module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.MvPolynomial.Equiv | {
"line": 740,
"column": 4
} | {
"line": 740,
"column": 67
} | {
"line": 740,
"column": 68
} | [
{
"pp": "case mpr\nR : Type u\ninst✝ : CommSemiring R\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) R\ni : ℕ\nm : Fin n →₀ ℕ\nh : cons i m ∈ f.support\n⊢ m ∈ (((finSuccEquiv R n) f).coeff i).support",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
... | [
"case mpr\nR : Type u\ninst✝ : CommSemiring R\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) R\ni : ℕ\nm : Fin n →₀ ℕ\nh : cons i m ∈ f.support\n⊢ ¬coeff m (((finSuccEquiv R n) f).coeff i) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.BigOperators | {
"line": 235,
"column": 4
} | {
"line": 235,
"column": 15
} | {
"line": 235,
"column": 16
} | [
{
"pp": "R : Type u\nι : Type w\ninst✝ : CommSemiring R\nf : ι → R[X]\nn : ℕ\nl : Multiset ι\nhl : l.Nodup\nh : ∀ p ∈ { val := l, nodup := hl }, (f p).natDegree ≤ n\n⊢ ∀ p ∈ Multiset.map f l, p.natDegree ≤ n",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Multiset.ma... | [
"R : Type u\nι : Type w\ninst✝ : CommSemiring R\nf : ι → R[X]\nn : ℕ\nl : Multiset ι\nhl : l.Nodup\nh : ∀ p ∈ { val := l, nodup := hl }, (f p).natDegree ≤ n\n⊢ ∀ a ∈ l, (f a).natDegree ≤ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Monic | {
"line": 529,
"column": 4
} | {
"line": 529,
"column": 15
} | {
"line": 529,
"column": 16
} | [
{
"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\nm : ℕ\nhm : ∀ (m_1 : ℕ), m ≤ m_1 → (k • p).coeff m_1 = 0\n⊢ (fun x ↦ k • x) (p.coeff m) = (fun x ↦ k • x) 0",
"ppTerm": "?refine_2",
"assigned": true,
"usedConsta... | [
"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 : ∀ (m_1 : ℕ), m ≤ m_1 → (k • p).coeff m_1 = 0\n⊢ k • p.coeff m = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.BigOperators | {
"line": 242,
"column": 2
} | {
"line": 242,
"column": 13
} | {
"line": 242,
"column": 14
} | [
{
"pp": "R : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf : ι → R[X]\n⊢ (∏ i ∈ s, f i).coeff 0 = ∏ i ∈ s, (f i).coeff 0",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommSemiring R\nf : ι → R[X]\n⊢ (∏ i ∈ s, f i).coeff 0 = ∏ i ∈ s, (f i).coeff 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.BigOperators | {
"line": 264,
"column": 2
} | {
"line": 264,
"column": 13
} | {
"line": 264,
"column": 14
} | [
{
"pp": "R : Type u\nι : Type w\ninst✝ : CommRing R\ns : Finset ι\nf : ι → R\n⊢ (∏ i ∈ s, (X - C (f i))).nextCoeff = -∑ i ∈ s, f i",
"ppTerm": "?m.32",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\nι : Type w\ninst✝ : CommRing R\ns : Finset ι\nf : ι → R\n⊢ (∏ i ∈ s, (X - C (f i))).nextCoeff = -∑ i ∈ s, f i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Equiv | {
"line": 775,
"column": 22
} | {
"line": 775,
"column": 33
} | {
"line": 775,
"column": 34
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) R\nm : Fin (n + 1) →₀ ℕ\nh : ¬coeff m f = 0\n⊢ ¬coeff (cons (m 0) m.tail) f = 0",
"ppTerm": "?m.90",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"instNeZeroNatHAdd_1",
... | [
"R : Type u\ninst✝ : CommSemiring R\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) R\nm : Fin (n + 1) →₀ ℕ\nh : ¬coeff m f = 0\n⊢ ¬coeff m f = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MvPolynomial.Equiv | {
"line": 779,
"column": 2
} | {
"line": 779,
"column": 13
} | {
"line": 779,
"column": 14
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) R\nx : ℕ\n⊢ x ∈ ((finSuccEquiv R n) f).support ↔ x ∈ (fun m ↦ m 0) '' ↑f.support",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"SetLike.mem_coe._simp_1",
... | [
"R : Type u\ninst✝ : CommSemiring R\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) R\nx : ℕ\n⊢ ¬((finSuccEquiv R n) f).coeff x = 0 ↔ ∃ x_1, ¬coeff x_1 f = 0 ∧ x_1 0 = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.BigOperators | {
"line": 284,
"column": 2
} | {
"line": 284,
"column": 13
} | {
"line": 284,
"column": 14
} | [
{
"pp": "R : Type u\nι : Type w\ninst✝ : CommRing R\ns : Finset ι\nf : ι → R\nhs : 0 < #s\n⊢ (∏ i ∈ s, (X - C (f i))).coeff (#s - 1) = -∑ i ∈ s, f i",
"ppTerm": "?m.46",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\nι : Type w\ninst✝ : CommRing R\ns : Finset ι\nf : ι → R\nhs : 0 < #s\n⊢ (∏ i ∈ s, (X - C (f i))).coeff (#s - 1) = -∑ i ∈ s, f i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.BigOperators | {
"line": 284,
"column": 68
} | {
"line": 284,
"column": 79
} | {
"line": 284,
"column": 80
} | [
{
"pp": "R : Type u\nι : Type w\ninst✝ : CommRing R\ns : Finset ι\nf : ι → R\nhs : 0 < #s\n⊢ 0 < (Multiset.map f s.val).card",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Multiset.map",
"congrArg",
"id",
"instOfNatNat",
"Finset.val",
"... | [
"R : Type u\nι : Type w\ninst✝ : CommRing R\ns : Finset ι\nf : ι → R\nhs : 0 < #s\n⊢ s.Nonempty"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.BigOperators | {
"line": 303,
"column": 4
} | {
"line": 303,
"column": 79
} | {
"line": 307,
"column": 0
} | [
{
"pp": "case neg\nR : Type u\nι : Type w\ns : Finset ι\ninst✝² : CommRing R\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nf : ι → R[X]\nv : ι → A\nh : LinearIndepOn R v ↑s\ni : ι\nhi : i ∈ s\nhf : ¬f i = 0\nH : ∑ x ∈ s, (f x).coeff (f i).natDegree • v x = 0\n⊢ False",
"ppTerm": "?neg✝",
"ass... | [] | exact hf (leadingCoeff_eq_zero.mp (linearIndepOn_finset_iff.mp h _ H i hi)) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.MvPolynomial.Equiv | {
"line": 799,
"column": 2
} | {
"line": 799,
"column": 13
} | {
"line": 799,
"column": 14
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) R\ni : ℕ\nx : Fin (n + 1) →₀ ℕ\n⊢ x ∈ cons i '' ↑(((finSuccEquiv R n) f).coeff i).support ↔ x ∈ f.support ∧ x 0 = i",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr"... | [
"R : Type u\ninst✝ : CommSemiring R\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) R\ni : ℕ\nx : Fin (n + 1) →₀ ℕ\n⊢ (∃ x_1, ¬coeff x_1 (((finSuccEquiv R n) f).coeff i) = 0 ∧ cons i x_1 = x) ↔ ¬coeff x f = 0 ∧ x 0 = i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.BigOperators | {
"line": 323,
"column": 4
} | {
"line": 323,
"column": 79
} | {
"line": 325,
"column": 0
} | [
{
"pp": "case neg\nR : Type u\nι : Type w\ns : Finset ι\ninst✝² : CommRing R\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nf : ι → R[X]\nv : ι → A\nh : LinearIndepOn R v ↑s\ni : ι\nhi : i ∈ s\nhf : ¬f i = 0\nH : ∑ x ∈ s, (f x).coeff (f i).natDegree • v x = 0\n⊢ False",
"ppTerm": "?neg✝",
"ass... | [] | exact hf (leadingCoeff_eq_zero.mp (linearIndepOn_finset_iff.mp h _ H i hi)) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Polynomial.BigOperators | {
"line": 414,
"column": 2
} | {
"line": 414,
"column": 13
} | {
"line": 414,
"column": 14
} | [
{
"pp": "R : Type u\nι : Type w\ns : Finset ι\ninst✝¹ : CommSemiring R\ninst✝ : NoZeroDivisors R\nf : ι → R[X]\n⊢ (∏ i ∈ s, f i).leadingCoeff = ∏ i ∈ s, (f i).leadingCoeff",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\nι : Type w\ns : Finset ι\ninst✝¹ : CommSemiring R\ninst✝ : NoZeroDivisors R\nf : ι → R[X]\n⊢ (∏ i ∈ s, f i).leadingCoeff = ∏ i ∈ s, (f i).leadingCoeff"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Basic | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 44
} | {
"line": 89,
"column": 45
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nn : ℕ\nf : R[X]\n⊢ f ∈ degreeLT R n ↔ f.degree < ↑n",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"Polynomial.degreeLT",
"Submodule",
"WithBot",
"Preorder.toLT",
"iInf",
... | [
"R : Type u\ninst✝ : Semiring R\nn : ℕ\nf : R[X]\n⊢ (∀ (i : ℕ), n ≤ i → f.coeff i = 0) ↔ f.degree < ↑n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.GeomSum | {
