module R N✝\ninst✝² : Module R S\ninst✝¹ : IsSimpleModule R S\nι : Type ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Maximal.Topology | {
"line": 49,
"column": 6
} | {
"line": 49,
"column": 42
} | {
"line": 49,
"column": 43
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nx : MaximalSpectrum R\n⊢ toPrimeSpectrum ⁻¹' {x.toPrimeSpectrum} = {x}",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.instSingletonSet",
"id",
"MaximalSpectrum",
"MaximalSpectrum.toPrim... | [
"R : Type u\ninst✝ : CommRing R\nx : MaximalSpectrum R\n⊢ toPrimeSpectrum ⁻¹' toPrimeSpectrum '' {x} = {x}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.IsOpenComapC | {
"line": 53,
"column": 78
} | {
"line": 64,
"column": 42
} | {
"line": 66,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf : R[X]\n⊢ imageOfDf f = PrimeSpectrum.comap C '' (zeroLocus {f})ᶜ",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Set.ext",
"Set.singleton_subset_iff",
"Eq.mpr",
"Polynomial.C",
"PrimeSpectrum.mk",
"Ideal.sub... | [] | by
ext x
refine ⟨fun hx => ⟨⟨map C x.asIdeal, isPrime_map_C_of_isPrime⟩, ⟨?_, ?_⟩⟩, ?_⟩
· rw [mem_compl_iff, mem_zeroLocus, singleton_subset_iff]
obtain ⟨i, hi⟩ := hx
exact fun a => hi (mem_map_C_iff.mp a i)
· ext x
refine ⟨fun h => ?_, fun h => subset_span (mem_image_of_mem C.1 h)⟩
rw [← @coeff... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Spectrum.Prime.IsOpenComapC | {
"line": 70,
"column": 23
} | {
"line": 70,
"column": 27
} | {
"line": 70,
"column": 28
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nU : Set (PrimeSpectrum R[X])\ns : Set R[X]\nz : zeroLocus s = Uᶜ\n⊢ IsOpen (PrimeSpectrum.comap C '' Uᶜᶜ)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"PrimeSpectrum.zeroLocus",
"congrArg",
... | [
"R : Type u_1\ninst✝ : CommRing R\nU : Set (PrimeSpectrum R[X])\ns : Set R[X]\nz : zeroLocus s = Uᶜ\n⊢ IsOpen (PrimeSpectrum.comap C '' (zeroLocus s)ᶜ)"
] | ← z, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.TwoSidedIdeal.BigOperators | {
"line": 65,
"column": 6
} | {
"line": 65,
"column": 54
} | {
"line": 65,
"column": 55
} | [
{
"pp": "case cons.inl\nR : Type u_1\ninst✝ : Ring R\nI : TwoSidedIdeal R\nι : Type u_2\nf : ι → R\nx : ι\nl : List ι\nih : (∃ x ∈ l, f x ∈ I) → (List.map f l).prod ∈ I\nh : f x ∈ I\n⊢ (List.map f (x :: l)).prod ∈ I",
"ppTerm": "?cons.inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"case cons.inl\nR : Type u_1\ninst✝ : Ring R\nI : TwoSidedIdeal R\nι : Type u_2\nf : ι → R\nx : ι\nl : List ι\nih : (∃ x ∈ l, f x ∈ I) → (List.map f l).prod ∈ I\nh : f x ∈ I\n⊢ f x * (List.map f l).prod ∈ I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.TwoSidedIdeal.BigOperators | {
"line": 66,
"column": 6
} | {
"line": 66,
"column": 54
} | {
"line": 66,
"column": 55
} | [
{
