module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.LinearAlgebra.Reflection | {
"line": 212,
"column": 4
} | {
"line": 213,
"column": 52
} | [
{
"pp": "case succ\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y : M\nf g : Dual R M\nhf : f x = 2\nhg : g y = 2\nz : M\nt : R\nht : t = f y * g x - 2\nm : ℕ\nS_eval_t_sub_two :\n ∀ (k : ℤ), Polynomial.eval t (S R (k - 2)) = t * Polynomial.eval t (S R (k - 1... | simp_rw [add_assoc (2 * k), add_sub_assoc (2 * k), add_comm (2 * k),
add_mul_ediv_left _ k (by simp : (2 : ℤ) ≠ 0)] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.LinearAlgebra.PerfectPairing.Restrict | {
"line": 68,
"column": 6
} | {
"line": 68,
"column": 17
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\np : M →ₗ[R] N →ₗ[R] R\ninst✝⁴ : p.IsPerfPair\nM' : Type u_4\nN' : Type u_5\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : AddCommGroup N'... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 586,
"column": 23
} | {
"line": 586,
"column": 59
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝⁷ : LieRing L\ninst✝⁶ : Field K\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : IsKilling K L\ninst✝¹ : IsTriangularizable K (↥H) L\ninst✝ : CharZero K\nα : Weight K (↥H) L\nhα : α.IsNonZero\nh e f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.PerfectPairing.Restrict | {
"line": 103,
"column": 6
} | {
"line": 103,
"column": 21
} | [
{
"pp": "case mem\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝¹⁸ : CommRing R\ninst✝¹⁷ : AddCommGroup M\ninst✝¹⁶ : Module R M\ninst✝¹⁵ : AddCommGroup N\ninst✝¹⁴ : Module R N\np : M →ₗ[R] N →ₗ[R] R\ninst✝¹³ : p.IsPerfPair\nS : Type u_4\nM' : Type u_5\nN' : Type u_6\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 594,
"column": 31
} | {
"line": 594,
"column": 70
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝⁷ : LieRing L\ninst✝⁶ : Field K\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : IsKilling K L\ninst✝¹ : IsTriangularizable K (↥H) L\ninst✝ : CharZero K\nα : Weight K (↥H) L\nhα : α.IsNonZero\ne f :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 604,
"column": 4
} | {
"line": 604,
"column": 22
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝⁷ : LieRing L\ninst✝⁶ : Field K\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : IsKilling K L\ninst✝¹ : IsTriangularizable K (↥H) L\ninst✝ : CharZero K\nα : Weight K (↥H) L\nhα : α.IsNonZero\ne f :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 612,
"column": 6
} | {
"line": 612,
"column": 17
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝⁷ : LieRing L\ninst✝⁶ : Field K\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : IsKilling K L\ninst✝¹ : IsTriangularizable K (↥H) L\ninst✝ : CharZero K\nα : Weight K (↥H) L\nhα : α.IsNonZero\nthis ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 620,
"column": 41
} | {
"line": 620,
"column": 72
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝⁷ : LieRing L\ninst✝⁶ : Field K\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : IsKilling K L\ninst✝¹ : IsTriangularizable K (↥H) L\ninst✝ : CharZero K\nα : Weight K (↥H) L\nhα : α.IsNonZero\ncontr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 621,
"column": 4
} | {
"line": 621,
"column": 22
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝⁷ : LieRing L\ninst✝⁶ : Field K\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : IsKilling K L\ninst✝¹ : IsTriangularizable K (↥H) L\ninst✝ : CharZero K\nα : Weight K (↥H) L\nhα : α.IsNonZero\ncontr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.PerfectPairing.Restrict | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 13
} | [
{
"pp": "case h\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝¹⁸ : CommRing R\ninst✝¹⁷ : AddCommGroup M\ninst✝¹⁶ : Module R M\ninst✝¹⁵ : AddCommGroup N\ninst✝¹⁴ : Module R N\np : M →ₗ[R] N →ₗ[R] R\ninst✝¹³ : p.IsPerfPair\nS : Type u_4\nM' : Type u_5\nN' : Type u_6\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 623,
"column": 20
} | {
"line": 623,
"column": 57
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝⁷ : LieRing L\ninst✝⁶ : Field K\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : IsKilling K L\ninst✝¹ : IsTriangularizable K (↥H) L\ninst✝ : CharZero K\nα : Weight K (↥H) L\nhα : α.IsNonZero\ncontr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.PerfectPairing.Restrict | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 13
} | [
{
"pp": "case h.a\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝¹⁸ : CommRing R\ninst✝¹⁷ : AddCommGroup M\ninst✝¹⁶ : Module R M\ninst✝¹⁵ : AddCommGroup N\ninst✝¹⁴ : Module R N\np : M →ₗ[R] N →ₗ[R] R\ninst✝¹³ : p.IsPerfPair\nS : Type u_4\nM' : Type u_5\nN' : Type u_6\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 643,
"column": 67
} | {
"line": 643,
"column": 78
