module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Data.Nat.Prime.Defs | {
"line": 173,
"column": 36
} | {
"line": 173,
"column": 42
} | {
"line": 175,
"column": 0
} | [
{
"pp": "⊢ Prime 3",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Nat.Prime",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"Bool",
"Eq.refl",
"OfNat.ofNat",
"Decidable.decide",
"Eq",
"Nat.decidablePr... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Data.Nat.Prime.Defs | {
"line": 173,
"column": 36
} | {
"line": 173,
"column": 42
} | {
"line": 175,
"column": 0
} | [
{
"pp": "⊢ Prime 3",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Nat.Prime",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"Bool",
"Eq.refl",
"OfNat.ofNat",
"Decidable.decide",
"Eq",
"Nat.decidablePr... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Prime.Defs | {
"line": 173,
"column": 36
} | {
"line": 173,
"column": 42
} | {
"line": 175,
"column": 0
} | [
{
"pp": "⊢ Prime 3",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Nat.Prime",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"Bool",
"Eq.refl",
"OfNat.ofNat",
"Decidable.decide",
"Eq",
"Nat.decidablePr... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.Prime.Defs | {
"line": 175,
"column": 35
} | {
"line": 175,
"column": 41
} | {
"line": 177,
"column": 0
} | [
{
"pp": "⊢ Prime 5",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Nat.Prime",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"Bool",
"Eq.refl",
"OfNat.ofNat",
"Decidable.decide",
"Eq",
"Nat.decidablePr... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Data.Nat.Prime.Defs | {
"line": 175,
"column": 35
} | {
"line": 175,
"column": 41
} | {
"line": 177,
"column": 0
} | [
{
"pp": "⊢ Prime 5",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Nat.Prime",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"Bool",
"Eq.refl",
"OfNat.ofNat",
"Decidable.decide",
"Eq",
"Nat.decidablePr... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Prime.Defs | {
"line": 175,
"column": 35
} | {
"line": 175,
"column": 41
} | {
"line": 177,
"column": 0
} | [
{
"pp": "⊢ Prime 5",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Nat.Prime",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"Bool",
"Eq.refl",
"OfNat.ofNat",
"Decidable.decide",
"Eq",
"Nat.decidablePr... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.Prime.Defs | {
"line": 177,
"column": 36
} | {
"line": 177,
"column": 42
} | {
"line": 179,
"column": 0
} | [
{
"pp": "⊢ Prime 7",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Nat.Prime",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"Bool",
"Eq.refl",
"OfNat.ofNat",
"Decidable.decide",
"Eq",
"Nat.decidablePr... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Data.Nat.Prime.Defs | {
"line": 177,
"column": 36
} | {
"line": 177,
"column": 42
} | {
"line": 179,
"column": 0
} | [
{
"pp": "⊢ Prime 7",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Nat.Prime",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"Bool",
"Eq.refl",
"OfNat.ofNat",
"Decidable.decide",
"Eq",
"Nat.decidablePr... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Prime.Defs | {
"line": 177,
"column": 36
} | {
"line": 177,
"column": 42
} | {
"line": 179,
"column": 0
} | [
{
"pp": "⊢ Prime 7",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Nat.Prime",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"Bool",
"Eq.refl",
"OfNat.ofNat",
"Decidable.decide",
"Eq",
"Nat.decidablePr... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.Prime.Defs | {
"line": 179,
"column": 38
} | {
"line": 179,
"column": 44
} | {
"line": 181,
"column": 0
} | [
{
"pp": "⊢ Prime 11",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Nat.Prime",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"Bool",
"Eq.refl",
"OfNat.ofNat",
"Decidable.decide",
"Eq",
"Nat.decidableP... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Data.Nat.Prime.Defs | {
"line": 179,
"column": 38
} | {
"line": 179,
"column": 44
} | {
"line": 181,
"column": 0
} | [
{
"pp": "⊢ Prime 11",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Nat.Prime",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"Bool",
"Eq.refl",
"OfNat.ofNat",
"Decidable.decide",
"Eq",
"Nat.decidableP... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Prime.Defs | {
"line": 179,
"column": 38
} | {
"line": 179,
"column": 44
} | {
"line": 181,
"column": 0
} | [
{
"pp": "⊢ Prime 11",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Nat.Prime",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"Bool",
"Eq.refl",
"OfNat.ofNat",
"Decidable.decide",
"Eq",
"Nat.decidableP... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.Prime.Defs | {
"line": 197,
"column": 87
} | {
"line": 197,
"column": 93
} | {
"line": 197,
"column": 93
} | [
{
"pp": "n k : ℕ\nh : ¬n < k * k\n⊢ 0 < 2",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"LT.lt",
"Bool",
"Nat.decLt",
"Eq.refl",
"instLTNat",
"OfNat.ofNat",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Data.Nat.Prime.Defs | {
"line": 197,
"column": 87
} | {
"line": 197,
"column": 93
} | {
"line": 197,
"column": 93
} | [
{
"pp": "n k : ℕ\nh : ¬n < k * k\n⊢ 0 < 2",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"LT.lt",
"Bool",
"Nat.decLt",
"Eq.refl",
"instLTNat",
"OfNat.ofNat",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Prime.Defs | {
"line": 197,
"column": 87
} | {
"line": 197,
"column": 93
} | {
"line": 197,
"column": 93
} | [
{
"pp": "n k : ℕ\nh : ¬n < k * k\n⊢ 0 < 2",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"LT.lt",
"Bool",
"Nat.decLt",
"Eq.refl",
"instLTNat",
"OfNat.ofNat",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.Prime.Defs | {
"line": 321,
"column": 64
} | {
"line": 321,
"column": 70
} | {
"line": 321,
"column": 70
} | [
{
"pp": "m p : ℕ\npp : 1 < p\nh : 1 = 1 ∨ m ≤ minFac 1\nd : p ∣ 1\n⊢ 0 < 1",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"LT.lt",
"Bool",
"Nat.decLt",
"Eq.refl",
"in... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Data.Nat.Prime.Defs | {
"line": 321,
"column": 64
} | {
"line": 321,
"column": 70
} | {
"line": 321,
"column": 70
} | [
{
"pp": "m p : ℕ\npp : 1 < p\nh : 1 = 1 ∨ m ≤ minFac 1\nd : p ∣ 1\n⊢ 0 < 1",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"LT.lt",
"Bool",
"Nat.decLt",
"Eq.refl",
"in... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Prime.Defs | {
"line": 321,
"column": 64
} | {
"line": 321,
"column": 70
} | {
"line": 321,
"column": 70
} | [
{
"pp": "m p : ℕ\npp : 1 < p\nh : 1 = 1 ∨ m ≤ minFac 1\nd : p ∣ 1\n⊢ 0 < 1",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"LT.lt",
"Bool",
"Nat.decLt",
"Eq.refl",
"in... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.Prime.Defs | {
"line": 353,
"column": 51
} | {
"line": 353,