"pp": "case cons.inr\nR : Type u_1\ninst✝ : Ring R\nI : TwoSidedIdeal R\nι : Type u_2\nf : ι → R\nx : ι\nl : List ι\nih : (∃ x ∈ l, f x ∈ I) → (List.map f l).prod ∈ I\nhal : ∃ a ∈ l, f a ∈ I\n⊢ (List.map f (x :: l)).prod ∈ I",
"ppTerm": "?cons.inr",
"assigned": true,
"usedConstants": [
"Eq.m... | [
"case cons.inr\nR : Type u_1\ninst✝ : Ring R\nI : TwoSidedIdeal R\nι : Type u_2\nf : ι → R\nx : ι\nl : List ι\nih : (∃ x ∈ l, f x ∈ I) → (List.map f l).prod ∈ I\nhal : ∃ a ∈ l, f a ∈ I\n⊢ f x * (List.map f l).prod ∈ I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.TwoSidedIdeal.BigOperators | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 13
} | {
"line": 77,
"column": 14
} | [
{
"pp": "case mk\nR : Type u_1\ninst✝ : CommRing R\nI : TwoSidedIdeal R\nι : Type u_2\ns : Multiset ι\nf : ι → R\na✝ : List ι\nhs : ∃ x ∈ Quot.mk (⇑(List.isSetoid ι)) a✝, f x ∈ I\n⊢ (Multiset.map f (Quot.mk (⇑(List.isSetoid ι)) a✝)).prod ∈ I",
"ppTerm": "?mk",
"assigned": true,
"usedConstants": [
... | [
"case mk\nR : Type u_1\ninst✝ : CommRing R\nI : TwoSidedIdeal R\nι : Type u_2\ns : Multiset ι\nf : ι → R\na✝ : List ι\nhs : ∃ x ∈ Quot.mk (⇑(List.isSetoid ι)) a✝, f x ∈ I\n⊢ (List.map f a✝).prod ∈ I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.TwoSidedIdeal.BigOperators | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 13
} | {
"line": 82,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : TwoSidedIdeal R\nι : Type u_2\nf : ι → R\nval✝ : Multiset ι\nnodup✝ : val✝.Nodup\nhs : ∃ x ∈ { val := val✝, nodup := nodup✝ }, f x ∈ I\n⊢ { val := val✝, nodup := nodup✝ }.prod f ∈ I",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"NonUni... | [
"R : Type u_1\ninst✝ : CommRing R\nI : TwoSidedIdeal R\nι : Type u_2\nf : ι → R\nval✝ : Multiset ι\nnodup✝ : val✝.Nodup\nhs : ∃ x ∈ { val := val✝, nodup := nodup✝ }, f x ∈ I\n⊢ (Multiset.map f val✝).prod ∈ I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.TensorProduct.IsBaseChangeRightExact | {
"line": 60,
"column": 4
} | {
"line": 60,
"column": 15
} | {
"line": 60,
"column": 16
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝²⁰ : CommRing R\nS : Type u_2\ninst✝¹⁹ : CommRing S\ninst✝¹⁸ : Algebra R S\nM₁ : Type u_3\nM₂ : Type u_4\nM₃ : Type u_5\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\ninst✝¹⁷ : AddCommGroup M₁\ninst✝¹⁶ : AddCommGroup M₂\ninst✝¹⁵ : AddCommGroup M₃\ninst✝¹⁴ : AddCommGroup... | [
"case refine_1\nR : Type u_1\ninst✝²⁰ : CommRing R\nS : Type u_2\ninst✝¹⁹ : CommRing S\ninst✝¹⁸ : Algebra R S\nM₁ : Type u_3\nM₂ : Type u_4\nM₃ : Type u_5\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\ninst✝¹⁷ : AddCommGroup M₁\ninst✝¹⁶ : AddCommGroup M₂\ninst✝¹⁵ : AddCommGroup M₃\ninst✝¹⁴ : AddCommGroup N₁\ninst✝¹³... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.TensorProduct.IsBaseChangeRightExact | {
"line": 62,
"column": 4
} | {
"line": 62,
"column": 15
} | {
"line": 62,
"column": 16