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝⁷ : LieRing L\ninst✝⁶ : Field K\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : IsKilling K L\ninst✝¹ : IsTriangularizable K (↥H) L\ninst✝ : CharZero K\nα : Weight K (↥H) L\nhα : α.IsNonZero\nh e f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 643,
"column": 64
} | {
"line": 643,
"column": 90
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝⁷ : LieRing L\ninst✝⁶ : Field K\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : IsKilling K L\ninst✝¹ : IsTriangularizable K (↥H) L\ninst✝ : CharZero K\nα : Weight K (↥H) L\nhα : α.IsNonZero\nh e f... | by simpa using t.e_ne_zero | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 645,
"column": 17
} | {
"line": 645,
"column": 28
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝⁷ : LieRing L\ninst✝⁶ : Field K\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : IsKilling K L\ninst✝¹ : IsTriangularizable K (↥H) L\ninst✝ : CharZero K\nα : Weight K (↥H) L\nhα : α.IsNonZero\nh e f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 647,
"column": 67
} | {
"line": 647,
"column": 78
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝⁷ : LieRing L\ninst✝⁶ : Field K\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : IsKilling K L\ninst✝¹ : IsTriangularizable K (↥H) L\ninst✝ : CharZero K\nα : Weight K (↥H) L\nhα : α.IsNonZero\nh e f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 649,
"column": 17
} | {
"line": 649,
"column": 28
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝⁷ : LieRing L\ninst✝⁶ : Field K\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : IsKilling K L\ninst✝¹ : IsTriangularizable K (↥H) L\ninst✝ : CharZero K\nα : Weight K (↥H) L\nhα : α.IsNonZero\nh e f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.PerfectPairing.Restrict | {
"line": 199,
"column": 23
} | {
"line": 199,
"column": 38
} | [
{
"pp": "K : Type u_1\nL : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : Field K\ninst✝⁹ : Field L\ninst✝⁸ : Algebra K L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module L M\ninst✝⁴ : Module L N\ninst✝³ : Module K M\ninst✝² : Module K N\ninst✝¹ : IsScalarTower K L M\np : M →ₗ[L] N →ₗ[L] L\ni... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.PerfectPairing.Restrict | {
"line": 199,
"column": 59
} | {
"line": 199,
"column": 75
} | [
{
"pp": "K : Type u_1\nL : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : Field K\ninst✝⁹ : Field L\ninst✝⁸ : Algebra K L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module L M\ninst✝⁴ : Module L N\ninst✝³ : Module K M\ninst✝² : Module K N\ninst✝¹ : IsScalarTower K L M\np : M →ₗ[L] N →ₗ[L] L\ni... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 715,
"column": 19
} | {
"line": 715,
"column": 30
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝⁷ : LieRing L\ninst✝⁶ : Field K\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : IsKilling K L\ninst✝¹ : IsTriangularizable K (↥H) L\ninst✝ : CharZero K\nα : Weight K (↥H) L\nhα : α.IsNonZero\nx h' ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Reflection | {
"line": 241,
"column": 2
} | {
"line": 241,
"column": 13
} | [
{
"pp": "case h\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y : M\nf g : Dual R M\nhf : f x = 2\nhg : g y = 2\nm : ℕ\nt : R\nht : t = f y * g x - 2\nz : M\n⊢ ↑((reflection hf * reflection hg) ^ m) z =\n (LinearMap.id +\n (Polynomial.eval t (S R ((... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 744,
"column": 33
} | {
"line": 744,
"column": 65
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝⁷ : LieRing L\ninst✝⁶ : Field K\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : IsKilling K L\ninst✝¹ : IsTriangularizable K (↥H) L\ninst✝ : CharZero K\nχ : Weight K (↥H) L\nhχ : χ ∈ LieSubalgebra.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Span.TensorProduct | {
"line": 58,
"column": 2
} | {
"line": 58,
"column": 13
} | [
{
"pp": "case h\nR : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring A\ninst✝⁴ : Algebra R A\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module A M\ninst✝ : IsScalarTower R A M\np : Submodule R M\nv : ↥(span A ↑p)\nf : ↑↑p →₀ A\nhf : (Finsupp.linearCombination A... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Span.TensorProduct | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 61
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁸ : CommSemiring R\ninst✝⁷ : CommSemiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : Module A M\ninst✝² : IsScalarTower R A M\np : Submodule R M\ninst✝¹ : Algebra.IsEpi R A\ninst✝ : Module.Flat R A\nf : A ⊗[R] ↥(span A... | have hf : Injective f := Algebra.injective_lift_lsmul R A _ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Localization.NumDen | {
"line": 72,
"column": 15
} | {
"line": 72,
"column": 45
} | [
{
"pp": "A : Type u_1\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx y : K\nh : x * (algebraMap A K) ↑(den A y) = (algebraMap A K) (num A y)\n⊢ x = y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.NumDen | {