"column": 57
} | {
"line": 354,
"column": 2
} | [
{
"pp": "n : ℕ\npos : 0 < n\nnp : ¬Prime n\nh0 : n = n.minFac * 0\n⊢ ¬0 < 0",
"ppTerm": "?m.80",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"Nat.instMulZeroClass",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"LT.lt",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Algebra.CharP.Defs | {
"line": 295,
"column": 48
} | {
"line": 295,
"column": 59
} | {
"line": 295,
"column": 59
} | [
{
"pp": "R : Type u_1\ninst✝ : NonAssocSemiring R\nv : ℕ\nhv : v ≠ 1\nhr : CharP R v\nh : v ∣ 1\n⊢ v = 1",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Dvd.dvd",
"congrArg",
"Nat.dvd_one",
"Eq.mp",
"instOfNatNat",
"Nat.instDvd",
"Nat",
"pro... | [
"R : Type u_1\ninst✝ : NonAssocSemiring R\nv : ℕ\nhv : v ≠ 1\nhr : CharP R v\nh : v = 1\n⊢ v = 1"
] | Nat.dvd_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.CharP.Defs | {
"line": 362,
"column": 4
} | {
"line": 362,
"column": 10
} | {
"line": 363,
"column": 2
} | [
{
"pp": "case zero\nR : Type u_1\ninst✝¹ : AddMonoidWithOne R\np : ℕ\nhp : CharP R p\ninst✝ : CharZero R\n⊢ 0 = 1 ↔ Nat.Prime 0",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Nat.Prime",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Iff",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.LinearAlgebra.LinearPMap | {
"line": 645,
"column": 2
} | {
"line": 645,
"column": 6
} | {
"line": 646,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : Ring R\ninst✝⁴ : Ring S\nσ : R →+* S\nE : Type u_4\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\nF : Type u_5\ninst✝¹ : AddCommGroup F\ninst✝ : Module S F\nc : Set (E →ₛₗ.[σ] F)\nhc : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) c\nl : E →ₛₗ.[σ] F\nhl : l ∈ c\nx : ↥l.domain\n⊢... | [
"R : Type u_1\nS : Type u_2\ninst✝⁵ : Ring R\ninst✝⁴ : Ring S\nσ : R →+* S\nE : Type u_4\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\nF : Type u_5\ninst✝¹ : AddCommGroup F\ninst✝ : Module S F\nc : Set (E →ₛₗ.[σ] F)\nhc : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) c\nl : E →ₛₗ.[σ] F\nhl : l ∈ c\nx : ↥l.domain\n⊢ ↑l x = ↑(Li... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Algebra.Polynomial.Degree.Defs | {
"line": 457,
"column": 2
} | {
"line": 458,
"column": 18
} | {
"line": 460,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nh : 0 ≠ 1\np : R[X]\nhp : p.Monic\n⊢ p ≠ 0",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"nontrivial_of_ne",
"AddCommMonoidWithOne.toAddMonoidWithOne",
"AddMonoidWithOne.toOne",
... | [] | nontriviality R
exact hp.ne_zero | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Degree.Defs | {
"line": 457,
"column": 2
} | {
"line": 458,
"column": 18
} | {
"line": 460,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nh : 0 ≠ 1\np : R[X]\nhp : p.Monic\n⊢ p ≠ 0",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"nontrivial_of_ne",
"AddCommMonoidWithOne.toAddMonoidWithOne",
"AddMonoidWithOne.toOne",
... | [] | nontriviality R
exact hp.ne_zero | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 237,
"column": 4
} | {
"line": 237,
"column": 23
} | {
"line": 239,
"column": 0
} | [
{
"pp": "case inr\nR : Type u\ninst✝ : Semiring R\np q : R[X]\nH : p.natDegree < p.natDegree\nh✝ : ¬p.natDegree = q.natDegree\nh : q.natDegree < p.natDegree\n⊢ False",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Nat.instPreorder",
"Nat",
"LT.lt.false",
"Polynomial... | [] | exact LT.lt.false H | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Polynomial.Eval.Coeff | {
"line": 61,
"column": 6
} | {
"line": 61,
"column": 10
} | {
"line": 62,
"column": 6
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\n⊢ p.coeff 0 * 0 ^ 0 = eval 0 p",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Polynomial.eval",
"HMul.hMul",
"instOfNatNat",
"NPow.toPow",
"Polynomial.coeff",
"instDistribOfSemiring",
"HPow.hPow"... | [
"R : Type u\ninst✝ : Semiring R\np : R[X]\n⊢ eval 0 p = p.coeff 0 * 0 ^ 0"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Algebra.Polynomial.Eval.Degree | {
"line": 69,
"column": 2
} | {
"line": 69,
"column": 44
} | {
"line": 69,
"column": 44
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\nn : ℕ\nhn : p.natDegree < n\nx : R\n⊢ eval x p = ∑ i ∈ range n, p.coeff i * x ^ i",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
"HMul.hMul",
"congrArg",
"Polynomial.sum",
"... | [
"R : Type u\ninst✝ : Semiring R\np : R[X]\nn : ℕ\nhn : p.natDegree < n\nx : R\n⊢ ∀ (n : ℕ), 0 * x ^ n = 0"
] | rw [eval_eq_sum, p.sum_over_range' _ _ hn] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.Eval.Degree | {
"line": 238,
"column": 2
} | {
"line": 238,
"column": 50
} | {
"line": 239,
"column": 2
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : Semiring R\ninst✝¹ : CommRing S\ninst✝ : IsDomain S\nφ : R →+* S\nf : R[X]\nhf : IsUnit f.leadingCoeff\nH : IsUnit (map φ f)\ndz : (map φ f).degree = 0\n⊢ IsUnit f",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
... | [
"R : Type u\nS : Type v\ninst✝² : Semiring R\ninst✝¹ : CommRing S\ninst✝ : IsDomain S\nφ : R →+* S\nf : R[X]\nhf : IsUnit f.leadingCoeff\nH : IsUnit (map φ f)\ndz : f.degree = 0\n⊢ IsUnit f",
"case hf\nR : Type u\nS : Type v\ninst✝² : Semiring R\ninst✝¹ : CommRing S\ninst✝ : IsDomain S\nφ : R →+* S\nf : R[X]\nhf ... | rw [degree_map_eq_of_leadingCoeff_ne_zero] at dz | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.Eval.Degree | {
"line": 239,
"column": 8
} | {
"line": 239,
"column": 33
} | {
"line": 239,
"column": 33
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : Semiring R\ninst✝¹ : CommRing S\ninst✝ : IsDomain S\nφ : R →+* S\nf : R[X]\nhf : IsUnit f.leadingCoeff\nH : IsUnit (map φ f)\ndz : f.degree = 0\n⊢ IsUnit f",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
... | [
"R : Type u\nS : Type v\ninst✝² : Semiring R\ninst✝¹ : CommRing S\ninst✝ : IsDomain S\nφ : R →+* S\nf : R[X]\nhf : IsUnit f.leadingCoeff\nH : IsUnit (map φ f)\ndz : f.degree = 0\n⊢ IsUnit (C (f.coeff 0))"
] | eq_C_of_degree_eq_zero dz | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.AlgebraMap | {
"line": 753,
"column": 40
} | {
"line": 753,
"column": 48
} | {
"line": 753,
"column": 49
} | [
{
"pp": "case inr.inr\nR : Type u\ninst✝ : CommSemiring R\nP : R[X]\nh : ∀ (r : R), r • P = 0 → r = 0\nQ : R[X]\nhQ : P * Q = 0\nl : ℕ\nIH : ∀ m > l, P.coeff m • Q = 0\nhl : (P.coeff l • Q).natDegree = Q.natDegree\nm : ℕ := Q.natDegree\ni j : ℕ\nhij : i + j = l + m\nH : i = l → ¬j = m\nhi : l < i\n⊢ (P.coeff i ... | [
"case inr.inr\nR : Type u\ninst✝ : CommSemiring R\nP : R[X]\nh : ∀ (r : R), r • P = 0 → r = 0\nQ : R[X]\nhQ : P * Q = 0\nl : ℕ\nIH : ∀ m > l, P.coeff m • Q = 0\nhl : (P.coeff l • Q).natDegree = Q.natDegree\nm : ℕ := Q.natDegree\ni j : ℕ\nhij : i + j = l + m\nH : i = l → ¬j = m\nhi : l < i\n⊢ coeff 0 j = 0"
