module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Data.Nat.Factorization.Basic | {
"line": 573,
"column": 2
} | {
"line": 573,
"column": 13
} | {
"line": 573,
"column": 14
} | [
{
"pp": "case h\na b m n : ℕ\nhmn : m.Coprime n\nh : a ^ m = b ^ n\nha0 : ¬a = 0\nhn0 : ¬n = 0\nfactors : ℕ →₀ ℕ := mapRange (fun x ↦ x / n) ⋯ a.factorization\nc : ℕ := factors.prod fun x1 x2 ↦ x1 ^ x2\nhc : c = factors.prod fun x1 x2 ↦ x1 ^ x2\np : ℕ\nfoo : ∀ p ∈ factors.support, Prime p\n⊢ m * a.factorization... | [
"case h\na b m n : ℕ\nhmn : m.Coprime n\nh : a ^ m = b ^ n\nha0 : ¬a = 0\nhn0 : ¬n = 0\nfactors : ℕ →₀ ℕ := mapRange (fun x ↦ x / n) ⋯ a.factorization\nc : ℕ := factors.prod fun x1 x2 ↦ x1 ^ x2\nhc : c = factors.prod fun x1 x2 ↦ x1 ^ x2\np : ℕ\nfoo : ∀ p ∈ factors.support, Prime p\n⊢ m * a.factorization p = n * b.f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.OrderOfElement | {
"line": 558,
"column": 2
} | {
"line": 558,
"column": 34
} | {
"line": 558,
"column": 35
} | [
{
"pp": "case hg1\nR : Type u_6\ninst✝¹ : Ring R\ninst✝ : Nontrivial R\nh : ¬ringChar R = 2\n⊢ -1 ≠ 1",
"ppTerm": "?hg1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"AddGroupWithOne.toAd... | [
"case hg1\nR : Type u_6\ninst✝¹ : Ring R\ninst✝ : Nontrivial R\nh : ¬ringChar R = 2\n⊢ ¬ringChar R = 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.CharP.Lemmas | {
"line": 60,
"column": 2
} | {
"line": 60,
"column": 13
} | {
"line": 60,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\np : ℕ\nhp : Nat.Prime p\nx y : R\nh : Commute x y\n⊢ (x + y) ^ p = x ^ p + y ^ p + ↑p * ∑ k ∈ Ioo 0 p, x ^ k * y ^ (p - k) * ↑(p.choose k / p)",
"ppTerm": "?m.70",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : Semiring R\np : ℕ\nhp : Nat.Prime p\nx y : R\nh : Commute x y\n⊢ (x + y) ^ p = x ^ p + y ^ p + ↑p * ∑ k ∈ Ioo 0 p, x ^ k * y ^ (p - k) * ↑(p.choose k / p)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.CharP.Lemmas | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 13
} | {
"line": 66,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\np : ℕ\nhp : Nat.Prime p\nx y : R\nh : Commute x y\n⊢ (x + y) ^ p = x ^ p + y ^ p + ↑p * x * y * ∑ k ∈ Ioo 0 p, x ^ (k - 1) * y ^ (p - k - 1) * ↑(p.choose k / p)",
"ppTerm": "?m.90",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals... | [
"R : Type u_1\ninst✝ : Semiring R\np : ℕ\nhp : Nat.Prime p\nx y : R\nh : Commute x y\n⊢ (x + y) ^ p = x ^ p + y ^ p + ↑p * x * y * ∑ k ∈ Ioo 0 p, x ^ (k - 1) * y ^ (p - k - 1) * ↑(p.choose k / p)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.Choose.Factorization | {
"line": 240,
"column": 17
} | {
"line": 240,
"column": 37
} | {
"line": 240,
"column": 37
} | [
{
"pp": "case refine_2\np n : ℕ\nn_big : 2 < n\np_le_n : p ≤ n\nbig : 2 * n < 3 * p\n⊢ n ≤ n + n - n",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instOrderedSub",
"Nat.instIsOrderedAddMonoid",
"AddLeftCancelSemigroup.toIsLeftCancelAdd",
"... | [
"case refine_2\np n : ℕ\nn_big : 2 < n\np_le_n : p ≤ n\nbig : 2 * n < 3 * p\n⊢ n ≤ n"
] | add_tsub_cancel_left | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.OrderOfElement | {
"line": 685,
"column": 13
} | {
"line": 685,
"column": 24
} | {
"line": 685,
"column": 25
} | [
{
"pp": "G : Type u_1\ninst✝ : RightCancelMonoid G\nx : G\nm✝ n m k : ℕ\nhmn : m ≤ m + k\nh : x ^ (m + k) = x ^ m\nhk : x ^ k = 1\n⊢ m + k ≡ m [MOD orderOf x]",
"ppTerm": "?m.166",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"id",
"orderOf",
"instHAdd",
... | [
"G : Type u_1\ninst✝ : RightCancelMonoid G\nx : G\nm✝ n m k : ℕ\nhmn : m ≤ m + k\nh : x ^ (m + k) = x ^ m\nhk : x ^ k = 1\n⊢ orderOf x ∣ k"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.Choose.Factorization | {
"line": 247,
"column": 52
} | {
"line": 247,
"column": 81
} | {
"line": 247,
"column": 81
} | [
{
"pp": "case succ\np n✝ n : ℕ\nhn : n < p → n !.factorization p = 0\nh : n + 1 < p\n⊢ n.succ.factorization p = 0",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Nat.instMulZeroClass",
"congrArg",
"AddMonoid.toAddZeroClass",
... | [
"case succ\np n✝ n : ℕ\nhn : n < p → n !.factorization p = 0\nh : n + 1 < p\n⊢ 0 = 0"
] | factorization_eq_zero_of_lt h | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.OrderOfElement | {
"line": 689,
"column": 45
} | {
"line": 689,
"column": 56
} | {
"line": 689,
"column": 57
} | [
{
"pp": "G : Type u_1\ninst✝ : RightCancelMonoid G\nx : G\nm✝ n m k : ℕ\nhmn : m ≤ m + k\nh : m + k ≡ m [MOD orderOf x]\n⊢ m + k ≡ m + 0 [MOD orderOf x]",
"ppTerm": "?m.208",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"congrArg",
"AddMonoid.toAddZeroClass",
... | [
"G : Type u_1\ninst✝ : RightCancelMonoid G\nx : G\nm✝ n m k : ℕ\nhmn : m ≤ m + k\nh : m + k ≡ m [MOD orderOf x]\n⊢ orderOf x ∣ k"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.OrderOfElement | {
"line": 884,
"column": 2
} | {
"line": 884,
"column": 42
} | {
"line": 885,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nx : G\nhx : IsOfFinOrder x\na : ↥(zpowers x)\n⊢ x ^ ↑((finEquivZPowers hx).symm a) = ↑a",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\ninst✝ : Group G\nx : G\nhx : IsOfFinOrder x\na : ↥(zpowers x)\n⊢ x ^ ↑((finEquivZPowers hx).symm a) = ↑a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.OrderOfElement | {
"line": 943,
"column": 2
} | {
"line": 943,
"column": 87
} | {
"line": 944,
"column": 2
} | [
{
"pp": "case refine_1\nG : Type u_1\ninst✝² : Monoid G\nn : ℕ\ninst✝¹ : Fintype G\ninst✝ : DecidableEq G\nhn : n ≠ 0\n⊢ (↑n.divisors).PairwiseDisjoint fun m ↦ {x | orderOf x = m}",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Nat.mem_divisors._simp_1",
"SetLike.mem_coe._... | [
"case refine_2\nG : Type u_1\ninst✝² : Monoid G\nn : ℕ\ninst✝¹ : Fintype G\ninst✝ : DecidableEq G\nhn : n ≠ 0\n⊢ #(n.divisors.biUnion fun m ↦ {x | orderOf x = m}) = #{x | x ^ n = 1}"
] | · simp +contextual [Set.PairwiseDisjoint, Set.Pairwise, disjoint_iff, Finset.ext_iff] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.OrderOfElement | {
"line": 953,
"column": 2
} | {
"line": 953,
"column": 13
} | {
"line": 953,
"column": 14
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Monoid G\nx : G\ninst✝ : Finite G\nval✝ : Fintype G\n⊢ orderOf x ≤ Nat.card G",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Fintype.card",
"id",
"Nat.card",
"LE.le",
"instLENat",
"order... | [
"G : Type u_1\ninst✝¹ : Monoid G\nx : G\ninst✝ : Finite G\nval✝ : Fintype G\n⊢ orderOf x ≤ card G"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.OrderOfElement | {
"line": 1064,
"column": 45
} | {
"line": 1064,
"column": 56
} | {
"line": 1064,
"column": 57
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nx : G\nh : Injective fun n ↦ x ^ n\nn : ℕ\nhn : 0 < n\nhx : x ^ n = 1\n⊢ (fun n ↦ x ^ n) ↑n = (fun n ↦ x ^ n) 0",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOneClass",
... | [
"G : Type u_1\ninst✝ : Group G\nx : G\nh : Injective fun n ↦ x ^ n\nn : ℕ\nhn : 0 < n\nhx : x ^ n = 1\n⊢ x ^ n = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.OrderOfElement | {
"line": 1172,
"column": 2
} | {
"line": 1172,
"column": 13
} | {
"line": 1172,
"column": 14
} | [
{
"pp": "G : Type u_6\ninst✝ : Group G\ns : Subgroup G\nx : G\nhx : x ∈ s\n⊢ orderOf x ∣ Nat.card ↥s",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_6\ninst✝ : Group G\ns : Subgroup G\nx : G\nhx : x ∈ s\n⊢ orderOf x ∣ Nat.card ↥s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MonoidAlgebra.Division | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 28