"line": 77,
"column": 14
} | {
"line": 77,
"column": 53
} | [
{
"pp": "A : Type u_1\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx y : K\nh : y * (algebraMap A K) ↑(den A x) = (algebraMap A K) (num A x)\n⊢ x = y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.NumDen | {
"line": 82,
"column": 14
} | {
"line": 82,
"column": 44
} | [
{
"pp": "A : Type u_1\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx y : K\nh : num A y * ↑(den A x) = num A x * ↑(den A y)\n⊢ x = y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Reflection | {
"line": 273,
"column": 2
} | {
"line": 273,
"column": 13
} | [
{
"pp": "case h\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y : M\nf g : Dual R M\nhf : f x = 2\nhg : g y = 2\nm : ℤ\nt : R\nht : t = f y * g x - 2\nz : M\n⊢ ↑((reflection hf * reflection hg) ^ m) z =\n (LinearMap.id +\n (Polynomial.eval t (S R ((... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.RationalRoot | {
"line": 106,
"column": 4
} | {
"line": 106,
"column": 15
} | [
{
"pp": "case pos.convert_4\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\np : A[X]\nr : K\nhr : (aeval r) p = 0\nj : ℕ\nhj : j ≠ p.natDegree\nh : j < p.natDegree\n⊢ j + 0 < p.natDeg... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.RationalRoot | {
"line": 125,
"column": 37
} | {
"line": 125,
"column": 59
} | [
{
"pp": "A : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\np : A[X]\nhp : p.Monic\nr : K\nhr : (aeval r) p = 0\ninv : A\nh_inv : 1 = ↑(den A r) * inv\n⊢ 1 = inv * ↑(den A r)",
"usedC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.RationalRoot | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 15
} | [
{
"pp": "case right\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\np : A[X]\nhp : p.Monic\nr : K\nhr : (aeval r) p = 0\ninv : A\nh_inv : 1 = ↑(den A r) * inv\nh : inv ∣ 1\n⊢ num A r ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 86,
"column": 4
} | {
"line": 86,
"column": 46
} | [
{
"pp": "motive : ℤ → Prop\nzero : motive 0\none : motive 1\nadd_two : ∀ (n : ℕ), motive (↑n + 1) → motive ↑n → motive (↑n + 2)\nneg_add_one : ∀ (n : ℕ), motive (-↑n) → motive (-↑n + 1) → motive (-↑n - 1)\nn : ℕ\nhn : motive (-↑n)\nhnm : motive (-↑n + 1)\n⊢ motive (Int.negSucc n)",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Reflection | {
"line": 367,
"column": 15
} | {
"line": 367,
"column": 38
} | [
{
"pp": "case succ\nR : Type u_1\nM : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\nx : M\nΦ : Set M\nhΦ₁ : Φ.Finite\nhΦ₂ : span R Φ = ⊤\nf g : Dual R M\nhf₁ : f x = 2\nhf₂ : MapsTo (⇑(preReflection x f)) Φ Φ\nhg... | Module.End.mul_eq_comp, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.Reflection | {
"line": 372,
"column": 4
} | {
"line": 372,
"column": 47
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\nx : M\nΦ : Set M\nhΦ₁ : Φ.Finite\nhΦ₂ : span R Φ = ⊤\nf g : Dual R M\nhf₁ : f x = 2\nhf₂ : MapsTo (⇑(preReflection x f)) Φ Φ\nhg₁ : g x = 2... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 227,
"column": 49
} | {
"line": 241,
"column": 25
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NeZero 2\nn : ℤ\n⊢ (T R n).leadingCoeff = 2 ^ (n.natAbs - 1)",
"usedConstants": [
"Nat.cast_mul._simp_1",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"WithBot.instPreorder",
"Polynomial.monic_on... | by
induction n using Chebyshev.induct' with
| zero => simp
| one => simp
| add_two n ih1 ih2 =>
have : leadingCoeff (2 : R[X]) = 2 := by
change leadingCoeff (C 2) = 2
rw [leadingCoeff_C]
rw [T_add_two, leadingCoeff_sub_of_degree_lt, leadingCoeff_mul, ih1,
leadingCoeff_mul, leadingCoeff... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Reflection | {
"line": 454,
"column": 31
} | {
"line": 454,
"column": 50
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι : Type u_3\ninst✝ : IsAddTorsionFree M\nr : ι ↪ M\nc : ι → Dual R M\nhfin : (range ⇑r).Finite\nh_two : ∀ (i : ι), (c i) (r i) = 2\nh_mapsTo : ∀ (i : ι), MapsTo (⇑(preReflection (r i) (c i))) (range ⇑r) (ran... | by rw [this, h_two] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Reflection | {
"line": 456,
"column": 2
} | {
"line": 456,
"column": 13
} | [
{
"pp": "case e_a\nR : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι : Type u_3\ninst✝ : IsAddTorsionFree M\nr : ι ↪ M\nc : ι → Dual R M\nhfin : (range ⇑r).Finite\nh_two : ∀ (i : ι), (c i) (r i) = 2\nh_mapsTo : ∀ (i : ι), MapsTo (⇑(preReflection (r i) (c i))) (rang... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 296,
"column": 39
} | {
"line": 296,
"column": 50
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\n⊢ U R (-1) = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 305,
"column": 2
} | {
"line": 305,
"column": 68