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝²⁰ : CommRing R\nS : Type u_2\ninst✝¹⁹ : CommRing S\ninst✝¹⁸ : Algebra R S\nM₁ : Type u_3\nM₂ : Type u_4\nM₃ : Type u_5\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\ninst✝¹⁷ : AddCommGroup M₁\ninst✝¹⁶ : AddCommGroup M₂\ninst✝¹⁵ : AddCommGroup M₃\ninst✝¹⁴ : AddCommGroup... | [
"case refine_2\nR : Type u_1\ninst✝²⁰ : CommRing R\nS : Type u_2\ninst✝¹⁹ : CommRing S\ninst✝¹⁸ : Algebra R S\nM₁ : Type u_3\nM₂ : Type u_4\nM₃ : Type u_5\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\ninst✝¹⁷ : AddCommGroup M₁\ninst✝¹⁶ : AddCommGroup M₂\ninst✝¹⁵ : AddCommGroup M₃\ninst✝¹⁴ : AddCommGroup N₁\ninst✝¹³... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.Quotient | {
"line": 36,
"column": 6
} | {
"line": 36,
"column": 17
} | {
"line": 36,
"column": 18
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\nM : Type u_2\nN : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\nI : Ideal R\nf : M →ₗ[R] N\nsurj : Function.Surjective ⇑f\ninst✝ : Module.Flat R N\nx : M\nhx : x ∈ f.ker ⊓ I • ⊤\ny : TensorProduct R (↥I) M\nhy : ... | [
"R : Type u_1\ninst✝⁵ : CommRing R\nM : Type u_2\nN : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\nI : Ideal R\nf : M →ₗ[R] N\nsurj : Function.Surjective ⇑f\ninst✝ : Module.Flat R N\nx : M\nhx : x ∈ f.ker ⊓ I • ⊤\ny : TensorProduct R (↥I) M\nhy : (↑(TensorPro... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.ValuativeRel.Trivial | {
"line": 45,
"column": 27
} | {
"line": 45,
"column": 38
} | {
"line": 45,
"column": 39
} | [
{
"pp": "R✝ Γ : Type\ninst✝⁶ : Ring R✝\ninst✝⁵ : DecidableEq R✝\ninst✝⁴ : IsDomain R✝\ninst✝³ : LinearOrderedCommGroupWithZero Γ\nR : Type\ninst✝² : Semiring R\ninst✝¹ : DecidableEq R\ninst✝ : IsDomain R\nx✝¹ x✝ : R\n⊢ if x✝ * x✝¹ = 0 then x✝¹ * x✝ = 0 else True",
"ppTerm": "?m.186",
"assigned": true,
... | [
"R✝ Γ : Type\ninst✝⁶ : Ring R✝\ninst✝⁵ : DecidableEq R✝\ninst✝⁴ : IsDomain R✝\ninst✝³ : LinearOrderedCommGroupWithZero Γ\nR : Type\ninst✝² : Semiring R\ninst✝¹ : DecidableEq R\ninst✝ : IsDomain R\nx✝¹ x✝ : R\n⊢ x✝ = 0 ∨ x✝¹ = 0 → x✝¹ = 0 ∨ x✝ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.Quotient | {
"line": 57,
"column": 6
} | {
"line": 57,
"column": 43
} | {
"line": 57,
"column": 44
} | [
{
"pp": "case refine_2.smul\nR : Type u_1\ninst✝⁵ : CommRing R\nM : Type u_2\nN : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\nI : Ideal R\nf : M →ₗ[R] N\nsurj : Function.Surjective ⇑f\ninst✝ : Module.Flat R N\nx : M\nr : R\nhr : r ∈ I\nm : M\nhm : m ∈ f.... | [
"case refine_2.smul\nR : Type u_1\ninst✝⁵ : CommRing R\nM : Type u_2\nN : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\nI : Ideal R\nf : M →ₗ[R] N\nsurj : Function.Surjective ⇑f\ninst✝ : Module.Flat R N\nx : M\nr : R\nhr : r ∈ I\nm : M\nhm : m ∈ f.ker\n⊢ r • m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Extension | {