"line": 150,
"column": 2
} | {
"line": 150,
"column": 46
} | {
"line": 150,
"column": 47
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommSemiring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : AddLeftReflectLE R\ninst✝³ : AddLeftMono R\ninst✝² : ExistsAddOfLE R\ninst✝¹ : Sub R\ninst✝ : OrderedSub R\nx y : R\nhxy : y ≤ x\nn : ℕ\n⊢ (∑ i ∈ range n, x ^ i * y ^ (n - 1 - i)) * (x - y) + y ^ n = x ^ n",
"ppTerm": "?m.70",... | [
"R : Type u_1\ninst✝⁶ : CommSemiring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : AddLeftReflectLE R\ninst✝³ : AddLeftMono R\ninst✝² : ExistsAddOfLE R\ninst✝¹ : Sub R\ninst✝ : OrderedSub R\nx y : R\nhxy : y ≤ x\nn : ℕ\n⊢ (∑ i ∈ range n, x ^ i * y ^ (n - 1 - i)) * (x - y) + y ^ n = x ^ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.GeomSum | {
"line": 162,
"column": 55
} | {
"line": 162,
"column": 66
} | {
"line": 162,
"column": 67
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommSemiring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : AddLeftReflectLE R\ninst✝³ : AddLeftMono R\ninst✝² : ExistsAddOfLE R\ninst✝¹ : Sub R\ninst✝ : OrderedSub R\nx : R\nhx : 1 ≤ x\nn : ℕ\n⊢ (∑ i ∈ range n, x ^ i) * (x - 1) = x ^ n - 1",
"ppTerm": "?m.44",
"assigned": false,
... | [
"R : Type u_1\ninst✝⁶ : CommSemiring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : AddLeftReflectLE R\ninst✝³ : AddLeftMono R\ninst✝² : ExistsAddOfLE R\ninst✝¹ : Sub R\ninst✝ : OrderedSub R\nx : R\nhx : 1 ≤ x\nn : ℕ\n⊢ (∑ i ∈ range n, x ^ i) * (x - 1) = x ^ n - 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.GeomSum | {
"line": 165,
"column": 55
} | {
"line": 165,
"column": 66
} | {
"line": 165,
"column": 67
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommSemiring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : AddLeftReflectLE R\ninst✝³ : AddLeftMono R\ninst✝² : ExistsAddOfLE R\ninst✝¹ : Sub R\ninst✝ : OrderedSub R\nx : R\nhx : x ≤ 1\nn : ℕ\n⊢ (∑ i ∈ range n, x ^ i) * (1 - x) = 1 - x ^ n",
"ppTerm": "?m.44",
"assigned": false,
... | [
"R : Type u_1\ninst✝⁶ : CommSemiring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : AddLeftReflectLE R\ninst✝³ : AddLeftMono R\ninst✝² : ExistsAddOfLE R\ninst✝¹ : Sub R\ninst✝ : OrderedSub R\nx : R\nhx : x ≤ 1\nn : ℕ\n⊢ (∑ i ∈ range n, x ^ i) * (1 - x) = 1 - x ^ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Basic | {
"line": 143,
"column": 4
} | {
"line": 144,
"column": 40
} | {
"line": 146,
"column": 0
} | [
{
"pp": "case h₁\nR✝ : Type u\nS : Type u_1\ninst✝¹ : Semiring R✝\nR : Type ?u.10\ninst✝ : Semiring R\nn : ℕ\nf : Fin n → R\ni : Fin n\n⊢ i ∉ univ → ((monomial ↑i) (f i)).coeff ↑i = 0",
"ppTerm": "?h₁",
"assigned": true,
"usedConstants": [
"Finset.mem_univ",
"Semiring.toModule",
"F... | [] | · intro h
exact (h (Finset.mem_univ _)).elim | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Ring.GeomSum | {
"line": 238,
"column": 21
} | {
"line": 238,
"column": 32
} | {
"line": 238,
"column": 33
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\nx : R\nn : ℕ\n⊢ op ((x - 1) * ∑ i ∈ range n, x ^ i) = op (x ^ n - 1)",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"AddGroupWithOne.toAddGroup",
"congrArg",
"Finset",
"MulOpposite",
"A... | [
"R : Type u_1\ninst✝ : Ring R\nx : R\nn : ℕ\n⊢ (∑ x_1 ∈ range n, op x ^ x_1) * (op x - 1) = op x ^ n - 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.GeomSum | {
"line": 241,
"column": 2
} | {
"line": 241,
"column": 46
} | {
"line": 242,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\nx : R\nn : ℕ\n⊢ (∑ i ∈ range n, x ^ i) * (1 - x) = 1 - x ^ n",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"HMul.hMul",
"AddGroupWithOne.toAddGroup",
"AddGroupWithOne.toAddMonoidWithOne",
"HSub.hSu... | [
"R : Type u_1\ninst✝ : Ring R\nx : R\nn : ℕ\nthis : -((∑ i ∈ range n, x ^ i) * (x - 1)) = -(x ^ n - 1)\n⊢ (∑ i ∈ range n, x ^ i) * (1 - x) = 1 - x ^ n"
] | have := congr_arg Neg.neg (geom_sum_mul x n) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Ring.GeomSum | {
"line": 246,
"column": 21
} | {
"line": 246,
"column": 32
} | {
"line": 246,
"column": 33
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\nx : R\nn : ℕ\n⊢ op ((1 - x) * ∑ i ∈ range n, x ^ i) = op (1 - x ^ n)",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"AddGroupWithOne.toAddGroup",
"congrArg",
"Finset",
"MulOpposite",
"A... | [
"R : Type u_1\ninst✝ : Ring R\nx : R\nn : ℕ\n⊢ (∑ x_1 ∈ range n, op x ^ x_1) * (1 - op x) = 1 - op x ^ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.GeomSum | {
"line": 285,
"column": 4
} | {
"line": 285,
"column": 37
} | {
"line": 285,
"column": 38
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\nx y : R\nh : Commute x y\nm n : ℕ\nhmn : m ≤ n\nk : ℕ\nx✝ : k ∈ Ico m n\nhp : Commute (MulOpposite.op y ^ (n - 1 - k)) (MulOpposite.op x ^ k)\n⊢ MulOpposite.op y ^ (n - 1 - k) * MulOpposite.op x ^ k = MulOpposite.op x ^ k * MulOpposite.op y ^ (n - 1 - k)",
"ppTerm": "?... | [
"R : Type u_1\ninst✝ : Ring R\nx y : R\nh : Commute x y\nm n : ℕ\nhmn : m ≤ n\nk : ℕ\nx✝ : k ∈ Ico m n\nhp : Commute (MulOpposite.op y ^ (n - 1 - k)) (MulOpposite.op x ^ k)\n⊢ MulOpposite.op y ^ (n - 1 - k) * MulOpposite.op x ^ k = MulOpposite.op x ^ k * MulOpposite.op y ^ (n - 1 - k)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.GeomSum | {
"line": 339,
"column": 2
} | {
"line": 339,
"column": 13
} | {
"line": 339,
"column": 14
} | [
{
"pp": "x n : ℕ\n⊢ x - 1 ∣ x ^ n - 1",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x n : ℕ\n⊢ x - 1 ∣ x ^ n - 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.GeomSum | {
"line": 342,
"column": 26
} | {
"line": 342,
"column": 47
} | {
"line": 342,
"column": 48
} | [
{
"pp": "m x y n : ℕ\n⊢ x ^ m - y ^ m ∣ x ^ (m * n) - y ^ (m * n)",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"HMul.hMul",
"congrArg",
"Nat.instMonoid",
"HSub.hSub",
"id",
"instSubNat",
"instMulNat",
"pow_... | [
"m x y n : ℕ\n⊢ x ^ m - y ^ m ∣ (x ^ m) ^ n - (y ^ m) ^ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.GeomSum | {
"line": 345,
"column": 2
} | {
"line": 345,
"column": 13
} | {
"line": 345,
"column": 14
} | [
{
"pp": "m k x : ℕ\nhmk : m ∣ k\n⊢ x ^ m - 1 ∣ x ^ k - 1",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"m k x : ℕ\nhmk : m ∣ k\n⊢ x ^ m - 1 ∣ x ^ k - 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.GeomSum | {
"line": 354,
"column": 2
} | {
"line": 354,
"column": 68
} | {
"line": 354,
"column": 69
} | [
{
"pp": "m : ℕ\nhm : 2 ≤ m\nn : ℕ\n⊢ m ^ n = (∑ k ∈ range n, m ^ k) * (m - 1) + 1",
"ppTerm": "?m.58",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"m : ℕ\nhm : 2 ≤ m\nn : ℕ\n⊢ m ^ n = (∑ k ∈ range n, m ^ k) * (m - 1) + 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Adjoin.FG | {
"line": 167,
"column": 4
} | {
"line": 167,
"column": 15
} | {
"line": 167,
"column": 16
} | [
{
"pp": "case refine_1\nR : Type u\nA : Type v\ninst✝³ : CommSemiring R\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : IsNoetherian R A\nP : Subalgebra R A → Prop\nbase : P ⊥\nih : ∀ (S : Subalgebra R A) (x : A), P S → P (Algebra.adjoin R (insert x ↑S))\nt : Finset A\n⊢ P (Algebra.adjoin R ↑∅)",
"ppTer... | [
"case refine_1\nR : Type u\nA : Type v\ninst✝³ : CommSemiring R\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : IsNoetherian R A\nP : Subalgebra R A → Prop\nbase : P ⊥\nih : ∀ (S : Subalgebra R A) (x : A), P S → P (Algebra.adjoin R (insert x ↑S))\nt : Finset A\n⊢ P ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Basic | {
"line": 423,
"column": 8
} | {
"line": 423,
"column": 23
} | {
"line": 423,
"column": 24
} | [
{