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\n⊢ U R (-2) = -1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 320,
"column": 54
} | {
"line": 320,
"column": 75
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn : ℤ\n⊢ U R (-n) = -U R (n - 2)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 324,
"column": 2
} | {
"line": 324,
"column": 40
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn : ℤ\n⊢ U R (-n - 2) = -U R n",
"usedConstants": [
"Eq.mpr",
"Polynomial.instNeg",
"Polynomial.Chebyshev.U",
"congrArg",
"CommSemiring.toSemiring",
"AddMonoid.toAddZeroClass",
"sub_eq_add_neg",
"HSub.hSub",
"Ad... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Quotient.PowTransition | {
"line": 55,
"column": 4
} | {
"line": 55,
"column": 54
} | [
{
"pp": "case h.refine_2\nR : Type u_1\ninst✝² : Ring R\nI J : Ideal R\nH : I ≤ J\ninst✝¹ : I.IsTwoSided\ninst✝ : J.IsTwoSided\nx : R ⧸ I\nh : x ∈ map (mk I) J\nr : R\nhr : r ∈ ↑J\neq : (mk I) r = x\n⊢ x ∈ RingHom.ker (factor H)",
"usedConstants": [
"Eq.mpr",
"RingHom.instRingHomClass",
"S... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Quotient.PowTransition | {
"line": 89,
"column": 4
} | {
"line": 89,
"column": 27
} | [
{
"pp": "case h.refine_2\nR : Type u_1\ninst✝² : Ring R\nI J K : Ideal R\ninst✝¹ : J.IsTwoSided\ninst✝ : K.IsTwoSided\nhIJ : J ≤ I\nhJK : K ≤ J\nx : R ⧸ K\nh : x ∈ map (mk K) I\nr : R\nhr : r ∈ ↑I\neq : (mk K) r = x\n⊢ x ∈ comap (factor hJK) (map (mk J) I)",
"usedConstants": [
"Eq.mpr",
"RingHom... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Quotient.PowTransition | {
"line": 182,
"column": 6
} | {
"line": 182,
"column": 59
} | [
{
"pp": "R : Type u_3\ninst✝ : CommRing R\nI : Ideal R\nn : ℕ\nnpos : n > 0\na : R ⧸ I ^ (n + 1)\nh : IsUnit ((factorPow I ⋯) a)\nb : R ⧸ I ^ n\nright✝ : b * (factorPow I ⋯) a = 1\nb' : R ⧸ I ^ n.succ\nhb' : (factor ⋯) b' = b\nhb : a * b' - 1 ∈ map (mk (I ^ n.succ)) (I ^ n)\nc : R\nhc : c ∈ ↑(I ^ n)\neq : (mk (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.GroupWithZero.WithZero | {
"line": 38,
"column": 37
} | {
"line": 38,
"column": 64
} | [
{
"pp": "α : Type u_1\ninst✝² : Mul α\ninst✝¹ : Preorder α\ninst✝ : MulLeftStrictMono α\nx : α\nhx : 0 < ↑x\nb : α\nx✝ : 0 < ↑b\n⊢ ↑x * 0 < ↑x * ↑b",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"HMul.hMul",
"MulZeroClass.toMul",
"congrArg",
"id",
"MulZeroClass.mul... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.GroupWithZero.WithZero | {
"line": 45,
"column": 37
} | {
"line": 45,
"column": 64
} | [
{
"pp": "α : Type u_1\ninst✝² : Mul α\ninst✝¹ : Preorder α\ninst✝ : MulRightStrictMono α\nx : α\nhx : 0 < ↑x\nb : α\nx✝ : 0 < ↑b\n⊢ 0 * ↑x < ↑b * ↑x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.GroupWithZero.Range | {
"line": 44,
"column": 2
} | {
"line": 44,
"column": 13
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹ : MonoidWithZero A\ninst✝ : LinearOrderedCommGroupWithZero B\nf : A →*₀ B\nx y : ValueGroup₀ f\nhxy : (orderMonoidWithZeroHom f) x < (orderMonoidWithZeroHom f) y\n⊢ embedding x < embedding y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Length | {
"line": 81,
"column": 40
} | {
"line": 81,
"column": 51
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ns : CompositionSeries (Submodule R M)\nh₁ : RelSeries.head s = ⊥\nh₂ : RelSeries.last s = ⊤\nH : IsFiniteLength R M\nthis✝¹ : IsNoetherian R M\nthis✝ : IsArtinian R M\nt : LTSeries (Submodule R M)\nt' : RelSeries ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Length | {
"line": 82,
"column": 41
} | {
"line": 82,
"column": 52
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ns : CompositionSeries (Submodule R M)\nh₁ : RelSeries.head s = ⊥\nh₂ : RelSeries.last s = ⊤\nH : IsFiniteLength R M\nthis✝¹ : IsNoetherian R M\nthis✝ : IsArtinian R M\nt : LTSeries (Submodule R M)\nt' : RelSeries ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Length | {
"line": 83,
"column": 4
} | {
"line": 83,
"column": 15
} | [
{
"pp": "case a\nR : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ns : CompositionSeries (Submodule R M)\nh₁ : RelSeries.head s = ⊥\nh₂ : RelSeries.last s = ⊤\nH : IsFiniteLength R M\nthis✝¹ : IsNoetherian R M\nthis✝ : IsArtinian R M\nt : LTSeries (Submodule R M)\nt' : Re... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 558,
"column": 2
} | {
"line": 558,
"column": 34
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\n⊢ C R 2 = X ^ 2 - 2",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"pow_two",
"Monoid.toMulOneClass",
"congrArg",
"CommSemiring.toSemiring",
"Nat.instAtLeastTwoHAddOfNat",
"HSub.hSub",
"id",
"MulOne.toMul",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Length | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 46
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsArtinian R M\ninst✝ : IsNoetherian R M\nN : Submodule R M\n⊢ Order.height N < ⊤",
"usedConstants": [
"Eq.mpr",
"Submodule",
"instTopENat",
"congrArg",