"line": 80,
"column": 2
} | {
"line": 80,
"column": 27
} | {
"line": 80,
"column": 28
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nΓR : Type u_3\nΓA : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : Ring A\ninst✝³ : LinearOrderedCommMonoidWithZero ΓR\ninst✝² : LinearOrderedCommMonoidWithZero ΓA\ninst✝¹ : Algebra R A\nvR : Valuation R ΓR\nvA : Valuation A ΓA\ninst✝ : vR.HasExtension vA\nx y : R\n⊢ vA ((algebraMa... | [
"R : Type u_1\nA : Type u_2\nΓR : Type u_3\nΓA : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : Ring A\ninst✝³ : LinearOrderedCommMonoidWithZero ΓR\ninst✝² : LinearOrderedCommMonoidWithZero ΓA\ninst✝¹ : Algebra R A\nvR : Valuation R ΓR\nvA : Valuation A ΓA\ninst✝ : vR.HasExtension vA\nx y : R\n⊢ vA ((algebraMap R A) x) < ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Extension | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 28
} | {
"line": 86,
"column": 29
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nΓR : Type u_3\nΓA : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : Ring A\ninst✝³ : LinearOrderedCommMonoidWithZero ΓR\ninst✝² : LinearOrderedCommMonoidWithZero ΓA\ninst✝¹ : Algebra R A\nvR : Valuation R ΓR\nvA : Valuation A ΓA\ninst✝ : vR.HasExtension vA\nx : R\n⊢ vA ((algebraMap ... | [
"R : Type u_1\nA : Type u_2\nΓR : Type u_3\nΓA : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : Ring A\ninst✝³ : LinearOrderedCommMonoidWithZero ΓR\ninst✝² : LinearOrderedCommMonoidWithZero ΓA\ninst✝¹ : Algebra R A\nvR : Valuation R ΓR\nvA : Valuation A ΓA\ninst✝ : vR.HasExtension vA\nx : R\n⊢ vA ((algebraMap R A) x) ≤ 1 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Extension | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 36
} | {
"line": 89,
"column": 37
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nΓR : Type u_3\nΓA : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : Ring A\ninst✝³ : LinearOrderedCommMonoidWithZero ΓR\ninst✝² : LinearOrderedCommMonoidWithZero ΓA\ninst✝¹ : Algebra R A\nvR : Valuation R ΓR\nvA : Valuation A ΓA\ninst✝ : vR.HasExtension vA\nx : R\n⊢ vA ((algebraMap ... | [
"R : Type u_1\nA : Type u_2\nΓR : Type u_3\nΓA : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : Ring A\ninst✝³ : LinearOrderedCommMonoidWithZero ΓR\ninst✝² : LinearOrderedCommMonoidWithZero ΓA\ninst✝¹ : Algebra R A\nvR : Valuation R ΓR\nvA : Valuation A ΓA\ninst✝ : vR.HasExtension vA\nx : R\n⊢ vA ((algebraMap R A) x) < 1 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.Quotient | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 20
} | {
"line": 95,
"column": 21
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nJ I : Ideal R\nsq : I * I = ⊥\nf : J.Cotangent →ₗ[R] J.Cotangent\nle : f.range ≤ Submodule.map J.toCotangent (Submodule.comap (Submodule.subtype J) (I * J))\nthis : ∀ {x : R} (h : x ∈ I * J), f (J.toCotangent ⟨x, ⋯⟩) = 0\nx : J.Cotangent\nhx : x ∈ Submodule.map J.toCot... | [