"pp": "case pos\nR : Type u\ninst✝ : CommSemiring R\nI : Ideal R\nf✝ : R[X]\nhf✝ : f✝ ∈ map C I\nf : R[X]\nhf : f ∈ ⇑C '' ↑I\nn : ℕ\nx : R\nhx : x ∈ ↑I ∧ C x = f\nh : n = 0\n⊢ (if n = 0 then x else 0) ∈ I",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toM... | [
"case pos\nR : Type u\ninst✝ : CommSemiring R\nI : Ideal R\nf✝ : R[X]\nhf✝ : f✝ ∈ map C I\nf : R[X]\nhf : f ∈ ⇑C '' ↑I\nn : ℕ\nx : R\nhx : x ∈ ↑I ∧ C x = f\nh : n = 0\n⊢ x ∈ I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Adjoin.FG | {
"line": 170,
"column": 2
} | {
"line": 170,
"column": 49
} | {
"line": 170,
"column": 50
} | [
{
"pp": "case refine_2\nR : Type u\nA : Type v\ninst✝³ : CommSemiring R\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : IsNoetherian R A\nP : Subalgebra R A → Prop\nbase : P ⊥\nih : ∀ (S : Subalgebra R A) (x : A), P S → P (Algebra.adjoin R (insert x ↑S))\nt✝ : Finset A\nx : A\nt : Finset A\na✝ : x ∉ t\nh : ... | [
"case refine_2\nR : Type u\nA : Type v\ninst✝³ : CommSemiring R\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : IsNoetherian R A\nP : Subalgebra R A → Prop\nbase : P ⊥\nih : ∀ (S : Subalgebra R A) (x : A), P S → P (Algebra.adjoin R (insert x ↑S))\nt✝ : Finset A\nx : A\nt : Finset A\na✝ : x ∉ t\nh : P (Algebra.a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Basic | {
"line": 523,
"column": 74
} | {
"line": 523,
"column": 90
} | {
"line": 523,
"column": 90
} | [
{
"pp": "R : Type u\ninst✝¹ : CommSemiring R\ninst✝ : NoZeroDivisors R\nI J : Ideal R[X]\na : R\nha : a ∈ I.leadingCoeff\nb : R\nhb : b ∈ J.leadingCoeff\np : R[X]\nhpI : p ∈ I\nhp : p.leadingCoeff = a\nq : R[X]\nhqJ : q ∈ J\nhq : q.leadingCoeff = b\n⊢ (p * q).leadingCoeff = a * b",
"ppTerm": "?m.110",
"... | [] | by simp [hp, hq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.Basic | {
"line": 530,
"column": 2
} | {
"line": 530,
"column": 18
} | {
"line": 530,
"column": 19
} | [
{
"pp": "R : Type u\ninst✝¹ : CommSemiring R\ninst✝ : NoZeroDivisors R\nι : Type u_2\ns✝ : Finset ι\nf : ι → Ideal R[X]\ni : ι\ns : Finset ι\nhi : i ∉ s\nhs : ∏ i ∈ s, (f i).leadingCoeff ≤ (s.prod f).leadingCoeff\n⊢ ∏ i ∈ insert i s, (f i).leadingCoeff ≤ ((insert i s).prod f).leadingCoeff",
"ppTerm": "?m.35... | [
"R : Type u\ninst✝¹ : CommSemiring R\ninst✝ : NoZeroDivisors R\nι : Type u_2\ns✝ : Finset ι\nf : ι → Ideal R[X]\ni : ι\ns : Finset ι\nhi : i ∉ s\nhs : ∏ i ∈ s, (f i).leadingCoeff ≤ (s.prod f).leadingCoeff\n⊢ (f i).leadingCoeff * ∏ i ∈ s, (f i).leadingCoeff ≤ (f i * ∏ x ∈ s, f x).leadingCoeff"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Basic | {
"line": 534,
"column": 2
} | {
"line": 534,
"column": 13
} | {
"line": 534,
"column": 14
} | [
{
"pp": "R : Type u\ninst✝¹ : CommSemiring R\nI : Ideal R[X]\ninst✝ : NoZeroDivisors R\nn : ℕ\n⊢ I.leadingCoeff ^ n ≤ (I ^ n).leadingCoeff",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝¹ : CommSemiring R\nI : Ideal R[X]\ninst✝ : NoZeroDivisors R\nn : ℕ\n⊢ I.leadingCoeff ^ n ≤ (I ^ n).leadingCoeff"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Basic | {
"line": 605,
"column": 6
} | {
"line": 605,
"column": 37
} | {
"line": 605,
"column": 38
} | [
{
"pp": "case mpr\nR : Type u\ninst✝ : CommRing R\nP : Ideal R\nH : (map C P).IsPrime\nthis : (comap C (map C P)).IsPrime\nx : R\nh : ∀ (n : ℕ), (C x).coeff n ∈ P\n⊢ x ∈ P",
"ppTerm": "?mpr",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case mpr\nR : Type u\ninst✝ : CommRing R\nP : Ideal R\nH : (map C P).IsPrime\nthis : (comap C (map C P)).IsPrime\nx : R\nh : ∀ (n : ℕ), (C x).coeff n ∈ P\n⊢ x ∈ P"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Basic | {
"line": 654,
"column": 6
} | {
"line": 654,
"column": 22
} | {
"line": 654,
"column": 23
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nI : Ideal R[X]\nhI : comap C I = I.leadingCoeff\nn✝ : ℕ\nih : ∀ m < n✝, ∀ f ∈ I, f.natDegree = m → f ∈ map C (comap C I)\nf : R[X]\nhfI : f ∈ I\nhn : f.natDegree = n✝\n⊢ f.leadingCoeff ∈ comap C I",
"ppTerm": "?m.145",
"assigned": true,
"usedConstants": [
... | [
"R : Type u\ninst✝ : CommRing R\nI : Ideal R[X]\nhI : comap C I = I.leadingCoeff\nn✝ : ℕ\nih : ∀ m < n✝, ∀ f ∈ I, f.natDegree = m → f ∈ map C (comap C I)\nf : R[X]\nhfI : f ∈ I\nhn : f.natDegree = n✝\n⊢ f.leadingCoeff ∈ I.leadingCoeff"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Basic | {
"line": 656,
"column": 36
} | {
"line": 656,
"column": 47
} | {
"line": 656,
"column": 48
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nI : Ideal R[X]\nhI : comap C I = I.leadingCoeff\nn✝ : ℕ\nih : ∀ m < n✝, ∀ f ∈ I, f.natDegree = m → f ∈ map C (comap C I)\nf : R[X]\nhfI : f ∈ I\nhn : f.natDegree = n✝\nh : C f.leadingCoeff * X ^ f.natDegree ∈ map C (comap C I)\nhlt : f.eraseLead.natDegree < f.natDegree\n... | [
"R : Type u\ninst✝ : CommRing R\nI : Ideal R[X]\nhI : comap C I = I.leadingCoeff\nn✝ : ℕ\nih : ∀ m < n✝, ∀ f ∈ I, f.natDegree = m → f ∈ map C (comap C I)\nf : R[X]\nhfI : f ∈ I\nhn : f.natDegree = n✝\nh : C f.leadingCoeff * X ^ f.natDegree ∈ map C (comap C I)\nhlt : f.eraseLead.natDegree < f.natDegree\n⊢ f.eraseLea... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Basic | {
"line": 657,
"column": 4
} | {
"line": 657,
"column": 15
} | {
"line": 657,
"column": 16
} | [
{
"pp": "case h.inl\nR : Type u\ninst✝ : CommRing R\nI : Ideal R[X]\nhI : comap C I = I.leadingCoeff\nn✝ : ℕ\nih : ∀ m < n✝, ∀ f ∈ I, f.natDegree = m → f ∈ map C (comap C I)\nf : R[X]\nhfI : f ∈ I\nhn : f.natDegree = n✝\nh : C f.leadingCoeff * X ^ f.natDegree ∈ map C (comap C I)\nhlt : f.eraseLead.natDegree < f... | [
"case h.inl\nR : Type u\ninst✝ : CommRing R\nI : Ideal R[X]\nhI : comap C I = I.leadingCoeff\nn✝ : ℕ\nih : ∀ m < n✝, ∀ f ∈ I, f.natDegree = m → f ∈ map C (comap C I)\nf : R[X]\nhfI : f ∈ I\nhn : f.natDegree = n✝\nh : C f.leadingCoeff * X ^ f.natDegree ∈ map C (comap C I)\nhlt : f.eraseLead.natDegree < f.natDegree\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Basic | {
"line": 657,
"column": 54
} | {
"line": 657,
"column": 70
} | {
"line": 657,
"column": 71
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nI : Ideal R[X]\nhI : comap C I = I.leadingCoeff\nn✝ : ℕ\nih : ∀ m < n✝, ∀ f ∈ I, f.natDegree = m → f ∈ map C (comap C I)\nf : R[X]\nhfI : f ∈ I\nhn : f.natDegree = n✝\nh : C f.leadingCoeff * X ^ f.natDegree ∈ map C (comap C I)\nhlt : f.eraseLead.natDegree < f.natDegree\n... | [
"R : Type u\ninst✝ : CommRing R\nI : Ideal R[X]\nhI : comap C I = I.leadingCoeff\nn✝ : ℕ\nih : ∀ m < n✝, ∀ f ∈ I, f.natDegree = m → f ∈ map C (comap C I)\nf : R[X]\nhfI : f ∈ I\nhn : f.natDegree = n✝\nh : C f.leadingCoeff * X ^ f.natDegree ∈ map C (comap C I)\nhlt : f.eraseLead.natDegree < f.natDegree\nhe : f.erase... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Basic | {
"line": 729,
"column": 14
} | {
"line": 729,
"column": 25
} | {
"line": 729,
"column": 26
} | [
{
"pp": "case convert_3.zero\nR : Type u\nσ : Type v\nr : R\ninst✝¹ : CommRing R\ninst✝ : Finite σ\nthis : Fintype σ\n⊢ Prime (C r) ↔ Prime r",
"ppTerm": "?convert_3.zero",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case convert_3.zero\nR : Type u\nσ : Type v\nr : R\ninst✝¹ : CommRing R\ninst✝ : Finite σ\nthis : Fintype σ\n⊢ Prime (C r) ↔ Prime r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Basic | {
"line": 761,
"column": 4
} | {
"line": 763,
"column": 80
} | {
"line": 764,
"column": 4
} | [
{
"pp": "R : Type u\nσ : Type v\ninst✝ : CommRing R\ns : Set σ\np : MvPolynomial (↑s) R\n⊢ Prime p ↔ Prime ((rename Subtype.val) p)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"AlgEquiv.symm",
"CommSemiring.to... | [
"R : Type u\nσ : Type v\ninst✝ : CommRing R\ns : Set σ\np : MvPolynomial (↑s) R\neqv : MvPolynomial (↑sᶜ) (MvPolynomial (↑s) R) ≃ₐ[R] MvPolynomial σ R :=\n (sumAlgEquiv R ↑sᶜ ↑s).symm.trans (renameEquiv R ((Equiv.sumComm ↑sᶜ ↑s).trans (Equiv.Set.sumCompl s)))\n⊢ Prime p ↔ Prime ((rename Subtype.val) p)"
