module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.Polynomial.Roots | {
"line": 293,
"column": 63
} | {
"line": 294,
"column": 36
} | {
"line": 296,
"column": 0
} | [
{
"pp": "R : Type u\na : R\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nha : a ≠ 0\nn : ℕ\n⊢ (C a * X ^ n).roots = n • {0}",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"instHSMul",
"Polynomial.roots",
"HMul.hMul",
"Polynomial.ro... | [] | by
rw [roots_C_mul _ ha, roots_X_pow] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Roots | {
"line": 338,
"column": 4
} | {
"line": 338,
"column": 15
} | {
"line": 338,
"column": 16
} | [
{
"pp": "case refine_2\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : R[X]\nS : Finset R\nhS : ∀ x ∈ S, eval x p = 0\nhcard : p.degree ≤ ↑(#S)\nhp : p ≠ 0\n⊢ p.roots.card ≤ S.val.card",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Polynomial.roots",
"id",
... | [
"case refine_2\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : R[X]\nS : Finset R\nhS : ∀ x ∈ S, eval x p = 0\nhcard : p.degree ≤ ↑(#S)\nhp : p ≠ 0\n⊢ p.roots.card ≤ #S"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Content | {
"line": 127,
"column": 10
} | {
"line": 127,
"column": 33
} | {
"line": 127,
"column": 33
} | [
{
"pp": "case succ.mpr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : NormalizedGCDMonoid R\np : R[X]\na b : ℕ\nh1 : ¬p.coeff b = 0\nh2 : b.succ = a + 1\n⊢ ¬p.coeff a = 0",
"ppTerm": "?succ.mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"CommSemiring.toSemiring",
... | [
"case succ.mpr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : NormalizedGCDMonoid R\np : R[X]\na b : ℕ\nh1 : ¬p.coeff b = 0\nh2 : b.succ = a + 1\n⊢ ¬p.coeff b = 0"
] | ← Nat.succ_injective h2 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Content | {
"line": 114,
"column": 66
} | {
"line": 130,
"column": 6
} | {
"line": 132,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : NormalizedGCDMonoid R\np : R[X]\n⊢ (X * p).content = p.content",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"False",
"Nat.instMulZeroClass",... | [] | by
rw [content, content, Finset.gcd_def, Finset.gcd_def]
refine congr rfl ?_
have h : (X * p).support = p.support.map ⟨Nat.succ, Nat.succ_injective⟩ := by
ext a
simp only [Finset.mem_map, Function.Embedding.coeFn_mk, Ne, mem_support_iff]
rcases a with - | a
· simp
rw [mul_comm, coeff_mul_X]
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Roots | {
"line": 438,
"column": 2
} | {
"line": 444,
"column": 20
} | {
"line": 446,
"column": 0
} | [
{
"pp": "R : Type u\nn : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nη a : R\nha : a ≠ 0\nhη : η ∈ nthRootsFinset n a\n⊢ η ≠ 0",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"False",
"congrArg",
"CommSemiring.toSemiring",
"Finset",
"False.elim",
"Member... | [] | rintro rfl
cases n with
| zero =>
simp only [nthRootsFinset_zero, notMem_empty] at hη
| succ n =>
rw [mem_nthRootsFinset n.succ_pos, zero_pow n.succ_ne_zero] at hη
exact ha hη.symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Roots | {
"line": 438,
"column": 2
} | {
"line": 444,
"column": 20
} | {
"line": 446,
"column": 0
} | [
{
"pp": "R : Type u\nn : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nη a : R\nha : a ≠ 0\nhη : η ∈ nthRootsFinset n a\n⊢ η ≠ 0",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"False",
"congrArg",
"CommSemiring.toSemiring",
"Finset",
"False.elim",
"Member... | [] | rintro rfl
cases n with
| zero =>
simp only [nthRootsFinset_zero, notMem_empty] at hη
| succ n =>
rw [mem_nthRootsFinset n.succ_pos, zero_pow n.succ_ne_zero] at hη
exact ha hη.symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Roots | {
"line": 451,
"column": 51
} | {
"line": 451,
"column": 62
} | {
"line": 451,
"column": 63
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nh₁ : X ^ 2 - C 1 = (X + C 1) * (X - C 1)\nh : X ^ 2 - C 1 = 0\n⊢ False",
"ppTerm": "?m.115",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nh₁ : X ^ 2 - C 1 = (X + C 1) * (X - C 1)\nh : X ^ 2 - C 1 = 0\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.FieldDivision | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 42
} | {
"line": 95,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\np : R[X]\nt : R\nn : ℕ\nh : p ≠ 0\nhnzd : ↑n ! ∈ nonZeroDivisors R\nh' : rootMultiplicity t p ≤ n\nhroot : (rootMultiplicity t p)! • eval t (p /ₘ (X - C t) ^ rootMultiplicity t p) = 0\nq : R\nhq : ↑n ! = ↑(rootMultiplicity t p)! * q\n⊢ False",
"ppTerm": "?m.66",
... | [
"R : Type u\ninst✝ : CommRing R\np : R[X]\nt : R\nn : ℕ\nh : p ≠ 0\nh' : rootMultiplicity t p ≤ n\nhroot : (rootMultiplicity t p)! • eval t (p /ₘ (X - C t) ^ rootMultiplicity t p) = 0\nq : R\nhnzd : ↑(rootMultiplicity t p)! ∈ nonZeroDivisors R ∧ q ∈ nonZeroDivisors R\nhq : ↑n ! = ↑(rootMultiplicity t p)! * q\n⊢ Fal... | rw [hq, mul_mem_nonZeroDivisors] at hnzd | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.UniqueFactorizationDomain.Basic | {
"line": 418,
"column": 6
} | {
"line": 419,
"column": 81
} | {
"line": 419,
"column": 81
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : IsCancelMulZero α\neif : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Irreducible b) ∧ f.prod ~ᵤ a\nuif :\n ∀ (f g : Multiset α),\n (∀ x ∈ f, Irreducible x) → (∀ x ∈ g, Irreducible x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g\n⊢ ∀ (a : α), a ≠ 0 → ∃ ... | [] | convert! eif using 7
simp_rw [irreducible_iff_prime_of_existsUnique_irreducible_factors eif uif] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.UniqueFactorizationDomain.Basic | {
"line": 418,
"column": 6
} | {
"line": 419,
"column": 81
} | {
"line": 419,
"column": 81
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : IsCancelMulZero α\neif : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Irreducible b) ∧ f.prod ~ᵤ a\nuif :\n ∀ (f g : Multiset α),\n (∀ x ∈ f, Irreducible x) → (∀ x ∈ g, Irreducible x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g\n⊢ ∀ (a : α), a ≠ 0 → ∃ ... | [] | convert! eif using 7
simp_rw [irreducible_iff_prime_of_existsUnique_irreducible_factors eif uif] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.UniqueFactorizationDomain.Basic | {
"line": 438,
"column": 2
} | {
"line": 438,
"column": 28
} | {
"line": 438,
"column": 29
} | [
{
"pp": "R : Type u_2\ninst✝¹ : CommMonoidWithZero R\ninst✝ : UniqueFactorizationMonoid R\na b c : R\nha : a ≠ 0\nno_factors : ∀ {d : R}, d ∣ a → d ∣ b → ¬Prime d\n⊢ a ∣ b * c → a ∣ c",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Eq.mpr",
... | [
"R : Type u_2\ninst✝¹ : CommMonoidWithZero R\ninst✝ : UniqueFactorizationMonoid R\na b c : R\nha : a ≠ 0\nno_factors : ∀ {d : R}, d ∣ a → d ∣ b → ¬Prime d\n⊢ a ∣ c * b → a ∣ c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.FieldDivision | {
"line": 158,
"column": 2
} | {
"line": 159,
"column": 93
} | {
"line": 161,
"column": 0
} | [
{
"pp": "case neg\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\np : R[X]\nt : R\nhpt : p.IsRoot t\nh : ¬p = 0\n⊢ rootMultiplicity t (derivative p) = rootMultiplicity t p - 1",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"CommSemir... | [] | exact derivative_rootMultiplicity_of_root_of_mem_nonZeroDivisors hpt <|
mem_nonZeroDivisors_of_ne_zero <| Nat.cast_ne_zero.2 ((rootMultiplicity_pos h).2 hpt).ne' | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Polynomial.Content | {
"line": 242,
"column": 59
} | {
"line": 242,
"column": 75
} | {
"line": 242,
"column": 75
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : NormalizedGCDMonoid R\np : R[X]\nh : ¬p.content = 0\n⊢ normalize p.primPart.content = 1",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"congrArg",