"R : Type u_1\ninst✝ : CommRing R\nJ I : Ideal R\nsq : I * I = ⊥\nf : J.Cotangent →ₗ[R] J.Cotangent\nle : f.range ≤ Submodule.map J.toCotangent (Submodule.comap (Submodule.subtype J) (I * J))\nthis : ∀ {x : R} (h : x ∈ I * J), f (J.toCotangent ⟨x, ⋯⟩) = 0\nx : J.Cotangent\nhx : x ∈ Submodule.map J.toCotangent (Subm... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Extension | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 45
} | {
"line": 93,
"column": 4
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nΓR : Type u_3\nΓA : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : Ring A\ninst✝³ : LinearOrderedCommMonoidWithZero ΓR\ninst✝² : LinearOrderedCommMonoidWithZero ΓA\ninst✝¹ : Algebra R A\nvR : Valuation R ΓR\nvA : Valuation A ΓA\ninst✝ : vR.HasExtension vA\nx : R\n⊢ vA ((algebraMap ... | [
"R : Type u_1\nA : Type u_2\nΓR : Type u_3\nΓA : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : Ring A\ninst✝³ : LinearOrderedCommMonoidWithZero ΓR\ninst✝² : LinearOrderedCommMonoidWithZero ΓA\ninst✝¹ : Algebra R A\nvR : Valuation R ΓR\nvA : Valuation A ΓA\ninst✝ : vR.HasExtension vA\nx : R\n⊢ vA ((algebraMap R A) x) ≤ 1 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.DiscreteValuationRing | {
"line": 97,
"column": 4
} | {
"line": 97,
"column": 73
} | {
"line": 98,
"column": 6
} | [
{
"pp": "p : ℕ\nhp✝ : Fact (Nat.Prime p)\nk : Type u_1\ninst✝¹ : Field k\ninst✝ : CharP k p\nhp : IsUnit ↑p\n⊢ False",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : ℕ\nhp✝ : Fact (Nat.Prime p)\nk : Type u_1\ninst✝¹ : Field k\ninst✝ : CharP k p\nhp : IsUnit ↑p\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.DiscreteValuationRing | {
"line": 101,
"column": 8
} | {
"line": 101,
"column": 25
} | {
"line": 101,
"column": 25
} | [
{
"pp": "p : ℕ\nhp✝ : Fact (Nat.Prime p)\nk : Type u_1\ninst✝¹ : Field k\ninst✝ : CharP k p\nhp : ¬IsUnit ↑p\na b : 𝕎 k\nhab : ↑p = a * b\n⊢ a ≠ 0 ∧ b ≠ 0",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"WittVector.instZero",
"Eq.mpr",
"IsDomain.to_noZeroDivisors",
... | [
"p : ℕ\nhp✝ : Fact (Nat.Prime p)\nk : Type u_1\ninst✝¹ : Field k\ninst✝ : CharP k p\nhp : ¬IsUnit ↑p\na b : 𝕎 k\nhab : ↑p = a * b\n⊢ a * b ≠ 0"
] | ← mul_ne_zero_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.WittVector.MulCoeff | {
"line": 82,
"column": 4
} | {
"line": 82,
"column": 15
} | {
"line": 82,
"column": 16
} | [
{
"pp": "case refine_2\np : ℕ\nhp : Fact (Nat.Prime p)\nn x : ℕ\nhx : x ∈ range n\n⊢ range (x + 1) ⊆ range n",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Nat.instOne",
"Finset",
"PartialOrder.toPreorder",
"Preorder.toLE... | [
"case refine_2\np : ℕ\nhp : Fact (Nat.Prime p)\nn x : ℕ\nhx : x ∈ range n\n⊢ x < n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.MulCoeff | {