] | let eqv :=
(sumAlgEquiv R (↥sᶜ) s).symm.trans
(renameEquiv R <| (Equiv.sumComm (↥sᶜ) s).trans <| Equiv.Set.sumCompl s) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.Polynomial.Basic | {
"line": 769,
"column": 4
} | {
"line": 769,
"column": 58
} | {
"line": 769,
"column": 59
} | [
{
"pp": "R : Type u\nσ : Type v\ninst✝ : CommRing R\ns : Set σ\np : MvPolynomial (↑s) R\neqv : MvPolynomial (↑sᶜ) (MvPolynomial (↑s) R) ≃ₐ[R] MvPolynomial σ R :=\n (sumAlgEquiv R ↑sᶜ ↑s).symm.trans (renameEquiv R ((Equiv.sumComm ↑sᶜ ↑s).trans (Equiv.Set.sumCompl s)))\nthis :\n (rename Subtype.val) p =\n ((... | [
"R : Type u\nσ : Type v\ninst✝ : CommRing R\ns : Set σ\np : MvPolynomial (↑s) R\neqv : MvPolynomial (↑sᶜ) (MvPolynomial (↑s) R) ≃ₐ[R] MvPolynomial σ R :=\n (sumAlgEquiv R ↑sᶜ ↑s).symm.trans (renameEquiv R ((Equiv.sumComm ↑sᶜ ↑s).trans (Equiv.Set.sumCompl s)))\nthis :\n (rename Subtype.val) p =\n ((↑eqv).comp (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Basic | {
"line": 862,
"column": 2
} | {
"line": 863,
"column": 57
} | {
"line": 864,
"column": 4
} | [
{
"pp": "R : Type u\nM : Type w\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : M →ₗ[R] M\np q : R[X]\nv : M\nhv : v ∈ ((aeval f) p).ker ⊓ ((aeval f) q).ker\np' q' : R[X]\nhpq' : p' * p + q' * q = 1\n⊢ v ∈ ⊥",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"Eq.m... | [
"R : Type u\nM : Type w\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : M →ₗ[R] M\np q : R[X]\nv : M\nhv : v ∈ ((aeval f) p).ker ⊓ ((aeval f) q).ker\np' q' : R[X]\nhpq' : p' * p + q' * q = 1\n⊢ v = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Basic | {
"line": 979,
"column": 2
} | {
"line": 979,
"column": 31
} | {
"line": 979,
"column": 32
} | [
{
"pp": "R : Type u\nσ : Type v\ninst✝ : CommRing R\nI : Ideal (MvPolynomial σ R)\np : MvPolynomial σ R\nhcoe : ∀ (m : σ →₀ ℕ), coeff m p ∈ Ideal.comap C I\nm : σ →₀ ℕ\na✝ : m ∈ p.support\n⊢ C (coeff m p) ∈ I",
"ppTerm": "?m.89",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"used... | [
"R : Type u\nσ : Type v\ninst✝ : CommRing R\nI : Ideal (MvPolynomial σ R)\np : MvPolynomial σ R\nhcoe : ∀ (m : σ →₀ ℕ), coeff m p ∈ Ideal.comap C I\nm : σ →₀ ℕ\na✝ : m ∈ p.support\n⊢ C (coeff m p) ∈ I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.FiniteType | {
"line": 180,
"column": 65
} | {
"line": 180,
"column": 76
} | {
"line": 180,
"column": 77
} | [
{
"pp": "R : Type uR\nS : Type uS\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : Algebra R S\nι : Type uS\nhfintype : Fintype ι\nf : MvPolynomial ι R →ₐ[R] S\nhsur : Surjective ⇑f\nequiv : MvPolynomial ι R ≃ₐ[R] MvPolynomial (Fin (Fintype.card ι)) R\n⊢ Surjective ⇑(f.comp ↑equiv.symm)",
"ppTerm"... | [
"R : Type uR\nS : Type uS\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : Algebra R S\nι : Type uS\nhfintype : Fintype ι\nf : MvPolynomial ι R →ₐ[R] S\nhsur : Surjective ⇑f\nequiv : MvPolynomial ι R ≃ₐ[R] MvPolynomial (Fin (Fintype.card ι)) R\n⊢ Surjective ⇑f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Basic | {
"line": 994,
"column": 8
} | {
"line": 994,
"column": 23
} | {
"line": 994,
"column": 24
} | [
{
"pp": "case pos\nR : Type u\nσ : Type v\ninst✝ : CommRing R\nI : Ideal R\nf✝ : MvPolynomial σ R\nhf✝ : f✝ ∈ Ideal.map C I\nf : MvPolynomial σ R\nhf : f ∈ ⇑C '' ↑I\nn : σ →₀ ℕ\nx : R\nhx : x ∈ ↑I ∧ C x = f\nh : n = 0\n⊢ (if 0 = n then x else 0) ∈ I",
"ppTerm": "?pos✝",
"assigned": true,
"usedConsta... | [
"case pos\nR : Type u\nσ : Type v\ninst✝ : CommRing R\nI : Ideal R\nf✝ : MvPolynomial σ R\nhf✝ : f✝ ∈ Ideal.map C I\nf : MvPolynomial σ R\nhf : f ∈ ⇑C '' ↑I\nn : σ →₀ ℕ\nx : R\nhx : x ∈ ↑I ∧ C x = f\nh : n = 0\n⊢ x ∈ I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.FiniteType | {
"line": 465,
"column": 2
} | {
"line": 465,
"column": 44
} | {
"line": 465,
"column": 45
} | [
{
"pp": "R : Type u_1\nG : Type u_3\ninst✝² : AddGroup G\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\n⊢ FiniteType R R[G] ↔ AddGroup.FG G",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AddMonoid.FG",
"AddMonoidAlgebra.semiring",
"congrArg",
"CommSemi... | [
"R : Type u_1\nG : Type u_3\ninst✝² : AddGroup G\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\n⊢ FiniteType R R[G] ↔ AddMonoid.FG G"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.FiniteType | {
"line": 596,
"column": 2
} | {
"line": 596,
"column": 38
} | {
"line": 596,
"column": 39
} | [
{
"pp": "R : Type u_1\nG : Type u_3\ninst✝² : Group G\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\n⊢ FiniteType R R[G] ↔ Group.FG G",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"MonoidAlgebra.semiring",
"Eq.mpr",
"congrArg",
"CommSemiring.toSemiring",
"Group.... | [
"R : Type u_1\nG : Type u_3\ninst✝² : Group G\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\n⊢ FiniteType R R[G] ↔ Monoid.FG G"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Quotient.Operations | {
"line": 229,
"column": 19
} | {
"line": 229,
"column": 59
} | {
"line": 229,
"column": 60
} | [
{
"pp": "R : Type u_2\ninst✝¹ : CommRing R\nι : Type u_3\ninst✝ : Finite ι\nI : ι → Ideal R\nhI : Pairwise (IsCoprime on I)\nval✝ : Fintype ι\ng : (i : ι) → R ⧸ I i\nf : ι → R\nhf : ∀ (i : ι), (Quotient.mk (I i)) (f i) = g i\ni j : ι\nhj : j ∈ {i}ᶜ\n⊢ i ≠ j",
"ppTerm": "?m.93",
"assigned": true,
"us... | [
"R : Type u_2\ninst✝¹ : CommRing R\nι : Type u_3\ninst✝ : Finite ι\nI : ι → Ideal R\nhI : Pairwise (IsCoprime on I)\nval✝ : Fintype ι\ng : (i : ι) → R ⧸ I i\nf : ι → R\nhf : ∀ (i : ι), (Quotient.mk (I i)) (f i) = g i\ni j : ι\nhj : j ∈ {i}ᶜ\n⊢ ¬i = j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Quotient.Operations | {
"line": 231,
"column": 44
} | {
"line": 231,
"column": 55
} | {
"line": 231,
"column": 56
} | [
{
"pp": "R : Type u_2\ninst✝¹ : CommRing R\nι : Type u_3\ninst✝ : Finite ι\nI : ι → Ideal R\nhI : Pairwise (IsCoprime on I)\nval✝ : Fintype ι\ng : (i : ι) → R ⧸ I i\nf : ι → R\nhf : ∀ (i : ι), (Quotient.mk (I i)) (f i) = g i\ni : ι\nhI' : ∀ j ∈ {i}ᶜ, IsCoprime (I i) (I j)\nu : R\nhu : u ∈ I i\ne : R\nhe : e ∈ ⨅... | [
"R : Type u_2\ninst✝¹ : CommRing R\nι : Type u_3\ninst✝ : Finite ι\nI : ι → Ideal R\nhI : Pairwise (IsCoprime on I)\nval✝ : Fintype ι\ng : (i : ι) → R ⧸ I i\nf : ι → R\nhf : ∀ (i : ι), (Quotient.mk (I i)) (f i) = g i\ni : ι\nhI' : ∀ j ∈ {i}ᶜ, IsCoprime (I i) (I j)\nu : R\nhu : u ∈ I i\ne : R\nhe : e ∈ ⨅ j ∈ {i}ᶜ, I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Subalgebra.Unitization | {
"line": 111,
"column": 8
} | {
"line": 111,
"column": 24
} | {
"line": 111,
"column": 25
} | [
{
"pp": "case pos.snd\nF : Type u_1\nR : Type u_2\nS : Type u_3\nA : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : Ring A\ninst✝³ : Algebra R A\ninst✝² : SetLike S A\nhSA : NonUnitalSubringClass S A\nhSRA : SMulMemClass S R A\ns : S\nh : ∀ (r : R), r ≠ 0 → (algebraMap R A) r ∉ s\ninst✝¹ : FunLike F (Unitization R ↥s)... | [
"case pos.snd\nF : Type u_1\nR : Type u_2\nS : Type u_3\nA : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : Ring A\ninst✝³ : Algebra R A\ninst✝² : SetLike S A\nhSA : NonUnitalSubringClass S A\nhSRA : SMulMemClass S R A\ns : S\nh : ∀ (r : R), r ≠ 0 → (algebraMap R A) r ∉ s\ninst✝¹ : FunLike F (Unitization R ↥s) A\ninst✝ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Int.ModEq | {
"line": 55,
"column": 2
} | {
"line": 55,
"column": 13
} | {
"line": 55,
"column": 14
} | [
{
"pp": "G : Type u_1\ninst✝¹ : AddCommGroupWithOne G\ninst✝ : CharZero G\na b : ℤ\nn : ℕ\n⊢ ↑a ≡ ↑b [PMOD ↑n] ↔ a ≡ b [PMOD ↑n]",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Int.instAddCommMonoid",
"AddCommGroup.ModEq",
"AddCommGroup.m... | [