] | IH _ hi, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.MvPolynomial.Eval | {
"line": 89,
"column": 41
} | {
"line": 89,
"column": 61
} | {
"line": 89,
"column": 61
} | [
{
"pp": "R : Type u\nS₁ : Type v\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf : R →+* S₁\ng : σ → S₁\n⊢ ∀ a ∈ (coeffEquiv.toFun p).support ∪ (coeffEquiv.toFun q).support, (f 0 * a.prod fun n e ↦ g n ^ e) = 0",
"ppTerm": "?m.41",
"assigned": true,
"usedCo... | [] | by simp [f.map_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.MvPolynomial.Eval | {
"line": 93,
"column": 28
} | {
"line": 93,
"column": 48
} | {
"line": 93,
"column": 48
} | [
{
"pp": "R : Type u\nS₁ : Type v\nσ : Type u_1\na : R\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\nf : R →+* S₁\ng : σ → S₁\n⊢ (f 0 * s.prod fun n e ↦ g n ^ e) = 0",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"HMul.hMul",
"cong... | [] | by simp [f.map_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 322,
"column": 83
} | {
"line": 333,
"column": 46
} | {
"line": 335,
"column": 0
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\nmotive : MvPolynomial σ R → Prop\nC : ∀ (a : R), motive (MvPolynomial.C a)\nmul_X : ∀ (p : MvPolynomial σ R) (n : σ), motive p → motive (p * X n)\n⊢ ∀ (s : σ →₀ ℕ) (a : R), motive ((monomial s) a)",
"ppTerm": "?m.34",
"assigned": true,
"used... | [] | by
intro s a
apply @Finsupp.induction σ ℕ _ _ s
· change motive (monomial 0 a)
exact C a
· intro n e p _hpn _he ih
have : ∀ e : ℕ, motive (monomial p a * X n ^ e) := by
intro e
induction e with
| zero => simp [ih]
| succ e e_ih => simp [pow_succ, (mul_assoc _ _ _).symm, mul_X, e_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 450,
"column": 2
} | {
"line": 450,
"column": 8
} | {
"line": 450,
"column": 9
} | [
{
"pp": "case C\nR : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\nS : Subalgebra R (MvPolynomial σ R) := Algebra.adjoin R (range X)\na✝ : R\n⊢ C a✝ ∈ S",
"ppTerm": "?C",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"CommSemiring.toSemir... | [] | | C => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 703,
"column": 2
} | {
"line": 703,
"column": 12
} | {
"line": 704,
"column": 2
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : DecidableEq σ\np q : MvPolynomial σ R\n⊢ p.support \\ q.support ⊆ (p + q).support",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"Finset",
"Finset.instSDiff",
"Membership.me... | [
"R : Type u\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : DecidableEq σ\np q : MvPolynomial σ R\nm : σ →₀ ℕ\nhm : m ∈ p.support \\ q.support\n⊢ m ∈ (p + q).support"
] | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Algebra.MvPolynomial.Degrees | {
"line": 283,
"column": 50
} | {
"line": 287,
"column": 7
} | {
"line": 289,
"column": 0
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\np : MvPolynomial σ R\ni : σ\ns : σ →₀ ℕ\nh : s ∈ p.support\n⊢ s i ≤ degreeOf i p",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"Nat.zero_le... | [] | by
obtain si | si := eq_or_lt_of_le <| Nat.zero_le (s i)
· simp [← si]
rw [degreeOf_eq_sup, Finset.le_sup_iff si]
use s | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 1063,
"column": 6
} | {
"line": 1063,
"column": 41
} | {
"line": 1064,
"column": 2
} | [
{
"pp": "case refine_1.add\nR : Type u_2\nS : Type u_3\nσ : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : Algebra R S\nM N : Submodule R S\nr : S\ns : σ →₀ ℕ\nx : S\nhx✝ : x ∈ M * N\ny : S\nhy✝ : y ∈ M * N\nhx : (monomial s) x ∈ coeffsIn σ M * coeffsIn σ N\nhy : (monomial s) y ∈ coeffsIn σ... | [] | simpa [map_add] using add_mem hx hy | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 1063,
"column": 6
} | {
"line": 1063,
"column": 41
} | {
"line": 1064,
"column": 2
} | [
{
"pp": "case refine_1.add\nR : Type u_2\nS : Type u_3\nσ : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : Algebra R S\nM N : Submodule R S\nr : S\ns : σ →₀ ℕ\nx : S\nhx✝ : x ∈ M * N\ny : S\nhy✝ : y ∈ M * N\nhx : (monomial s) x ∈ coeffsIn σ M * coeffsIn σ N\nhy : (monomial s) y ∈ coeffsIn σ... | [] | simpa [map_add] using add_mem hx hy | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 1063,
"column": 6
} | {
"line": 1063,
"column": 41
} | {
"line": 1064,
"column": 2
} | [
{
"pp": "case refine_1.add\nR : Type u_2\nS : Type u_3\nσ : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : Algebra R S\nM N : Submodule R S\nr : S\ns : σ →₀ ℕ\nx : S\nhx✝ : x ∈ M * N\ny : S\nhy✝ : y ∈ M * N\nhx : (monomial s) x ∈ coeffsIn σ M * coeffsIn σ N\nhy : (monomial s) y ∈ coeffsIn σ... | [] | simpa [map_add] using add_mem hx hy | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.Degrees | {
"line": 406,
"column": 4
} | {
"line": 406,
"column": 64
} | {
"line": 408,
"column": 0
} | [
{
"pp": "case inr\nR : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\ni : σ\nh✝ : degreeOf i p ≠ degreeOf i q\nh : degreeOf i q < degreeOf i p\n⊢ max (degreeOf i p) (degreeOf i q) ≤ degreeOf i (p + q)",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Nat.instMulZ... | [] | simp [degreeOf_add_eq_of_degreeOf_lt h, max_eq_left_of_lt h] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.MvPolynomial.Degrees | {
"line": 406,
"column": 4
} | {
"line": 406,
"column": 64
} | {
"line": 408,
"column": 0
} | [
{
"pp": "case inr\nR : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\ni : σ\nh✝ : degreeOf i p ≠ degreeOf i q\nh : degreeOf i q < degreeOf i p\n⊢ max (degreeOf i p) (degreeOf i q) ≤ degreeOf i (p + q)",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Nat.instMulZ... | [] | simp [degreeOf_add_eq_of_degreeOf_lt h, max_eq_left_of_lt h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Degrees | {
"line": 406,
"column": 4
} | {
"line": 406,
"column": 64
} | {
"line": 408,
"column": 0
} | [
{
"pp": "case inr\nR : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\ni : σ\nh✝ : degreeOf i p ≠ degreeOf i q\nh : degreeOf i q < degreeOf i p\n⊢ max (degreeOf i p) (degreeOf i q) ≤ degreeOf i (p + q)",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Nat.instMulZ... | [] | simp [degreeOf_add_eq_of_degreeOf_lt h, max_eq_left_of_lt h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.CommRing | {
"line": 159,
"column": 6
} | {
"line": 159,
"column": 32
} | {
"line": 159,
"column": 32
} | [
{
"pp": "S : Type v\ninst✝ : CommRing S\nR : Type u\nc : ℤ →+* S\nf : MvPolynomial R ℤ →+* S\nx p : MvPolynomial R ℤ\nn : R\nhp : eval₂ c (⇑f ∘ X) p = f p\n⊢ f p * (⇑f ∘ X) n = f (p * X n)",
"ppTerm": "?m.104",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Nat.instMul... | [] | exact (f.map_mul _ _).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.MvPolynomial.Degrees | {
"line": 596,
"column": 4
} | {
"line": 596,
"column": 25
} | {
"line": 598,
"column": 0
} | [