"AddCommGroup.to... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Length | {
"line": 162,
"column": 2
} | {
"line": 162,
"column": 69
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsArtinian R M\ninst✝ : IsNoetherian R M\nN : Submodule R M\nh : N ≠ ⊤\n⊢ Module.length R ↥N < Module.length R M",
"usedConstants": [
"Eq.mpr",
"Submodule",
"congrArg",
"AddCo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Length | {
"line": 183,
"column": 31
} | {
"line": 183,
"column": 61
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nN : Type u_3\nP : Type u_4\ninst✝³ : AddCommGroup N\ninst✝² : AddCommGroup P\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : N →ₗ[R] M\ng : M →ₗ[R] P\nhf : Function.Injective ⇑f\nhg : Function.Surjective ⇑g\nH : Fu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Length | {
"line": 187,
"column": 14
} | {
"line": 187,
"column": 58
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nN : Type u_3\nP : Type u_4\ninst✝³ : AddCommGroup N\ninst✝² : AddCommGroup P\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : N →ₗ[R] M\ng : M →ₗ[R] P\nhf : Function.Injective ⇑f\nhg : Function.Surjective ⇑g\nH : Fu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Length | {
"line": 188,
"column": 14
} | {
"line": 188,
"column": 60
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nN : Type u_3\nP : Type u_4\ninst✝³ : AddCommGroup N\ninst✝² : AddCommGroup P\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : N →ₗ[R] M\ng : M →ₗ[R] P\nhf : Function.Injective ⇑f\nhg : Function.Surjective ⇑g\nH : Fu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 663,
"column": 39
} | {
"line": 663,
"column": 50
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\n⊢ S R (-1) = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 672,
"column": 2
} | {
"line": 672,
"column": 68
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\n⊢ S R (-2) = -1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 683,
"column": 4
} | {
"line": 685,
"column": 73
} | [
{
"pp": "case neg_add_one\nR : Type u_1\ninst✝ : CommRing R\nn : ℕ\nih1 : S R (- -↑n - 1) = -S R (-↑n - 1)\nih2 : S R (-(-↑n + 1) - 1) = -S R (-↑n + 1 - 1)\n⊢ S R (-(-↑n - 1) - 1) = -S R (-↑n - 1 - 1)",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_... | have h₁ := S_eq R n
have h₂ := S_sub_two R (-n)
linear_combination (norm := ring_nf) (X : R[X]) * ih1 - ih2 + h₁ + h₂ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 683,
"column": 4
} | {
"line": 685,
"column": 73
} | [
{
"pp": "case neg_add_one\nR : Type u_1\ninst✝ : CommRing R\nn : ℕ\nih1 : S R (- -↑n - 1) = -S R (-↑n - 1)\nih2 : S R (-(-↑n + 1) - 1) = -S R (-↑n + 1 - 1)\n⊢ S R (-(-↑n - 1) - 1) = -S R (-↑n - 1 - 1)",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_... | have h₁ := S_eq R n
have h₂ := S_sub_two R (-n)
linear_combination (norm := ring_nf) (X : R[X]) * ih1 - ih2 + h₁ + h₂ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 687,
"column": 54
} | {
"line": 687,
"column": 75
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn : ℤ\n⊢ S R (-n) = -S R (n - 2)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 691,
"column": 2
} | {
"line": 691,
"column": 40
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn : ℤ\n⊢ S R (-n - 2) = -S R n",
"usedConstants": [
"Eq.mpr",
"Polynomial.instNeg",
"congrArg",
"CommSemiring.toSemiring",
"AddMonoid.toAddZeroClass",
"sub_eq_add_neg",
"HSub.hSub",
"Polynomial.Chebyshev.S",
"Ad... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Integers | {
"line": 228,
"column": 68
} | {
"line": 228,
"column": 79
} | [
{
"pp": "F : Type u\nΓ₀ : Type v\ninst✝³ : Field F\ninst✝² : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation F Γ₀\nO : Type w\ninst✝¹ : CommRing O\ninst✝ : Algebra O F\nhv : v.Integers O\nI : Ideal O\nx : Γ₀\nhx : IsGreatest (⇑v ∘ ⇑(algebraMap O F) '' ↑I) x\n⊢ ∃ a ∈ I, (⇑v ∘ ⇑(algebraMap O F)) a = x",
"use... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Integers | {
"line": 238,
"column": 68
} | {
"line": 238,
"column": 79
} | [
{
"pp": "F : Type u\nΓ₀ : Type v\ninst✝³ : Field F\ninst✝² : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation F Γ₀\nO : Type w\ninst✝¹ : CommRing O\ninst✝ : Algebra O F\nhv : v.Integers O\nI : Ideal O\nx : Γ₀\nhx : IsGreatest (⇑v ∘ ⇑(algebraMap O F) '' ↑I) x\n⊢ ∃ a ∈ I, (⇑v ∘ ⇑(algebraMap O F)) a = x",
"use... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Integers | {
"line": 256,
"column": 37
} | {
"line": 256,
"column": 48
} | [
{
"pp": "F : Type u\nΓ₀ : Type v\ninst✝⁴ : Field F\ninst✝³ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation F Γ₀\nO : Type w\ninst✝² : CommRing O\ninst✝¹ : Algebra O F\ninst✝ : IsPrincipalIdealRing O\nhv : v.Integers O\nH : DenselyOrdered ↑(range ⇑v)\na b : O\nha : a ∈ ⇑v ∘ ⇑(algebraMap O F) ⁻¹' Iio 1\nhb : b... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Integers | {