"G : Type u_1\ninst✝¹ : AddCommGroupWithOne G\ninst✝ : CharZero G\na b : ℤ\nn : ℕ\n⊢ ↑a ≡ ↑b [PMOD ↑n] ↔ a ≡ b [ZMOD ↑n]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Quotient.Operations | {
"line": 595,
"column": 4
} | {
"line": 595,
"column": 81
} | {
"line": 595,
"column": 82
} | [
{
"pp": "R : Type u\nS✝ : Type v\nF : Type w\ninst✝⁹ : Ring R\ninst✝⁸ : Semiring S✝\nR₁ : Type u_1\nR₂ : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁷ : CommSemiring R₁\ninst✝⁶ : CommSemiring R₂\ninst✝⁵ : Ring A\ninst✝⁴ : Algebra R₁ A\ninst✝³ : Algebra R₂ A\nS : Type v\ninst✝² : Ring S\nI : Ideal R\nJ : Ideal S\... | [
"R : Type u\nS✝ : Type v\nF : Type w\ninst✝⁹ : Ring R\ninst✝⁸ : Semiring S✝\nR₁ : Type u_1\nR₂ : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁷ : CommSemiring R₁\ninst✝⁶ : CommSemiring R₂\ninst✝⁵ : Ring A\ninst✝⁴ : Algebra R₁ A\ninst✝³ : Algebra R₂ A\nS : Type v\ninst✝² : Ring S\nI : Ideal R\nJ : Ideal S\ninst✝¹ : I.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Int.ModEq | {
"line": 220,
"column": 32
} | {
"line": 220,
"column": 54
} | {
"line": 220,
"column": 55
} | [
{
"pp": "m a b c : ℤ\nhm : 0 < m\nh : c * a ≡ c * b [ZMOD m]\n⊢ a * c ≡ b * c [ZMOD m]",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"m a b c : ℤ\nhm : 0 < m\nh : c * a ≡ c * b [ZMOD m]\n⊢ a * c ≡ b * c [ZMOD m]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Quotient.Operations | {
"line": 737,
"column": 4
} | {
"line": 737,
"column": 39
} | {
"line": 737,
"column": 40
} | [
{
"pp": "R✝ : Type u\nS : Type v\nF : Type w\ninst✝⁹ : Ring R✝\ninst✝⁸ : Semiring S\nR₁ : Type u_1\nR₂ : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁷ : CommSemiring R₁\ninst✝⁶ : CommSemiring R₂\ninst✝⁵ : Ring A\ninst✝⁴ : Algebra R₁ A\ninst✝³ : Algebra R₂ A\nR : Type ?u.26\ninst✝² : CommRing R\ninst✝¹ : Algebra ... | [
"R✝ : Type u\nS : Type v\nF : Type w\ninst✝⁹ : Ring R✝\ninst✝⁸ : Semiring S\nR₁ : Type u_1\nR₂ : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁷ : CommSemiring R₁\ninst✝⁶ : CommSemiring R₂\ninst✝⁵ : Ring A\ninst✝⁴ : Algebra R₁ A\ninst✝³ : Algebra R₂ A\nR : Type ?u.26\ninst✝² : CommRing R\ninst✝¹ : Algebra R A\np : Ide... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.GroupWithZero.Units.Fintype | {
"line": 45,
"column": 39
} | {
"line": 45,
"column": 59
} | {
"line": 45,
"column": 59
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝ : GroupWithZero α\nh✝ : Finite { a // a ≠ 0 }\n⊢ 1 + Nat.card { a // ¬a = 0 } - 1 = Nat.card { a // a ≠ 0 }",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Nat.instOrderedSub",
"Nat.i... | [
"case inl\nα : Type u_1\ninst✝ : GroupWithZero α\nh✝ : Finite { a // a ≠ 0 }\n⊢ Nat.card { a // ¬a = 0 } = Nat.card { a // a ≠ 0 }"
] | add_tsub_cancel_left | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.CharP.Basic | {
"line": 185,
"column": 14
} | {
"line": 185,
"column": 25
} | {
"line": 185,
"column": 26
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝² : NonAssocRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : Nontrivial R\nh : ∀ (p : ℕ), Nat.Prime p → ↑p ≠ 0\np : ℕ := ringChar R\nhp : Nat.Prime p\n⊢ CharZero R",
"ppTerm": "?inl",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}... | [
"case inl\nR : Type u_1\ninst✝² : NonAssocRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : Nontrivial R\nh : ∀ (p : ℕ), Nat.Prime p → ↑p ≠ 0\np : ℕ := ringChar R\nhp : Nat.Prime p\n⊢ CharZero R"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.CharP.Two | {
"line": 123,
"column": 47
} | {
"line": 123,
"column": 58
} | {
"line": 123,
"column": 59
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : CharP R 2\nn : ℕ\n⊢ ↑↑n = if Even ↑n then 0 else 1",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Int.cast_natCast",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne",
"Int.i... | [
"case inl\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : CharP R 2\nn : ℕ\n⊢ ↑n = if Even n then 0 else 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.CharP.Two | {
"line": 123,
"column": 47
} | {
"line": 123,
"column": 58
} | {
"line": 123,
"column": 59
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : CharP R 2\nn : ℕ\n⊢ ↑(-↑n) = if Even (-↑n) then 0 else 1",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"AddGroup.toSubtractionMonoid",
"Int.cast_neg",
"Int.cast",
"Eq.mpr",
... | [
"case inr\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : CharP R 2\nn : ℕ\n⊢ ↑n = if Even n then 0 else 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.Prime.Basic | {
"line": 205,
"column": 84
} | {
"line": 206,
"column": 39
} | {
"line": 208,
"column": 0
} | [
{
"pp": "p : ℕ\npp : Prime p\ni : ℕ\n⊢ p.Coprime i ∨ p ∣ i",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.Coprime",
"Dvd.dvd",
"congrArg",
"em",
"id",
"Nat.Prime.dvd_iff_not_coprime",
"Nat.instDvd",
"Nat",
"propext"... | [] | by
rw [pp.dvd_iff_not_coprime]; apply em | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Nat.Prime.Basic | {
"line": 237,
"column": 12
} | {
"line": 237,
"column": 23
} | {
"line": 237,
"column": 24
} | [
{
"pp": "⊢ 0 ≠ 1 ↔ ∃ p, Prime p ∧ p ∣ 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Nat.instMulZeroClass",
"Nat.Prime",
"Dvd.dvd",
"Nat.instOne",
"and_true",
"Nat.instSemigroupWithZero",
"congrArg",
"true_i... | [
"⊢ ∃ p, Prime p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.Prime.Basic | {
"line": 246,
"column": 2
} | {
"line": 246,
"column": 13
} | {
"line": 246,
"column": 14
} | [
{
"pp": "n : ℕ\n⊢ n = 1 ↔ ∀ (p : ℕ), Prime p → ¬p ∣ n",
"ppTerm": "?m.6",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\n⊢ n = 1 ↔ ∀ (p : ℕ), Prime p → ¬p ∣ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.Prime.Basic | {
"line": 255,
"column": 48
} | {
"line": 255,
"column": 69
} | {
"line": 255,
"column": 70
} | [
{
"pp": "p : ℕ\np_prime : Prime p\nm n k l : ℕ\nhpm : p ^ k ∣ m\nhpn : p ^ l ∣ n\nhpmn : p ^ (k + l + 1) ∣ m * n\nhpd : p ^ (k + l) * p ∣ m * n\nhpd2 : p ∣ m * n / p ^ (k + l)\n⊢ p ∣ m * n / (p ^ k * p ^ l)",
"ppTerm": "?m.146",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedG... | [
"p : ℕ\np_prime : Prime p\nm n k l : ℕ\nhpm : p ^ k ∣ m\nhpn : p ^ l ∣ n\nhpmn : p ^ (k + l + 1) ∣ m * n\nhpd : p ^ (k + l) * p ∣ m * n\nhpd2 : p ∣ m * n / p ^ (k + l)\n⊢ p ∣ m * n / (p ^ k * p ^ l)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.Prime.Basic | {
"line": 256,
"column": 48
} | {
"line": 256,
"column": 90
} | {
"line": 256,
"column": 91
} | [
{
"pp": "p : ℕ\np_prime : Prime p\nm n k l : ℕ\nhpm : p ^ k ∣ m\nhpn : p ^ l ∣ n\nhpmn : p ^ (k + l + 1) ∣ m * n\nhpd : p ^ (k + l) * p ∣ m * n\nhpd2 : p ∣ m * n / p ^ (k + l)\nhpd3 : p ∣ m * n / (p ^ k * p ^ l)\n⊢ p ∣ m / p ^ k * (n / p ^ l)",
"ppTerm": "?m.180",
"assigned": true,
"usedConstants": ... | [
"p : ℕ\np_prime : Prime p\nm n k l : ℕ\nhpm : p ^ k ∣ m\nhpn : p ^ l ∣ n\nhpmn : p ^ (k + l + 1) ∣ m * n\nhpd : p ^ (k + l) * p ∣ m * n\nhpd2 : p ∣ m * n / p ^ (k + l)\nhpd3 : p ∣ m * n / (p ^ k * p ^ l)\n⊢ p ∣ m * n / (p ^ k * p ^ l)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Fintype.List | {
"line": 54,
"column": 2
} | {
"line": 54,
"column": 13
} | {
"line": 54,
"column": 14
} | [
{
"pp": "case h\nα : Type u_1\nl a✝ : List α\n⊢ l ∈ lists ⟦a✝⟧ ↔ ⟦a✝⟧ = ⟦l⟧",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Multiset.mem_coe._simp_1",
"Membership.mem",
"Multiset",
"id",
"List.permutations",
"List.mem_perm... | [
"case h\nα : Type u_1\nl a✝ : List α\n⊢ l ~ a✝ ↔ a✝ ~ l"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.Subalgebra | {
"line": 69,
"column": 23
} | {
"line": 69,
"column": 34
} | {
"line": 69,
"column": 35
} | [
{