} | {
"line": 105,
"column": 29
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : AddCommMonoid G\ninst✝ : IsCancelAdd G\na : G\n⊢ of' k G a /ᵒᶠ a = 1",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"k : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : AddCommMonoid G\ninst✝ : IsCancelAdd G\na : G\n⊢ of' k G a /ᵒᶠ a = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MonoidAlgebra.Division | {
"line": 127,
"column": 12
} | {
"line": 127,
"column": 76
} | {
"line": 129,
"column": 0
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝¹ : Semiring k\ninst✝ : AddCommMonoid G\nx : k[G]\ng g' : G\nh : ∃ d, g' = g + d\n⊢ (x %ᵒᶠ g).coeff g' = 0",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instDecidableNot",
"Finsupp.filter_apply_neg",
"Classical.... | [] | exact Finsupp.filter_apply_neg _ _ <| by rwa [Classical.not_not] | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.MonoidAlgebra.Division | {
"line": 127,
"column": 12
} | {
"line": 127,
"column": 76
} | {
"line": 129,
"column": 0
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝¹ : Semiring k\ninst✝ : AddCommMonoid G\nx : k[G]\ng g' : G\nh : ∃ d, g' = g + d\n⊢ (x %ᵒᶠ g).coeff g' = 0",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instDecidableNot",
"Finsupp.filter_apply_neg",
"Classical.... | [] | exact Finsupp.filter_apply_neg _ _ <| by rwa [Classical.not_not] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MonoidAlgebra.Division | {
"line": 127,
"column": 12
} | {
"line": 127,
"column": 76
} | {
"line": 129,
"column": 0
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝¹ : Semiring k\ninst✝ : AddCommMonoid G\nx : k[G]\ng g' : G\nh : ∃ d, g' = g + d\n⊢ (x %ᵒᶠ g).coeff g' = 0",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instDecidableNot",
"Finsupp.filter_apply_neg",
"Classical.... | [] | exact Finsupp.filter_apply_neg _ _ <| by rwa [Classical.not_not] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MonoidAlgebra.Division | {
"line": 148,
"column": 4
} | {
"line": 148,
"column": 25
} | {
"line": 148,
"column": 26
} | [
{
"pp": "case inr.h\nk : Type u_1\nG : Type u_2\ninst✝¹ : Semiring k\ninst✝ : AddCommMonoid G\ng : G\nx : k[G]\ng' : G\nh : ¬∃ d, g' = g + d\n⊢ ∀ (d : G), g + d ≠ g'",
"ppTerm": "?inr.h",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"Ne",
"instHAdd",... | [
"case inr.h\nk : Type u_1\nG : Type u_2\ninst✝¹ : Semiring k\ninst✝ : AddCommMonoid G\ng : G\nx : k[G]\ng' : G\nh : ¬∃ d, g' = g + d\n⊢ ∀ (d : G), ¬g' = g + d"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MonoidAlgebra.Division | {
"line": 156,
"column": 4
} | {
"line": 156,
"column": 35
} | {
"line": 156,
"column": 36
} | [
{
"pp": "case inr.h\nk : Type u_1\nG : Type u_2\ninst✝¹ : Semiring k\ninst✝ : AddCommMonoid G\nx : k[G]\ng g' : G\nh : ¬∃ d, g' = g + d\n⊢ ∀ (d : G), d + g ≠ g'",
"ppTerm": "?inr.h",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"Ne",
"add_comm",
... | [
"case inr.h\nk : Type u_1\nG : Type u_2\ninst✝¹ : Semiring k\ninst✝ : AddCommMonoid G\nx : k[G]\ng g' : G\nh : ¬∃ d, g' = g + d\n⊢ ∀ (d : G), ¬g' = g + d"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MonoidAlgebra.Division | {
"line": 159,
"column": 2
} | {
"line": 159,
"column": 28
} | {
"line": 159,
"column": 29
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝¹ : Semiring k\ninst✝ : AddCommMonoid G\ng : G\n⊢ of' k G g %ᵒᶠ g = 0",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"k : Type u_1\nG : Type u_2\ninst✝¹ : Semiring k\ninst✝ : AddCommMonoid G\ng : G\n⊢ of' k G g %ᵒᶠ g = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MonoidAlgebra.Division | {
"line": 168,
"column": 4
} | {
"line": 168,
"column": 25
} | {
"line": 168,
"column": 26
} | [
{
"pp": "case inr.h\nk : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : AddCommMonoid G\ninst✝ : IsCancelAdd G\nx : k[G]\ng g' : G\nh : ¬∃ d, g' = g + d\n⊢ ∀ (d : G), g + d ≠ g'",
"ppTerm": "?inr.h",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"Ne"... | [
"case inr.h\nk : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : AddCommMonoid G\ninst✝ : IsCancelAdd G\nx : k[G]\ng g' : G\nh : ¬∃ d, g' = g + d\n⊢ ∀ (d : G), ¬g' = g + d"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.OrderOfElement | {
"line": 1345,
"column": 33
} | {
"line": 1345,
"column": 68
} | {
"line": 1345,
"column": 69
} | [
{
"pp": "G : Type u_1\ninst✝² : Ring G\ninst✝¹ : LinearOrder G\ninst✝ : IsStrictOrderedRing G\nx : G\nh : |x| ≠ 1\nn : ℕ\nhn : 0 < n\nhx : x ^ n = 1\n⊢ |x| ^ n = 1",
"ppTerm": "?m.35",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\ninst✝² : Ring G\ninst✝¹ : LinearOrder G\ninst✝ : IsStrictOrderedRing G\nx : G\nh : |x| ≠ 1\nn : ℕ\nhn : 0 < n\nhx : x ^ n = 1\n⊢ |x| ^ n = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.OrderOfElement | {
"line": 1385,
"column": 2
} | {
"line": 1385,
"column": 52
} | {
"line": 1385,
"column": 53
} | [
{
"pp": "α : Type u_4\nβ : Type u_5\ninst✝¹ : Monoid α\ninst✝ : Monoid β\na : α\nb : β\n⊢ IsOfFinOrder a → IsOfFinOrder b → IsOfFinOrder (a, b)",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Nat.lcm",
"Eq.mpr",
"congrArg",
"id",
"_private.Mathlib.GroupTheory.... | [
"α : Type u_4\nβ : Type u_5\ninst✝¹ : Monoid α\ninst✝ : Monoid β\na : α\nb : β\n⊢ 0 < orderOf a → 0 < orderOf b → 0 < (orderOf a).lcm (orderOf b)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Inductions | {
"line": 59,
"column": 15
} | {
"line": 59,
"column": 39
} | {
"line": 59,
"column": 40
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\nh : p.divX = 0\n⊢ p = C (p.coeff 0)",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝ : Semiring R\np : R[X]\nh : p.divX = 0\n⊢ p = C (p.coeff 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Inductions | {
"line": 95,
"column": 4
} | {
"line": 95,
"column": 44
} | {
"line": 95,
"column": 45
} | [
{
"pp": "case φ_k\nR : Type u\ninst✝ : Semiring R\np f : R[X]\n⊢ f.natDegree < 1 → divX_hom f = 0",
"ppTerm": "?φ_k",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Preorder.toLT",
"LinearOrderedCommMonoidWithZ... | [
"case φ_k\nR : Type u\ninst✝ : Semiring R\np f : R[X]\n⊢ f.natDegree = 0 → f = C (f.coeff 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Inductions | {
"line": 115,
"column": 71
} | {
"line": 115,
"column": 82
} | {
"line": 115,
"column": 83
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\nhp0 : p ≠ 0\nthis : Nontrivial R\nh : p.degree ≤ 0\nh' : C (p.coeff 0) ≠ 0\n⊢ ¬C ((C (p.coeff 0)).coeff 0) = 0",
"ppTerm": "?m.137",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"congrArg",
"RingHom",
... | [
"R : Type u\ninst✝ : Semiring R\np : R[X]\nhp0 : p ≠ 0\nthis : Nontrivial R\nh : p.degree ≤ 0\nh' : C (p.coeff 0) ≠ 0\n⊢ ¬p.coeff 0 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Inductions | {
"line": 118,
"column": 10
} | {
"line": 118,
"column": 69
} | {
"line": 118,
"column": 70
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\nhp0 : p ≠ 0\nthis : Nontrivial R\nh : ¬p.degree ≤ 0\n⊢ p.divX ≠ 0",
"ppTerm": "?m.153",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"congrArg",
"RingHom",
"id",
"Ne",
"instOfNatNat",
... | [
"R : Type u\ninst✝ : Semiring R\np : R[X]\nhp0 : p ≠ 0\nthis : Nontrivial R\nh : ¬p.degree ≤ 0\n⊢ ¬p = C (p.coeff 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.RingDivision | {