"CommSemiring.toSemiring",
... | [
"case neg\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : NormalizedGCDMonoid R\np : R[X]\nh : ¬p.content = 0\n⊢ IsUnit p.primPart.content"
] | normalize_eq_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.FieldDivision | {
"line": 229,
"column": 2
} | {
"line": 229,
"column": 17
} | {
"line": 231,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : NormalizationMonoid R\np : R[X]\n⊢ ↑(normUnit p) = C ↑(normUnit p.leadingCoeff)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Units.val",
"Polynomial.C",
"CommSemiring.toSemiring",
"Ring... | [] | simp [normUnit] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Polynomial.FieldDivision | {
"line": 229,
"column": 2
} | {
"line": 229,
"column": 17
} | {
"line": 231,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : NormalizationMonoid R\np : R[X]\n⊢ ↑(normUnit p) = C ↑(normUnit p.leadingCoeff)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Units.val",
"Polynomial.C",
"CommSemiring.toSemiring",
"Ring... | [] | simp [normUnit] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.FieldDivision | {
"line": 229,
"column": 2
} | {
"line": 229,
"column": 17
} | {
"line": 231,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : NormalizationMonoid R\np : R[X]\n⊢ ↑(normUnit p) = C ↑(normUnit p.leadingCoeff)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Units.val",
"Polynomial.C",
"CommSemiring.toSemiring",
"Ring... | [] | simp [normUnit] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.FieldDivision | {
"line": 261,
"column": 67
} | {
"line": 261,
"column": 78
} | {
"line": 261,
"column": 79
} | [
{
"pp": "R : Type u\ninst✝ : DivisionRing R\np : R[X]\nhp0 : C (p.coeff 0) ≠ 0\nhp : ¬IsUnit (C (p.coeff 0))\nh : p.degree ≤ 0\n⊢ ¬C (p.coeff 0) = C 0",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"GroupWithZero.toMonoidWithZero",
"RingHo... | [
"R : Type u\ninst✝ : DivisionRing R\np : R[X]\nhp0 : C (p.coeff 0) ≠ 0\nhp : ¬IsUnit (C (p.coeff 0))\nh : p.degree ≤ 0\n⊢ ¬p.coeff 0 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.FieldDivision | {
"line": 304,
"column": 10
} | {
"line": 304,
"column": 38
} | {
"line": 304,
"column": 39
} | [
{
"pp": "R : Type u\ninst✝ : Field R\np : R[X]\nh : p.degree = 0\nthis : p.degree ≤ 0\nhc : p.coeff 0 = 0\n⊢ False",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Polynomial.C",
"Nat.instMulZeroClass",
"WithBot",
"congrArg",
"Wit... | [
"R : Type u\ninst✝ : Field R\np : R[X]\nh : (C (p.coeff 0)).degree = 0\nthis : p.degree ≤ 0\nhc : p.coeff 0 = 0\n⊢ False"
] | eq_C_of_degree_le_zero this, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Roots | {
"line": 769,
"column": 4
} | {
"line": 769,
"column": 46
} | {
"line": 769,
"column": 47
} | [
{
"pp": "case inl\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf : R[X]\nhf : ∀ (r : R), eval r f = 0\nhfR : ↑f.natDegree < #R\nhR : Finite R\nthis : Fintype R\n⊢ f.natDegree < Fintype.card R",
"ppTerm": "?inl",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": [... | [
"case inl\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf : R[X]\nhf : ∀ (r : R), eval r f = 0\nhfR : ↑f.natDegree < #R\nhR : Finite R\nthis : Fintype R\n⊢ f.natDegree < Fintype.card R"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Roots | {
"line": 857,
"column": 24
} | {
"line": 857,
"column": 58
} | {
"line": 857,
"column": 59
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\na p : R[X]\nhp : p.Monic\nhap : a ∣ p\n⊢ a.leadingCoeff ∣ 1",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\na p : R[X]\nhp : p.Monic\nhap : a ∣ p\n⊢ a.leadingCoeff ∣ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.FieldDivision | {
"line": 551,
"column": 37
} | {
"line": 551,
"column": 53
} | {
"line": 551,
"column": 53
} | [
{
"pp": "R : Type u\ninst✝¹ : Field R\np : R[X]\ninst✝ : DecidableEq R\nhp0 : IsUnit p.leadingCoeff\n⊢ normalize p.leadingCoeff = 1",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"congrArg",
... | [
"R : Type u\ninst✝¹ : Field R\np : R[X]\ninst✝ : DecidableEq R\nhp0 : IsUnit p.leadingCoeff\n⊢ IsUnit p.leadingCoeff"
] | normalize_eq_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.FieldDivision | {
"line": 575,
"column": 2
} | {
"line": 575,
"column": 24
} | {
"line": 575,
"column": 25
} | [
{
"pp": "R : Type u\na : R\ninst✝ : Field R\np : R[X]\n⊢ p / C a = p * C a⁻¹",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\na : R\ninst✝ : Field R\np : R[X]\n⊢ p / C a = p * C a⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Roots | {
"line": 990,
"column": 2
} | {
"line": 990,
"column": 46
} | {
"line": 990,
"column": 47
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : IsDomain A\ninst✝ : IsDomain B\np : A[X]\nf : A →+* B\nhf : Function.Injective ⇑f\nhroots : p.roots.card = p.natDegree\n⊢ (map f p).roots.card ≤ (Multiset.map (⇑f) p.roots).card",
"ppTerm": "?m.41",
"assigned": true,... | [
"A : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : IsDomain A\ninst✝ : IsDomain B\np : A[X]\nf : A →+* B\nhf : Function.Injective ⇑f\nhroots : p.roots.card = p.natDegree\n⊢ (map f p).roots.card ≤ p.natDegree"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Roots | {
"line": 996,
"column": 4
} | {
"line": 996,
"column": 48
} | {
"line": 996,
"column": 49
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : IsDomain A\ninst✝ : IsDomain B\np : A[X]\nf : A →+* B\nh : map f p ≠ 0\nhroots : p.roots.card = p.natDegree\n⊢ (map f p).roots.card ≤ (Multiset.map (⇑f) p.roots).card",
"ppTerm": "?m.45",
"assigned": true,
"usedC... | [
"A : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : IsDomain A\ninst✝ : IsDomain B\np : A[X]\nf : A →+* B\nh : map f p ≠ 0\nhroots : p.roots.card = p.natDegree\n⊢ (map f p).roots.card ≤ p.natDegree"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet | {
"line": 393,
"column": 2
} | {
"line": 397,
"column": 28
} | {
"line": 398,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\na b : α\nha : a ≠ 0\nhb : b ≠ 0\nh : Associates.mk a ⊓ Associates.mk b ≠ 1\n⊢ ∃ p, Prime p ∧ p ∣ a ∧ p ∣ b",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Multiset.prod_zero",
"CommMonoi... | [
"α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\na b : α\nha : a ≠ 0\nhb : b ≠ 0\nh : Associates.mk a ⊓ Associates.mk b ≠ 1\nhz : (Associates.mk a).factors ⊓ (Associates.mk b).factors ≠ 0\n⊢ ∃ p, Prime p ∧ p ∣ a ∧ p ∣ b"
] | have hz : factors (Associates.mk a) ⊓ factors (Associates.mk b) ≠ 0 := by
contrapose h with hf
change (factors (Associates.mk a) ⊓ factors (Associates.mk b)).prod = 1
rw [hf]
exact Multiset.prod_zero | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Polynomial.FieldDivision | {
"line": 640,
"column": 2
} | {
"line": 641,
"column": 39
} | {
"line": 641,
"column": 40
} | [
{
"pp": "K : Type u_1\ninst✝ : CommRing K\nf : K[X]\na : K\nkey : derivative ((X - C a) * (f /ₘ (X - C a))) = derivative (f - f %ₘ (X - C a))\n⊢ f /ₘ (X - C a) + (X - C a) * derivative (f /ₘ (X - C a)) = derivative f",
"ppTerm": "?m.59",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"K : Type u_1\ninst✝ : CommRing K\nf : K[X]\na : K\nkey : derivative ((X - C a) * (f /ₘ (X - C a))) = derivative (f - f %ₘ (X - C a))\n⊢ f /ₘ (X - C a) + (X - C a) * derivative (f /ₘ (X - C a)) = derivative f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.FieldDivision | {
"line": 710,
"column": 4
} | {
"line": 710,
"column": 20
} | {
"line": 710,
"column": 21
} | [
{
"pp": "case pos\nR : Type u\ninst✝¹ : Field R\np q : R[X]\ninst✝ : DecidableEq R\nhq : q ≠ 0\nhp : p = 0\n⊢ p ∈ normalizedFactors q ↔ Irreducible p ∧ p.Monic ∧ p ∣ q",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Polynomial.instNormalizationMonoid",