"line": 103,
"column": 6
} | {
"line": 103,
"column": 17
} | {
"line": 103,
"column": 18
} | [
{
"pp": "case refine_1\np : ℕ\nhp : Fact (Nat.Prime p)\nn x : ℕ\nhx : x ∈ range (n + 1)\n⊢ {(0, x)} ⊆ univ ×ˢ range (n + 1)",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Finset.singleton_subset_iff._simp_1",
"Eq.mpr",
"Finset.mem_range._simp_1",
"Finset.univ"... | [
"case refine_1\np : ℕ\nhp : Fact (Nat.Prime p)\nn x : ℕ\nhx : x ∈ range (n + 1)\n⊢ x ≤ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.MulCoeff | {
"line": 103,
"column": 6
} | {
"line": 103,
"column": 17
} | {
"line": 103,
"column": 18
} | [
{
"pp": "case refine_2\np : ℕ\nhp : Fact (Nat.Prime p)\nn x : ℕ\nhx : x ∈ range (n + 1)\n⊢ {(1, x)} ⊆ univ ×ˢ range (n + 1)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Finset.singleton_subset_iff._simp_1",
"Eq.mpr",
"Finset.mem_range._simp_1",
"Finset.univ"... | [
"case refine_2\np : ℕ\nhp : Fact (Nat.Prime p)\nn x : ℕ\nhx : x ∈ range (n + 1)\n⊢ x ≤ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.FrobeniusFractionField | {
"line": 160,
"column": 4
} | {
"line": 160,
"column": 26
} | {
"line": 160,
"column": 27
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝¹ : Field k\ninst✝ : IsAlgClosed k\na₁ a₂ : 𝕎 k\nha₁ : a₁.coeff 0 ≠ 0\nha₂ : a₂.coeff 0 ≠ 0\nh : solution p a₁ a₂ = 0\nthis : 0 = a₂.coeff 0 / a₁.coeff 0\n⊢ False",
"ppTerm": "?m.44",
"assigned": false,
"usedConstants": [],
"usedFVars"... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝¹ : Field k\ninst✝ : IsAlgClosed k\na₁ a₂ : 𝕎 k\nha₁ : a₁.coeff 0 ≠ 0\nha₂ : a₂.coeff 0 ≠ 0\nh : solution p a₁ a₂ = 0\nthis : 0 = a₂.coeff 0 / a₁.coeff 0\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.FrobeniusFractionField | {
"line": 195,
"column": 2
} | {
"line": 195,
"column": 62
} | {
"line": 195,
"column": 63
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : Field k\ninst✝¹ : CharP k p\ninst✝ : IsAlgClosed k\na₁ a₂ : 𝕎 k\nha₁ : a₁.coeff 0 ≠ 0\nha₂ : a₂.coeff 0 ≠ 0\nh : frobeniusRotation p ha₁ ha₂ = 0\n⊢ solution p a₁ a₂ = 0",
"ppTerm": "?m.36",
"assigned": false,
"usedConstants": [],
"... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : Field k\ninst✝¹ : CharP k p\ninst✝ : IsAlgClosed k\na₁ a₂ : 𝕎 k\nha₁ : a₁.coeff 0 ≠ 0\nha₂ : a₂.coeff 0 ≠ 0\nh : frobeniusRotation p ha₁ ha₂ = 0\n⊢ solution p a₁ a₂ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.FrobeniusFractionField | {
"line": 236,
"column": 4
} | {
"line": 236,
"column": 35
} | {
"line": 237,
"column": 6