{
"pp": "case mpr\nR : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\np : MvPolynomial σ R\nh : p = C (coeff 0 p)\n⊢ p.totalDegree = 0",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"Nat.instMulZeroClass",
"congrArg",
"Comm... | [] | rw [h, totalDegree_C] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.MvPolynomial.Degrees | {
"line": 596,
"column": 4
} | {
"line": 596,
"column": 25
} | {
"line": 598,
"column": 0
} | [
{
"pp": "case mpr\nR : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\np : MvPolynomial σ R\nh : p = C (coeff 0 p)\n⊢ p.totalDegree = 0",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"Nat.instMulZeroClass",
"congrArg",
"Comm... | [] | rw [h, totalDegree_C] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Degrees | {
"line": 596,
"column": 4
} | {
"line": 596,
"column": 25
} | {
"line": 598,
"column": 0
} | [
{
"pp": "case mpr\nR : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\np : MvPolynomial σ R\nh : p = C (coeff 0 p)\n⊢ p.totalDegree = 0",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"Nat.instMulZeroClass",
"congrArg",
"Comm... | [] | rw [h, totalDegree_C] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 97,
"column": 87
} | {
"line": 99,
"column": 74
} | {
"line": 101,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nf : R[X]\nh : f ≠ 0\n⊢ #f.eraseLead.support < #f.support",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.eraseLead_support",
"congrArg",
"Finset",
"id",
"Polynomial.eraseLead",
"Finse... | [] | by
rw [eraseLead_support]
exact card_lt_card (erase_ssubset <| natDegree_mem_support_of_nonzero h) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 163,
"column": 4
} | {
"line": 163,
"column": 82
} | {
"line": 164,
"column": 4
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\npq : q.degree < p.degree\nn : ℕ\nnd : ¬n = p.natDegree\n⊢ (p + q).eraseLead.coeff n = (p.eraseLead + q).coeff n",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Polynomial.eraseLead_coe... | [
"case neg.hnc\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\npq : q.degree < p.degree\nn : ℕ\nnd : ¬n = p.natDegree\n⊢ ¬n = (p + q).natDegree"
] | rw [eraseLead_coeff, coeff_add, coeff_add, eraseLead_coeff, if_neg, if_neg nd] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 177,
"column": 4
} | {
"line": 177,
"column": 82
} | {
"line": 178,
"column": 4
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\npq : p.degree < q.degree\nn : ℕ\nnd : ¬n = q.natDegree\n⊢ (p + q).eraseLead.coeff n = (p + q.eraseLead).coeff n",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Polynomial.eraseLead_coe... | [
"case neg.hnc\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\npq : p.degree < q.degree\nn : ℕ\nnd : ¬n = q.natDegree\n⊢ ¬n = (p + q).natDegree"
] | rw [eraseLead_coeff, coeff_add, coeff_add, eraseLead_coeff, if_neg, if_neg nd] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 269,
"column": 2
} | {
"line": 269,
"column": 31
} | {
"line": 270,
"column": 2
} | [
{
"pp": "case neg\nR : Type u_2\ninst✝² : Ring R\ninst✝¹ : NoZeroDivisors R\ninst✝ : Nontrivial R\nx : R\nP : R[X]\nhx : x ≠ 0\nh : P.nextCoeff = 0\nhp : ¬P = 0\n⊢ ((X - C x) * P).eraseLead.eraseLead = (X - C x) * P.eraseLead",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Polynomia... | [
"case pos\nR : Type u_2\ninst✝² : Ring R\ninst✝¹ : NoZeroDivisors R\ninst✝ : Nontrivial R\nx : R\nP : R[X]\nhx : x ≠ 0\nh : P.nextCoeff = 0\nhp : ¬P = 0\nhe : P.eraseLead = 0\n⊢ ((X - C x) * P).eraseLead.eraseLead = (X - C x) * P.eraseLead",
"case neg\nR : Type u_2\ninst✝² : Ring R\ninst✝¹ : NoZeroDivisors R\nins... | by_cases he : P.eraseLead = 0 | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 277,
"column": 4
} | {
"line": 279,
"column": 74
} | {
"line": 280,
"column": 4
} | [
{
"pp": "case neg\nR : Type u_2\ninst✝² : Ring R\ninst✝¹ : NoZeroDivisors R\ninst✝ : Nontrivial R\nx : R\nP : R[X]\nhx : x ≠ 0\nh : P.nextCoeff = 0\nhp : ¬P = 0\nhe : P.eraseLead = 0\nhe₂ : ¬((X - C x) * P).eraseLead = 0\n⊢ #((X - C x) * P).support ≤ 2",
"ppTerm": "?neg✝",
"assigned": true,
"usedCon... | [
"case neg\nR : Type u_2\ninst✝² : Ring R\ninst✝¹ : NoZeroDivisors R\ninst✝ : Nontrivial R\nx : R\nP : R[X]\nhx : x ≠ 0\nh : P.nextCoeff = 0\nhp : ¬P = 0\nhe : P.eraseLead = 0\nhe₂ : ¬((X - C x) * P).eraseLead = 0\nh₂ : #(X - C x).support = 2\n⊢ #((X - C x) * P).support ≤ 2"
] | have h₂ : #(X - C x).support = 2 := by
simpa [← sub_eq_add_neg] using!
card_support_binomial one_ne_zero one_ne_zero (neg_ne_zero.mpr hx) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Polynomial.Monic | {
"line": 46,
"column": 5
} | {
"line": 48,
"column": 40
} | {
"line": 48,
"column": 40
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\n⊢ Subsingleton R → (∀ (f g : R[X]), f = g) ∧ ∀ (a b : R), a = b",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"congrArg",
"and_self",
"_private.Mathlib.Algebra.Polynomial.Monic.0.Polynomial.monic_zero_iff_subsingleton'._simp_1_1"... | [] | by
intro
simp [eq_iff_true_of_subsingleton] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Monic | {
"line": 280,
"column": 25
} | {
"line": 280,
"column": 43
} | {
"line": 280,
"column": 43
} | [
{
"pp": "case refine_2\nR : Type u\nι : Type y\ninst✝ : CommSemiring R\nt✝ : Multiset ι\nf : ι → R[X]\na : ι\nt : Multiset ι\nih : (∀ i ∈ t, (f i).Monic) → (Multiset.map f t).prod.Monic\nht : ∀ i ∈ a ::ₘ t, (f i).Monic\n⊢ (f a ::ₘ Multiset.map f t).prod.Monic",
"ppTerm": "?refine_2",
"assigned": true,
... | [
"case refine_2\nR : Type u\nι : Type y\ninst✝ : CommSemiring R\nt✝ : Multiset ι\nf : ι → R[X]\na : ι\nt : Multiset ι\nih : (∀ i ∈ t, (f i).Monic) → (Multiset.map f t).prod.Monic\nht : ∀ i ∈ a ::ₘ t, (f i).Monic\n⊢ (f a * (Multiset.map f t).prod).Monic"
] | Multiset.prod_cons | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.BigOperators | {
"line": 187,
"column": 25
} | {
"line": 187,
"column": 43
} | {
"line": 187,
"column": 43
} | [
{
"pp": "case refine_2\nR : Type u\ninst✝ : CommSemiring R\nt✝ : Multiset R[X]\na : R[X]\nt : Multiset R[X]\nih : (Multiset.map (fun f ↦ f.leadingCoeff) t).prod ≠ 0 → t.prod.natDegree = (Multiset.map (fun f ↦ f.natDegree) t).sum\nht : (a.leadingCoeff ::ₘ Multiset.map (fun f ↦ f.leadingCoeff) t).prod ≠ 0\n⊢ (a :... | [
"case refine_2\nR : Type u\ninst✝ : CommSemiring R\nt✝ : Multiset R[X]\na : R[X]\nt : Multiset R[X]\nih : (Multiset.map (fun f ↦ f.leadingCoeff) t).prod ≠ 0 → t.prod.natDegree = (Multiset.map (fun f ↦ f.natDegree) t).sum\nht : a.leadingCoeff * (Multiset.map (fun f ↦ f.leadingCoeff) t).prod ≠ 0\n⊢ (a * t.prod).natDe... | Multiset.prod_cons | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Monic | {
"line": 522,
"column": 4
} | {
"line": 522,
"column": 14
} | {
"line": 523,
"column": 4
} | [