"line": 194,
"column": 2
} | {
"line": 194,
"column": 59
} | {
"line": 194,
"column": 60
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\np : R[X]\na b c : R\nha : IsUnit a\nthis✝ : Invertible a := ha.invertible\nthis :\n rootMultiplicity (a * c + b) ((p.comp (C a * X + C b)).comp (C ⅟a * X + C (-⅟a * b))) ≤\n rootMultiplicity (⅟a * (a * c + b) + -⅟a * b) (p.comp (C a * X + C b))\n⊢ rootMultiplicity (a... | [
"R : Type u\ninst✝ : CommRing R\np : R[X]\na b c : R\nha : IsUnit a\nthis✝ : Invertible a := ha.invertible\nthis :\n rootMultiplicity (a * c + b) ((p.comp (C a * X + C b)).comp (C ⅟a * X + C (-⅟a * b))) ≤\n rootMultiplicity (⅟a * (a * c + b) + -⅟a * b) (p.comp (C a * X + C b))\n⊢ rootMultiplicity (a * c + b) p ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.RingDivision | {
"line": 204,
"column": 48
} | {
"line": 212,
"column": 26
} | {
"line": 214,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\np : R[X]\nhp : p.Monic\n⊢ ((-1) ^ p.natDegree * p.comp (-X)).Monic",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"one_pow",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Polynomial.C",
"NegZeroClass.toNeg",
... | [] | by
simp only [Monic]
calc
((-1) ^ p.natDegree * p.comp (-X)).leadingCoeff =
(p.comp (-X) * C ((-1) ^ p.natDegree)).leadingCoeff := by
simp [mul_comm]
_ = 1 := by
apply monic_mul_C_of_leadingCoeff_mul_eq_one
simp [← pow_add, hp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.RingDivision | {
"line": 230,
"column": 38
} | {
"line": 230,
"column": 67
} | {
"line": 230,
"column": 68
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : R[X]\nhp1 : p.degree = 1\nhm : p.Monic\n⊢ p = X - C (-p.coeff 0)",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"NegZeroClass.toNeg",
"RingHom.instRingHomClass",
"Ring... | [
"R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : R[X]\nhp1 : p.degree = 1\nhm : p.Monic\n⊢ p = X + C (p.coeff 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.RingDivision | {
"line": 269,
"column": 4
} | {
"line": 269,
"column": 15
} | {
"line": 269,
"column": 16
} | [
{
"pp": "case hc\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\na b : R\nha : a ≠ 0\nhab : IsRelPrime a b\n⊢ IsRelPrime ((C a * X + C b).coeff 0) ((C a * X + C b).coeff 1)",
"ppTerm": "?hc",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"HMul.hMul",
"c... | [
"case hc\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\na b : R\nha : a ≠ 0\nhab : IsRelPrime a b\n⊢ IsRelPrime b a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.RingDivision | {
"line": 315,
"column": 8
} | {
"line": 315,
"column": 19
} | {
"line": 315,
"column": 20
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DecidableEq R\np : R[X]\nhp : p ≠ 0\nthis : Decidable (∃ x, p.IsRoot x)\nh : ∃ x, p.IsRoot x\nx : R\nhx : p.IsRoot x\nhpd : 0 < p.degree\nhd0 : p /ₘ (X - C x) ≠ 0\nwf : (p /ₘ (X - C x)).degree < p.degree\nt : Multiset R\nhtd : ↑t.card ≤ (p /... | [
"R : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DecidableEq R\np : R[X]\nhp : p ≠ 0\nthis : Decidable (∃ x, p.IsRoot x)\nh : ∃ x, p.IsRoot x\nx : R\nhx : p.IsRoot x\nhpd : 0 < p.degree\nhd0 : p /ₘ (X - C x) ≠ 0\nwf : (p /ₘ (X - C x)).degree < p.degree\nt : Multiset R\nhtd : ↑t.card ≤ (p /ₘ (X - C x))... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Div | {
"line": 91,
"column": 31
} | {
"line": 91,
"column": 65
} | {
"line": 91,
"column": 66
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\nhp : p.Monic\nr : R[X]\nhdeg : (p * r).natDegree ≤ p.natDegree\nhr : r ≠ 0\n⊢ r.natDegree = 0",
"ppTerm": "?m.72",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝ : Semiring R\np : R[X]\nhp : p.Monic\nr : R[X]\nhdeg : (p * r).natDegree ≤ p.natDegree\nhr : r ≠ 0\n⊢ r.natDegree = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Expand | {
"line": 194,
"column": 20
} | {
"line": 194,
"column": 46
} | {
"line": 194,
"column": 47
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\np : ℕ\nhp : p ≠ 0\nf : R[X]\nn : ℕ\nhn : f.natDegree + 1 ≤ n\n⊢ f.natDegree + 1 ≤ n * 1",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"Nat.instMulOneCl... | [
"R : Type u\ninst✝ : CommSemiring R\np : ℕ\nhp : p ≠ 0\nf : R[X]\nn : ℕ\nhn : f.natDegree + 1 ≤ n\n⊢ f.natDegree + 1 ≤ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Expand | {
"line": 220,
"column": 24
} | {
"line": 220,
"column": 91
} | {
"line": 220,
"column": 92
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\np : ℕ\nhp : p ≠ 0\nf g : R[X]\nn : ℕ\nx : ℕ × ℕ\nhx : x ∈ ↑(antidiagonal n)\ny : ℕ × ℕ\nhy : y ∈ ↑(antidiagonal n)\neq : (fun x ↦ (x.1 * p, x.2 * p)) x = (fun x ↦ (x.1 * p, x.2 * p)) y\n⊢ x = y",
"ppTerm": "?m.104",
"assigned": true,
"usedConstants": [
... | [
"R : Type u\ninst✝ : CommSemiring R\np : ℕ\nhp : p ≠ 0\nf g : R[X]\nn : ℕ\nx : ℕ × ℕ\nhx : x ∈ ↑(antidiagonal n)\ny : ℕ × ℕ\nhy : y ∈ ↑(antidiagonal n)\neq : (fun x ↦ (x.1 * p, x.2 * p)) x = (fun x ↦ (x.1 * p, x.2 * p)) y\n⊢ x.1 = y.1 ∧ x.2 = y.2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Div | {
"line": 269,
"column": 2
} | {
"line": 269,
"column": 13
} | {
"line": 269,
"column": 14
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\np q : R[X]\n⊢ q ∣ p %ₘ q ↔ q ∣ p",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝ : Ring R\np q : R[X]\n⊢ q ∣ p %ₘ q ↔ q ∣ p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Div | {
"line": 349,
"column": 14
} | {
"line": 349,
"column": 34
} | {
"line": 349,
"column": 34
} | [
{
"pp": "case neg\nR : Type u\ninst✝ : Ring R\nf g : R[X]\nhg : g.Monic\na✝ : Nontrivial R\nhfg : ¬f /ₘ g = 0\nhgf : g.degree ≤ f.degree\nthis : g.natDegree + (f /ₘ g).natDegree = f.natDegree\nhf : f ≠ 0\n⊢ (f /ₘ g).natDegree = g.natDegree + (f /ₘ g).natDegree - g.natDegree",
"ppTerm": "?neg✝",
"assigne... | [
"case neg\nR : Type u\ninst✝ : Ring R\nf g : R[X]\nhg : g.Monic\na✝ : Nontrivial R\nhfg : ¬f /ₘ g = 0\nhgf : g.degree ≤ f.degree\nthis : g.natDegree + (f /ₘ g).natDegree = f.natDegree\nhf : f ≠ 0\n⊢ (f /ₘ g).natDegree = (f /ₘ g).natDegree"
] | add_tsub_cancel_left | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Div | {
"line": 373,
"column": 77
} | {
"line": 373,
"column": 93
} | {
"line": 373,
"column": 94
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nf g q r : R[X]\nhg : g.Monic\nh : r + g * q = f ∧ r.degree < g.degree\na✝ : Nontrivial R\nh₁ : r - f %ₘ g = -g * (q - f /ₘ g)\nh₂ : (r - f %ₘ g).degree = (g * (q - f /ₘ g)).degree\nh₄ : (r - f %ₘ g).degree < g.degree\nh₅ : q - f /ₘ g = 0\n⊢ r - f %ₘ g = 0",
"ppTerm": "?m... | [
"R : Type u\ninst✝ : Ring R\nf g q r : R[X]\nhg : g.Monic\nh : r + g * q = f ∧ r.degree < g.degree\na✝ : Nontrivial R\nh₁ : r - f %ₘ g = -g * (q - f /ₘ g)\nh₂ : (r - f %ₘ g).degree = (g * (q - f /ₘ g)).degree\nh₄ : (r - f %ₘ g).degree < g.degree\nh₅ : q - f /ₘ g = 0\n⊢ r - f %ₘ g = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Derivative | {
"line": 344,
"column": 26
} | {
"line": 344,
"column": 47
} | {
"line": 344,
"column": 48
} | [
{
"pp": "case succ.e_a.refine_1\nR : Type u\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nIH : (⇑derivative)^[n] (p * q) = ∑ k ∈ range n.succ, n.choose k • ((⇑derivative)^[n - k] p * (⇑derivative)^[k] q)\n⊢ ∑ k ∈ range n, n.choose (k + 1) • ((⇑derivative)^[n - (k + 1) + 1] p * (⇑derivative)^[k + 1] q) =\n ∑ x ∈ ra... | [