"line": 267,
"column": 4
} | {
"line": 268,
"column": 11
} | [
{
"pp": "F : Type u\nΓ₀ : Type v\ninst✝⁴ : Field F\ninst✝³ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation F Γ₀\nO : Type w\ninst✝² : CommRing O\ninst✝¹ : Algebra O F\ninst✝ : IsPrincipalIdealRing O\nhv : v.Integers O\nH : DenselyOrdered ↑(range ⇑v)\nI : Ideal O := { carrier := ⇑v ∘ ⇑(algebraMap O F) ⁻¹' Iio... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Integers | {
"line": 270,
"column": 4
} | {
"line": 271,
"column": 11
} | [
{
"pp": "F : Type u\nΓ₀ : Type v\ninst✝⁴ : Field F\ninst✝³ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation F Γ₀\nO : Type w\ninst✝² : CommRing O\ninst✝¹ : Algebra O F\ninst✝ : IsPrincipalIdealRing O\nhv : v.Integers O\nH : DenselyOrdered ↑(range ⇑v)\nI : Ideal O := { carrier := ⇑v ∘ ⇑(algebraMap O F) ⁻¹' Iio... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Integers | {
"line": 311,
"column": 4
} | {
"line": 311,
"column": 34
} | [
{
"pp": "R : Type u\nΓ₀ : Type v\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nγ : Γ₀\nr : ↥v.integer\nx : R\nh : x ∈ (v.leAddSubgroup γ).carrier\n⊢ r • x ∈ (v.leAddSubgroup γ).carrier",
"usedConstants": [
"Valuation.mem_leAddSubgroup_iff._simp_1",
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Integers | {
"line": 317,
"column": 4
} | {
"line": 317,
"column": 34
} | [
{
"pp": "R : Type u\nΓ₀ : Type v\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nγ : Γ₀ˣ\nr : ↥v.integer\nx : R\nh : x ∈ (v.ltAddSubgroup γ).carrier\n⊢ r • x ∈ (v.ltAddSubgroup γ).carrier",
"usedConstants": [
"Units.val",
"Eq.mpr",
"GroupWithZero.toMonoidWi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Integers | {
"line": 353,
"column": 2
} | {
"line": 353,
"column": 55
} | [
{
"pp": "case inr\nΓ₀ : Type v\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\nK : Type u_1\ninst✝ : Field K\nv : Valuation K Γ₀\nS : Submodule (↥v.integer) K\nx : K\nhx : x ∈ S\nhx0 : x ≠ 0\ny : K\nhy : y ∈ v.leSubmodule (v x)\nthis : v (y / x) ≤ 1\n⊢ y ∈ S",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Integers | {
"line": 370,
"column": 6
} | {
"line": 370,
"column": 17
} | [
{
"pp": "R : Type u\nΓ₀ : Type v\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nγ : Γ₀\nr x : ↥v.integer\nh : x ∈ ((v.leAddSubgroup γ).addSubgroupOf v.integer.toAddSubgroup).carrier\n⊢ ↑(r • x) ∈ v.leAddSubgroup γ",
"usedConstants": [
"Valuation.mem_leAddSubgroup_iff.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Integers | {
"line": 377,
"column": 4
} | {
"line": 377,
"column": 34
} | [
{
"pp": "R : Type u\nΓ₀ : Type v\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nγ : Γ₀ˣ\nr x : ↥v.integer\nh : x ∈ ((v.ltAddSubgroup γ).addSubgroupOf v.integer.toAddSubgroup).carrier\n⊢ v (↑r * ↑x) < ↑γ",
"usedConstants": [
"Units.val",
"Eq.mpr",
"GroupWit... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 30
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nI : Ideal R\nM : Type u_4\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsHausdorff I M\nx y : M\nh : ∀ (n : ℕ), x ≡ y [SMOD I ^ n • ⊤]\n⊢ ∀ (n : ℕ), x - y ≡ 0 [SMOD I ^ n • ⊤]",
"usedConstants": [
"Eq.mpr",
"Submodule",
"instHSMul",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Integers | {
"line": 435,
"column": 35
} | {
"line": 435,
"column": 46
} | [
{
"pp": "Γ₀ : Type v\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\nK : Type u_1\ninst✝ : Field K\nv : Valuation K Γ₀\nI : Ideal ↥v.integer\nx : ↥v.integer\nhx : x ∈ I\nhx0 : x ≠ 0\ny : ↥v.integer\nhy : y ∈ v.leIdeal (v ↑x)\n⊢ v (↑y / ↑x) ≤ 1",
"usedConstants": [
"Eq.mpr",
"LinearOrderedCommGroupW... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 13
} | [
{
"pp": "case h\nR : Type u_6\nS : Type u_7\ninst✝¹ : CommRing S\nI : Ideal S\ninst✝ : IsHausdorff I S\nf g : R → S\nh : ∀ (n : ℕ) (r : R), (Ideal.Quotient.mk (I ^ n)) (f r) = (Ideal.Quotient.mk (I ^ n)) (g r)\nr : R\nn : ℕ\n⊢ f r ≡ g r [SMOD I ^ n • ⊤]",
"usedConstants": [
"Eq.mpr",
"Submodule"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 13
} | [
{
"pp": "case h\nR : Type u_6\nS : Type u_7\ninst✝¹ : CommRing S\nI : Ideal S\ninst✝ : IsHausdorff I S\nf g : R → S\na : ℕ → ℕ\nha : StrictMono a\nh : ∀ (n : ℕ) (r : R), (Ideal.Quotient.mk (I ^ a n)) (f r) = (Ideal.Quotient.mk (I ^ a n)) (g r)\nm : R\nn : ℕ\n⊢ f m ≡ g m [SMOD I ^ a n • ⊤]",
"usedConstants":... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.ValuationRing | {
"line": 89,
"column": 10
} | {
"line": 89,
"column": 32
} | [
{
"pp": "case h\nA : Type u\ninst✝² : CommRing A\nK : Type v\ninst✝¹ : Field K\ninst✝ : Algebra A K\nx y : ValueGroup A K\na b : K\nc d : Aˣ\ne : A\nhe : c⁻¹ • e • (fun m ↦ m • b) d = c⁻¹ • (fun m ↦ m • a) c\n⊢ (↑c⁻¹ * e * ↑d) • b = a",
"usedConstants": [
"Units.val",
"Eq.mpr",
"Semigroup.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 178,
"column": 18
} | {
"line": 178,