{
"pp": "case refine_1\nR : Type u\ninst✝¹ : Semiring R\nS : Type u_1\ninst✝ : SMulZeroClass S R\nk : S\np : R[X]\nh : IsSMulRegular R k\n⊢ ∀ (m : ℕ), p.degree < ↑m → (k • p).coeff m = 0",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"WithBot",
... | [
"case refine_1\nR : Type u\ninst✝¹ : Semiring R\nS : Type u_1\ninst✝ : SMulZeroClass S R\nk : S\np : R[X]\nh : IsSMulRegular R k\nm : ℕ\nhm : p.degree < ↑m\n⊢ (k • p).coeff m = 0"
] | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Algebra.Polynomial.Monic | {
"line": 526,
"column": 4
} | {
"line": 526,
"column": 14
} | {
"line": 527,
"column": 4
} | [
{
"pp": "case refine_2\nR : Type u\ninst✝¹ : Semiring R\nS : Type u_1\ninst✝ : SMulZeroClass S R\nk : S\np : R[X]\nh : IsSMulRegular R k\n⊢ ∀ (m : ℕ), (k • p).degree < ↑m → p.coeff m = 0",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"WithBot",
... | [
"case refine_2\nR : Type u\ninst✝¹ : Semiring R\nS : Type u_1\ninst✝ : SMulZeroClass S R\nk : S\np : R[X]\nh : IsSMulRegular R k\nm : ℕ\nhm : (k • p).degree < ↑m\n⊢ p.coeff m = 0"
] | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Algebra.Polynomial.BigOperators | {
"line": 274,
"column": 6
} | {
"line": 274,
"column": 34
} | {
"line": 275,
"column": 6
} | [
{
"pp": "case hnc.h\nR : Type u\ninst✝ : CommRing R\nt : Multiset R\nht : 0 < t.card\na✝ : Nontrivial R\n⊢ ∀ f ∈ Multiset.map (fun x ↦ X - C x) t, f.Monic",
"ppTerm": "?hnc.h",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Multiset.map",
"CommSemiring.toSemi... | [
"case hnc.h\nR : Type u\ninst✝ : CommRing R\nt : Multiset R\nht : 0 < t.card\na✝ : Nontrivial R\n⊢ ∀ (f : R[X]), (∃ a ∈ t, X - C a = f) → f.Monic"
] | simp only [Multiset.mem_map] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.MvPolynomial.Equiv | {
"line": 841,
"column": 2
} | {
"line": 841,
"column": 12
} | {
"line": 842,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nn : ℕ\np : MvPolynomial (Fin (n + 1)) R\nj : Fin n\ni : ℕ\n⊢ ∀ b ∈ (((finSuccEquiv R n) p).coeff i).support, b j ≤ p.support.sup fun m ↦ m j.succ",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"AddMonoidAlgeb... | [
"R : Type u\ninst✝ : CommSemiring R\nn : ℕ\np : MvPolynomial (Fin (n + 1)) R\nj : Fin n\ni : ℕ\nm : Fin n →₀ ℕ\nhm : m ∈ (((finSuccEquiv R n) p).coeff i).support\n⊢ m j ≤ p.support.sup fun m ↦ m j.succ"
] | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Algebra.Ring.GeomSum | {
"line": 223,
"column": 39
} | {
"line": 223,
"column": 61
} | {
"line": 223,
"column": 61
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\nr : R\np b : ℕ\nhp₀ : p = 0\nhb₀ : ¬b = 0\nh : 0 ∣ b\n⊢ False",
"ppTerm": "?m.85",
"assigned": true,
"usedConstants": [
"Dvd.dvd",
"Nat.instSemigroupWithZero",
"congrArg",
"semigroupDvd",
"zero_dvd_iff",
"SemigroupWithZero.to... | [
"R : Type u_1\ninst✝ : Ring R\nr : R\np b : ℕ\nhp₀ : p = 0\nhb₀ : ¬b = 0\nh : b = 0\n⊢ False"
] | rw [zero_dvd_iff] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Polynomial.Basic | {
"line": 228,
"column": 50
} | {
"line": 230,
"column": 51
} | {
"line": 232,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\ns : Set R[X]\ns_fin : s.Finite\n⊢ ∃ n, Submodule.span R s ≤ degreeLT R n",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.degreeLT",
"Submodule",
"WithBot",
"Semiring.toModule",
"congrArg",
... | [] | by
rcases span_le_degreeLE_of_finite s_fin with ⟨n, _⟩
exact ⟨n + 1, by rwa [degreeLT_succ_eq_degreeLE]⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.Basic | {
"line": 235,
"column": 2
} | {
"line": 243,
"column": 24
} | {
"line": 245,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : Semiring R\ninst✝ : Nontrivial R\n⊢ ¬Module.Finite R R[X]",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_exists._simp_1",
"WithBot.addMonoidWithOne",
"WithBot.instPreorder",
"Eq.mpr",
"WithBot.zeroLEOneCla... | [] | rw [Module.finite_def, Submodule.fg_def]
push Not
intro s hs contra
rcases span_le_degreeLE_of_finite hs with ⟨n, hn⟩
have : ((X : R[X]) ^ (n + 1)) ∈ Polynomial.degreeLE R ↑n := by
rw [contra] at hn
exact hn Submodule.mem_top
rw [mem_degreeLE, degree_X_pow, Nat.cast_le, add_le_iff_nonpos_right, nonpos... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Basic | {
"line": 235,
"column": 2
} | {
"line": 243,
"column": 24
} | {
"line": 245,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : Semiring R\ninst✝ : Nontrivial R\n⊢ ¬Module.Finite R R[X]",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_exists._simp_1",
"WithBot.addMonoidWithOne",
"WithBot.instPreorder",
"Eq.mpr",
"WithBot.zeroLEOneCla... | [] | rw [Module.finite_def, Submodule.fg_def]
push Not
intro s hs contra
rcases span_le_degreeLE_of_finite hs with ⟨n, hn⟩
have : ((X : R[X]) ^ (n + 1)) ∈ Polynomial.degreeLE R ↑n := by
rw [contra] at hn
exact hn Submodule.mem_top
rw [mem_degreeLE, degree_X_pow, Nat.cast_le, add_le_iff_nonpos_right, nonpos... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 437,
"column": 6
} | {
"line": 443,
"column": 28
} | {
"line": 444,
"column": 4
} | [
{
"pp": "case succ.refine_2\nR : Type u_1\ninst✝ : Semiring R\nn : ℕ\nhn : ∀ {f : R[X]}, #f.support = n → ∃ k x, ∃ (_ : StrictMono k) (_ : ∀ (i : Fin n), x i ≠ 0), f = ∑ i, C (x i) * X ^ k i\nf : R[X]\nh : #f.support = n + 1\nk : Fin n → ℕ\nx : Fin n → R\nhk : StrictMono k\nhx : ∀ (i : Fin n), x i ≠ 0\nhf : f.e... | [] | intro i
by_cases hi : ∃ i₀, Fin.castSucc i₀ = i
· obtain ⟨i, rfl⟩ := hi
rw [Fin.strictMono_castSucc.injective.extend_apply]
exact hx i
· rw [Function.extend_apply' _ _ _ hi, Ne, leadingCoeff_eq_zero, ← card_support_eq_zero, h]
exact n.succ_ne_zero | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 437,
"column": 6
} | {
"line": 443,
"column": 28
} | {
"line": 444,
"column": 4
} | [
{
"pp": "case succ.refine_2\nR : Type u_1\ninst✝ : Semiring R\nn : ℕ\nhn : ∀ {f : R[X]}, #f.support = n → ∃ k x, ∃ (_ : StrictMono k) (_ : ∀ (i : Fin n), x i ≠ 0), f = ∑ i, C (x i) * X ^ k i\nf : R[X]\nh : #f.support = n + 1\nk : Fin n → ℕ\nx : Fin n → R\nhk : StrictMono k\nhx : ∀ (i : Fin n), x i ≠ 0\nhf : f.e... | [] | intro i
by_cases hi : ∃ i₀, Fin.castSucc i₀ = i
· obtain ⟨i, rfl⟩ := hi
rw [Fin.strictMono_castSucc.injective.extend_apply]
exact hx i
· rw [Function.extend_apply' _ _ _ hi, Ne, leadingCoeff_eq_zero, ← card_support_eq_zero, h]
exact n.succ_ne_zero | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Adjoin.FG | {
"line": 122,
"column": 33
} | {
"line": 122,
"column": 44
} | {
"line": 122,
"column": 45
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nh : ⊤.FG\ns : Finset A\nhs : span R ↑s = ⊤\n⊢ toSubmodule (Algebra.adjoin R ↑s) = ⊤",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"eq_top_iff",
... | [
"R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nh : ⊤.FG\ns : Finset A\nhs : span R ↑s = ⊤\n⊢ ⊤ ≤ toSubmodule (Algebra.adjoin R ↑s)"
] | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Adjoin.Tower | {
"line": 126,
"column": 27
} | {
"line": 126,
"column": 38
} | {
"line": 126,
"column": 39
} | [
{
"pp": "case convert_3\nA : Type w\nB : Type u₁\nC : Type u_1\ninst✝⁶ : CommSemiring A\ninst✝⁵ : CommSemiring B\ninst✝⁴ : Semiring C\ninst✝³ : Algebra A B\ninst✝² : Algebra B C\ninst✝¹ : Algebra A C\ninst✝ : IsScalarTower A B C\nx : Finset C\nhx : Algebra.adjoin A ↑x = ⊤\ny : Finset C\nhy : span B ↑y = ⊤\nf : ... | [
"case convert_3\nA : Type w\nB : Type u₁\nC : Type u_1\ninst✝⁶ : CommSemiring A\ninst✝⁵ : CommSemiring B\ninst✝⁴ : Semiring C\ninst✝³ : Algebra A B\ninst✝² : Algebra B C\ninst✝¹ : Algebra A C\ninst✝ : IsScalarTower A B C\nx : Finset C\nhx : Algebra.adjoin A ↑x = ⊤\ny : Finset C\nhy : span B ↑y = ⊤\nf : C → C → B\nh... | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Basic | {
"line": 760,
"column": 4
} | {
"line": 760,
"column": 8
} | {
"line": 761,
"column": 4
} | [
{
"pp": "R : Type u\nσ : Type v\ninst✝ : CommRing R\ns : Set σ\np : MvPolynomial (↑s) R\n⊢ Prime ((rename Subtype.val) p) ↔ Prime p",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"CommSemiring.toSemiring",
"AlgHo... | [
"R : Type u\nσ : Type v\ninst✝ : CommRing R\ns : Set σ\np : MvPolynomial (↑s) R\n⊢ Prime p ↔ Prime ((rename Subtype.val) p)"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.RingTheory.Polynomial.Basic | {
"line": 940,
"column": 2
} | {
"line": 940,
"column": 6
} | {
"line": 940,
"column": 6
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\n⊢ R ≃+* MvPolynomial (Fin 0) R",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"CommSemiring.toSemiring",
"Finsupp.instAddMonoid",
"Nat.i... | [
"R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\n⊢ MvPolynomial (Fin 0) R ≃+* R"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Data.Nat.Prime.Basic | {
"line": 43,
"column": 74
} | {
"line": 43,
"column": 80
} | {
"line": 43,
"column": 80
} | [
{
"pp": "p : ℕ\nhp : Prime p\nh : p % 2 = 0\n⊢ ¬2 = 1",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"Bool",
"Eq.refl",
"instDecidableEqNat",
"OfN... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Data.Nat.Prime.Basic | {
"line": 43,
"column": 74
} | {
"line": 43,
"column": 80
} | {
"line": 43,
"column": 80
} | [
{
"pp": "p : ℕ\nhp : Prime p\nh : p % 2 = 0\n⊢ ¬2 = 1",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"Bool",
"Eq.refl",
"instDecidableEqNat",
"OfN... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Prime.Basic | {
"line": 43,
"column": 74
} | {
"line": 43,
"column": 80
} | {
"line": 43,
"column": 80
} | [
{
"pp": "p : ℕ\nhp : Prime p\nh : p % 2 = 0\n⊢ ¬2 = 1",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"Bool",
"Eq.refl",
"instDecidableEqNat",
"OfN... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Star.Subalgebra | {
"line": 499,
"column": 27
} | {
"line": 499,
"column": 49
} | {
"line": 499,
"column": 49
} | [
{
"pp": "R : Type u_2\nA : Type u_3\ninst✝⁵ : CommSemiring R\ninst✝⁴ : StarRing R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : StarRing A\ninst✝ : StarModule R A\ns : Set A\n⊢ Subalgebra.toSubmodule (Algebra.adjoin R (s ∪ star s)) = span R ↑(Submonoid.closure (s ∪ star s))",
"ppTerm": "?m.57",
"... | [
"R : Type u_2\nA : Type u_3\ninst✝⁵ : CommSemiring R\ninst✝⁴ : StarRing R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : StarRing A\ninst✝ : StarModule R A\ns : Set A\n⊢ span R ↑(Submonoid.closure (s ∪ star s)) = span R ↑(Submonoid.closure (s ∪ star s))"
] | Algebra.adjoin_eq_span | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Dynamics.PeriodicPts.Defs | {
"line": 295,
"column": 53
} | {
"line": 303,
"column": 37
} | {
"line": 305,
"column": 0
} | [
{
"pp": "α : Type u_1\nf : α → α\nx : α\nhx : x ∈ periodicPts f\nn : ℕ\n⊢ minimalPeriod f (f^[n] x) = minimalPeriod f x",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Function.IsPeriodicPt.minimalPeriod_le",
"Function.isPeriodicPt_minimalPeriod",
"Membership.mem",
... | [] | by
apply
(IsPeriodicPt.minimalPeriod_le (minimalPeriod_pos_of_mem_periodicPts hx) _).antisymm
((isPeriodicPt_of_mem_periodicPts_of_isPeriodicPt_iterate hx
(isPeriodicPt_minimalPeriod f _)).minimalPeriod_le
(minimalPeriod_pos_of_mem_periodicPts _))
· exact (isPeriodicPt_minimalPeriod f ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.List.Permutation | {
"line": 166,
"column": 39
} | {
"line": 166,
"column": 47
} | {
"line": 166,
"column": 47
} | [
{
"pp": "α : Type u_1\nt : α\nts ys l : List α\n⊢ id = fun x ↦ [] ++ x",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"id",
"funext",
"instHAppendOfAppend",
"List",
"List.instAppend",
"HAppend.hAppend",
"List.nil"
],
"usedFVars": [
"... | [
"α : Type u_1\nt : α\nts ys l x✝ : List α\n⊢ id x✝ = [] ++ x✝"
] | funext _ | _aux_Init_NotationExtra___macroRules_tacticFunext____1 | tacticFunext___ |
Mathlib.Data.List.Permutation | {
"line": 248,
"column": 2
} | {
"line": 248,
"column": 10
} | {
"line": 249,
"column": 2
} | [
{
"pp": "case cons.e_a.e_b\nα : Type u_1\nts : List α\nt : α\nis : List α\nih :\n ∀ (is' : List α),\n (is ++ ts).permutationsAux is' =\n map (fun x ↦ x ++ ts) (is.permutationsAux is') ++ ts.permutationsAux (is.reverse ++ is')\nis' : List α\n⊢ (fun y ↦ (permutationsAux2 t (is ++ ts) [] y id).snd) = fun ... | [
"case cons.e_a.e_b\nα : Type u_1\nts : List α\nt : α\nis : List α\nih :\n ∀ (is' : List α),\n (is ++ ts).permutationsAux is' =\n map (fun x ↦ x ++ ts) (is.permutationsAux is') ++ ts.permutationsAux (is.reverse ++ is')\nis' x✝ : List α\n⊢ (permutationsAux2 t (is ++ ts) [] x✝ id).snd = map (fun x ↦ x ++ ts) ... | funext _ | _aux_Init_NotationExtra___macroRules_tacticFunext____1 | tacticFunext___ |
Mathlib.Data.List.Permutation | {
"line": 280,
"column": 69
} | {
"line": 280,
"column": 84
} | {
"line": 281,
"column": 2
} | [
{
"pp": "α : Type u_1\nt : α\nts is : List α\nIH1 : (ts.permutationsAux (t :: is)).length + (t :: is).length ! = (ts.length + (t :: is).length)!\nIH2 : (is.permutationsAux []).length + [].length ! = (is.length + [].length)!\n⊢ (is.permutationsAux []).length + 1 = is.length !",
"ppTerm": "?m.43",
"assign... | [] | simpa using IH2 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Data.List.Permutation | {
"line": 311,
"column": 2
} | {
"line": 311,
"column": 27
} | {
"line": 313,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_1\nt : α\nts is : List α\nIH1 : ∀ (l : List α), l ~ t :: is ++ ts → (∃ is' x, l = is' ++ ts) ∨ l ∈ ts.permutationsAux (t :: is)\nIH2 : ∀ (l : List α), l ~ [] ++ is → (∃ is' x, l = is' ++ is) ∨ l ∈ is.permutationsAux []\nl : List α\np : l ~ is ++ t :: ts\nm : l ∈ ts.permutationsAux ... | [] | · exact Or.inr (Or.inl m) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Star.Subalgebra | {