"case succ.e_a.refine_1\nR : Type u\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nIH : (⇑derivative)^[n] (p * q) = ∑ k ∈ range n.succ, n.choose k • ((⇑derivative)^[n - k] p * (⇑derivative)^[k] q)\n⊢ ∑ k ∈ range n, n.choose (k + 1) • ((⇑derivative)^[n - (k + 1) + 1] p * (⇑derivative)^[k + 1] q) =\n ∑ x ∈ range n, n.cho... | Nat.choose_succ_self, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Derivative | {
"line": 379,
"column": 2
} | {
"line": 379,
"column": 13
} | {
"line": 379,
"column": 14
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\n⊢ derivativeFinsupp 1 = Finsupp.single 0 1",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝ : Semiring R\n⊢ derivativeFinsupp 1 = Finsupp.single 0 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Div | {
"line": 491,
"column": 2
} | {
"line": 491,
"column": 13
} | {
"line": 491,
"column": 14
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\na✝ : Nontrivial R\nh : IsField R[X]\nthis : Field R[X] := h.toField\n⊢ False",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝ : Ring R\na✝ : Nontrivial R\nh : IsField R[X]\nthis : Field R[X] := h.toField\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Finite | {
"line": 157,
"column": 4
} | {
"line": 157,
"column": 97
} | {
"line": 157,
"column": 98
} | [
{
"pp": "α : Type u\nι : Type u_2\ninst✝ : Finite ι\nl : ι → Filter α\nhd : Pairwise (Disjoint on l)\n⊢ Pairwise fun i j ↦ ∃ s t, Disjoint ↑s ↑t",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"CompleteBooleanAlgebra.toCompleteDistribLat... | [
"α : Type u\nι : Type u_2\ninst✝ : Finite ι\nl : ι → Filter α\nhd : Pairwise (Disjoint on l)\n⊢ ∀ ⦃i j : ι⦄, i ≠ j → ∃ a ∈ l i, ∃ a_1 ∈ l j, Disjoint a a_1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Pi | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 13
} | {
"line": 88,
"column": 14
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\nf : (i : ι) → Filter (α i)\ns : (i : ι) → Set (α i)\ninst✝ : ∀ (i : ι), (f i).NeBot\nI : Set ι\nh : I.pi s ∈ pi f\ni : ι\nhi : i ∈ I\nI' : Set ι\nt : (i : ι) → Set (α i)\nhtf : ∀ (i : ι), t i ∈ f i\nhts : I'.pi t ⊆ I.pi s\nx : α i\nhx : x ∈ t i\ng : (i : ι) → α i\nhg : ∀... | [
"ι : Type u_1\nα : ι → Type u_2\nf : (i : ι) → Filter (α i)\ns : (i : ι) → Set (α i)\ninst✝ : ∀ (i : ι), (f i).NeBot\nI : Set ι\nh : I.pi s ∈ pi f\ni : ι\nhi : i ∈ I\nI' : Set ι\nt : (i : ι) → Set (α i)\nhtf : ∀ (i : ι), t i ∈ f i\nhts : I'.pi t ⊆ I.pi s\nx : α i\nhx : x ∈ t i\ng : (i : ι) → α i\nhg : ∀ (i : ι), g ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Div | {
"line": 596,
"column": 6
} | {
"line": 596,
"column": 34
} | {
"line": 596,
"column": 35
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\np : R[X]\na : R\na✝ : Nontrivial R\nh : eval a (p %ₘ (X - C a)) = eval a p\nthis✝ : (p %ₘ (X - C a)).degree < 1\nthis : (p %ₘ (X - C a)).degree ≤ 0\n⊢ p %ₘ (X - C a) = C (eval a p)",
"ppTerm": "?m.147",
"assigned": true,
"usedConstants": [
"WithBot.inst... | [
"R : Type u\ninst✝ : CommRing R\np : R[X]\na : R\na✝ : Nontrivial R\nh : eval a (C ((p %ₘ (X - C a)).coeff 0)) = eval a p\nthis✝ : (p %ₘ (X - C a)).degree < 1\nthis : (p %ₘ (X - C a)).degree ≤ 0\n⊢ p %ₘ (X - C a) = C (eval a p)"
] | eq_C_of_degree_le_zero this, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Filter.Finite | {
"line": 198,
"column": 27
} | {
"line": 198,
"column": 43
} | {
"line": 198,
"column": 44
} | [
{
"pp": "α : Type u\nβ : Type v\ns : Finset α\nf : α → Filter β\np : Subtype (Membership.mem s) → Set β\nhp : ∀ (i : Subtype (Membership.mem s)), p i ∈ f ↑i\nh : ⋂ i, p i ∈ ⨅ x, f ↑x\na : α\nha : a ∈ s\n⊢ (fun a ↦ if h : a ∈ s then p ⟨a, h⟩ else univ) a ∈ f a",
"ppTerm": "?m.92",
"assigned": true,
"... | [
"α : Type u\nβ : Type v\ns : Finset α\nf : α → Filter β\np : Subtype (Membership.mem s) → Set β\nhp : ∀ (i : Subtype (Membership.mem s)), p i ∈ f ↑i\nh : ⋂ i, p i ∈ ⨅ x, f ↑x\na : α\nha : a ∈ s\n⊢ p ⟨a, ⋯⟩ ∈ f a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Div | {
"line": 597,
"column": 6
} | {
"line": 597,
"column": 34
} | {
"line": 597,
"column": 35
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\np : R[X]\na : R\na✝ : Nontrivial R\nh : (p %ₘ (X - C a)).coeff 0 = eval a p\nthis✝ : (p %ₘ (X - C a)).degree < 1\nthis : (p %ₘ (X - C a)).degree ≤ 0\n⊢ p %ₘ (X - C a) = C (eval a p)",
"ppTerm": "?m.237",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"R : Type u\ninst✝ : CommRing R\np : R[X]\na : R\na✝ : Nontrivial R\nh : (p %ₘ (X - C a)).coeff 0 = eval a p\nthis✝ : (p %ₘ (X - C a)).degree < 1\nthis : (p %ₘ (X - C a)).degree ≤ 0\n⊢ C ((p %ₘ (X - C a)).coeff 0) = C (eval a p)"
] | eq_C_of_degree_le_zero this, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Div | {
"line": 625,
"column": 4
} | {
"line": 626,
"column": 11
} | {
"line": 626,
"column": 12
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\na b : R\np : R[X]\nthis : X - C b ∣ p - C (eval b p)\n⊢ a - b ∣ eval a p - eval b p",
"ppTerm": "?m.44",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝ : CommRing R\na b : R\np : R[X]\nthis : X - C b ∣ p - C (eval b p)\n⊢ a - b ∣ eval a p - eval b p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Div | {
"line": 630,
"column": 2
} | {
"line": 630,
"column": 50
} | {
"line": 630,
"column": 51
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\np : R[X]\nx : R\nh : p.IsRoot x\n⊢ x ∣ p.coeff 0",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
"Dvd.dvd",
"Polynomial.coeff_zero_eq_eval_zero",
"CommRing.toNonUnitalCommRing",
"congr... | [
"R : Type u\ninst✝ : CommRing R\np : R[X]\nx : R\nh : p.IsRoot x\n⊢ x ∣ eval 0 p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Finite | {
"line": 218,
"column": 2
} | {
"line": 218,
"column": 13
} | {
"line": 218,
"column": 14
} | [
{
"pp": "case intro\nα : Type u\nι : Sort w\ninst✝ : Finite ι\nf : ι → Set α\nval✝ : Fintype (PLift ι)\n⊢ ⨅ i, 𝓟 (f i.down) = 𝓟 (⋂ i, f i.down)",
"ppTerm": "?intro",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case intro\nα : Type u\nι : Sort w\ninst✝ : Finite ι\nf : ι → Set α\nval✝ : Fintype (PLift ι)\n⊢ ⨅ i, 𝓟 (f i.down) = 𝓟 (⋂ i, f i.down)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Finite | {
"line": 249,
"column": 2
} | {
"line": 249,
"column": 52
} | {
"line": 249,
"column": 53
} | [
{
"pp": "α : Type u\nι : Sort u_2\ninst✝ : Finite ι\nl : Filter α\np : ι → α → Prop\n⊢ (∀ᶠ (x : α) in l, ∀ (i : ι), p i x) ↔ ∀ (i : ι), ∀ᶠ (x : α) in l, p i x",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"congrArg",
"Set.iInter",... | [
"α : Type u\nι : Sort u_2\ninst✝ : Finite ι\nl : Filter α\np : ι → α → Prop\n⊢ ⋂ i, {x | p i x} ∈ l ↔ ∀ (i : ι), {x | p i x} ∈ l"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Finite | {
"line": 254,
"column": 2
} | {
"line": 254,
"column": 52
} | {
"line": 254,
"column": 53
} | [
{
"pp": "α : Type u\nι : Type u_2\nI : Set ι\nhI : I.Finite\nl : Filter α\np : ι → α → Prop\n⊢ (∀ᶠ (x : α) in l, ∀ i ∈ I, p i x) ↔ ∀ i ∈ I, ∀ᶠ (x : α) in l, p i x",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"congrArg",
"Set.iIn... | [
"α : Type u\nι : Type u_2\nI : Set ι\nhI : I.Finite\nl : Filter α\np : ι → α → Prop\n⊢ ⋂ i ∈ I, {x | p i x} ∈ l ↔ ∀ i ∈ I, {x | p i x} ∈ l"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Finite | {
"line": 286,
"column": 2
} | {
"line": 286,
"column": 56
} | {
"line": 286,
"column": 57
} | [