"column": 68
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\nI : Ideal R\nM : Type u_4\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_5\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nx : M\nhx : ∀ (n : ℕ), x ≡ 0 [SMOD ⊥ ^ n • ⊤]\n⊢ x = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 983,
"column": 2
} | {
"line": 984,
"column": 58
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn : ℤ\nk : ℕ\nx : R\n⊢ (1 - x ^ 2) * eval x ((⇑derivative)^[k + 2] (U R n)) =\n (2 * ↑k + 3) * x * eval x ((⇑derivative)^[k + 1] (U R n)) -\n ((↑n + 1) ^ 2 - (↑k + 1) ^ 2) * eval x ((⇑derivative)^[k] (U R n))",
"usedConstants": [
"Polynomial.derivativ... | have h := congr_arg (fun (p : R[X]) => p.eval x) <|
one_sub_X_sq_mul_iterate_derivative_U_eq_poly_in_U n k | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 248,
"column": 2
} | {
"line": 249,
"column": 20
} | [
{
"pp": "case zero\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\nI : Ideal R\nM : Type u_4\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_5\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : ℕ → M\nhf : ∀ {m n : ℕ}, m ≤ n → f m ≡ f n [SMOD ⊥ ^ m • ⊤]\n⊢ f 0 ≡ f 1 [SMOD ⊥ ^ 0 • ⊤]",
... | · rw [pow_zero, Ideal.one_eq_top, top_smul]
exact SModEq.top | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 538,
"column": 4
} | {
"line": 538,
"column": 31
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\nI : Ideal R\nM : Type u_4\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_5\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : AdicCauchySequence I M\nm n : ℕ\nhmn : m ≤ n\n⊢ (transitionMap I M hmn) (Submodule.Quotient.mk (↑f n)) =... | exact (f.property hmn).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 551,
"column": 2
} | {
"line": 551,
"column": 13
} | [
{
"pp": "case h\nR : Type u_1\ninst✝² : CommRing R\nI : Ideal R\nM : Type u_4\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : AdicCauchySequence I M\nk : ℕ\nh : ∀ n ≥ k, ∃ m ≥ n, ∃ l ≥ n, ↑f m ∈ I ^ l • ⊤\nn m : ℕ\nhnm : m ≥ n + k\nl : ℕ\nhnl : l ≥ n + k\nhl : ↑f m ∈ I ^ l • ⊤\n⊢ Submodule.Quotient.mk (↑f m) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.ValuationRing | {
"line": 366,
"column": 6
} | {
"line": 366,
"column": 25
} | [
{
"pp": "case neg.inl\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nH : ∀ (x : K), IsLocalization.IsInteger R x ∨ IsLocalization.IsInteger R x⁻¹\na b : R\nha : (algebraMap R K) a ≠ 0\nhb : (algebraMap R K) b ≠ 0\nc : R\... | exact ⟨c, Or.inr e⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 608,
"column": 4
} | {
"line": 608,
"column": 33
} | [
{
"pp": "case mp.h\nR : Type u_1\ninst✝² : CommRing R\nI : Ideal R\nM : Type u_4\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nh : Function.Injective ⇑(of I M)\nx : M\nhx : ∀ (n : ℕ), x ≡ 0 [SMOD I ^ n • ⊤]\nn : ℕ\n⊢ ↑((of I M) x) n = ↑((of I M) 0) n",
"usedConstants": [
"Eq.mpr",
"Submodule",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 615,
"column": 4
} | {
"line": 615,
"column": 29
} | [
{
"pp": "case mpr.h\nR : Type u_1\ninst✝² : CommRing R\nI : Ideal R\nM : Type u_4\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nh : IsHausdorff I M\nx : M\nhx : ↑((of I M) x) = ↑0\nn : ℕ\n⊢ x ≡ 0 [SMOD I ^ n • ⊤]",
"usedConstants": [
"Eq.mpr",
"Submodule",
"instHSMul",
"Semiring.toMo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 639,
"column": 4
} | {
"line": 639,
"column": 45
} | [
{
"pp": "case h.h\nR : Type u_1\ninst✝² : CommRing R\nI : Ideal R\nM : Type u_4\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nh : IsPrecomplete I M\nu : AdicCompletion I M\nx : ℕ → M\nhx : ∀ (n : ℕ), Submodule.Quotient.mk (x n) = ↑u n\na : M\nha : ∀ (n : ℕ), x n ≡ a [SMOD I ^ n • ⊤]\nn : ℕ\n⊢ ↑((of I M) a) n = ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 691,
"column": 2
} | {
"line": 691,
"column": 55
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝² : CommRing R\nI : Ideal R\nM : Type u_4\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nm n : ℕ\nx : AdicCompletion I M\nm_ge : n ≤ m\nh : ↑x n = 0\n⊢ ↑x m ∈ I ^ n • ⊤",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Submodule",
"S... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 802,
"column": 4
} | {
"line": 802,
"column": 15
} | [
{
"pp": "case h.refine_2\nR : Type u_1\ninst✝⁴ : CommRing R\nI : Ideal R\nM : Type u_4\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_5\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\na : ℕ → ℕ\nha : StrictMono a\nf : (n : ℕ) → M →ₗ[R] N ⧸ I ^ a n • ⊤\nm n : ℕ\nhle : m ≤ n\nx : M\ns : ℕ\nhf : ∀ {m : ℕ}... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.ValuationRing | {
"line": 447,
"column": 40
} | {
"line": 447,
"column": 51
} | [
{