"UniqueFactorizationMono... | [
"case pos\nR : Type u\ninst✝¹ : Field R\np q : R[X]\ninst✝ : DecidableEq R\nhq : q ≠ 0\nhp : p = 0\n⊢ 0 ∉ normalizedFactors q"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Content | {
"line": 375,
"column": 59
} | {
"line": 375,
"column": 75
} | {
"line": 375,
"column": 75
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\n⊢ normalize (p * q).content = 1",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"congrArg",
"C... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\n⊢ IsUnit (p * q).content"
] | normalize_eq_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.FieldDivision | {
"line": 727,
"column": 4
} | {
"line": 727,
"column": 29
} | {
"line": 727,
"column": 30
} | [
{
"pp": "case inl\nR : Type u\ninst✝ : Field R\np₁ p₂ : R[X]\nh : 0 ∣ p₁ - p₂\n⊢ p₁ % 0 = p₂ % 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"EuclideanDomain.mod_zero",
"Polynomial.instMod",
"congrArg",
"id",
"instHMod",
"Field.toSemif... | [
"case inl\nR : Type u\ninst✝ : Field R\np₁ p₂ : R[X]\nh : 0 ∣ p₁ - p₂\n⊢ p₁ = p₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.FieldDivision | {
"line": 731,
"column": 34
} | {
"line": 731,
"column": 45
} | {
"line": 731,
"column": 46
} | [
{
"pp": "R : Type u\ninst✝ : Field R\np₁ p₂ q : R[X]\nh : q ∣ p₁ - p₂\nhq : q ≠ 0\n⊢ q.leadingCoeff⁻¹ ≠ 0",
"ppTerm": "?m.75",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"inv_eq_zero._simp_1",
"GroupWithZero.toMonoidWithZero",
"GroupWithZero.toDivisionMonoid",
"Divi... | [
"R : Type u\ninst✝ : Field R\np₁ p₂ q : R[X]\nh : q ∣ p₁ - p₂\nhq : q ≠ 0\n⊢ ¬q = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Content | {
"line": 427,
"column": 8
} | {
"line": 427,
"column": 24
} | {
"line": 427,
"column": 24
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nh : ∃ n r, r.natDegree = n ∧ r.IsPrimitive ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : r.natDegree = Nat.find h\nrprim : normalize (r.coeff 0) = 1\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, ... | [
"case pos\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : NormalizedGCDMonoid R\np q : R[X]\nhp : p.IsPrimitive\nhq : q.IsPrimitive\nh : ∃ n r, r.natDegree = n ∧ r.IsPrimitive ∧ p ∣ r ∧ q ∣ r\nr : R[X]\nrdeg : r.natDegree = Nat.find h\nrprim : IsUnit (r.coeff 0)\npr : p ∣ r\nqr : q ∣ r\ncon : ∃ n s, s.natDegree = n ∧ (... | normalize_eq_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet | {
"line": 532,
"column": 4
} | {
"line": 532,
"column": 22
} | {
"line": 534,
"column": 0
} | [
{
"pp": "case neg.inr\nα : Type u_1\ninst✝³ : CommMonoidWithZero α\ninst✝² : UniqueFactorizationMonoid α\ninst✝¹ : DecidableEq (Associates α)\ninst✝ : (p : Associates α) → Decidable (Irreducible p)\na b p : Associates α\nhp : Irreducible p\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d\nha : ¬a = 0\nhb :... | [] | rw [hb0, add_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet | {
"line": 538,
"column": 4
} | {
"line": 538,
"column": 19
} | {
"line": 538,
"column": 20
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝³ : CommMonoidWithZero α\ninst✝² : UniqueFactorizationMonoid α\ninst✝¹ : DecidableEq (Associates α)\ninst✝ : (p : Associates α) → Decidable (Irreducible p)\na b : Associates α\nhb : b ≠ 0\np : Associates α\nhp : Irreducible p\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Pr... | [
"case pos\nα : Type u_1\ninst✝³ : CommMonoidWithZero α\ninst✝² : UniqueFactorizationMonoid α\ninst✝¹ : DecidableEq (Associates α)\ninst✝ : (p : Associates α) → Decidable (Irreducible p)\na b : Associates α\nhb : b ≠ 0\np : Associates α\nhp : Irreducible p\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d\nk : ℕ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Central.Basic | {
"line": 55,
"column": 2
} | {
"line": 55,
"column": 61
} | {
"line": 56,
"column": 2
} | [
{
"pp": "k : Type u_1\nK : Type u_2\nD : Type u_3\ninst✝⁸ : Field k\ninst✝⁷ : Field K\ninst✝⁶ : Ring D\ninst✝⁵ : Nontrivial D\ninst✝⁴ : Algebra k K\ninst✝³ : Algebra K D\ninst✝² : Algebra k D\ninst✝¹ : IsScalarTower k K D\ninst✝ : IsCentral k D\nthis : IsCentral K D\n⊢ Function.Bijective ⇑(algebraMap k K)",
... | [
"k : Type u_1\nK : Type u_2\nD : Type u_3\ninst✝⁸ : Field k\ninst✝⁷ : Field K\ninst✝⁶ : Ring D\ninst✝⁵ : Nontrivial D\ninst✝⁴ : Algebra k K\ninst✝³ : Algebra K D\ninst✝² : Algebra k D\ninst✝¹ : IsScalarTower k K D\ninst✝ : IsCentral k D\nthis : IsCentral K D\nx : K\n⊢ ∃ a, (algebraMap k K) a = x"
] | refine ⟨FaithfulSMul.algebraMap_injective k K, fun x => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Central.Basic | {
"line": 65,
"column": 11
} | {
"line": 65,
"column": 29
} | {
"line": 65,
"column": 30
} | [
{
"pp": "K : Type u\ninst✝⁴ : CommSemiring K\nD D' : Type v\ninst✝³ : Semiring D\ninst✝² : Algebra K D\nh : IsCentral K D\ninst✝¹ : Semiring D'\ninst✝ : Algebra K D'\ne : D ≃ₐ[K] D'\nx : D'\nhx : x ∈ Subalgebra.center K D'\nk : K\nhk : (ofId K D).toRingHom k = e.symm x\n⊢ (ofId K D').toRingHom k = x",
"ppTe... | [
"K : Type u\ninst✝⁴ : CommSemiring K\nD D' : Type v\ninst✝³ : Semiring D\ninst✝² : Algebra K D\nh : IsCentral K D\ninst✝¹ : Semiring D'\ninst✝ : Algebra K D'\ne : D ≃ₐ[K] D'\nx : D'\nhx : x ∈ Subalgebra.center K D'\nk : K\nhk : (ofId K D).toRingHom k = e.symm x\n⊢ (algebraMap K D') k = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Central.Basic | {
"line": 75,
"column": 11
} | {
"line": 75,
"column": 22
} | {
"line": 75,
"column": 23
} | [
{
"pp": "K : Type u\ninst✝⁴ : CommSemiring K\nD D' : Type v\ninst✝³ : Semiring D\ninst✝² : Algebra K D\nh : IsCentral K D\ninst✝¹ : Semiring D'\ninst✝ : Algebra K D'\nz : Dᵐᵒᵖ\nhz : z ∈ Subalgebra.center K Dᵐᵒᵖ\nk : K\nhk : (ofId K D).toRingHom k = unop z\n⊢ (ofId K Dᵐᵒᵖ).toRingHom k = z",
"ppTerm": "?m.36"... | [
"K : Type u\ninst✝⁴ : CommSemiring K\nD D' : Type v\ninst✝³ : Semiring D\ninst✝² : Algebra K D\nh : IsCentral K D\ninst✝¹ : Semiring D'\ninst✝ : Algebra K D'\nz : Dᵐᵒᵖ\nhz : z ∈ Subalgebra.center K Dᵐᵒᵖ\nk : K\nhk : (ofId K D).toRingHom k = unop z\n⊢ op ((algebraMap K D) k) = z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Algebraic.Basic | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 13
} | {
"line": 97,
"column": 14
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nf : R[X]\nhf : f.natDegree ≠ 0\nhf' : f.leadingCoeff ∈ R⁰\n⊢ Transcendental R f",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝ : CommRing R\nf : R[X]\nhf : f.natDegree ≠ 0\nhf' : f.leadingCoeff ∈ R⁰\n⊢ Transcendental R f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Algebraic.Basic | {
"line": 145,
"column": 74
} | {
"line": 147,
"column": 43
} | {
"line": 149,
"column": 0
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝³ : CommRing R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : Nontrivial R\nn : ℕ\n⊢ IsAlgebraic R ↑n",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClass",
... | [] | by
rw [← map_natCast (_ : R →+* A) n]
exact isAlgebraic_algebraMap (Nat.cast n) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Algebraic.Basic | {
"line": 169,
"column": 48
} | {
"line": 169,
"column": 85
} | {
"line": 169,
"column": 86
} | [
{
"pp": "R : Type u\nS : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Nontrivial R\nM : Submonoid R\ninst✝ : IsLocalization M S\nx : S\nhx : x ≠ 0\nr : R\nm : ↥M\nh : x * (algebraMap R S) ↑(r, m).2 = (algebraMap R S) (r, m).1\n⊢ (aeval x) (C ↑m * X - C r) = 0",
"ppTerm"... | [
