module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.Localization.Defs | {
"line": 320,
"column": 26
} | {
"line": 320,
"column": 37
} | {
"line": 320,
"column": 38
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nh : 0 ∈ M\n⊢ 0 = 1",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"u... | [
"R : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nh : 0 ∈ M\n⊢ 0 = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.Defs | {
"line": 354,
"column": 24
} | {
"line": 354,
"column": 35
} | {
"line": 354,
"column": 36
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nhM : ∀ m ∈ M, IsRegular m\nr : R\nm : ↥M\nthis : ∀ (n : ↥M) (x y : R), ↑n * x = ↑n * y ↔ x = y\nh : ∀ (a a_1 : R), a_1 ∈ M → ∀ (a_2 a_3 : R), a_3 ∈ M → r * (a... | [
"R : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nhM : ∀ m ∈ M, IsRegular m\nr : R\nm : ↥M\nthis : ∀ (n : ↥M) (x y : R), ↑n * x = ↑n * y ↔ x = y\nh : ∀ (a a_1 : R), a_1 ∈ M → ∀ (a_2 a_3 : R), a_3 ∈ M → r * (a_3 * a) = r ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.Basic | {
"line": 309,
"column": 15
} | {
"line": 309,
"column": 34
} | {
"line": 309,
"column": 34
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM N : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\ng : R →+* P\nhg : ∀ (y : ↥M), IsUnit (g ↑y)\nH : M ≤ IsUnit.submonoid R\nx y : R\nhxy : (Algebra.ofId ... | [
"case refine_1\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM N : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\ng : R →+* P\nhg : ∀ (y : ↥M), IsUnit (g ↑y)\nH : M ≤ IsUnit.submonoid R\nx y : R\nhxy : (Algebra.ofId R S) x = (Al... | Units.mul_right_inj | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Localization.FractionRing | {
"line": 410,
"column": 39
} | {
"line": 410,
"column": 91
} | {
"line": 412,
"column": 0
} | [
{
"pp": "A : Type u_4\ninst✝⁴ : CommRing A\nK : Type u_5\ninst✝³ : Field K\nL : Type u_7\ninst✝² : Field L\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\ng : A →+* L\nhg : Injective ⇑g\nx : A\ny : ↥A⁰\n⊢ (lift hg) (mk' K x y) = g x / g ↑y",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [... | [] | by simp only [mk'_eq_div, map_div₀, lift_algebraMap] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Localization.FractionRing | {
"line": 474,
"column": 2
} | {
"line": 474,
"column": 13
} | {
"line": 474,
"column": 14
} | [
{
"pp": "A : Type u_8\nK : Type u_9\ninst✝³ : CommRing A\ninst✝² : CommRing K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nf g : A ≃+* A\nh : (ringEquivOfRingEquivHom A K) f = (ringEquivOfRingEquivHom A K) g\nb : A\n⊢ f b = g b",
"ppTerm": "?m.49",
"assigned": false,
"usedConstants": [],
"... | [
"A : Type u_8\nK : Type u_9\ninst✝³ : CommRing A\ninst✝² : CommRing K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nf g : A ≃+* A\nh : (ringEquivOfRingEquivHom A K) f = (ringEquivOfRingEquivHom A K) g\nb : A\n⊢ f b = g b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.FractionRing | {
"line": 619,
"column": 2
} | {
"line": 619,
"column": 13
} | {
"line": 619,
"column": 14
} | [
{
"pp": "A : Type u_8\nB : Type u_9\ninst✝¹² : CommRing A\ninst✝¹¹ : CommRing B\ninst✝¹⁰ : Algebra A B\nK : Type u_10\nL : Type u_11\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : Algebra A K\ninst✝⁶ : Algebra B L\ninst✝⁵ : IsFractionRing A K\ninst✝⁴ : IsFractionRing B L\ninst✝³ : Algebra A L\ninst✝² : IsScalarT... | [
"A : Type u_8\nB : Type u_9\ninst✝¹² : CommRing A\ninst✝¹¹ : CommRing B\ninst✝¹⁰ : Algebra A B\nK : Type u_10\nL : Type u_11\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : Algebra A K\ninst✝⁶ : Algebra B L\ninst✝⁵ : IsFractionRing A K\ninst✝⁴ : IsFractionRing B L\ninst✝³ : Algebra A L\ninst✝² : IsScalarTower A B L\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.FractionRing | {
"line": 670,
"column": 41
} | {
"line": 670,
"column": 77
} | {
"line": 670,
"column": 78
} | [
{
"pp": "G : Type u_10\nB : Type u_12\nL : Type u_14\ninst✝⁸ : Group G\ninst✝⁷ : CommRing B\ninst✝⁶ : MulSemiringAction G B\ninst✝⁵ : Field L\ninst✝⁴ : Algebra B L\ninst✝³ : IsFractionRing B L\ninst✝² : MulSemiringAction G L\ninst✝¹ : SMulDistribClass G B L\ninst✝ : FaithfulSMul G B\nm₁✝ m₂✝ : G\nh : ∀ (a : L),... | [