{
"pp": "α : Type u\nι : Type u_2\nf : Filter ι\ns : ι → Set α\nt : Set α\nht : t.Finite\nhs : ∀ a ∈ t, ∀ᶠ (i : ι) in f, a ∈ s i\n⊢ ∀ᶠ (i : ι) in f, t ⊆ s i",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Filter.Eventually",
"Membership.mem",
"id",
... | [
"α : Type u\nι : Type u_2\nf : Filter ι\ns : ι → Set α\nt : Set α\nht : t.Finite\nhs : ∀ a ∈ t, ∀ᶠ (i : ι) in f, a ∈ s i\n⊢ ∀ i ∈ t, ∀ᶠ (x : ι) in f, i ∈ s x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Pi | {
"line": 223,
"column": 2
} | {
"line": 223,
"column": 13
} | {
"line": 223,
"column": 14
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\nf : (i : ι) → Filter (α i)\n⊢ pi f = ⊥ ↔ ∃ i, f i = ⊥",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_1\nα : ι → Type u_2\nf : (i : ι) → Filter (α i)\n⊢ pi f = ⊥ ↔ ∃ i, f i = ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Pi | {
"line": 293,
"column": 2
} | {
"line": 293,
"column": 26
} | {
"line": 293,
"column": 27
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\nf : (i : ι) → Filter (α i)\ninst✝ : ∀ (i : ι), Nonempty (α i)\n⊢ Filter.coprodᵢ f = ⊥ ↔ f = ⊥",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Filter.coprodᵢ",
"_private.Mathlib.Order.Filter.Pi.0.Filter.coprodᵢ_eq_bot_iff._simp_1_1",
... | [
"ι : Type u_1\nα : ι → Type u_2\nf : (i : ι) → Filter (α i)\ninst✝ : ∀ (i : ι), Nonempty (α i)\n⊢ Filter.coprodᵢ f = ⊥ ↔ ∀ (x : ι), f x = ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Div | {
"line": 773,
"column": 4
} | {
"line": 774,
"column": 43
} | {
"line": 774,
"column": 44
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\np q : R[X]\nhmonic : q.Monic\nhdegree : q.degree ≤ p.degree\na✝ : Nontrivial R\n⊢ q.leadingCoeff * (p /ₘ q).leadingCoeff ≠ 0",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"NonAssocSemiring.toAddC... | [
"R : Type u\ninst✝ : CommRing R\np q : R[X]\nhmonic : q.Monic\nhdegree : q.degree ≤ p.degree\na✝ : Nontrivial R\n⊢ q.degree ≤ p.degree"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Derivative | {
"line": 559,
"column": 30
} | {
"line": 559,
"column": 47
} | {
"line": 559,
"column": 48
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nn m : ℕ\np : R[X]\nhsum :\n (⇑derivative)^[n] (p * X ^ m) =\n ∑ k ∈ range n.succ, (n.choose k * m.descFactorial k) • ((⇑derivative)^[n - k] p * X ^ (m - k))\nk : ℕ\nhk : k ∈ range n.succ\nhk' : k ∉ range (min m n).succ\nhkm : k ≤ m\n⊢ n < k",
"ppTerm": "?m.24... | [
"R : Type u\ninst✝ : CommSemiring R\nn m : ℕ\np : R[X]\nhsum :\n (⇑derivative)^[n] (p * X ^ m) =\n ∑ k ∈ range n.succ, (n.choose k * m.descFactorial k) • ((⇑derivative)^[n - k] p * X ^ (m - k))\nk : ℕ\nhk : k ∈ range n.succ\nhk' : k ∉ range (min m n).succ\nhkm : k ≤ m\n⊢ n < k"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Div | {
"line": 818,
"column": 4
} | {
"line": 818,
"column": 15
} | {
"line": 818,
"column": 16
} | [
{
"pp": "case inl\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : R[X]\nhpq : p ∣ 0\nh₁ : natDegree 0 ≤ p.natDegree\nh₂ : p.leadingCoeff = leadingCoeff 0\n⊢ p = 0",
"ppTerm": "?inl",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case inl\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : R[X]\nhpq : p ∣ 0\nh₁ : natDegree 0 ≤ p.natDegree\nh₂ : p.leadingCoeff = leadingCoeff 0\n⊢ p = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Div | {
"line": 833,
"column": 39
} | {
"line": 833,
"column": 68
} | {
"line": 833,
"column": 69
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\nhpq : p ∣ q\nh₁ : q.natDegree ≤ p.natDegree\nh₂ : q.leadingCoeff ∣ p.leadingCoeff\nr : R[X]\nhr : q = p * r\nu : Rˣ\nhu : p.leadingCoeff * ↑u = q.leadingCoeff\n⊢ q.natDegree ≤ (p * ↑((Units.map ↑C) u)).natDegree",
"ppTerm": "?m.65",
... | [
"R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\nhpq : p ∣ q\nh₁ : q.natDegree ≤ p.natDegree\nh₂ : q.leadingCoeff ∣ p.leadingCoeff\nr : R[X]\nhr : q = p * r\nu : Rˣ\nhu : p.leadingCoeff * ↑u = q.leadingCoeff\n⊢ q.natDegree ≤ p.natDegree"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Div | {
"line": 833,
"column": 77
} | {
"line": 833,
"column": 88
} | {
"line": 833,
"column": 89
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\nhpq : p ∣ q\nh₁ : q.natDegree ≤ p.natDegree\nh₂ : q.leadingCoeff ∣ p.leadingCoeff\nr : R[X]\nhr : q = p * r\nu : Rˣ\nhu : p.leadingCoeff * ↑u = q.leadingCoeff\n⊢ (p * ↑((Units.map ↑C) u)).leadingCoeff = q.leadingCoeff",
"ppTerm": "?m.... | [
"R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\nhpq : p ∣ q\nh₁ : q.natDegree ≤ p.natDegree\nh₂ : q.leadingCoeff ∣ p.leadingCoeff\nr : R[X]\nhr : q = p * r\nu : Rˣ\nhu : p.leadingCoeff * ↑u = q.leadingCoeff\n⊢ p.leadingCoeff * ↑u = q.leadingCoeff"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Div | {
"line": 838,
"column": 20
} | {
"line": 838,
"column": 76
} | {
"line": 838,
"column": 76
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\np q : K[X]\nhpq : p ∣ q\nhq : q ≠ 0\nh₁ : q.natDegree ≤ p.natDegree\n⊢ IsUnit q.leadingCoeff",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"congrArg",
"IsUnit",
"DivisionSemiring.toGroupWith... | [] | rwa [← leadingCoeff_ne_zero, ← isUnit_iff_ne_zero] at hq | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Algebra.Polynomial.Div | {
"line": 838,
"column": 20
} | {
"line": 838,
"column": 76
} | {
"line": 838,
"column": 76
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\np q : K[X]\nhpq : p ∣ q\nhq : q ≠ 0\nh₁ : q.natDegree ≤ p.natDegree\n⊢ IsUnit q.leadingCoeff",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"congrArg",
"IsUnit",
"DivisionSemiring.toGroupWith... | [] | rwa [← leadingCoeff_ne_zero, ← isUnit_iff_ne_zero] at hq | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Div | {
"line": 838,
"column": 20
} | {
"line": 838,
"column": 76
} | {
"line": 838,
"column": 76
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\np q : K[X]\nhpq : p ∣ q\nhq : q ≠ 0\nh₁ : q.natDegree ≤ p.natDegree\n⊢ IsUnit q.leadingCoeff",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"congrArg",
"IsUnit",
"DivisionSemiring.toGroupWith... | [] | rwa [← leadingCoeff_ne_zero, ← isUnit_iff_ne_zero] at hq | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Div | {
"line": 842,
"column": 55
} | {
"line": 842,
"column": 71
} | {
"line": 842,
"column": 72
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\np q : K[X]\nhpq : p ∣ q\nh₁ : p.degree = q.degree\nhq : q = 0\n⊢ p = q",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"Field.toSemifield",
"Polynomial",
"Semifield.toDivisionSemiring",
... | [
"K : Type u_1\ninst✝ : Field K\np q : K[X]\nhpq : p ∣ q\nh₁ : p.degree = q.degree\nhq : q = 0\n⊢ p = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Cofinite | {
"line": 177,
"column": 6
} | {
"line": 177,
"column": 22
} | {
"line": 177,
"column": 23
} | [
{
"pp": "case pos\nι : Type u_1\nα : ι → Type u_4\nβ : ι → Type u_5\nf : (i : ι) → α i → β i\nhf : ∀ᶠ (i : ι) in cofinite, Surjective (f i)\nl : (i : ι) → Filter (α i)\nI : Set ι\ns : (i : ι) → Set (α i)\nhI : I.Finite\nhs : ∀ i ∈ I, s i ∈ l i\ni : ι\nhi : i ∈ I\n⊢ f i '' (I, s).1.piecewise (fun i ↦ id ((I, s).... | [
"case pos\nι : Type u_1\nα : ι → Type u_4\nβ : ι → Type u_5\nf : (i : ι) → α i → β i\nhf : ∀ᶠ (i : ι) in cofinite, Surjective (f i)\nl : (i : ι) → Filter (α i)\nI : Set ι\ns : (i : ι) → Set (α i)\nhI : I.Finite\nhs : ∀ i ∈ I, s i ∈ l i\ni : ι\nhi : i ∈ I\n⊢ f i ⁻¹' f i '' s i ∈ l i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Cofinite | {