"R : Type u\nS : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Nontrivial R\nM : Submonoid R\ninst✝ : IsLocalization M S\nx : S\nhx : x ≠ 0\nr : R\nm : ↥M\nh : x * (algebraMap R S) ↑(r, m).2 = (algebraMap R S) (r, m).1\n⊢ (algebraMap R S) ↑m * x = (algebraMap R S) r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Algebraic.Basic | {
"line": 170,
"column": 44
} | {
"line": 170,
"column": 55
} | {
"line": 170,
"column": 56
} | [
{
"pp": "R : Type u\nS : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Nontrivial R\nM : Submonoid R\ninst✝ : IsLocalization M S\nx : S\nhx : x ≠ 0\nr : R\nm : ↥M\nh : x = mk' S (r, m).1 (r, m).2\neq : C ↑m * X - C r = 0\n⊢ r = 0",
"ppTerm": "?m.148",
"assigned": fal... | [
"R : Type u\nS : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Nontrivial R\nM : Submonoid R\ninst✝ : IsLocalization M S\nx : S\nhx : x ≠ 0\nr : R\nm : ↥M\nh : x = mk' S (r, m).1 (r, m).2\neq : C ↑m * X - C r = 0\n⊢ r = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Algebraic.Basic | {
"line": 187,
"column": 10
} | {
"line": 187,
"column": 58
} | {
"line": 187,
"column": 58
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝⁴ : CommRing R\ninst✝³ : Ring A\ninst✝² : Algebra R A\nB : Type u_2\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\na : A\nh : IsAlgebraic R a\np : R[X]\nhp : p ≠ 0\nha : (aeval a) p = 0\n⊢ (aeval (f a)) p = 0",
"ppTerm": "?m.37",
"assigned": true,
"usedCo... | [] | by rw [aeval_algHom, f.comp_apply, ha, map_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Algebraic.Basic | {
"line": 374,
"column": 24
} | {
"line": 374,
"column": 35
} | {
"line": 374,
"column": 36
} | [
{
"pp": "R : Type u\nS : Type u_1\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nx : S\ninst✝ : Invertible x\np : R[X]\nhp : p ≠ 0\nhp' : (aeval x) p = 0\n⊢ p.reverse ≠ 0",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"CommSemiring... | [
"R : Type u\nS : Type u_1\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nx : S\ninst✝ : Invertible x\np : R[X]\nhp : p ≠ 0\nhp' : (aeval x) p = 0\n⊢ ¬p = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Algebraic.Basic | {
"line": 442,
"column": 4
} | {
"line": 442,
"column": 49
} | {
"line": 442,
"column": 50
} | [
{
"pp": "R : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Ring A\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra S A\ninst✝² : Algebra R A\ninst✝¹ : IsScalarTower R S A\ninst✝ : Algebra.IsAlgebraic R A\nhinj : Function.Injective ⇑(algebraMap S A)\nx : S\n⊢ IsAlgebraic R x",
... | [
"R : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Ring A\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra S A\ninst✝² : Algebra R A\ninst✝¹ : IsScalarTower R S A\ninst✝ : Algebra.IsAlgebraic R A\nhinj : Function.Injective ⇑(algebraMap S A)\nx : S\n⊢ IsAlgebraic R x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Central.End | {
"line": 67,
"column": 45
} | {
"line": 67,
"column": 72
} | {
"line": 67,
"column": 73
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\ninst✝¹³ : Free R M\nS : Type u_4\nM₂ : Type u_5\ninst✝¹² : CommRing S\ninst✝¹¹ : IsCancelMulZero S\ninst✝¹⁰ : Module S M\ninst✝⁹ : SMulCommClass R S M\ninst✝⁸ : Algebra S R\ninst✝⁷ : IsScalarTower S R M\n... | [
"R : Type u_1\nM : Type u_3\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\ninst✝¹³ : Free R M\nS : Type u_4\nM₂ : Type u_5\ninst✝¹² : CommRing S\ninst✝¹¹ : IsCancelMulZero S\ninst✝¹⁰ : Module S M\ninst✝⁹ : SMulCommClass R S M\ninst✝⁸ : Algebra S R\ninst✝⁷ : IsScalarTower S R M\ninst✝⁶ : Add... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Central.End | {
"line": 69,
"column": 70
} | {
"line": 69,
"column": 81
} | {
"line": 69,
"column": 82
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\ninst✝¹³ : Free R M\nS : Type u_4\nM₂ : Type u_5\ninst✝¹² : CommRing S\ninst✝¹¹ : IsCancelMulZero S\ninst✝¹⁰ : Module S M\ninst✝⁹ : SMulCommClass R S M\ninst✝⁸ : Algebra S R\ninst✝⁷ : IsScalarTower S R M\n... | [
"R : Type u_1\nM : Type u_3\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\ninst✝¹³ : Free R M\nS : Type u_4\nM₂ : Type u_5\ninst✝¹² : CommRing S\ninst✝¹¹ : IsCancelMulZero S\ninst✝¹⁰ : Module S M\ninst✝⁹ : SMulCommClass R S M\ninst✝⁸ : Algebra S R\ninst✝⁷ : IsScalarTower S R M\ninst✝⁶ : Add... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.DirectedInverseSystem | {
"line": 210,
"column": 98
} | {
"line": 214,
"column": 52
} | {
"line": 216,
"column": 0
} | [
{
"pp": "ι : Type u_1\ninst✝⁷ : Preorder ι\nF₁ : ι → Type u_2\nF₂ : ι → Type u_3\nF : ι → Type u_4\nX : ι → Type u_5\nT₁ : ⦃i j : ι⦄ → i ≤ j → Sort u_6\nf₁ : (i j : ι) → (h : i ≤ j) → T₁ h\ninst✝⁶ : ⦃i j : ι⦄ → (h : i ≤ j) → FunLike (T₁ h) (F₁ i) (F₁ j)\ninst✝⁵ : DirectedSystem F₁ fun x1 x2 x3 ↦ ⇑(f₁ x1 x2 x3)\... | [] | by
choose j hzj hwj using exists_ge_ge z.1 w.1
refine ⟨ih j (f₁ _ _ hzj z.2) (f₂ _ _ hwj w.2), fun k hzk hwk ↦ ?_⟩
have ⟨i, hji, hki⟩ := exists_ge_ge j k
simp_rw [compat _ _ hji, compat _ _ hki, map_map'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Colimit.Module | {
"line": 56,
"column": 52
} | {
"line": 56,
"column": 81
} | {
"line": 56,
"column": 82
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : Semiring R\nι : Type u_2\ninst✝³ : Preorder ι\nG : ι → Type u_3\ninst✝² : (i : ι) → AddCommMonoid (G i)\ninst✝¹ : (i : ι) → Module R (G i)\nf : (i j : ι) → i ≤ j → G i →ₗ[R] G j\ninst✝ : DecidableEq ι\ns✝ : R\ni✝ j✝ : ι\nh✝ : i✝ ≤ j✝\nx✝ : G i✝\n⊢ Eqv f (s✝ • (DirectSum.lof R ι G... | [
"R : Type u_1\ninst✝⁴ : Semiring R\nι : Type u_2\ninst✝³ : Preorder ι\nG : ι → Type u_3\ninst✝² : (i : ι) → AddCommMonoid (G i)\ninst✝¹ : (i : ι) → Module R (G i)\nf : (i j : ι) → i ≤ j → G i →ₗ[R] G j\ninst✝ : DecidableEq ι\ns✝ : R\ni✝ j✝ : ι\nh✝ : i✝ ≤ j✝\nx✝ : G i✝\n⊢ Eqv f ((DirectSum.lof R ι G i✝) (s✝ • x✝)) (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Colimit.Module | {
"line": 133,
"column": 46
} | {
"line": 133,
"column": 57
} | {
"line": 133,
"column": 58
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : Semiring R\nι : Type u_2\ninst✝⁵ : Preorder ι\nG : ι → Type u_3\ninst✝⁴ : (i : ι) → AddCommMonoid (G i)\ninst✝³ : (i : ι) → Module R (G i)\nf : (i j : ι) → i ≤ j → G i →ₗ[R] G j\ninst✝² : DecidableEq ι\nP : Type u_4\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\ng : (i : ι) → G i... | [
"R : Type u_1\ninst✝⁶ : Semiring R\nι : Type u_2\ninst✝⁵ : Preorder ι\nG : ι → Type u_3\ninst✝⁴ : (i : ι) → AddCommMonoid (G i)\ninst✝³ : (i : ι) → Module R (G i)\nf : (i j : ι) → i ≤ j → G i →ₗ[R] G j\ninst✝² : DecidableEq ι\nP : Type u_4\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\ng : (i : ι) → G i →ₗ[R] P\nHg... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.DirectedInverseSystem | {
"line": 400,
"column": 4
} | {
"line": 400,
"column": 66
} | {
"line": 400,
"column": 66
} | [
{
"pp": "case inr\nι : Type u_6\nF : ι → Type u_7\nX : ι → Type u_8\ni : ι\ninst✝¹ : LinearOrder ι\nf : ⦃i j : ι⦄ → i ≤ j → F j → F i\nequiv : (j : ↑(Iio i)) → F ↑j ≃ piLT X ↑j\nnat : IsNatEquiv f equiv\nequivLim : F i ≃ ↑(limit f i)\nhi : IsSuccPrelimit i\ninst✝ : InverseSystem f\nH : ∀ (x : F i) (l : ↑(Iio i)... | [
"case inr\nι : Type u_6\nF : ι → Type u_7\nX : ι → Type u_8\ni : ι\ninst✝¹ : LinearOrder ι\nf : ⦃i j : ι⦄ → i ≤ j → F j → F i\nequiv : (j : ↑(Iio i)) → F ↑j ≃ piLT X ↑j\nnat : IsNatEquiv f equiv\nequivLim : F i ≃ ↑(limit f i)\nhi : IsSuccPrelimit i\ninst✝ : InverseSystem f\nH : ∀ (x : F i) (l : ↑(Iio i)), ↑(equivLi... | rw [piEquivLim, piSplitLE_lt (h.trans_lt hj), piSplitLE_lt hj] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.MvPolynomial.Basic | {
"line": 180,
"column": 2
} | {
"line": 180,
"column": 37
} | {
"line": 181,
"column": 2
} | [
{