"pp": "F : Type u_1\nR : Type u_2\nA : Type u_3\nB : Type u_4\nC : Type u_5\ninst✝¹³ : CommSemiring R\ninst✝¹² : StarRing R\ninst✝¹¹ : Semiring A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : Algebra R A\ninst✝⁸ : StarModule R A\ninst✝⁷ : Semiring B\ninst✝⁶ : StarRing B\ninst✝⁵ : Algebra R B\ninst✝⁴ : StarModule R B\ninst✝... | [
"F : Type u_1\nR : Type u_2\nA : Type u_3\nB : Type u_4\nC : Type u_5\ninst✝¹³ : CommSemiring R\ninst✝¹² : StarRing R\ninst✝¹¹ : Semiring A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : Algebra R A\ninst✝⁸ : StarModule R A\ninst✝⁷ : Semiring B\ninst✝⁶ : StarRing B\ninst✝⁵ : Algebra R B\ninst✝⁴ : StarModule R B\ninst✝³ : Semiring... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.Subalgebra | {
"line": 67,
"column": 24
} | {
"line": 67,
"column": 35
} | {
"line": 67,
"column": 36
} | [
{
"pp": "F : Type u_1\nR : Type u_2\nA : Type u_3\nB : Type u_4\nC : Type u_5\ninst✝¹³ : CommSemiring R\ninst✝¹² : StarRing R\ninst✝¹¹ : Semiring A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : Algebra R A\ninst✝⁸ : StarModule R A\ninst✝⁷ : Semiring B\ninst✝⁶ : StarRing B\ninst✝⁵ : Algebra R B\ninst✝⁴ : StarModule R B\ninst✝... | [
"F : Type u_1\nR : Type u_2\nA : Type u_3\nB : Type u_4\nC : Type u_5\ninst✝¹³ : CommSemiring R\ninst✝¹² : StarRing R\ninst✝¹¹ : Semiring A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : Algebra R A\ninst✝⁸ : StarModule R A\ninst✝⁷ : Semiring B\ninst✝⁶ : StarRing B\ninst✝⁵ : Algebra R B\ninst✝⁴ : StarModule R B\ninst✝³ : Semiring... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.Subalgebra | {
"line": 161,
"column": 43
} | {
"line": 161,
"column": 59
} | {
"line": 161,
"column": 60
} | [
{
"pp": "F : Type u_1\nR : Type u_2\nA : Type u_3\nB : Type u_4\nC : Type u_5\ninst✝¹³ : CommSemiring R\ninst✝¹² : StarRing R\ninst✝¹¹ : Semiring A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : Algebra R A\ninst✝⁸ : StarModule R A\ninst✝⁷ : Semiring B\ninst✝⁶ : StarRing B\ninst✝⁵ : Algebra R B\ninst✝⁴ : StarModule R B\ninst✝... | [
"F : Type u_1\nR : Type u_2\nA : Type u_3\nB : Type u_4\nC : Type u_5\ninst✝¹³ : CommSemiring R\ninst✝¹² : StarRing R\ninst✝¹¹ : Semiring A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : Algebra R A\ninst✝⁸ : StarModule R A\ninst✝⁷ : Semiring B\ninst✝⁶ : StarRing B\ninst✝⁵ : Algebra R B\ninst✝⁴ : StarModule R B\ninst✝³ : Semiring... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.Subalgebra | {
"line": 161,
"column": 43
} | {
"line": 161,
"column": 59
} | {
"line": 161,
"column": 60
} | [
{
"pp": "F : Type u_1\nR : Type u_2\nA : Type u_3\nB : Type u_4\nC : Type u_5\ninst✝¹³ : CommSemiring R\ninst✝¹² : StarRing R\ninst✝¹¹ : Semiring A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : Algebra R A\ninst✝⁸ : StarModule R A\ninst✝⁷ : Semiring B\ninst✝⁶ : StarRing B\ninst✝⁵ : Algebra R B\ninst✝⁴ : StarModule R B\ninst✝... | [
"F : Type u_1\nR : Type u_2\nA : Type u_3\nB : Type u_4\nC : Type u_5\ninst✝¹³ : CommSemiring R\ninst✝¹² : StarRing R\ninst✝¹¹ : Semiring A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : Algebra R A\ninst✝⁸ : StarModule R A\ninst✝⁷ : Semiring B\ninst✝⁶ : StarRing B\ninst✝⁵ : Algebra R B\ninst✝⁴ : StarModule R B\ninst✝³ : Semiring... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.Subalgebra | {
"line": 362,
"column": 8
} | {
"line": 362,
"column": 87
} | {
"line": 363,
"column": 10
} | [
{
"pp": "F : Type u_1\nR : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : StarRing A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : StarRing B\ninst✝ : StarModule R B\nS : Subalgebra R A\nr : ... | [
"F : Type u_1\nR : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : StarRing A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : StarRing B\ninst✝ : StarModule R B\nS : Subalgebra R A\nr : R\n⊢ (algebr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.Subalgebra | {
"line": 386,
"column": 18
} | {
"line": 386,
"column": 46
} | {
"line": 386,
"column": 47
} | [
{
"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\nthis : ∀ (t : Set A), Algebra.adjoin R (star t) ≤ star (Algebra.adjoin R t)\n⊢ star (Algebra.adjoin R s) ≤ Algebra.adjoin R (star ... | [
"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\nthis : ∀ (t : Set A), Algebra.adjoin R (star t) ≤ star (Algebra.adjoin R t)\n⊢ star (Algebra.adjoin R s) ≤ Algebra.adjoin R (star s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.Subalgebra | {
"line": 395,
"column": 4
} | {
"line": 395,
"column": 15
} | {
"line": 395,
"column": 16
} | [
{
"pp": "F : Type u_1\nR : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : StarRing A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : StarRing B\ninst✝ : StarModule R B\nS : Subalgebra R A\na : ... | [
"F : Type u_1\nR : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : StarRing A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : StarRing B\ninst✝ : StarModule R B\nS : Subalgebra R A\na : A\nha : a ∈ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.List.Permutation | {
"line": 135,
"column": 20
} | {
"line": 135,
"column": 92
} | {
"line": 137,
"column": 0
} | [
{
"pp": "case cons\nα : Type u_1\nβ : Type u_2\nf : α → β\nt a : α\nts : List α\nih : map (map f) (permutations'Aux t ts) = permutations'Aux (f t) (map f ts)\n⊢ map (map f) (permutations'Aux t (a :: ts)) = permutations'Aux (f t) (map f (a :: ts))",
"ppTerm": "?cons",
"assigned": true,
"usedConstants... | [] | simp only [permutations'Aux, map_cons, map_map, ← ih, Function.comp_def] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.List.Permutation | {
"line": 135,
"column": 20
} | {
"line": 135,
"column": 92
} | {
"line": 137,
"column": 0
} | [
{
"pp": "case cons\nα : Type u_1\nβ : Type u_2\nf : α → β\nt a : α\nts : List α\nih : map (map f) (permutations'Aux t ts) = permutations'Aux (f t) (map f ts)\n⊢ map (map f) (permutations'Aux t (a :: ts)) = permutations'Aux (f t) (map f (a :: ts))",
"ppTerm": "?cons",
"assigned": true,
"usedConstants... | [] | simp only [permutations'Aux, map_cons, map_map, ← ih, Function.comp_def] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.List.Permutation | {
"line": 135,
"column": 20
} | {
"line": 135,
"column": 92
} | {
"line": 137,
"column": 0
} | [
{
"pp": "case cons\nα : Type u_1\nβ : Type u_2\nf : α → β\nt a : α\nts : List α\nih : map (map f) (permutations'Aux t ts) = permutations'Aux (f t) (map f ts)\n⊢ map (map f) (permutations'Aux t (a :: ts)) = permutations'Aux (f t) (map f (a :: ts))",
"ppTerm": "?cons",
"assigned": true,
"usedConstants... | [] | simp only [permutations'Aux, map_cons, map_map, ← ih, Function.comp_def] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Dynamics.PeriodicPts.Defs | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 57
} | {
"line": 134,
"column": 2
} | [
{
"pp": "α : Type u_1\nf : α → α\nx : α\nn : ℕ\nhf : IsPeriodicPt f n x\nm : ℕ\n⊢ IsPeriodicPt f^[m] n x",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"id",
"CommMagma.toMul",
"instMulNat",
"Nat.iterate",
"... | [
"α : Type u_1\nf : α → α\nx : α\nn : ℕ\nhf : IsPeriodicPt f n x\nm : ℕ\n⊢ IsFixedPt f^[n]^[m] x"
] | rw [IsPeriodicPt, ← iterate_mul, mul_comm, iterate_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Dynamics.PeriodicPts.Defs | {
"line": 162,
"column": 4
} | {
"line": 162,
"column": 20
} | {
"line": 163,
"column": 4
} | [
{
"pp": "case refine_2\nα : Type u_1\nf : α → α\nx : α\nm✝ n✝ m n : ℕ\nx✝ : 0 < m\nih : IsPeriodicPt f (n % m) x → IsPeriodicPt f m x → IsPeriodicPt f ((n % m).gcd m) x\nhm : IsPeriodicPt f m x\nhn : IsPeriodicPt f n x\n⊢ IsPeriodicPt f (m.gcd n) x",
"ppTerm": "?refine_2",
"assigned": true,
"usedCon... | [