"line": 199,
"column": 2
} | {
"line": 199,
"column": 13
} | {
"line": 199,
"column": 14
} | [
{
"pp": "α : Type u_2\ns : Set α\nhs : s.Finite\n⊢ cofinite ⊓ 𝓟 sᶜ = cofinite",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"compl_compl",
"congrArg",
"Compl.compl",
"Filter.instCompleteLatticeFilter",
"Partial... | [
"α : Type u_2\ns : Set α\nhs : s.Finite\n⊢ s.Finite"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Derivative | {
"line": 685,
"column": 2
} | {
"line": 685,
"column": 13
} | {
"line": 685,
"column": 14
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nS : Multiset R\nr : R\nhr : r ∈ S\nthis :\n ↑(evalRingHom r) (Multiset.map (fun a ↦ X - C a) (S.erase r)).prod =\n (Multiset.map (⇑↑(evalRingHom r)) (Multiset.map (fun a ↦ X - C a) (S.erase r))).prod\n⊢ eval r (derivative (Multiset.map (fun a ... | [
"R : Type u\ninst✝¹ : CommRing R\ninst✝ : DecidableEq R\nS : Multiset R\nr : R\nhr : r ∈ S\nthis :\n ↑(evalRingHom r) (Multiset.map (fun a ↦ X - C a) (S.erase r)).prod =\n (Multiset.map (⇑↑(evalRingHom r)) (Multiset.map (fun a ↦ X - C a) (S.erase r))).prod\n⊢ eval r (Multiset.map (fun a ↦ X - C a) (S.erase r)).... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Cofinite | {
"line": 294,
"column": 46
} | {
"line": 294,
"column": 57
} | {
"line": 294,
"column": 58
} | [
{
"pp": "α : Type u_2\nf : Filter α\nx : α\nx✝ : ∃ i, ∃ (_ : i ∈ f), x ∉ i\ns : Set α\nhs : s ∈ f\nhx : x ∉ s\n⊢ s ⊆ {x}ᶜ",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Compl.compl",
"Membership.mem",
"Set.instSingletonSet",
"id",
"LE.le",
... | [
"α : Type u_2\nf : Filter α\nx : α\nx✝ : ∃ i, ∃ (_ : i ∈ f), x ∉ i\ns : Set α\nhs : s ∈ f\nhx : x ∉ s\n⊢ x ∉ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Cofinite | {
"line": 327,
"column": 4
} | {
"line": 327,
"column": 80
} | {
"line": 328,
"column": 4
} | [
{
"pp": "case refine_4\nα : Type u_2\nf : Filter α\nq : Set α × Filter α\nhq : (fun p ↦ p.2 ≤ cofinite ∧ Disjoint (𝓟 p.1) p.2 ∧ f = 𝓟 p.1 ⊔ p.2) q\nhqk : f.ker = (𝓟 q.1 ⊔ q.2).ker\n⊢ q = (f.ker, Coheyting.boundary f)",
"ppTerm": "?refine_4",
"assigned": true,
"usedConstants": [
"Set.union_e... | [
"case refine_4\nα : Type u_2\nf : Filter α\nq : Set α × Filter α\nhq : (fun p ↦ p.2 ≤ cofinite ∧ Disjoint (𝓟 p.1) p.2 ∧ f = 𝓟 p.1 ⊔ p.2) q\nhqk : f.ker = q.1\n⊢ q = (f.ker, Coheyting.boundary f)"
] | rw [ker_sup, ker_principal, le_cofinite_iff_ker.mp hq.1, union_empty] at hqk | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.Filter.TendstoCofinite | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 35
} | {
"line": 52,
"column": 36
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\ninst✝ : TendstoCofinite f\ns : Set β\nhs : s.Finite\n⊢ (f ⁻¹' s).Finite",
"ppTerm": "?m.7",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\nf : α → β\ninst✝ : TendstoCofinite f\ns : Set β\nhs : s.Finite\n⊢ (f ⁻¹' s).Finite"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.TendstoCofinite | {
"line": 56,
"column": 33
} | {
"line": 56,
"column": 44
} | {
"line": 56,
"column": 45
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\ninst✝ : TendstoCofinite f\nb : β\n⊢ (f ⁻¹' {b}).Finite",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\nf : α → β\ninst✝ : TendstoCofinite f\nb : β\n⊢ (f ⁻¹' {b}).Finite"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.TendstoCofinite | {
"line": 123,
"column": 4
} | {
"line": 123,
"column": 74
} | {
"line": 124,
"column": 6
} | [
{
"pp": "case refine_2\nα : Type u_1\nβ : Type u_2\nf : α → β\ninst✝ : TendstoCofinite f\nx : β →₀ ℕ\ns : Finset α := x.support.sup fun t ↦ ⋯.toFinset\ne : ↥s ↪ α := Function.Embedding.subtype fun u ↦ u ∈ s\ny : α →₀ ℕ\nhy : mapDomain f y = x\n⊢ y.support ⊆ s",
"ppTerm": "?refine_2",
"assigned": true,
... | [
"case refine_2\nα : Type u_1\nβ : Type u_2\nf : α → β\ninst✝ : TendstoCofinite f\nx : β →₀ ℕ\ns : Finset α := x.support.sup fun t ↦ ⋯.toFinset\ne : ↥s ↪ α := Function.Embedding.subtype fun u ↦ u ∈ s\ny : α →₀ ℕ\nhy : mapDomain f y = x\n⊢ ∀ ⦃x : α⦄, ¬y x = 0 → ∃ x_1, ¬y x_1 = 0 ∧ f x_1 = f x ∧ ¬y x_1 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Prod | {
"line": 534,
"column": 4
} | {
"line": 534,
"column": 15
} | {
"line": 534,
"column": 16
} | [
{
"pp": "case h\nα : Type u_6\nβ : Type u_7\nι : Type u_8\na : α\nb : β\ni : ι\nb' : β\ni' : ι\nh₁ : (b', i').1 ∈ {b}\nright✝ : (b', i').2 ∈ univ\n⊢ (a, i') ∈ ({a}ᶜ ×ˢ {i}ᶜ)ᶜ ∧ Prod.map (fun x ↦ b) id (a, i') = (b', i')",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Set.instSProd",
... | [
"case h\nα : Type u_6\nβ : Type u_7\nι : Type u_8\na : α\nb : β\ni : ι\nb' : β\ni' : ι\nh₁ : (b', i').1 ∈ {b}\nright✝ : (b', i').2 ∈ univ\n⊢ b = b'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.Associated | {
"line": 35,
"column": 32
} | {
"line": 35,
"column": 43
} | {
"line": 35,
"column": 44
} | [
{
"pp": "M₀ : Type u_3\ninst✝ : CommMonoidWithZero M₀\np : M₀\nhp : Prime p\ns✝ : Multiset M₀\na : M₀\ns : Multiset M₀\nih : p ∣ s.prod → ∃ a ∈ s, p ∣ a\nh : p ∣ (a ::ₘ s).prod\n⊢ p ∣ a * s.prod",
"ppTerm": "?m.35",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}... | [
"M₀ : Type u_3\ninst✝ : CommMonoidWithZero M₀\np : M₀\nhp : Prime p\ns✝ : Multiset M₀\na : M₀\ns : Multiset M₀\nih : p ∣ s.prod → ∃ a ∈ s, p ∣ a\nh : p ∣ (a ::ₘ s).prod\n⊢ p ∣ a * s.prod"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.Associated | {
"line": 44,
"column": 2
} | {
"line": 44,
"column": 76
} | {
"line": 45,
"column": 4
} | [
{
"pp": "ι : Type u_1\nM₀ : Type u_3\ninst✝ : CommMonoidWithZero M₀\np : M₀\nhp : Prime p\ns : Multiset ι\nf : ι → M₀\nh : p ∣ (Multiset.map f s).prod\n⊢ ∃ a ∈ s, p ∣ f a",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_1\nM₀ : Type u_3\ninst✝ : CommMonoidWithZero M₀\np : M₀\nhp : Prime p\ns : Multiset ι\nf : ι → M₀\nh : p ∣ (Multiset.map f s).prod\n⊢ ∃ a ∈ s, p ∣ f a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.Associated | {
"line": 110,
"column": 14
} | {
"line": 110,
"column": 25
} | {
"line": 111,
"column": 4
} | [
{
"pp": "case cons.inl.refine_2\nM₀ : Type u_3\ninst✝¹ : CommMonoidWithZero M₀\ninst✝ : IsCancelMulZero M₀\nc : M₀\ns : Multiset M₀\nhind :\n ∀ (x y : M₀),\n x * y ∈ closure {r | IsUnit r ∨ Prime r} →\n (∀ y ∈ s, IsUnit y ∨ Prime y) → s.prod = x * y → x ∈ closure {r | IsUnit r ∨ Prime r}\nx y : M₀\nhxy... | [] | exact hprod | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Finsupp.Weight | {
"line": 134,
"column": 2
} | {
"line": 142,
"column": 17
} | {
"line": 144,
"column": 0
} | [
{
"pp": "σ : Type u_1\nw : σ → ℕ\ns : σ\nhs : w s ≠ 0\nf : σ →₀ ℕ\n⊢ f s ≤ (weight w) f",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"zero_le",
"Finsupp.instAddZeroClass",
"Finsupp.instFunLike",
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"NonAssocSemi... | [] | simp only [weight_apply, Finsupp.sum]
by_cases h : s ∈ f.support
· rw [Finset.sum_eq_add_sum_sdiff_singleton_of_mem h]
refine le_trans ?_ (Nat.le_add_right _ _)
apply Nat.le_mul_of_pos_right
exact Nat.zero_lt_of_ne_zero hs
· simp only [notMem_support_iff] at h
rw [h]
apply zero_le | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finsupp.Weight | {
"line": 134,
"column": 2
} | {
"line": 142,
"column": 17
} | {
"line": 144,
"column": 0
} | [
{