"pp": "σ : Type u\nR : Type v\ninst✝ : CommSemiring R\nm : ℕ\np : MvPolynomial σ R\n⊢ p ∈ restrictTotalDegree σ R m ↔ p.totalDegree ≤ m",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"Nat.instMulZeroClass",
"Nat.instLattice",
"Lattice... | [
"σ : Type u\nR : Type v\ninst✝ : CommSemiring R\nm : ℕ\np : MvPolynomial σ R\n⊢ p ∈ restrictTotalDegree σ R m ↔ ∀ b ∈ p.support, (b.sum fun x e ↦ e) ≤ m"
] | rw [totalDegree, Finset.sup_le_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.MvPolynomial.Basic | {
"line": 180,
"column": 2
} | {
"line": 181,
"column": 5
} | {
"line": 183,
"column": 0
} | [
{
"pp": "σ : Type u\nR : Type v\ninst✝ : CommSemiring R\nm : ℕ\np : MvPolynomial σ R\n⊢ p ∈ restrictTotalDegree σ R m ↔ p.totalDegree ≤ m",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"Nat.instMulZeroClass",
"Nat.instLattice",
"Lattice... | [] | rw [totalDegree, Finset.sup_le_iff]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPolynomial.Basic | {
"line": 180,
"column": 2
} | {
"line": 181,
"column": 5
} | {
"line": 183,
"column": 0
} | [
{
"pp": "σ : Type u\nR : Type v\ninst✝ : CommSemiring R\nm : ℕ\np : MvPolynomial σ R\n⊢ p ∈ restrictTotalDegree σ R m ↔ p.totalDegree ≤ m",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"Nat.instMulZeroClass",
"Nat.instLattice",
"Lattice... | [] | rw [totalDegree, Finset.sup_le_iff]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.TensorProduct.RightExactness | {
"line": 431,
"column": 6
} | {
"line": 431,
"column": 30
} | {
"line": 431,
"column": 30
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : AddCommGroup N\ninst✝⁹ : AddCommGroup P\ninst✝⁸ : Module R M\ninst✝⁷ : Module R N\ninst✝⁶ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nhfg : Exact ⇑f ⇑g\nhg : Surjective ⇑g\nM' : Type u_6\nN... | [
"R : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : AddCommGroup N\ninst✝⁹ : AddCommGroup P\ninst✝⁸ : Module R M\ninst✝⁷ : Module R N\ninst✝⁶ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nhfg : Exact ⇑f ⇑g\nhg : Surjective ⇑g\nM' : Type u_6\nN' : Type u_7... | ← Submodule.comap_map_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Flat.Basic | {
"line": 253,
"column": 2
} | {
"line": 253,
"column": 13
} | {
"line": 253,
"column": 14
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nS : Type u_4\ninst✝² : Semiring S\ninst✝¹ : Algebra R S\ninst✝ : Flat R S\nι : Type u_5\nv : ι → M\nhv : LinearIndependent R v\n⊢ Injective ⇑(lTensor S (Finsupp.linearCombination R v) ∘ₗ ↑(finsuppScalarRight... | [
"R : Type u\nM : Type v\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nS : Type u_4\ninst✝² : Semiring S\ninst✝¹ : Algebra R S\ninst✝ : Flat R S\nι : Type u_5\nv : ι → M\nhv : LinearIndependent R v\n⊢ Injective ⇑(lTensor S (Finsupp.linearCombination R v))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.Basic | {
"line": 270,
"column": 2
} | {
"line": 270,
"column": 13
} | {
"line": 270,
"column": 14
} | [
{
"pp": "R : Type u\nM : Type v\nN : Type u_1\ninst✝⁸ : CommSemiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : Module R N\nS : Type u_4\ninst✝³ : Semiring S\ninst✝² : Algebra R S\ninst✝¹ : FaithfulSMul R S\ninst✝ : Flat R N\nf g : M →ₗ[R] N\nh : (baseChangeHom R S M N)... | [
"R : Type u\nM : Type v\nN : Type u_1\ninst✝⁸ : CommSemiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : Module R N\nS : Type u_4\ninst✝³ : Semiring S\ninst✝² : Algebra R S\ninst✝¹ : FaithfulSMul R S\ninst✝ : Flat R N\nf g : M →ₗ[R] N\nh : (baseChangeHom R S M N) f = (baseCh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.TensorProduct.RightExactness | {
"line": 612,
"column": 2
} | {
"line": 617,
"column": 5
} | {
"line": 619,
"column": 0
} | [
{
"pp": "R : Type u_4\nS : Type u_5\ninst✝¹² : CommRing R\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R S\nA : Type u_6\nB : Type u_7\nC : Type u_8\ninst✝⁹ : Ring A\ninst✝⁸ : Ring B\ninst✝⁷ : Ring C\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : Algebra R C\ninst✝³ : Algebra S A\ninst✝² : Algebra S B\ninst✝... | [] | rw [← Submodule.restrictScalars_inj R]
have : (RingHom.ker (map f (AlgHom.id R C))).restrictScalars R =
LinearMap.ker (LinearMap.rTensor C (f.restrictScalars R).toLinearMap) := rfl
rw [this, Ideal.map_includeLeft_eq]
rw [(rTensor_exact C (f.restrictScalars R).toLinearMap.exact_subtype_ker_map hf).linearMap_ke... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.TensorProduct.RightExactness | {
"line": 612,
"column": 2
} | {
"line": 617,
"column": 5
} | {
"line": 619,
"column": 0
} | [
{
"pp": "R : Type u_4\nS : Type u_5\ninst✝¹² : CommRing R\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R S\nA : Type u_6\nB : Type u_7\nC : Type u_8\ninst✝⁹ : Ring A\ninst✝⁸ : Ring B\ninst✝⁷ : Ring C\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : Algebra R C\ninst✝³ : Algebra S A\ninst✝² : Algebra S B\ninst✝... | [] | rw [← Submodule.restrictScalars_inj R]
have : (RingHom.ker (map f (AlgHom.id R C))).restrictScalars R =
LinearMap.ker (LinearMap.rTensor C (f.restrictScalars R).toLinearMap) := rfl
rw [this, Ideal.map_includeLeft_eq]
rw [(rTensor_exact C (f.restrictScalars R).toLinearMap.exact_subtype_ker_map hf).linearMap_ke... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Flat.Basic | {
"line": 335,
"column": 6
} | {
"line": 335,
"column": 45
} | {
"line": 335,
"column": 46
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : Flat R M\nN : Type u_1\nN' : Type u_2\nN'' : Type u_3\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : AddCommGroup N'\ninst✝³ : AddCommGroup N''\ninst✝² : Module R N\ninst✝¹ : Module R N'\ninst✝ : Module R N''\nf : N... | [
"R : Type u\nM : Type v\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : Flat R M\nN : Type u_1\nN' : Type u_2\nN'' : Type u_3\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : AddCommGroup N'\ninst✝³ : AddCommGroup N''\ninst✝² : Module R N\ninst✝¹ : Module R N'\ninst✝ : Module R N''\nf : N →ₗ[R] N'\ng... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.Basic | {
"line": 352,
"column": 6
} | {
"line": 352,
"column": 45
} | {
"line": 352,
"column": 46
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : Flat R M\nN : Type u_1\nN' : Type u_2\nN'' : Type u_3\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : AddCommGroup N'\ninst✝³ : AddCommGroup N''\ninst✝² : Module R N\ninst✝¹ : Module R N'\ninst✝ : Module R N''\nf : N... | [
"R : Type u\nM : Type v\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : Flat R M\nN : Type u_1\nN' : Type u_2\nN'' : Type u_3\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : AddCommGroup N'\ninst✝³ : AddCommGroup N''\ninst✝² : Module R N\ninst✝¹ : Module R N'\ninst✝ : Module R N''\nf : N →ₗ[R] N'\ng... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.Basic | {
"line": 364,
"column": 2
} | {
"line": 364,
"column": 23
} | {
"line": 364,
"column": 24
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Small.{v', u} R\nH :\n ∀ ⦃N N' N'' : Type v'⦄ [inst : AddCommGroup N] [inst_1 : AddCommGroup N'] [inst_2 : AddCommGroup N'']\n [inst_3 : Module R N] [inst_4 : Module R N'] [inst_5 : Module R N''] ⦃f :... | [
"R : Type u\nM : Type v\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Small.{v', u} R\nH :\n ∀ ⦃N N' N'' : Type v'⦄ [inst : AddCommGroup N] [inst_1 : AddCommGroup N'] [inst_2 : AddCommGroup N'']\n [inst_3 : Module R N] [inst_4 : Module R N'] [inst_5 : Module R N''] ⦃f : N →ₗ[R] N'⦄... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.Basic | {
"line": 386,
"column": 2
} | {
"line": 386,
"column": 23
} | {
"line": 386,
"column": 24