"case refine_2\nα : Type u_1\nf : α → α\nx : α\nm✝ n✝ m n : ℕ\nx✝ : 0 < m\nih : IsPeriodicPt f (n % m) x → IsPeriodicPt f m x → IsPeriodicPt f ((n % m).gcd m) x\nhm : IsPeriodicPt f m x\nhn : IsPeriodicPt f n x\n⊢ IsPeriodicPt f ((n % m).gcd m) x"
] | rw [Nat.gcd_rec] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Star.Subalgebra | {
"line": 526,
"column": 4
} | {
"line": 526,
"column": 15
} | {
"line": 526,
"column": 16
} | [
{
"pp": "case inr\nR : 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\np : (x : A) → x ∈ adjoin R s → Prop\nmem : ∀ (x : A) (h : x ∈ s), p x ⋯\nalgebraMap : ∀ (r : R), p ((Algebra.algebraMap ... | [
"case inr\nR : 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\np : (x : A) → x ∈ adjoin R s → Prop\nmem : ∀ (x : A) (h : x ∈ s), p x ⋯\nalgebraMap : ∀ (r : R), p ((Algebra.algebraMap R A) r) ⋯\na... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.Subalgebra | {
"line": 526,
"column": 50
} | {
"line": 526,
"column": 61
} | {
"line": 526,
"column": 62
} | [
{
"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\np : (x : A) → x ∈ adjoin R s → Prop\nmem : ∀ (x : A) (h : x ∈ s), p x ⋯\nalgebraMap : ∀ (r : R), p ((Algebra.algebraMap R A) r) ⋯\... | [
"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\np : (x : A) → x ∈ adjoin R s → Prop\nmem : ∀ (x : A) (h : x ∈ s), p x ⋯\nalgebraMap : ∀ (r : R), p ((Algebra.algebraMap R A) r) ⋯\nadd : ∀ (x ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.Subalgebra | {
"line": 521,
"column": 46
} | {
"line": 526,
"column": 84
} | {
"line": 528,
"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\np : (x : A) → x ∈ adjoin R s → Prop\nmem : ∀ (x : A) (h : x ∈ s), p x ⋯\nalgebraMap : ∀ (r : R), p ((Algebra.algebraMap R A) r) ⋯\... | [] | by
refine Algebra.adjoin_induction (fun x hx ↦ ?_) algebraMap add mul ha
push _ ∈ _ at hx
obtain (hx | hx) := hx
· exact mem x hx
· simpa using star _ (Algebra.subset_adjoin (by simpa using Or.inl hx)) (mem _ hx) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.List.Permutation | {
"line": 280,
"column": 69
} | {
"line": 280,
"column": 80
} | {
"line": 280,
"column": 81
} | [
{
"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... | [
"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 !"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.List.Permutation | {
"line": 292,
"column": 39
} | {
"line": 292,
"column": 75
} | {
"line": 292,
"column": 76
} | [
{
"pp": "α : Type u_1\nis l : List α\nH : l ~ [] ++ is → (∃ ts' x, l = ts' ++ is) ∨ l ∈ is.permutationsAux []\n⊢ l ~ is → l ∈ is.permutations",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Membership.mem",
"id",
"List.permutations",
"List.Perm",
... | [
"α : Type u_1\nis l : List α\nH : l ~ [] ++ is → (∃ ts' x, l = ts' ++ is) ∨ l ∈ is.permutationsAux []\n⊢ l ~ is → l = is ∨ l ∈ is.permutationsAux []"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.List.Permutation | {
"line": 309,
"column": 27
} | {
"line": 309,
"column": 38
} | {
"line": 309,
"column": 39
} | [
{
"pp": "α : 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 []\nl₁ : List α\np' : l₁ ++ [t] ~ t :: is\np : l₁ ++ [] ~ is\n⊢ ∃ x, l₁ ++ [... | [
"α : 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 []\nl₁ : List α\np' : l₁ ++ [t] ~ t :: is\np : l₁ ++ [] ~ is\n⊢ l₁ ~ is"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.PeriodicPts.Lemmas | {
"line": 96,
"column": 4
} | {
"line": 96,
"column": 27
} | {
"line": 96,
"column": 28
} | [
{
"pp": "α : Type u_1\nf : α → α\ninst✝ : Finite α\nh : Injective f\nx : α\n⊢ ∃ m n, f^[m] x = f^[n] x ∧ m ≠ n",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Exists",
"id",
"Ne",
"Nat.iterate",
"And",
"Nat",
"Eq"
],
"usedFVars": [
"... | [
"α : Type u_1\nf : α → α\ninst✝ : Finite α\nh : Injective f\nx : α\n⊢ ∃ m n, f^[m] x = f^[n] x ∧ ¬m = n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Coset.Card | {
"line": 62,
"column": 6
} | {
"line": 62,
"column": 37
} | {
"line": 62,
"column": 37
} | [
{
"pp": "α : Type u_1\ninst✝ : Group α\ns : Subgroup α\nt : Set α\n⊢ ↑(t * ↑s) ≃ ↑(QuotientGroup.mk ⁻¹' QuotientGroup.mk '' t)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"QuotientGroup.preimage_image_mk",
"HMul.hMul",
"Monoid.toMulOneClass",
"co... | [
"α : Type u_1\ninst✝ : Group α\ns : Subgroup α\nt : Set α\n⊢ ↑(t * ↑s) ≃ ↑(⋃ x, (fun x_1 ↦ x_1 * ↑x) ⁻¹' t)"
] | QuotientGroup.preimage_image_mk | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.List.Cycle | {
"line": 71,
"column": 4
} | {
"line": 71,
"column": 19
} | {
"line": 71,
"column": 20
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝ : DecidableEq α\nx d d' y z : α\nzs : List α\nIH : x ∈ (z :: zs).dropLast → (z :: zs).nextOr x d = (z :: zs).nextOr x d'\nx_mem : x ∈ (y :: z :: zs).dropLast\nh : ¬x = y\n⊢ x ∈ (z :: zs).dropLast",
"ppTerm": "?neg✝",
"assigned": false,
"usedConstants": [],
... | [
"case neg\nα : Type u_1\ninst✝ : DecidableEq α\nx d d' y z : α\nzs : List α\nIH : x ∈ (z :: zs).dropLast → (z :: zs).nextOr x d = (z :: zs).nextOr x d'\nx_mem : x ∈ (y :: z :: zs).dropLast\nh : ¬x = y\n⊢ x ∈ (z :: zs).dropLast"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.List.Cycle | {
"line": 86,
"column": 6
} | {
"line": 86,
"column": 22
} | {
"line": 86,
"column": 23
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝ : DecidableEq α\nx d y z : α\nzs : List α\nIH : (z :: zs).nextOr x d ≠ d → x ∈ z :: zs\nh : (z :: zs).nextOr x d ≠ d\nhx : ¬x = y\n⊢ x ∈ y :: z :: zs",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"eq_false",
... | [
"case neg\nα : Type u_1\ninst✝ : DecidableEq α\nx d y z : α\nzs : List α\nIH : (z :: zs).nextOr x d ≠ d → x ∈ z :: zs\nh : (z :: zs).nextOr x d ≠ d\nhx : ¬x = y\n⊢ x = z ∨ x ∈ zs"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.Subalgebra | {
"line": 746,
"column": 2
} | {
"line": 746,
"column": 13
} | {
"line": 746,
"column": 14
} | [
{
"pp": "R : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : StarRing R\ninst✝⁸ : Semiring A\ninst✝⁷ : Algebra R A\ninst✝⁶ : StarRing A\ninst✝⁵ : StarModule R A\ninst✝⁴ : Semiring B\ninst✝³ : Algebra R B\ninst✝² : StarRing B\ninst✝¹ : StarModule R B\nι : Sort u_5\ninst✝ : Nonempty ι\nf ... | [
"R : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : StarRing R\ninst✝⁸ : Semiring A\ninst✝⁷ : Algebra R A\ninst✝⁶ : StarRing A\ninst✝⁵ : StarModule R A\ninst✝⁴ : Semiring B\ninst✝³ : Algebra R B\ninst✝² : StarRing B\ninst✝¹ : StarModule R B\nι : Sort u_5\ninst✝ : Nonempty ι\nf : A →⋆ₐ[R] B... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.List.Permutation | {
"line": 410,
"column": 6
} | {
"line": 410,
"column": 23
} | {
"line": 410,
"column": 24
} | [
{
"pp": "case cons.succ\nα : Type u_1\nx y : α\ns : List α\nIH : ∀ (n : ℕ) (hn : n < (permutations'Aux x s).length), (permutations'Aux x s)[n] = s.insertIdx n x\nn✝ : ℕ\nhn : n✝ + 1 < (permutations'Aux x (y :: s)).length\n⊢ (permutations'Aux x (y :: s))[n✝ + 1] = (y :: s).insertIdx (n✝ + 1) x",
"ppTerm": "?... | [
"case cons.succ\nα : Type u_1\nx y : α\ns : List α\nIH : ∀ (n : ℕ) (hn : n < (permutations'Aux x s).length), (permutations'Aux x s)[n] = s.insertIdx n x\nn✝ : ℕ\nhn : n✝ + 1 < (permutations'Aux x (y :: s)).length\n⊢ (permutations'Aux x s)[n✝] = s.insertIdx n✝ x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.List.Cycle | {
"line": 139,
"column": 51
} | {
"line": 139,
"column": 67
} | {
"line": 139,
"column": 68
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\ny z : α\nxs : List α\nx : α\nh : x ∈ y :: z :: xs\nhx : ¬x = y\n⊢ x ∈ z :: xs",
"ppTerm": "?m.73",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Membership.mem",
"id",
"List.cons",
"List",
"List.instMembership",
... | [
"α : Type u_1\ninst✝ : DecidableEq α\ny z : α\nxs : List α\nx : α\nh : x ∈ y :: z :: xs\nhx : ¬x = y\n⊢ x = z ∨ x ∈ xs"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.List.Permutation | {