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : Field k\ninst✝¹ : CharP k p\ninst✝ : IsAlgClosed k\nm n : ℕ\nr' q' : 𝕎 k\nhr' : r'.coeff 0 ≠ 0\nhq' : q'.coeff 0 ≠ 0\nhq : ↑p ^ n * q' ∈ nonZeroDivisors (𝕎 k)\nb : 𝕎 k := frobeniusRotation p hr' hq'\nkey : frobenius b * r' = q' * b\nhq''' : q' ≠... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : Field k\ninst✝¹ : CharP k p\ninst✝ : IsAlgClosed k\nm n : ℕ\nr' q' : 𝕎 k\nhr' : r'.coeff 0 ≠ 0\nhq' : q'.coeff 0 ≠ 0\nhq : ↑p ^ n * q' ∈ nonZeroDivisors (𝕎 k)\nb : 𝕎 k := frobeniusRotation p hr' hq'\nkey : frobenius b * r' = q' * b\nhq''' : q' ≠ 0\n⊢ ¬(alge... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.FrobeniusFractionField | {
"line": 255,
"column": 4
} | {
"line": 255,
"column": 31
} | {
"line": 256,
"column": 6
} | [
{
"pp": "case refine_1\np : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : Field k\ninst✝¹ : CharP k p\ninst✝ : IsAlgClosed k\na : FractionRing (𝕎 k)\nm : ℕ\nr' : 𝕎 k\nhr' : r'.coeff 0 ≠ 0\nhr0 : ↑p ^ m * r' ≠ 0\nn : ℕ\nq' : 𝕎 k\nhq' : q'.coeff 0 ≠ 0\nhq : ↑p ^ n * q' ∈ nonZeroDivisors (𝕎 k)\nhq0 : ↑p ^... | [
"case refine_1\np : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : Field k\ninst✝¹ : CharP k p\ninst✝ : IsAlgClosed k\na : FractionRing (𝕎 k)\nm : ℕ\nr' : 𝕎 k\nhr' : r'.coeff 0 ≠ 0\nhr0 : ↑p ^ m * r' ≠ 0\nn : ℕ\nq' : 𝕎 k\nhq' : q'.coeff 0 ≠ 0\nhq : ↑p ^ n * q' ∈ nonZeroDivisors (𝕎 k)\nhq0 : ↑p ^ n * q' ≠ 0\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZMod.Torsion | {
"line": 25,
"column": 2
} | {
"line": 25,
"column": 29
} | {
"line": 25,
"column": 30
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nx✝ : (ZMod p)ˣ\n⊢ x✝ ∈ rootsOfUnity (p - 1) (ZMod p) ↔ x✝ ∈ ⊤",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"MulOne.toOne",
"ZMod.commRing",
"Monoid.toMulOneClass",
"congrArg",
"H... | [
"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nx✝ : (ZMod p)ˣ\n⊢ ↑x✝ ^ (p - 1) = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.Isocrystal | {
"line": 188,
"column": 29
} | {
"line": 188,
"column": 56
} | {
"line": 188,
"column": 57
} | [
{
"pp": "p : ℕ\ninst✝⁵ : Fact (Nat.Prime p)\nk : Type u_2\ninst✝⁴ : Field k\ninst✝³ : IsAlgClosed k\ninst✝² : CharP k p\nV : Type u_3\ninst✝¹ : AddCommGroup V\ninst✝ : Isocrystal p k V\nh_dim : finrank K(p, k) V = 1\nthis : Nontrivial V\nx : V\nhx : x ≠ 0\n⊢ Φ(p, k) x ≠ 0",
"ppTerm": "?m.67",
"assigned"... | [
"p : ℕ\ninst✝⁵ : Fact (Nat.Prime p)\nk : Type u_2\ninst✝⁴ : Field k\ninst✝³ : IsAlgClosed k\ninst✝² : CharP k p\nV : Type u_3\ninst✝¹ : AddCommGroup V\ninst✝ : Isocrystal p k V\nh_dim : finrank K(p, k) V = 1\nthis : Nontrivial V\nx : V\nhx : x ≠ 0\n⊢ Φ(p, k) x ≠ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Cofinality.Club | {
"line": 109,
"column": 4
} | {
"line": 109,
"column": 15
} | {
"line": 109,
"column": 16
} | [
{
"pp": "α : Type v\ninst✝¹ : LinearOrder α\ninst✝ : WellFoundedLT α\ns : Set (Set α)\nhα✝ : cof α ≠ ℵ₀\nhsα : #↑s < cof α\nhs : ∀ x ∈ s, IsClub x\nh✝ : Nonempty α\nhα : ℵ₀ < cof α\na : α\nf : ↑s → α → α\nhf : ∀ (x : ↑s) (x_1 : α), f x x_1 ∈ ↑x ∧ x_1 ≤ f x x_1\ng : ℕ → α := fun t ↦ Nat.rec a (fun x IH ↦ sSup (r... | [