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Small.{v', u} R\nH :\n ∀ ⦃N N' N'' : Type v'⦄ [inst : AddCommGroup N] [inst_1 : AddCommGroup N'] [inst_2 : AddCommGroup N'']\n [inst_3 : Module R N] [inst_4 : Module R N'] [inst_5 : Module R N''] ⦃f :... | [
"R : Type u\nM : Type v\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Small.{v', u} R\nH :\n ∀ ⦃N N' N'' : Type v'⦄ [inst : AddCommGroup N] [inst_1 : AddCommGroup N'] [inst_2 : AddCommGroup N'']\n [inst_3 : Module R N] [inst_4 : Module R N'] [inst_5 : Module R N''] ⦃f : N →ₗ[R] N'⦄... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.Basic | {
"line": 592,
"column": 2
} | {
"line": 592,
"column": 17
} | {
"line": 592,
"column": 18
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : AddCommMonoid N\ninst✝⁸ : Module R M\ninst✝⁷ : Module R N\nM₁ : Type u_5\nM₂ : Type u_6\nN₂ : Type u_8\ninst✝⁶ : AddCommMonoid M₁\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₁\ninst✝³ : Module R M... | [
"R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : AddCommMonoid N\ninst✝⁸ : Module R M\ninst✝⁷ : Module R N\nM₁ : Type u_5\nM₂ : Type u_6\nN₂ : Type u_8\ninst✝⁶ : AddCommMonoid M₁\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₁\ninst✝³ : Module R M₂\ninst✝² : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.Basic | {
"line": 599,
"column": 2
} | {
"line": 599,
"column": 17
} | {
"line": 599,
"column": 18
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : AddCommMonoid N\ninst✝⁸ : Module R M\ninst✝⁷ : Module R N\nM₁ : Type u_5\nM₂ : Type u_6\nN₁ : Type u_7\ninst✝⁶ : AddCommMonoid M₁\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₁\ninst✝³ : Module R M... | [
"R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : AddCommMonoid N\ninst✝⁸ : Module R M\ninst✝⁷ : Module R N\nM₁ : Type u_5\nM₂ : Type u_6\nN₁ : Type u_7\ninst✝⁶ : AddCommMonoid M₁\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₁\ninst✝³ : Module R M₂\ninst✝² : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.Basic | {
"line": 609,
"column": 2
} | {
"line": 609,
"column": 17
} | {
"line": 609,
"column": 18
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝¹⁴ : CommSemiring R\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : AddCommMonoid N\ninst✝¹¹ : Module R M\ninst✝¹⁰ : Module R N\nM₁ : Type u_5\nM₂ : Type u_6\nN₁ : Type u_7\nN₂ : Type u_8\ninst✝⁹ : AddCommMonoid M₁\ninst✝⁸ : AddCommMonoid M₂\ninst✝⁷ : Module R M₁\ni... | [
"R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝¹⁴ : CommSemiring R\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : AddCommMonoid N\ninst✝¹¹ : Module R M\ninst✝¹⁰ : Module R N\nM₁ : Type u_5\nM₂ : Type u_6\nN₁ : Type u_7\nN₂ : Type u_8\ninst✝⁹ : AddCommMonoid M₁\ninst✝⁸ : AddCommMonoid M₂\ninst✝⁷ : Module R M₁\ninst✝⁶ : Modu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.Basic | {
"line": 619,
"column": 2
} | {
"line": 619,
"column": 17
} | {
"line": 619,
"column": 18
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝¹⁴ : CommSemiring R\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : AddCommMonoid N\ninst✝¹¹ : Module R M\ninst✝¹⁰ : Module R N\nM₁ : Type u_5\nM₂ : Type u_6\nN₁ : Type u_7\nN₂ : Type u_8\ninst✝⁹ : AddCommMonoid M₁\ninst✝⁸ : AddCommMonoid M₂\ninst✝⁷ : Module R M₁\ni... | [
"R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝¹⁴ : CommSemiring R\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : AddCommMonoid N\ninst✝¹¹ : Module R M\ninst✝¹⁰ : Module R N\nM₁ : Type u_5\nM₂ : Type u_6\nN₁ : Type u_7\nN₂ : Type u_8\ninst✝⁹ : AddCommMonoid M₁\ninst✝⁸ : AddCommMonoid M₂\ninst✝⁷ : Module R M₁\ninst✝⁶ : Modu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.Basic | {
"line": 655,
"column": 59
} | {
"line": 655,
"column": 74
} | {
"line": 655,
"column": 75
} | [
{
"pp": "R : Type u_1\nC : Type u_2\nA : Type u_3\ninst✝⁵ : CommSemiring R\ninst✝⁴ : CommSemiring C\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : Algebra R C\ninst✝ : Module.Flat R C\nh : ∀ (B : Subalgebra R A), B.FG → IsReduced (C ⊗[R] ↥B)\nh_contra : ¬IsReduced (C ⊗[R] A)\nD : Subalgebra R A\nh_inj : F... | [
"R : Type u_1\nC : Type u_2\nA : Type u_3\ninst✝⁵ : CommSemiring R\ninst✝⁴ : CommSemiring C\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : Algebra R C\ninst✝ : Module.Flat R C\nh : ∀ (B : Subalgebra R A), B.FG → IsReduced (C ⊗[R] ↥B)\nh_contra : ¬IsReduced (C ⊗[R] A)\nD : Subalgebra R A\nh_inj : Function.Inje... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.Basic | {
"line": 653,
"column": 2
} | {
"line": 655,
"column": 81
} | {
"line": 656,
"column": 2
} | [
{
"pp": "R : Type u_1\nC : Type u_2\nA : Type u_3\ninst✝⁵ : CommSemiring R\ninst✝⁴ : CommSemiring C\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : Algebra R C\ninst✝ : Module.Flat R C\nh : ∀ (B : Subalgebra R A), B.FG → IsReduced (C ⊗[R] ↥B)\nh_contra : ¬IsReduced (C ⊗[R] A)\nD : Subalgebra R A\nh_inj : F... | [
"R : Type u_1\nC : Type u_2\nA : Type u_3\ninst✝⁵ : CommSemiring R\ninst✝⁴ : CommSemiring C\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : Algebra R C\ninst✝ : Module.Flat R C\nh : ∀ (B : Subalgebra R A), B.FG → IsReduced (C ⊗[R] ↥B)\nh_contra : ¬IsReduced (C ⊗[R] A)\nD : Subalgebra R A\nh_inj : Function.Inje... | have h_notReduced : ¬IsReduced (C ⊗[R] D) := by
simp_rw [isReduced_iff, not_forall]
exact ⟨z, (IsNilpotent.map_iff h_inj).mp hx.right, (by simpa [·] using hx.1)⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.BigOperators.Group.Finset.Indicator | {
"line": 55,
"column": 4
} | {
"line": 55,
"column": 77
} | {
"line": 57,
"column": 0
} | [
{
"pp": "case refine_2\nι : Type u_1\nκ : Type u_2\nβ : Type u_4\ninst✝¹ : CommMonoid β\ns : Finset ι\nf : ι → κ → β\nt : ι → Set κ\ng : ι → κ\ninst✝ : DecidablePred fun i ↦ g i ∈ t i\n⊢ ∏ x ∈ s with g x ∉ t x, (t x).mulIndicator (f x) (g x) = 1",
"ppTerm": "?refine_2",
"assigned": true,
"usedConsta... | [] | exact prod_eq_one fun x hx ↦ mulIndicator_of_notMem (mem_filter.1 hx).2 _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.BigOperators.Group.Finset.Indicator | {
"line": 55,
"column": 4
} | {
"line": 55,
"column": 77
} | {
"line": 57,
"column": 0
} | [
{
"pp": "case refine_2\nι : Type u_1\nκ : Type u_2\nβ : Type u_4\ninst✝¹ : CommMonoid β\ns : Finset ι\nf : ι → κ → β\nt : ι → Set κ\ng : ι → κ\ninst✝ : DecidablePred fun i ↦ g i ∈ t i\n⊢ ∏ x ∈ s with g x ∉ t x, (t x).mulIndicator (f x) (g x) = 1",
"ppTerm": "?refine_2",
"assigned": true,
"usedConsta... | [] | exact prod_eq_one fun x hx ↦ mulIndicator_of_notMem (mem_filter.1 hx).2 _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.BigOperators.Group.Finset.Indicator | {
"line": 55,
"column": 4
} | {
"line": 55,
"column": 77
} | {
"line": 57,
"column": 0
} | [
{
"pp": "case refine_2\nι : Type u_1\nκ : Type u_2\nβ : Type u_4\ninst✝¹ : CommMonoid β\ns : Finset ι\nf : ι → κ → β\nt : ι → Set κ\ng : ι → κ\ninst✝ : DecidablePred fun i ↦ g i ∈ t i\n⊢ ∏ x ∈ s with g x ∉ t x, (t x).mulIndicator (f x) (g x) = 1",
"ppTerm": "?refine_2",
"assigned": true,
"usedConsta... | [] | exact prod_eq_one fun x hx ↦ mulIndicator_of_notMem (mem_filter.1 hx).2 _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Expect | {
"line": 137,
"column": 54
} | {
"line": 137,
"column": 65
} | {
"line": 137,
"column": 66
} | [
{
"pp": "ι : Type u_1\nM : Type u_4\ninst✝³ : AddCommMonoid M\ninst✝² : Module ℚ≥0 M\ns t : Finset ι\nf g : ι → M\np q : ι → Prop\ninst✝¹ : DecidablePred p\ninst✝ : DecidablePred q\nhst : s = t\nhpq : ∀ i ∈ t, p i ↔ q i\nh : ∀ i ∈ t, q i → f i = g i\n⊢ ∀ i ∈ {i ∈ t | q i}, f i = g i",
"ppTerm": "?m.45",
... | [
"ι : Type u_1\nM : Type u_4\ninst✝³ : AddCommMonoid M\ninst✝² : Module ℚ≥0 M\ns t : Finset ι\nf g : ι → M\np q : ι → Prop\ninst✝¹ : DecidablePred p\ninst✝ : DecidablePred q\nhst : s = t\nhpq : ∀ i ∈ t, p i ↔ q i\nh : ∀ i ∈ t, q i → f i = g i\n⊢ ∀ i ∈ t, q i → f i = g i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.Expect | {