"line": 581,
"column": 52
} | {
"line": 585,
"column": 6
} | {
"line": 587,
"column": 0
} | [
{
"pp": "R : Type u_2\nA : Type u_3\ninst✝⁵ : CommSemiring R\ninst✝⁴ : StarRing R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : StarRing A\ninst✝ : StarModule R A\ns : Set A\n⊢ adjoin R s ≤ centralizer R ↑(centralizer R s)",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Subal... | [] | by
rw [← toSubalgebra_le_iff, centralizer_toSubalgebra, adjoin_toSubalgebra]
convert! Algebra.adjoin_le_centralizer_centralizer R (s ∪ star s)
rw [StarMemClass.star_coe_eq]
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.List.Permutation | {
"line": 379,
"column": 2
} | {
"line": 379,
"column": 10
} | {
"line": 380,
"column": 2
} | [
{
"pp": "case succ.refine_2.e_b\nα : Type u_1\nn : ℕ\nIH : ∀ (ts : List α), ts.length < n → ts.permutations ~ ts.permutations'\nts✝ : List α\nh✝ : ts✝.length < n + 1\nts : List α\nt : α\nx✝ : ts.length < n + 1 → ts.permutations ~ ts.permutations'\nh : ts.length < n\nIH₂ : ts.reverse.permutations ~ ts.permutatio... | [
"case succ.refine_2.e_b\nα : Type u_1\nn : ℕ\nIH : ∀ (ts : List α), ts.length < n → ts.permutations ~ ts.permutations'\nts✝ : List α\nh✝ : ts✝.length < n + 1\nts : List α\nt : α\nx✝¹ : ts.length < n + 1 → ts.permutations ~ ts.permutations'\nh : ts.length < n\nIH₂ : ts.reverse.permutations ~ ts.permutations'\nx✝ : L... | funext _ | _aux_Init_NotationExtra___macroRules_tacticFunext____1 | tacticFunext___ |
Mathlib.Data.List.Cycle | {
"line": 97,
"column": 2
} | {
"line": 98,
"column": 29
} | {
"line": 99,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nxs : List α\nx d : α\n⊢ d ∈ xs → xs.nextOr x d ∈ xs",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"List.nextOr",
"Membership.mem",
"id",
"List",
"List.instMembership"
],
"usedFVars": [
"α",
"x... | [
"α : Type u_1\ninst✝ : DecidableEq α\nxs : List α\nx d : α\n⊢ ∀ (xs' : List α), (∀ x ∈ xs, x ∈ xs') → d ∈ xs' → xs.nextOr x d ∈ xs'"
] | suffices ∀ xs' : List α, (∀ x ∈ xs, x ∈ xs') → d ∈ xs' → nextOr xs x d ∈ xs' by
exact this xs fun _ => id | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Data.List.Permutation | {
"line": 446,
"column": 2
} | {
"line": 452,
"column": 35
} | {
"line": 454,
"column": 0
} | [
{
"pp": "α : Type u_1\nx : α\n⊢ Injective (permutations'Aux x)",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"List.get",
"List.insertIdx",
"GetElem.getElem.congr_simp",
"List.insertIdx_injective",
"List.get_permutations'Aux",... | [] | intro s t h
apply insertIdx_injective s.length x
dsimp
have hl : s.length = t.length := by simpa using congr_arg length h
rw [← get_permutations'Aux s x s.length (by simp),
← get_permutations'Aux t x s.length (by simp [hl])]
simp only [get_eq_getElem, h, hl] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.List.Permutation | {
"line": 446,
"column": 2
} | {
"line": 452,
"column": 35
} | {
"line": 454,
"column": 0
} | [
{
"pp": "α : Type u_1\nx : α\n⊢ Injective (permutations'Aux x)",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"List.get",
"List.insertIdx",
"GetElem.getElem.congr_simp",
"List.insertIdx_injective",
"List.get_permutations'Aux",... | [] | intro s t h
apply insertIdx_injective s.length x
dsimp
have hl : s.length = t.length := by simpa using congr_arg length h
rw [← get_permutations'Aux s x s.length (by simp),
← get_permutations'Aux t x s.length (by simp [hl])]
simp only [get_eq_getElem, h, hl] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.GroupAction.Basic | {
"line": 217,
"column": 6
} | {
"line": 217,
"column": 44
} | {
"line": 217,
"column": 44
} | [
{
"pp": "G : Type u_1\nα : Type u_2\ninst✝³ : Group G\ninst✝² : MulAction G α\ninst✝¹ : Finite G\ninst✝ : Finite Ω\n⊢ Finite α",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"MulAction.orbitRel.Quotient.orbit",
"Finite",
"Equiv.finite_iff... | [
"G : Type u_1\nα : Type u_2\ninst✝³ : Group G\ninst✝² : MulAction G α\ninst✝¹ : Finite G\ninst✝ : Finite Ω\n⊢ Finite ((ω : Ω) × ↑ω.orbit)"
] | (selfEquivSigmaOrbits' G _).finite_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.List.Cycle | {
"line": 211,
"column": 34
} | {
"line": 214,
"column": 36
} | {
"line": 216,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nl : List α\nx y z : α\nh : x ∈ y :: z :: l\nhy : x ≠ y\nhz : x = z\n⊢ (y :: z :: l).prev x h = y",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"List.getLast",
"Eq.mpr",
"dite_congr",
"ite_eq_right_iff._simp_1",
"... | [] | by
cases l
· simp [prev, hz]
· rw [prev, dif_neg hy, if_pos hz] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.List.Cycle | {
"line": 223,
"column": 4
} | {
"line": 223,
"column": 22
} | {
"line": 224,
"column": 2
} | [
{
"pp": "case nil\nα : Type u_1\ninst✝ : DecidableEq α\nx y z : α\nhy : x ≠ y\nhz : x ≠ z\nh : x ∈ [y, z]\n⊢ [y, z].prev x h = [z].prev x ⋯",
"ppTerm": "?nil",
"assigned": true,
"usedConstants": [
"List.prev_ne_cons_cons._proof_1",
"False",
"eq_false",
"congrArg",
"Fals... | [] | simp [hy, hz] at h | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.List.Cycle | {
"line": 223,
"column": 4
} | {
"line": 223,
"column": 22
} | {
"line": 224,
"column": 2
} | [
{
"pp": "case nil\nα : Type u_1\ninst✝ : DecidableEq α\nx y z : α\nhy : x ≠ y\nhz : x ≠ z\nh : x ∈ [y, z]\n⊢ [y, z].prev x h = [z].prev x ⋯",
"ppTerm": "?nil",
"assigned": true,
"usedConstants": [
"List.prev_ne_cons_cons._proof_1",
"False",
"eq_false",
"congrArg",
"Fals... | [] | simp [hy, hz] at h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.List.Cycle | {
"line": 223,
"column": 4
} | {
"line": 223,
"column": 22
} | {
"line": 224,
"column": 2
} | [
{
"pp": "case nil\nα : Type u_1\ninst✝ : DecidableEq α\nx y z : α\nhy : x ≠ y\nhz : x ≠ z\nh : x ∈ [y, z]\n⊢ [y, z].prev x h = [z].prev x ⋯",
"ppTerm": "?nil",
"assigned": true,
"usedConstants": [
"List.prev_ne_cons_cons._proof_1",
"False",
"eq_false",
"congrArg",
"Fals... | [] | simp [hy, hz] at h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.Factors | {
"line": 136,
"column": 61
} | {
"line": 137,
"column": 86
} | {
"line": 139,
"column": 0
} | [
{
"pp": "a b : ℕ\nha : a ≠ 0\nhb : b ≠ 0\nh : a.primeFactorsList ~ b.primeFactorsList\n⊢ a = b",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"List.Perm.prod_eq",
"Eq.mp",
"MulOne.toMul",
"MulOneC... | [] | by
simpa [prod_primeFactorsList ha, prod_primeFactorsList hb] using List.Perm.prod_eq h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Nat.Factors | {
"line": 183,
"column": 2
} | {
"line": 183,
"column": 6
} | {
"line": 184,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Prime p\nn : ℕ\n⊢ (p ^ n).primeFactorsList = replicate n p",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"List.replicate",