"α : Type v\ninst✝¹ : LinearOrder α\ninst✝ : WellFoundedLT α\ns : Set (Set α)\nhα✝ : cof α ≠ ℵ₀\nhsα : #↑s < cof α\nhs : ∀ x ∈ s, IsClub x\nh✝ : Nonempty α\nhα : ℵ₀ < cof α\na : α\nf : ↑s → α → α\nhf : ∀ (x : ↑s) (x_1 : α), f x x_1 ∈ ↑x ∧ x_1 ≤ f x x_1\ng : ℕ → α := fun t ↦ Nat.rec a (fun x IH ↦ sSup (range fun x ↦... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Descriptive.Tree | {
"line": 43,
"column": 11
} | {
"line": 43,
"column": 22
} | {
"line": 43,
"column": 23
} | [
{
"pp": "case nil\nA : Type u_1\nT : ↥(tree A)\nx : List A\nh : x ++ [] ∈ T\n⊢ x ∈ T",
"ppTerm": "?nil",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case nil\nA : Type u_1\nT : ↥(tree A)\nx : List A\nh : x ++ [] ∈ T\n⊢ x ∈ T"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Descriptive.Tree | {
"line": 57,
"column": 21
} | {
"line": 57,
"column": 32
} | {
"line": 57,
"column": 33
} | [
{
"pp": "A : Type u_1\nT : ↥(tree A)\nh : [] ∉ T\nx : List A\n⊢ x ∈ T ↔ x ∈ ⊥",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"iff_false",
"congrArg",
"Membership.mem",
"id",
"Subtype",
"Bot.bot",
"Descriptive.tree",
... | [
"A : Type u_1\nT : ↥(tree A)\nh : [] ∉ T\nx : List A\n⊢ x ∉ T"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Cofinality.Club | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 22
} | {
"line": 142,
"column": 23
} | [
{
"pp": "α : Type v\ns t : Set α\ninst✝¹ : LinearOrder α\ninst✝ : WellFoundedLT α\nhα : cof α ≠ ℵ₀\nhs : IsClub s\nht : IsClub t\n⊢ IsClub (s ∩ t)",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type v\ns t : Set α\ninst✝¹ : LinearOrder α\ninst✝ : WellFoundedLT α\nhα : cof α ≠ ℵ₀\nhs : IsClub s\nht : IsClub t\n⊢ IsClub (s ∩ t)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Descriptive.Tree | {
"line": 113,
"column": 4
} | {
"line": 113,
"column": 69
} | {
"line": 113,
"column": 70
} | [
{
"pp": "case right\nA : Type u_1\nT : ↥(tree A)\nx y : List A\nhl : x.length ≤ y.length\nh1 : List.take x.length y <+: x\nh2 : List.drop x.length y ∈ T\n⊢ List.take x.length y ++ List.drop x.length y = x ++ List.drop x.length y",
"ppTerm": "?right",
"assigned": true,
"usedConstants": [
"Eq.mp... | [
"case right\nA : Type u_1\nT : ↥(tree A)\nx y : List A\nhl : x.length ≤ y.length\nh1 : List.take x.length y <+: x\nh2 : List.drop x.length y ∈ T\n⊢ List.take x.length y = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Descriptive.Tree | {
"line": 119,
"column": 2
} | {
"line": 119,
"column": 13
} | {
"line": 119,
"column": 14
} | [
{
"pp": "A : Type u_1\nT : ↥(tree A)\nx : List A\n⊢ x ∈ pullSub T x ↔ [] ∈ T",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"A : Type u_1\nT : ↥(tree A)\nx : List A\n⊢ x ∈ pullSub T x ↔ [] ∈ T"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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