"line": 141,
"column": 2
} | {
"line": 141,
"column": 37
} | {
"line": 141,
"column": 38
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝¹ : AddCommMonoid M\ninst✝ : Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ 𝔼 i ∈ s, ∑ j ∈ t, f i j = ∑ j ∈ t, 𝔼 i ∈ s, f i j",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddC... | [
"ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝¹ : AddCommMonoid M\ninst✝ : Module ℚ≥0 M\ns : Finset ι\nt : Finset κ\nf : ι → κ → M\n⊢ ∑ x ∈ s, ∑ x_1 ∈ t, (↑(#s))⁻¹ • f x x_1 = ∑ x ∈ t, ∑ x_1 ∈ s, (↑(#s))⁻¹ • f x_1 x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finset.Density | {
"line": 117,
"column": 2
} | {
"line": 117,
"column": 45
} | {
"line": 117,
"column": 46
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝² : Fintype α\ninst✝¹ : Fintype β\ninst✝ : DecidableEq β\nf : α → β\nhf : Bijective f\ns : Finset α\n⊢ (image f s).dens = s.dens",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_2\nβ : Type u_3\ninst✝² : Fintype α\ninst✝¹ : Fintype β\ninst✝ : DecidableEq β\nf : α → β\nhf : Bijective f\ns : Finset α\n⊢ (image f s).dens = s.dens"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finset.Density | {
"line": 157,
"column": 4
} | {
"line": 157,
"column": 15
} | {
"line": 157,
"column": 16
} | [
{
"pp": "case inr\nα : Type u_2\ninst✝ : Fintype α\ns : Finset α\nh✝ : Nonempty α\n⊢ s.dens ≤ 1",
"ppTerm": "?inr",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case inr\nα : Type u_2\ninst✝ : Fintype α\ns : Finset α\nh✝ : Nonempty α\n⊢ s.dens ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.Group.Finset.Gaps | {
"line": 44,
"column": 2
} | {
"line": 44,
"column": 59
} | {
"line": 45,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : CommGroup β\nF : Finset (α × α)\nk : ℕ\nh : #F = k\na b : α\nf : α → α → β\np : Fin (k + 1) → α × α := ⋯\n⊢ ∏ i ∈ range k, f (p ↑i).2 (p ↑i.succ).1 = ∏ z ∈ F, f z.1 z.2",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
... | [
"case hi\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : CommGroup β\nF : Finset (α × α)\nk : ℕ\nh : #F = k\na b : α\nf : α → α → β\np : Fin (k + 1) → α × α := ⋯\n⊢ ∀ a ∈ range k, ((p ↑a).2, (p ↑a.succ).1) ∈ F",
"case i_inj\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : CommGroup β\nF :... | apply prod_bij (fun (i : ℕ) hi ↦ ((p i).2, (p i.succ).1)) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.BigOperators.Expect | {
"line": 275,
"column": 19
} | {
"line": 275,
"column": 42
} | {
"line": 275,
"column": 43
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝² : AddCommMonoid M\ninst✝¹ : Module ℚ≥0 M\nf : ι → M\nt : Finset κ\ninst✝ : DecidableEq ι\nm : κ → ι\nhm : Set.InjOn m ↑t\n⊢ (↑(#(image m t)))⁻¹ • ∑ x ∈ image m t, f x = (↑(#t))⁻¹ • ∑ x ∈ t, f (m x)",
"ppTerm": "?m.26",
"assigned": true,
"used... | [
"ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝² : AddCommMonoid M\ninst✝¹ : Module ℚ≥0 M\nf : ι → M\nt : Finset κ\ninst✝ : DecidableEq ι\nm : κ → ι\nhm : Set.InjOn m ↑t\n⊢ (↑(#t))⁻¹ • ∑ x ∈ image m t, f x = (↑(#t))⁻¹ • ∑ x ∈ t, f (m x)"
] | card_image_of_injOn hm, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Group.EvenFunction | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 42
} | {
"line": 155,
"column": 43
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\ninst✝² : AddCommGroup β\ninst✝¹ : IsAddTorsionFree β\ninst✝ : InvolutiveNeg α\nf : α → β\nhf : Function.Odd f\ns : Finset α\nhs : Finset.map (Equiv.toEmbedding (Equiv.neg α)) s = s\n⊢ s.sum f = 0",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"u... | [
"α : Type u_3\nβ : Type u_4\ninst✝² : AddCommGroup β\ninst✝¹ : IsAddTorsionFree β\ninst✝ : InvolutiveNeg α\nf : α → β\nhf : Function.Odd f\ns : Finset α\nhs : Finset.map (Equiv.toEmbedding (Equiv.neg α)) s = s\n⊢ s.sum f = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.Expect | {
"line": 411,
"column": 4
} | {
"line": 411,
"column": 15
} | {
"line": 411,
"column": 16
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝³ : Fintype ι\ninst✝² : Fintype κ\ninst✝¹ : AddCommMonoid M\ninst✝ : Module ℚ≥0 M\ne : ι → κ\nhe : Bijective e\nf : ι → M\ng : κ → M\nh : ∀ (i : ι), f i = g (e i)\n⊢ Set.SurjOn e ↑univ ↑univ",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants... | [
"ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝³ : Fintype ι\ninst✝² : Fintype κ\ninst✝¹ : AddCommMonoid M\ninst✝ : Module ℚ≥0 M\ne : ι → κ\nhe : Bijective e\nf : ι → M\ng : κ → M\nh : ∀ (i : ι), f i = g (e i)\n⊢ Surjective e"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.Expect | {
"line": 424,
"column": 35
} | {
"line": 424,
"column": 46
} | {
"line": 424,
"column": 47
} | [
{
"pp": "ι : Type u_1\nM : Type u_4\ninst✝³ : Fintype ι\ninst✝² : AddCommMonoid M\ninst✝¹ : Module ℚ≥0 M\np : ι → Prop\ninst✝ : DecidablePred p\nh : ∀ (i j : ι), p i → p j → i = j\na : M\n⊢ ∀ i ∈ univ, ∀ j ∈ univ, p i → p j → i = j",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Eq.... | [
"ι : Type u_1\nM : Type u_4\ninst✝³ : Fintype ι\ninst✝² : AddCommMonoid M\ninst✝¹ : Module ℚ≥0 M\np : ι → Prop\ninst✝ : DecidablePred p\nh : ∀ (i j : ι), p i → p j → i = j\na : M\n⊢ ∀ (i j : ι), p i → p j → i = j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.ModEq | {
"line": 37,
"column": 2
} | {
"line": 37,
"column": 13
} | {
"line": 37,
"column": 14
} | [
{
"pp": "n : ℕ\nl : List ℕ\nh : ∀ x ∈ l, x ≡ 1 [MOD n]\n⊢ l.prod ≡ 1 [MOD n]",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\nl : List ℕ\nh : ∀ x ∈ l, x ≡ 1 [MOD n]\n⊢ l.prod ≡ 1 [MOD n]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.ModEq | {
"line": 43,
"column": 2
} | {
"line": 43,
"column": 13
} | {
"line": 43,
"column": 14
} | [
{
"pp": "α : Type u_1\nn : ℕ\nl : List α\nf : α → ℕ\nh : ∀ x ∈ l, f x ≡ 0 [MOD n]\n⊢ (List.map f l).sum ≡ 0 [MOD n]",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nn : ℕ\nl : List α\nf : α → ℕ\nh : ∀ x ∈ l, f x ≡ 0 [MOD n]\n⊢ (List.map f l).sum ≡ 0 [MOD n]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.ModEq | {
"line": 46,
"column": 2
} | {
"line": 46,
"column": 13
} | {
"line": 46,
"column": 14
} | [
{
"pp": "n : ℕ\nl : List ℕ\nh : ∀ x ∈ l, x ≡ 0 [MOD n]\n⊢ l.sum ≡ 0 [MOD n]",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\nl : List ℕ\nh : ∀ x ∈ l, x ≡ 0 [MOD n]\n⊢ l.sum ≡ 0 [MOD n]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.ModEq | {
"line": 51,
"column": 2
} | {
"line": 51,
"column": 13
} | {
"line": 51,
"column": 14
} | [
{
"pp": "case mk\nα : Type u_1\nn : ℕ\nf g : α → ℕ\ns : Multiset α\nl : List α\nh : ∀ x ∈ Quot.mk (⇑(List.isSetoid α)) l, f x ≡ g x [MOD n]\n⊢ (Multiset.map f (Quot.mk (⇑(List.isSetoid α)) l)).prod ≡ (Multiset.map g (Quot.mk (⇑(List.isSetoid α)) l)).prod [MOD\n n]",
"ppTerm": "?mk",
"assigned": true,... | [
"case mk\nα : Type u_1\nn : ℕ\nf g : α → ℕ\ns : Multiset α\nl : List α\nh : ∀ x ∈ Quot.mk (⇑(List.isSetoid α)) l, f x ≡ g x [MOD n]\n⊢ (List.map f l).prod ≡ (List.map g l).prod [MOD n]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.ModEq | {
"line": 55,
"column": 2
} | {
"line": 55,
"column": 13
} | {
"line": 55,
"column": 14
} | [
{
"pp": "α : Type u_1\nn : ℕ\nf : α → ℕ\ns : Multiset α\nh : ∀ x ∈ s, f x ≡ 1 [MOD n]\n⊢ (Multiset.map f s).prod ≡ 1 [MOD n]",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nn : ℕ\nf : α → ℕ\ns : Multiset α\nh : ∀ x ∈ s, f x ≡ 1 [MOD n]\n⊢ (Multiset.map f s).prod ≡ 1 [MOD n]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.ModEq | {
"line": 58,
"column": 2
} | {