"line": 433,
"column": 6
} | {
"line": 433,
"column": 53
} | {
"line": 433,
"column": 54
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝ : DecidableEq α\nl : List α\nIH : ∀ (x : α), count (x :: l) (permutations'Aux x l) = (takeWhile (fun x_1 ↦ decide (x = x_1)) l).length + 1\nx : α\n⊢ count (x :: x :: l) (map (cons x) (permutations'Aux x l)) + 1 =\n (takeWhile (fun x_1 ↦ decide (x = x_1)) (x :: l)).lengt... | [
"case pos\nα : Type u_1\ninst✝ : DecidableEq α\nl : List α\nIH : ∀ (x : α), count (x :: l) (permutations'Aux x l) = (takeWhile (fun x_1 ↦ decide (x = x_1)) l).length + 1\nx : α\n⊢ count (x :: l) (permutations'Aux x l) = (takeWhile (fun x_1 ↦ decide (x = x_1)) l).length + 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.List.Permutation | {
"line": 443,
"column": 19
} | {
"line": 443,
"column": 30
} | {
"line": 443,
"column": 31
} | [
{
"pp": "case cons\nα : Type u_1\nx y : α\ns : List α\nIH : (permutations'Aux x s).length = s.length + 1\n⊢ (permutations'Aux x (y :: s)).length = (y :: s).length + 1",
"ppTerm": "?cons",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"List.map",
"id",
"inst... | [
"case cons\nα : Type u_1\nx y : α\ns : List α\nIH : (permutations'Aux x s).length = s.length + 1\n⊢ (permutations'Aux x s).length = s.length + 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.List.Permutation | {
"line": 449,
"column": 38
} | {
"line": 449,
"column": 49
} | {
"line": 449,
"column": 50
} | [
{
"pp": "α : Type u_1\nx : α\ns t : List α\nh : permutations'Aux x s = permutations'Aux x t\n⊢ s.length = t.length",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nx : α\ns t : List α\nh : permutations'Aux x s = permutations'Aux x t\n⊢ s.length = t.length"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.List.Permutation | {
"line": 471,
"column": 52
} | {
"line": 471,
"column": 81
} | {
"line": 471,
"column": 82
} | [
{
"pp": "α : Type u_1\ns : List α\nx : α\nh : Injective (permutations'Aux x s).get\nH : x ∈ s\nk : ℕ\nhk : k < s.length\nhk' : s.get ⟨k, hk⟩ = x\n⊢ k < (permutations'Aux x s).length",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"instOfN... | [
"α : Type u_1\ns : List α\nx : α\nh : Injective (permutations'Aux x s).get\nH : x ∈ s\nk : ℕ\nhk : k < s.length\nhk' : s.get ⟨k, hk⟩ = x\n⊢ k ≤ s.length"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.List.Permutation | {
"line": 472,
"column": 57
} | {
"line": 472,
"column": 68
} | {
"line": 472,
"column": 69
} | [
{
"pp": "α : Type u_1\ns : List α\nx : α\nh : Injective (permutations'Aux x s).get\nH : x ∈ s\nk : ℕ\nhk : k < s.length\nhk' : s.get ⟨k, hk⟩ = x\nkl : k < (permutations'Aux x s).length\n⊢ k + 1 < (permutations'Aux x s).length",
"ppTerm": "?m.78",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"α : Type u_1\ns : List α\nx : α\nh : Injective (permutations'Aux x s).get\nH : x ∈ s\nk : ℕ\nhk : k < s.length\nhk' : s.get ⟨k, hk⟩ = x\nkl : k < (permutations'Aux x s).length\n⊢ k < s.length"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.List.Cycle | {
"line": 187,
"column": 62
} | {
"line": 187,
"column": 73
} | {
"line": 187,
"column": 74
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\ny : α\nk : ℕ\nhd : α\ntl : List α\nhl : Function.Injective (hd :: tl).get\nh✝ : (y :: hd :: tl).getLast ⋯ ∈ hd :: tl\nh : (y :: hd :: tl).getLast ⋯ ∈ y :: hd :: tl\nhy : (y :: hd :: tl).getLast ⋯ ≠ y\nH : (y :: hd :: tl).getLast ⋯ ∈ (y :: hd :: tl).dropLast\nhk : k ... | [
"α : Type u_1\ninst✝ : DecidableEq α\ny : α\nk : ℕ\nhd : α\ntl : List α\nhl : Function.Injective (hd :: tl).get\nh✝ : (y :: hd :: tl).getLast ⋯ ∈ hd :: tl\nh : (y :: hd :: tl).getLast ⋯ ∈ y :: hd :: tl\nhy : (y :: hd :: tl).getLast ⋯ ≠ y\nH : (y :: hd :: tl).getLast ⋯ ∈ (y :: hd :: tl).dropLast\nhk : k + 1 < (y :: ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.List.Permutation | {
"line": 494,
"column": 6
} | {
"line": 494,
"column": 45
} | {
"line": 495,
"column": 6
} | [
{
"pp": "case cons.right\nα : Type u_1\ns : List α\nx : α\nl : List α\nh : ∀ (a' : α), a' ∈ l → x ≠ a'\nh' : Pairwise (fun x1 x2 ↦ x1 ≠ x2) l\nIH : l.permutations'.Nodup\nas : List α\nha : as ∈ l.permutations'\nbs : List α\nhb : bs ∈ l.permutations'\nH : as ≠ bs\na : List α\nha' : a ∈ permutations'Aux x as\nhb'... | [
"case cons.right\nα : Type u_1\ns : List α\nx : α\nl : List α\nh : ∀ (a' : α), a' ∈ l → x ≠ a'\nh' : Pairwise (fun x1 x2 ↦ x1 ≠ x2) l\nIH : l.permutations'.Nodup\nas : List α\nha : as ∈ l.permutations'\nbs : List α\nhb : bs ∈ l.permutations'\nH : as ≠ bs\na : List α\nha' : a ∈ permutations'Aux x as\nhb' : a ∈ permu... | obtain ⟨⟨n, hn⟩, hn'⟩ := get_of_mem ha' | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.GroupTheory.GroupAction.Basic | {
"line": 221,
"column": 4
} | {
"line": 221,
"column": 36
} | {
"line": 221,
"column": 37
} | [
{
"pp": "case h\nG : Type u_1\nα : Type u_2\ninst✝³ : Group G\ninst✝² : MulAction G α\ninst✝¹ : Finite G\ninst✝ : Finite Ω\na✝ : α\n⊢ Finite ↑(orbitRel.Quotient.orbit (Quotient.mk'' a✝))",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulAction.orbitRel.Quotient.orbit",... | [
"case h\nG : Type u_1\nα : Type u_2\ninst✝³ : Group G\ninst✝² : MulAction G α\ninst✝¹ : Finite G\ninst✝ : Finite Ω\na✝ : α\n⊢ (orbit G a✝).Finite"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.List.Cycle | {
"line": 221,
"column": 49
} | {
"line": 221,
"column": 65
} | {
"line": 221,
"column": 66
} | [
{
"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⊢ x ∈ z :: l",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Membership.mem",
"id",
"List.cons",
"List",
"List.instMembership",
... | [
"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⊢ x = z ∨ x ∈ l"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.Subalgebra | {
"line": 974,
"column": 2
} | {
"line": 975,
"column": 46
} | {
"line": 975,
"column": 47
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁶ : CommSemiring R\ninst✝⁵ : StarRing R\ninst✝⁴ : Semiring A\ninst✝³ : StarRing A\ninst✝² : Algebra R A\ninst✝¹ : StarModule R A\nι : Type u_3\ninst✝ : Nonempty ι\nS : ι → StarSubalgebra R A\nhS : ∀ (i : ι), IsMulCommutative ↥(S i)\ndir : Directed (fun x1 x2 ↦ x1 ≤ x2) ... | [
"R : Type u_1\nA : Type u_2\ninst✝⁶ : CommSemiring R\ninst✝⁵ : StarRing R\ninst✝⁴ : Semiring A\ninst✝³ : StarRing A\ninst✝² : Algebra R A\ninst✝¹ : StarModule R A\nι : Type u_3\ninst✝ : Nonempty ι\nS : ι → StarSubalgebra R A\nhS : ∀ (i : ι), IsMulCommutative ↥(S i)\ndir : Directed (fun x1 x2 ↦ x1 ≤ x2) S\n⊢ ∀ (a : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.GroupAction.Basic | {
"line": 322,
"column": 20
} | {
"line": 322,
"column": 42
} | {
"line": 322,
"column": 43
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nS : Set α\na b : α\nh : ∀ (x : α), (if x = a then b else if x = b then a else x) ∈ S ↔ x ∈ S\n⊢ a ∈ S ↔ b ∈ S",
"ppTerm": "?m.38",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝ : DecidableEq α\nS : Set α\na b : α\nh : ∀ (x : α), (if x = a then b else if x = b then a else x) ∈ S ↔ x ∈ S\n⊢ a ∈ S ↔ b ∈ S"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.List.Cycle | {
"line": 349,
"column": 4
} | {
"line": 350,
"column": 55
} | {
"line": 352,
"column": 0
} | [
{
"pp": "case h\nα : Type u_1\ninst✝ : DecidableEq α\nl : List α\nh : l.Nodup\n⊢ ∀ (i : ℕ) (h₁ : i < (pmap l.prev l ⋯).length) (h₂ : i < (l.rotate (l.length - 1)).length),\n (pmap l.prev l ⋯)[i] = (l.rotate (l.length - 1))[i]",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | intro n hn hn'
rw [getElem_rotate, getElem_pmap, prev_getElem _ h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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