"Nat.instMonoid",
"NPow.toPow",
"List",
"HPow.hPow",
"Nat",
"instHPow",
"Eq.symm",
"Monoid.... | [
"p : ℕ\nhp : Prime p\nn : ℕ\n⊢ replicate n p = (p ^ n).primeFactorsList"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.NumberTheory.Divisors | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 88
} | {
"line": 83,
"column": 2
} | [
{
"pp": "n : ℕ\nh : n ≠ 0\na✝ : ℕ\n⊢ a✝ ∈ {d ∈ range n.succ | d ∣ n} ↔ a✝ ∈ n.divisors",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Dvd.dvd",
"congrArg",
"Nat.decidable_dvd",
"Finset",
"Preorder.toLE",
"Nat.inst... | [
"n : ℕ\nh : n ≠ 0\na✝ : ℕ\n⊢ a✝ ∣ n → a✝ < n.succ → 1 ≤ a✝"
] | simp only [divisors, mem_filter, mem_range, mem_Ico, and_congr_left_iff, iff_and_self] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.ZMod.Basic | {
"line": 297,
"column": 8
} | {
"line": 297,
"column": 19
} | {
"line": 297,
"column": 19
} | [
{
"pp": "case succ.zero\nR : Type u_1\ninst✝¹ : Ring R\nm : ℕ\ninst✝ : CharP R m\nh : m ∣ 0 + 1\n⊢ ↑(1 % (0 + 1)) = 1",
"ppTerm": "?succ.zero",
"assigned": true,
"usedConstants": [
"Dvd.dvd",
"congrArg",
"Nat.dvd_one",
"Eq.mp",
"instOfNatNat",
"Nat.instDvd",
... | [
"case succ.zero\nR : Type u_1\ninst✝¹ : Ring R\nm : ℕ\ninst✝ : CharP R m\nh : m = 1\n⊢ ↑(1 % (0 + 1)) = 1"
] | Nat.dvd_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.ZMod.Basic | {
"line": 309,
"column": 2
} | {
"line": 309,
"column": 6
} | {
"line": 310,
"column": 2
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝¹ : Ring R\nm : ℕ\ninst✝ : CharP R m\nn✝ : ℕ\nh : m ∣ n✝ + 1\na b : ZMod (n✝ + 1)\n⊢ (a + b).cast = a.cast + b.cast",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"ZMod.cast",
"ZMod.commRing",
"CommSemiring.toSemiring",
"Distr... | [
"case succ\nR : Type u_1\ninst✝¹ : Ring R\nm : ℕ\ninst✝ : CharP R m\nn✝ : ℕ\nh : m ∣ n✝ + 1\na b : ZMod (n✝ + 1)\n⊢ a.cast + b.cast = (a + b).cast"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Data.ZMod.Basic | {
"line": 320,
"column": 2
} | {
"line": 320,
"column": 6
} | {
"line": 321,
"column": 2
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝¹ : Ring R\nm : ℕ\ninst✝ : CharP R m\nn✝ : ℕ\nh : m ∣ n✝ + 1\na b : ZMod (n✝ + 1)\n⊢ (a * b).cast = a.cast * b.cast",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"ZMod.cast",
"ZMod.commRing",
"CommSemiring.toSemi... | [
"case succ\nR : Type u_1\ninst✝¹ : Ring R\nm : ℕ\ninst✝ : CharP R m\nn✝ : ℕ\nh : m ∣ n✝ + 1\na b : ZMod (n✝ + 1)\n⊢ a.cast * b.cast = (a * b).cast"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.NumberTheory.Divisors | {
"line": 466,
"column": 6
} | {
"line": 466,
"column": 10
} | {
"line": 467,
"column": 6
} | [
{
"pp": "case succ.succ\nn : ℕ\nh : ∑ x ∈ (n + 1 + 1).properDivisors, x ∣ n + 1 + 1\nne_n : ¬∑ x ∈ (n + 1 + 1).properDivisors, x = n + 1 + 1\nhlt : ∑ x ∈ n.succ.succ.properDivisors, x < n.succ.succ\n⊢ ∑ x ∈ (n + 1 + 1).properDivisors, x = 1",
"ppTerm": "?succ.succ",
"assigned": true,
"usedConstants"... | [
"case succ.succ\nn : ℕ\nh : ∑ x ∈ (n + 1 + 1).properDivisors, x ∣ n + 1 + 1\nne_n : ¬∑ x ∈ (n + 1 + 1).properDivisors, x = n + 1 + 1\nhlt : ∑ x ∈ n.succ.succ.properDivisors, x < n.succ.succ\n⊢ 1 = ∑ x ∈ (n + 1 + 1).properDivisors, x"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.NumberTheory.Divisors | {
"line": 488,
"column": 2
} | {
"line": 499,
"column": 44
} | {
"line": 501,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ ∑ x ∈ n.properDivisors, x = 1 ↔ Prime n",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Nat.properDivisors_zero",
"Eq.mpr",
"False",
"Nat.Prime",
"Nat.one_mem_properDivisors_iff_one_lt",
"Nat.properDivisors_one",
... | [] | rcases n with - | n
· simp [Nat.not_prime_zero]
· cases n
· simp [Nat.not_prime_one]
· rw [← properDivisors_eq_singleton_one_iff_prime]
refine ⟨fun h => ?_, fun h => h.symm ▸ sum_singleton _ _⟩
rw [@eq_comm (Finset ℕ) _ _]
apply
eq_properDivisors_of_subset_of_sum_eq_sum
(... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Divisors | {
"line": 488,
"column": 2
} | {
"line": 499,
"column": 44
} | {
"line": 501,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ ∑ x ∈ n.properDivisors, x = 1 ↔ Prime n",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Nat.properDivisors_zero",
"Eq.mpr",
"False",
"Nat.Prime",
"Nat.one_mem_properDivisors_iff_one_lt",
"Nat.properDivisors_one",
... | [] | rcases n with - | n
· simp [Nat.not_prime_zero]
· cases n
· simp [Nat.not_prime_one]
· rw [← properDivisors_eq_singleton_one_iff_prime]
refine ⟨fun h => ?_, fun h => h.symm ▸ sum_singleton _ _⟩
rw [@eq_comm (Finset ℕ) _ _]
apply
eq_properDivisors_of_subset_of_sum_eq_sum
(... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.ZMod.Basic | {
"line": 706,
"column": 43
} | {
"line": 706,
"column": 54
} | {
"line": 706,
"column": 54
} | [
{
"pp": "n : ℕ\n⊢ n ∣ 1 ↔ n = 1",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"congrArg",
"Nat.dvd_one",
"id",
"instOfNatNat",
"Iff",
"Nat.instDvd",
"Nat",
"propext",
"OfNat.ofNat",
"Eq"
],
... | [
"n : ℕ\n⊢ n = 1 ↔ n = 1"
] | Nat.dvd_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Divisors | {
"line": 572,
"column": 84
} | {
"line": 582,
"column": 26
} | {
"line": 584,
"column": 0
} | [
{
"pp": "n N : ℕ\nn_ne_zero : n ≠ 0\nhn : n ≤ N\n⊢ n.divisorsAntidiagonal = {x ∈ Ioc 0 N ×ˢ Ioc 0 N | x.1 * x.2 = n}",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"le_refl",
"Nat.instMulZeroClass",
"Preorder.toLT",
"IsOrderedR... | [] | by
ext ⟨n1, n2⟩
rw [Nat.mem_divisorsAntidiagonal]
simp only [ne_eq, Finset.mem_filter, Finset.mem_product, Finset.mem_Ioc]
constructor
· intro ⟨rfl, hn2⟩
grw [← hn]
simp (disch := lia) only [le_mul_iff_one_le_right, le_mul_iff_one_le_left, and_true]
lia
· intro ⟨⟨hn1, hn2⟩, hn3⟩
exact ⟨hn3, ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Ring.GeomSum | {
"line": 190,
"column": 10
} | {
"line": 190,
"column": 23
} | {
"line": 190,
"column": 24
} | [
{
"pp": "b : ℕ\nhb : 2 ≤ b\na n : ℕ\n⊢ a / b ^ 0 + ∑ i ∈ Ico 1 n.succ, a / b ^ i = ∑ i ∈ range n.succ, a / b ^ i",
"ppTerm": "?m.140",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHDiv",
"congrArg",
"Finset",
"Nat.instMonoid",
"Nat.instLocallyFiniteOrder",
... | [
"b : ℕ\nhb : 2 ≤ b\na n : ℕ\n⊢ a / b ^ 0 + ∑ i ∈ Ico 1 n.succ, a / b ^ i = ∑ i ∈ Ico 0 n.succ, a / b ^ i"
] | range_eq_Ico, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.ZMod.Basic | {
"line": 939,
"column": 63
} | {
"line": 939,
"column": 69
} | {
"line": 939,
"column": 69
} | [
{
"pp": "m n : ℕ\n⊢ ∀ (a b : (ZMod 2)ˣ), a = b",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"ZMod.commRing",
"CommSemiring.toSemiring",
"ZMod.fintype",
"ZMod.decidableEq",
"Units",
"instFintypeUnitsOfDecidableEq",
"id",... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
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