"pp": "σ : Type u_1\nw : σ → ℕ\ns : σ\nhs : w s ≠ 0\nf : σ →₀ ℕ\n⊢ f s ≤ (weight w) f",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"zero_le",
"Finsupp.instAddZeroClass",
"Finsupp.instFunLike",
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"NonAssocSemi... | [] | simp only [weight_apply, Finsupp.sum]
by_cases h : s ∈ f.support
· rw [Finset.sum_eq_add_sum_sdiff_singleton_of_mem h]
refine le_trans ?_ (Nat.le_add_right _ _)
apply Nat.le_mul_of_pos_right
exact Nat.zero_lt_of_ne_zero hs
· simp only [notMem_support_iff] at h
rw [h]
apply zero_le | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Finsupp.Weight | {
"line": 133,
"column": 2
} | {
"line": 142,
"column": 17
} | {
"line": 144,
"column": 0
} | [
{
"pp": "σ : Type u_1\nw : σ → ℕ\ns : σ\nhs : w s ≠ 0\nf : σ →₀ ℕ\n⊢ f s ≤ (weight w) f",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"zero_le",
"Finsupp.instAddZeroClass",
"Finsupp.instFunLike",
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"NonAssocSemi... | [] | classical
simp only [weight_apply, Finsupp.sum]
by_cases h : s ∈ f.support
· rw [Finset.sum_eq_add_sum_sdiff_singleton_of_mem h]
refine le_trans ?_ (Nat.le_add_right _ _)
apply Nat.le_mul_of_pos_right
exact Nat.zero_lt_of_ne_zero hs
· simp only [notMem_support_iff] at h
rw [h]
apply zero_le | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Data.Finsupp.Weight | {
"line": 133,
"column": 2
} | {
"line": 142,
"column": 17
} | {
"line": 144,
"column": 0
} | [
{
"pp": "σ : Type u_1\nw : σ → ℕ\ns : σ\nhs : w s ≠ 0\nf : σ →₀ ℕ\n⊢ f s ≤ (weight w) f",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"zero_le",
"Finsupp.instAddZeroClass",
"Finsupp.instFunLike",
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"NonAssocSemi... | [] | classical
simp only [weight_apply, Finsupp.sum]
by_cases h : s ∈ f.support
· rw [Finset.sum_eq_add_sum_sdiff_singleton_of_mem h]
refine le_trans ?_ (Nat.le_add_right _ _)
apply Nat.le_mul_of_pos_right
exact Nat.zero_lt_of_ne_zero hs
· simp only [notMem_support_iff] at h
rw [h]
apply zero_le | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finsupp.Weight | {
"line": 133,
"column": 2
} | {
"line": 142,
"column": 17
} | {
"line": 144,
"column": 0
} | [
{
"pp": "σ : Type u_1\nw : σ → ℕ\ns : σ\nhs : w s ≠ 0\nf : σ →₀ ℕ\n⊢ f s ≤ (weight w) f",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"zero_le",
"Finsupp.instAddZeroClass",
"Finsupp.instFunLike",
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"NonAssocSemi... | [] | classical
simp only [weight_apply, Finsupp.sum]
by_cases h : s ∈ f.support
· rw [Finset.sum_eq_add_sum_sdiff_singleton_of_mem h]
refine le_trans ?_ (Nat.le_add_right _ _)
apply Nat.le_mul_of_pos_right
exact Nat.zero_lt_of_ne_zero hs
· simp only [notMem_support_iff] at h
rw [h]
apply zero_le | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Finsupp.Weight | {
"line": 157,
"column": 8
} | {
"line": 157,
"column": 55
} | {
"line": 157,
"column": 55
} | [
{
"pp": "σ : Type u_1\nM : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : PartialOrder M\ninst✝ : IsOrderedAddMonoid M\nw : σ → M\nhw : ∀ (s : σ), 0 ≤ w s\ns : σ\nf : σ →₀ ℕ\nhs : s ∈ f.support\n⊢ f s • w s ≤ ∑ x ∈ f.support, f x • w x",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"F... | [
"σ : Type u_1\nM : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : PartialOrder M\ninst✝ : IsOrderedAddMonoid M\nw : σ → M\nhw : ∀ (s : σ), 0 ≤ w s\ns : σ\nf : σ →₀ ℕ\nhs : s ∈ f.support\n⊢ f s • w s ≤ f s • w s + ∑ x ∈ f.support \\ {s}, f x • w x"
] | Finset.sum_eq_add_sum_sdiff_singleton_of_mem hs | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.Associated | {
"line": 148,
"column": 37
} | {
"line": 148,
"column": 88
} | {
"line": 148,
"column": 89
} | [
{
"pp": "M₀ : Type u_3\ninst✝² : CommMonoidWithZero M₀\ninst✝¹ : IsCancelMulZero M₀\ninst✝ : Subsingleton M₀ˣ\ns : Finset M₀\nn : M₀\nh : ∀ a ∈ s, Prime a\ndiv : ∀ a ∈ s, a ∣ n\n⊢ ∀ a ∈ Multiset.map (fun p ↦ p) s.val, Prime a",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"M₀ : Type u_3\ninst✝² : CommMonoidWithZero M₀\ninst✝¹ : IsCancelMulZero M₀\ninst✝ : Subsingleton M₀ˣ\ns : Finset M₀\nn : M₀\nh : ∀ a ∈ s, Prime a\ndiv : ∀ a ∈ s, a ∣ n\n⊢ ∀ a ∈ s.val, Prime a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.Associated | {
"line": 149,
"column": 12
} | {
"line": 149,
"column": 63
} | {
"line": 149,
"column": 64
} | [
{
"pp": "M₀ : Type u_3\ninst✝² : CommMonoidWithZero M₀\ninst✝¹ : IsCancelMulZero M₀\ninst✝ : Subsingleton M₀ˣ\ns : Finset M₀\nn : M₀\nh : ∀ a ∈ s, Prime a\ndiv : ∀ a ∈ s, a ∣ n\n⊢ ∀ a ∈ Multiset.map (fun p ↦ p) s.val, a ∣ n",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"M₀ : Type u_3\ninst✝² : CommMonoidWithZero M₀\ninst✝¹ : IsCancelMulZero M₀\ninst✝ : Subsingleton M₀ˣ\ns : Finset M₀\nn : M₀\nh : ∀ a ∈ s, Prime a\ndiv : ∀ a ∈ s, a ∣ n\n⊢ ∀ a ∈ s.val, a ∣ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.Associated | {
"line": 196,
"column": 32
} | {
"line": 196,
"column": 43
} | {
"line": 196,
"column": 44
} | [
{
"pp": "M₀ : Type u_3\ninst✝ : CommMonoidWithZero M₀\ns✝ : Multiset (Associates M₀)\np : Associates M₀\nhp : Prime p\na : Associates M₀\ns : Multiset (Associates M₀)\nih : p ≤ s.prod → ∃ a ∈ s, p ≤ a\nh : p ≤ (a ::ₘ s).prod\n⊢ p ≤ a * s.prod",
"ppTerm": "?m.64",
"assigned": false,
"usedConstants": ... | [
"M₀ : Type u_3\ninst✝ : CommMonoidWithZero M₀\ns✝ : Multiset (Associates M₀)\np : Associates M₀\nhp : Prime p\na : Associates M₀\ns : Multiset (Associates M₀)\nih : p ≤ s.prod → ∃ a ∈ s, p ≤ a\nh : p ≤ (a ::ₘ s).prod\n⊢ p ≤ a * s.prod"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finsupp.Weight | {
"line": 292,
"column": 2
} | {
"line": 292,
"column": 74
} | {
"line": 293,
"column": 4
} | [
{
"pp": "σ : Type u_1\nM : Type u_2\nτ : Type u_5\nf : σ → τ\ninst✝² : AddCommMonoid M\ninst✝¹ : PartialOrder M\ninst✝ : CanonicallyOrderedAdd M\nx : τ →₀ M\nhf : Set.InjOn f (f ⁻¹' ↑x.support)\n⊢ degree (comapDomain f x hf) ≤ degree x",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
... | [
"σ : Type u_1\nM : Type u_2\nτ : Type u_5\nf : σ → τ\ninst✝² : AddCommMonoid M\ninst✝¹ : PartialOrder M\ninst✝ : CanonicallyOrderedAdd M\nx : τ →₀ M\nhf : Set.InjOn f (f ⁻¹' ↑x.support)\n⊢ ∑ x_1 ∈ x.support with ∃ y, f y = x_1, x x_1 ≤ ∑ i ∈ x.support, x i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.UniqueFactorizationDomain.Basic | {
"line": 83,
"column": 35
} | {
"line": 83,
"column": 74
} | {
"line": 83,
"column": 75
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : IsCancelMulZero α\ng : Multiset α\na✝ : ∀ x ∈ 0, Prime x\nhg : ∀ x ∈ g, Prime x\nh : Multiset.prod 0 ~ᵤ g.prod\nx : α\nhx : x ∈ g\n⊢ IsUnit g.prod",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCommM... | [
"α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : IsCancelMulZero α\ng : Multiset α\na✝ : ∀ x ∈ 0, Prime x\nhg : ∀ x ∈ g, Prime x\nh : Multiset.prod 0 ~ᵤ g.prod\nx : α\nhx : x ∈ g\n⊢ ∀ a ∈ g, IsUnit a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finsupp.Weight | {
"line": 337,
"column": 30
} | {
"line": 337,
"column": 66
} | {
"line": 337,
"column": 67
} | [
{
"pp": "α : Type u_5\ns : Set α\nn : ℕ\nih : n • (fun x ↦ single x 1) '' s = {x | degree x = n ∧ ↑x.support ⊆ s}\nf : α →₀ ℕ\nx✝ : f ∈ {x | degree x = n + 1 ∧ ↑x.support ⊆ s}\nf_deg : degree f = n + 1\nf_supp : ↑f.support ⊆ s\ni : α\nhi : i ∈ f.support\n⊢ single i 1 ≤ f",
"ppTerm": "?m.117",
"assigned"... | [