"line": 58,
"column": 13
} | {
"line": 58,
"column": 14
} | [
{
"pp": "n : ℕ\ns : Multiset ℕ\nh : ∀ x ∈ s, x ≡ 1 [MOD n]\n⊢ s.prod ≡ 1 [MOD n]",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\ns : Multiset ℕ\nh : ∀ x ∈ s, x ≡ 1 [MOD n]\n⊢ s.prod ≡ 1 [MOD n]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.ModEq | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 13
} | {
"line": 63,
"column": 14
} | [
{
"pp": "case mk\nα : Type u_1\nn : ℕ\nf g : α → ℕ\ns : Multiset α\nl : List α\nh : ∀ x ∈ Quot.mk (⇑(List.isSetoid α)) l, f x ≡ g x [MOD n]\n⊢ (Multiset.map f (Quot.mk (⇑(List.isSetoid α)) l)).sum ≡ (Multiset.map g (Quot.mk (⇑(List.isSetoid α)) l)).sum [MOD n]",
"ppTerm": "?mk",
"assigned": true,
"u... | [
"case mk\nα : Type u_1\nn : ℕ\nf g : α → ℕ\ns : Multiset α\nl : List α\nh : ∀ x ∈ Quot.mk (⇑(List.isSetoid α)) l, f x ≡ g x [MOD n]\n⊢ (List.map f l).sum ≡ (List.map g l).sum [MOD n]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.ModEq | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 13
} | {
"line": 67,
"column": 14
} | [
{
"pp": "α : Type u_1\nn : ℕ\nf : α → ℕ\ns : Multiset α\nh : ∀ x ∈ s, f x ≡ 0 [MOD n]\n⊢ (Multiset.map f s).sum ≡ 0 [MOD n]",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nn : ℕ\nf : α → ℕ\ns : Multiset α\nh : ∀ x ∈ s, f x ≡ 0 [MOD n]\n⊢ (Multiset.map f s).sum ≡ 0 [MOD n]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.ModEq | {
"line": 70,
"column": 2
} | {
"line": 70,
"column": 13
} | {
"line": 70,
"column": 14
} | [
{
"pp": "n : ℕ\ns : Multiset ℕ\nh : ∀ x ∈ s, x ≡ 0 [MOD n]\n⊢ s.sum ≡ 0 [MOD n]",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\ns : Multiset ℕ\nh : ∀ x ∈ s, x ≡ 0 [MOD n]\n⊢ s.sum ≡ 0 [MOD n]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.ModEq | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 13
} | {
"line": 78,
"column": 14
} | [
{
"pp": "α : Type u_1\nn : ℕ\nf : α → ℕ\ns : Finset α\nh : ∀ x ∈ s, f x ≡ 1 [MOD n]\n⊢ ∏ x ∈ s, f x ≡ 1 [MOD n]",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nn : ℕ\nf : α → ℕ\ns : Finset α\nh : ∀ x ∈ s, f x ≡ 1 [MOD n]\n⊢ ∏ x ∈ s, f x ≡ 1 [MOD n]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.ModEq | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 13
} | {
"line": 86,
"column": 14
} | [
{
"pp": "α : Type u_1\nn : ℕ\nf : α → ℕ\ns : Finset α\nh : ∀ x ∈ s, f x ≡ 0 [MOD n]\n⊢ ∑ x ∈ s, f x ≡ 0 [MOD n]",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nn : ℕ\nf : α → ℕ\ns : Finset α\nh : ∀ x ∈ s, f x ≡ 0 [MOD n]\n⊢ ∑ x ∈ s, f x ≡ 0 [MOD n]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.Ring.Nat | {
"line": 49,
"column": 4
} | {
"line": 49,
"column": 37
} | {
"line": 49,
"column": 38
} | [
{
"pp": "case refine_1\nι : Type u_1\nM : Type u_2\nf : ι → M\ns : Finset M\nhb : ∀ b ∈ s, {a | f a = b}.Finite\nt : Finset M := ⋯.toFinset\nht : (f ⁻¹' ↑t).Finite\n⊢ image f ht.toFinset = t",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset... | [
"case refine_1\nι : Type u_1\nM : Type u_2\nf : ι → M\ns : Finset M\nhb : ∀ b ∈ s, {a | f a = b}.Finite\nt : Finset M := ⋯.toFinset\nht : (f ⁻¹' ↑t).Finite\n⊢ f '' f ⁻¹' ↑s = ↑s ∩ Set.range f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.ModEq | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 13
} | {
"line": 130,
"column": 14
} | [
{
"pp": "n : ℤ\nl : List ℤ\nh : ∀ x ∈ l, x ≡ 1 [ZMOD n]\n⊢ l.prod ≡ 1 [ZMOD n]",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℤ\nl : List ℤ\nh : ∀ x ∈ l, x ≡ 1 [ZMOD n]\n⊢ l.prod ≡ 1 [ZMOD n]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.ModEq | {
"line": 136,
"column": 2
} | {
"line": 136,
"column": 13
} | {
"line": 136,
"column": 14
} | [
{
"pp": "α : Type u_1\nn : ℤ\nl : List α\nf : α → ℤ\nh : ∀ x ∈ l, f x ≡ 0 [ZMOD n]\n⊢ (List.map f l).sum ≡ 0 [ZMOD n]",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nn : ℤ\nl : List α\nf : α → ℤ\nh : ∀ x ∈ l, f x ≡ 0 [ZMOD n]\n⊢ (List.map f l).sum ≡ 0 [ZMOD n]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.ModEq | {
"line": 139,
"column": 2
} | {
"line": 139,
"column": 13
} | {
"line": 139,
"column": 14
} | [
{
"pp": "n : ℤ\nl : List ℤ\nh : ∀ x ∈ l, x ≡ 0 [ZMOD n]\n⊢ l.sum ≡ 0 [ZMOD n]",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℤ\nl : List ℤ\nh : ∀ x ∈ l, x ≡ 0 [ZMOD n]\n⊢ l.sum ≡ 0 [ZMOD n]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.Ring.Nat | {
"line": 52,
"column": 63
} | {
"line": 52,
"column": 74
} | {
"line": 52,
"column": 75
} | [
{
"pp": "ι : Type u_1\nM : Type u_2\nf : ι → M\ns : Finset M\nhb : ∀ b ∈ s, {a | f a = b}.Finite\nt : Finset M := ⋯.toFinset\nht : (f ⁻¹' ↑t).Finite\nm : M\nhm : m ∈ t\nthis : {a | f a = m} ⊆ ↑ht.toFinset\na : ι\n⊢ a ∈ {a ∈ ht.toFinset | f a = m} ↔ a ∈ ⋯.toFinset",
"ppTerm": "?m.142",
"assigned": true,
... | [
"ι : Type u_1\nM : Type u_2\nf : ι → M\ns : Finset M\nhb : ∀ b ∈ s, {a | f a = b}.Finite\nt : Finset M := ⋯.toFinset\nht : (f ⁻¹' ↑t).Finite\nm : M\nhm : m ∈ t\nthis : {a | f a = m} ⊆ ↑ht.toFinset\na : ι\n⊢ f a = m → f a ∈ t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.ModEq | {
"line": 144,
"column": 2
} | {
"line": 144,
"column": 13
} | {
"line": 144,
"column": 14
} | [
{
"pp": "case mk\nα : Type u_1\nn : ℤ\nf g : α → ℤ\ns : Multiset α\nl : List α\nh : ∀ x ∈ Quot.mk (⇑(List.isSetoid α)) l, f x ≡ g x [ZMOD n]\n⊢ (Multiset.map f (Quot.mk (⇑(List.isSetoid α)) l)).prod ≡ (Multiset.map g (Quot.mk (⇑(List.isSetoid α)) l)).prod [ZMOD\n n]",
"ppTerm": "?mk",
"assigned": tru... | [
"case mk\nα : Type u_1\nn : ℤ\nf g : α → ℤ\ns : Multiset α\nl : List α\nh : ∀ x ∈ Quot.mk (⇑(List.isSetoid α)) l, f x ≡ g x [ZMOD n]\n⊢ (List.map f l).prod ≡ (List.map g l).prod [ZMOD n]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.ModEq | {
"line": 148,
"column": 2
} | {
"line": 148,
"column": 13
} | {
"line": 148,
"column": 14
} | [
{
"pp": "α : Type u_1\nn : ℤ\nf : α → ℤ\ns : Multiset α\nh : ∀ x ∈ s, f x ≡ 1 [ZMOD n]\n⊢ (Multiset.map f s).prod ≡ 1 [ZMOD n]",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nn : ℤ\nf : α → ℤ\ns : Multiset α\nh : ∀ x ∈ s, f x ≡ 1 [ZMOD n]\n⊢ (Multiset.map f s).prod ≡ 1 [ZMOD n]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.ModEq | {
"line": 151,
"column": 2
} | {
"line": 151,
"column": 13
} | {
"line": 151,
"column": 14
} | [
{
"pp": "n : ℤ\ns : Multiset ℤ\nh : ∀ x ∈ s, x ≡ 1 [ZMOD n]\n⊢ s.prod ≡ 1 [ZMOD n]",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℤ\ns : Multiset ℤ\nh : ∀ x ∈ s, x ≡ 1 [ZMOD n]\n⊢ s.prod ≡ 1 [ZMOD n]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.ModEq | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 13
} | {
"line": 156,
"column": 14
} | [
{
"pp": "case mk\nα : Type u_1\nn : ℤ\nf g : α → ℤ\ns : Multiset α\nl : List α\nh : ∀ x ∈ Quot.mk (⇑(List.isSetoid α)) l, f x ≡ g x [ZMOD n]\n⊢ (Multiset.map f (Quot.mk (⇑(List.isSetoid α)) l)).sum ≡ (Multiset.map g (Quot.mk (⇑(List.isSetoid α)) l)).sum [ZMOD n]",
"ppTerm": "?mk",
"assigned": true,
... | [
"case mk\nα : Type u_1\nn : ℤ\nf g : α → ℤ\ns : Multiset α\nl : List α\nh : ∀ x ∈ Quot.mk (⇑(List.isSetoid α)) l, f x ≡ g x [ZMOD n]\n⊢ (List.map f l).sum ≡ (List.map g l).sum [ZMOD n]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.ModEq | {
"line": 160,
"column": 2
} | {
"line": 160,
"column": 13
} | {
"line": 160,
"column": 14
} | [
{
"pp": "α : Type u_1\nn : ℤ\nf : α → ℤ\ns : Multiset α\nh : ∀ x ∈ s, f x ≡ 0 [ZMOD n]\n⊢ (Multiset.map f s).sum ≡ 0 [ZMOD n]",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nn : ℤ\nf : α → ℤ\ns : Multiset α\nh : ∀ x ∈ s, f x ≡ 0 [ZMOD n]\n⊢ (Multiset.map f s).sum ≡ 0 [ZMOD n]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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