"pp": "𝒪 : Type u\nK : Type v\ninst✝² : CommRing 𝒪\ninst✝¹ : Field K\ninst✝ : Algebra 𝒪 K\nh : ∀ (x : K), ∃ a, x = (algebraMap 𝒪 K) a ∨ x⁻¹ = (algebraMap 𝒪 K) a\nhinj : Function.Injective ⇑(algebraMap 𝒪 K)\nthis✝¹ : IsDomain 𝒪\nthis✝ : FaithfulSMul 𝒪 K\nthis : IsDomain 𝒪\nx✝ y✝ : 𝒪\nhab : (algebraMa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.ValuationRing | {
"line": 451,
"column": 23
} | {
"line": 451,
"column": 48
} | [
{
"pp": "𝒪 : Type u\nK : Type v\ninst✝² : CommRing 𝒪\ninst✝¹ : Field K\ninst✝ : Algebra 𝒪 K\nh : ∀ (x : K), ∃ a, x = (algebraMap 𝒪 K) a ∨ x⁻¹ = (algebraMap 𝒪 K) a\nhinj : Function.Injective ⇑(algebraMap 𝒪 K)\nthis✝¹ : IsDomain 𝒪\nthis✝ : FaithfulSMul 𝒪 K\nthis : IsDomain 𝒪\nx : K\na : 𝒪\nha : x = (alg... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.ValuationRing | {
"line": 452,
"column": 38
} | {
"line": 452,
"column": 49
} | [
{
"pp": "𝒪 : Type u\nK : Type v\ninst✝² : CommRing 𝒪\ninst✝¹ : Field K\ninst✝ : Algebra 𝒪 K\nh : ∀ (x : K), ∃ a, x = (algebraMap 𝒪 K) a ∨ x⁻¹ = (algebraMap 𝒪 K) a\nhinj : Function.Injective ⇑(algebraMap 𝒪 K)\nthis✝¹ : IsDomain 𝒪\nthis✝ : FaithfulSMul 𝒪 K\nthis : IsDomain 𝒪\nx : K\na : 𝒪\nha : x = (alg... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.ValuationRing | {
"line": 453,
"column": 27
} | {
"line": 453,
"column": 37
} | [
{
"pp": "case neg\n𝒪 : Type u\nK : Type v\ninst✝² : CommRing 𝒪\ninst✝¹ : Field K\ninst✝ : Algebra 𝒪 K\nh : ∀ (x : K), ∃ a, x = (algebraMap 𝒪 K) a ∨ x⁻¹ = (algebraMap 𝒪 K) a\nhinj : Function.Injective ⇑(algebraMap 𝒪 K)\nthis✝² : IsDomain 𝒪\nthis✝¹ : FaithfulSMul 𝒪 K\nthis✝ : IsDomain 𝒪\nx : K\na : 𝒪\nh... | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Valuation.ValuationRing | {
"line": 456,
"column": 71
} | {
"line": 456,
"column": 82
} | [
{
"pp": "𝒪 : Type u\nK : Type v\ninst✝² : CommRing 𝒪\ninst✝¹ : Field K\ninst✝ : Algebra 𝒪 K\nh : ∀ (x : K), ∃ a, x = (algebraMap 𝒪 K) a ∨ x⁻¹ = (algebraMap 𝒪 K) a\nhinj : Function.Injective ⇑(algebraMap 𝒪 K)\nthis✝² : IsDomain 𝒪\nthis✝¹ : FaithfulSMul 𝒪 K\nthis✝ : IsDomain 𝒪\nx : K\na : 𝒪\nh0 : ¬a = 0... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 171,
"column": 4
} | {
"line": 171,
"column": 48
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝ : CommRing R\nϖ : R\nhϖ : Irreducible ϖ\nhR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x\np : R\nhp : Irreducible p\nhn : Associated (ϖ ^ 0) p\nthis : Irreducible (ϖ ^ 0)\nH : 0 < 1\n⊢ Associated p ϖ",
"usedConstants": [
"not_irreducible_one._simp_2",
"Mu... | simp [not_irreducible_one, pow_zero] at this | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Jacobson.Polynomial | {
"line": 37,
"column": 2
} | {
"line": 37,
"column": 18
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nJ : Ideal R[X]\nj : Ideal R\nhj : j ∈ {J | J.IsMaximal} ∧ map C j = J\nthis : j.IsMaximal\nf : (R ⧸ j)[X]\nhf : f ∈ ⊥.jacobson\nr1 : X ≠ 0\n⊢ f ∈ ⊥",
"usedConstants": [
"Eq.mpr",
"Submodule",
"Semiring.toModule",
"CommSemiring.toSemiring",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 172,
"column": 4
} | {
"line": 172,
"column": 30
} | [
{
"pp": "case inr.inl\nR : Type u_1\ninst✝ : CommRing R\nϖ : R\nhϖ : Irreducible ϖ\nhR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x\np : R\nhp : Irreducible p\nhn : Associated (ϖ ^ 1) p\nthis : Irreducible (ϖ ^ 1)\n⊢ Associated p ϖ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Jacobson.Ring | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 12
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nh : ∀ (P : Ideal R), P.IsPrime → P.jacobson = P\nI : Ideal R\nhI : I.IsRadical\nx : R\nhx : x ∈ I.jacobson\n⊢ ∀ ⦃I_1 : Ideal R⦄, I_1 ∈ {J | I ≤ J ∧ J.IsPrime} → x ∈ I_1",
"usedConstants": [
"CommSemiring.toSemiring",
"Ideal",
"CommRing.toCommSemir... | intro P hP | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.Jacobson.Ring | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 12
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Algebra.IsIntegral R S\ninst✝ : IsJacobsonRing R\n⊢ ∀ (P : Ideal S), P.IsPrime → P.jacobson = P",
"usedConstants": [
"CommSemiring.toSemiring",
"Ideal",
"CommRing.toCommSemiring"
... | intro P hP | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.Jacobson.Ring | {
"line": 140,
"column": 27
} | {
"line": 140,
"column": 38
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Algebra.IsIntegral R S\ninst✝ : IsJacobsonRing R\nP : Ideal S\nhP : P.IsPrime\nhP_top : ¬comap (algebraMap R S) P = ⊤\nthis : Nontrivial (R ⧸ comap (algebraMap R S) P)\nJ : Ideal (R ⧸ comap (algebraMap ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.KrullDimension.Zero | {
"line": 78,
"column": 49
} | {
"line": 78,
"column": 60
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\ntfae_1_to_3 : KrullDimLE 0 R ∧ IsLocalRing R → ∀ (x : R), IsNilpotent x ↔ ¬IsUnit x\nH : ∀ (x : R), IsNilpotent x ↔ ¬IsUnit x\ne : nilradical R = ⊤\nn : ℕ\nhn : 1 ^ n ∈ 0\n⊢ 1 = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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