"α : Type u_5\ns : Set α\nn : ℕ\nih : n • (fun x ↦ single x 1) '' s = {x | degree x = n ∧ ↑x.support ⊆ s}\nf : α →₀ ℕ\nx✝ : f ∈ {x | degree x = n + 1 ∧ ↑x.support ⊆ s}\nf_deg : degree f = n + 1\nf_supp : ↑f.support ⊆ s\ni : α\nhi : i ∈ f.support\n⊢ ¬f i = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Roots | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 61
} | {
"line": 80,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : R[X]\nhp0 : p ≠ 0\n⊢ ↑(Classical.choose ⋯).card ≤ p.degree",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"WithBot",
"CommSemiring.toSemiring",
"WithBot.instNatCast",
"C... | [] | exact (Classical.choose_spec (exists_multiset_roots hp0)).1 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Polynomial.Roots | {
"line": 143,
"column": 2
} | {
"line": 144,
"column": 9
} | {
"line": 144,
"column": 10
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : R[X]\nhp : p ≠ 0\n⊢ {x | p.IsRoot x}.Finite",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : R[X]\nhp : p ≠ 0\n⊢ {x | p.IsRoot x}.Finite"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Roots | {
"line": 193,
"column": 61
} | {
"line": 193,
"column": 72
} | {
"line": 193,
"column": 73
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nr : R\n⊢ (X + C r).roots = {-r}",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nr : R\n⊢ (X + C r).roots = {-r}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors | {
"line": 268,
"column": 6
} | {
"line": 268,
"column": 41
} | {
"line": 268,
"column": 42
} | [
{
"pp": "α : Type u_1\ninst✝³ : CommMonoidWithZero α\ninst✝² : NormalizationMonoid α\ninst✝¹ : UniqueFactorizationMonoid α\ninst✝ : Subsingleton αˣ\nm : Multiset α\nh : ∀ p ∈ m, Prime p\nh✝ : Subsingleton α\nx : α\nhx : x ∈ m\n⊢ False",
"ppTerm": "?m.64",
"assigned": false,
"usedConstants": [],
... | [
"α : Type u_1\ninst✝³ : CommMonoidWithZero α\ninst✝² : NormalizationMonoid α\ninst✝¹ : UniqueFactorizationMonoid α\ninst✝ : Subsingleton αˣ\nm : Multiset α\nh : ∀ p ∈ m, Prime p\nh✝ : Subsingleton α\nx : α\nhx : x ∈ m\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.UniqueFactorizationDomain.Basic | {
"line": 263,
"column": 2
} | {
"line": 263,
"column": 13
} | {
"line": 263,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nhx : IsUnit x\n⊢ factors x = 0",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nhx : IsUnit x\n⊢ factors x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.UniqueFactorizationDomain.Basic | {
"line": 267,
"column": 29
} | {
"line": 267,
"column": 68
} | {
"line": 267,
"column": 69
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nhx : x ≠ 0\nh : ¬IsUnit x\n⊢ factors x ≠ 0",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Classical.not_not._simp_1",
"congrArg",
"Membership.mem",
"E... | [
"α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nhx : x ≠ 0\nh : ¬IsUnit x\n⊢ ∃ x_1, x_1 ∈ factors x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Roots | {
"line": 226,
"column": 53
} | {
"line": 227,
"column": 77
} | {
"line": 229,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : R[X]\n⊢ (-p).roots = p.roots",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"NegZeroClass.toNeg",
"instHSMul",
"Polynomial.roots",
"NeZero.one",
"Semiring.toMod... | [] | by
rw [← neg_one_smul R p, roots_smul_nonzero p (neg_ne_zero.mpr one_ne_zero)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors | {
"line": 270,
"column": 4
} | {
"line": 270,
"column": 60
} | {
"line": 271,
"column": 6
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝³ : CommMonoidWithZero α\ninst✝² : NormalizationMonoid α\ninst✝¹ : UniqueFactorizationMonoid α\ninst✝ : Subsingleton αˣ\nm : Multiset α\nh : ∀ p ∈ m, Prime p\nh✝ : Nontrivial α\n⊢ normalizedFactors m.prod = m",
"ppTerm": "?inr",
"assigned": true,
"usedConstants"... | [
"case inr\nα : Type u_1\ninst✝³ : CommMonoidWithZero α\ninst✝² : NormalizationMonoid α\ninst✝¹ : UniqueFactorizationMonoid α\ninst✝ : Subsingleton αˣ\nm : Multiset α\nh : ∀ p ∈ m, Prime p\nh✝ : Nontrivial α\n⊢ Multiset.Rel (fun x1 x2 ↦ x1 ~ᵤ x2) (normalizedFactors m.prod) m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Roots | {
"line": 278,
"column": 2
} | {
"line": 278,
"column": 60
} | {
"line": 278,
"column": 61
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nι : Type u_1\nf : ι → R[X]\nm : Multiset ι\nhm : m.Nodup\n⊢ { val := m, nodup := hm }.prod f ≠ 0 →\n ({ val := m, nodup := hm }.prod f).roots = { val := m, nodup := hm }.val.bind fun i ↦ (f i).roots",
"ppTerm": "?m.33",
"assigned": true,
... | [
"R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nι : Type u_1\nf : ι → R[X]\nm : Multiset ι\nhm : m.Nodup\n⊢ (∀ x ∈ m, ¬f x = 0) → (Multiset.map f m).prod.roots = m.bind fun i ↦ (f i).roots"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Roots | {
"line": 287,
"column": 67
} | {
"line": 287,
"column": 71
} | {
"line": 287,
"column": 72
} | [
{
"pp": "case succ.inr\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : R[X]\nn : ℕ\nihn : (p ^ n).roots = n • p.roots\nhp : p ≠ 0\n⊢ (p ^ n).roots + p.roots = (n + 1) • p.roots",
"ppTerm": "?succ.inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"Polynomia... | [
"case succ.inr\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : R[X]\nn : ℕ\nihn : (p ^ n).roots = n • p.roots\nhp : p ≠ 0\n⊢ n • p.roots + p.roots = (n + 1) • p.roots"
] | ihn, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Content | {
"line": 70,
"column": 24
} | {
"line": 70,
"column": 35
} | {
"line": 70,
"column": 36
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : NoZeroDivisors R\np : R[X]\nhp : Irreducible p\nhp' : p.natDegree ≠ 0\nr : R\nq : R[X]\nhq : p = C r * q\nthis : ¬IsUnit q\n⊢ IsUnit r",
"ppTerm": "?m.28",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : NoZeroDivisors R\np : R[X]\nhp : Irreducible p\nhp' : p.natDegree ≠ 0\nr : R\nq : R[X]\nhq : p = C r * q\nthis : ¬IsUnit q\n⊢ IsUnit r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Content | {
"line": 69,
"column": 2
} | {
"line": 74,
"column": 49
} | {
"line": 76,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : NoZeroDivisors R\np : R[X]\nhp : Irreducible p\nhp' : p.natDegree ≠ 0\n⊢ p.IsPrimitive",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"RingHom.instRingHomClass",
"False",
"Semigroup.toMul",
... | [] | rintro r ⟨q, hq⟩
suffices ¬IsUnit q by simpa using ((hp.2 hq).resolve_right this).map Polynomial.constantCoeff
intro H
have hr : r ≠ 0 := by rintro rfl; simp_all
obtain ⟨s, hs, rfl⟩ := Polynomial.isUnit_iff.mp H
simp [hq, Polynomial.natDegree_C_mul hr] at hp' | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Content | {
"line": 69,
"column": 2
} | {
"line": 74,
"column": 49
} | {
"line": 76,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : NoZeroDivisors R\np : R[X]\nhp : Irreducible p\nhp' : p.natDegree ≠ 0\n⊢ p.IsPrimitive",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"RingHom.instRingHomClass",
"False",
"Semigroup.toMul",
... | [] | rintro r ⟨q, hq⟩
suffices ¬IsUnit q by simpa using ((hp.2 hq).resolve_right this).map Polynomial.constantCoeff
intro H
have hr : r ≠ 0 := by rintro rfl; simp_all
obtain ⟨s, hs, rfl⟩ := Polynomial.isUnit_iff.mp H
simp [hq, Polynomial.natDegree_C_mul hr] at hp' | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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