"G : Type u_10\nB : Type u_12\nL : Type u_14\ninst✝⁸ : Group G\ninst✝⁷ : CommRing B\ninst✝⁶ : MulSemiringAction G B\ninst✝⁵ : Field L\ninst✝⁴ : Algebra B L\ninst✝³ : IsFractionRing B L\ninst✝² : MulSemiringAction G L\ninst✝¹ : SMulDistribClass G B L\ninst✝ : FaithfulSMul G B\nm₁✝ m₂✝ : G\nh : ∀ (a : L), m₁✝ • a = m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.Basic | {
"line": 442,
"column": 2
} | {
"line": 442,
"column": 13
} | {
"line": 442,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nM : Submonoid R\nm : ℕ\n⊢ mk (↑m) 1 = ↑m",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : CommSemiring R\nM : Submonoid R\nm : ℕ\n⊢ mk (↑m) 1 = ↑m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.Basic | {
"line": 651,
"column": 2
} | {
"line": 651,
"column": 13
} | {
"line": 651,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nM : Submonoid R\nm : ℤ\n⊢ mk (↑m) 1 = ↑m",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : CommRing R\nM : Submonoid R\nm : ℤ\n⊢ mk (↑m) 1 = ↑m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.Basic | {
"line": 657,
"column": 40
} | {
"line": 657,
"column": 78
} | {
"line": 657,
"column": 79
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nM : Submonoid R\nhM : M ≤ nonZeroDivisors R\na c : R\nb d : ↥M\nh : a * ↑d = ↑b * c\n⊢ 1 * (↑d * a) = 1 * (↑b * c)",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
... | [
"R : Type u_1\ninst✝ : CommRing R\nM : Submonoid R\nhM : M ≤ nonZeroDivisors R\na c : R\nb d : ↥M\nh : a * ↑d = ↑b * c\n⊢ ↑d * a = ↑b * c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Adjoin.Basic | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 19
} | {
"line": 69,
"column": 20
} | [
{
"pp": "case a\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ns : Set A\nt : Set B\na : A\nb : B\nha : (a, b).1 ∈ ↑(adjoin R s)\nhb : (a, b).2 ∈ ↑(adjoin R t)\nP : Subalgebra R (A × B) := adjoin R (⇑(LinearMa... | [
"case a\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\ns : Set A\nt : Set B\na : A\nb : B\nha : (a, b).1 ∈ ↑(adjoin R s)\nhb : (a, b).2 ∈ ↑(adjoin R t)\nP : Subalgebra R (A × B) := adjoin R (⇑(LinearMap.inl R A B)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Adjoin.Basic | {
"line": 127,
"column": 2
} | {
"line": 127,
"column": 13
} | {
"line": 127,
"column": 14
} | [
{
"pp": "R : Type uR\nA : Type uA\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\ns t : Set A\n⊢ adjoin R (s ∪ t) = Subalgebra.restrictScalars R (adjoin (↥(adjoin R s)) t)",
"ppTerm": "?m.27",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"R : Type uR\nA : Type uA\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\ns t : Set A\n⊢ adjoin R (s ∪ t) = Subalgebra.restrictScalars R (adjoin (↥(adjoin R s)) t)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Adjoin.Basic | {
"line": 191,
"column": 4
} | {
"line": 192,
"column": 74
} | {
"line": 194,
"column": 0
} | [
{
"pp": "F : Type u_1\nE : Type u_2\nK : Type u_3\ninst✝⁶ : CommSemiring E\ninst✝⁵ : Semiring K\ninst✝⁴ : SMul F E\ninst✝³ : Algebra E K\ninst✝² : CommSemiring F\ninst✝¹ : Algebra F K\ninst✝ : IsScalarTower F E K\nL : Subalgebra F K\nι : Type u_4\nbL : Basis ι F ↥L\n⊢ toSubmodule L = span F (Set.range fun i ↦ ↑... | [] | simpa only [← L.range_val, Submodule.map_span, Submodule.map_top, ← Set.range_comp]
using! congr_arg (Submodule.map (L.val : L →ₗ[F] K)) bL.span_eq.symm | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.RingTheory.Adjoin.Basic | {
"line": 191,
"column": 4
} | {
"line": 192,
"column": 74
} | {
"line": 194,
"column": 0
} | [
{
"pp": "F : Type u_1\nE : Type u_2\nK : Type u_3\ninst✝⁶ : CommSemiring E\ninst✝⁵ : Semiring K\ninst✝⁴ : SMul F E\ninst✝³ : Algebra E K\ninst✝² : CommSemiring F\ninst✝¹ : Algebra F K\ninst✝ : IsScalarTower F E K\nL : Subalgebra F K\nι : Type u_4\nbL : Basis ι F ↥L\n⊢ toSubmodule L = span F (Set.range fun i ↦ ↑... | [] | simpa only [← L.range_val, Submodule.map_span, Submodule.map_top, ← Set.range_comp]
using! congr_arg (Submodule.map (L.val : L →ₗ[F] K)) bL.span_eq.symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Adjoin.Basic | {
"line": 191,
"column": 4
} | {
"line": 192,
"column": 74
} | {
"line": 194,
"column": 0
} | [
{
"pp": "F : Type u_1\nE : Type u_2\nK : Type u_3\ninst✝⁶ : CommSemiring E\ninst✝⁵ : Semiring K\ninst✝⁴ : SMul F E\ninst✝³ : Algebra E K\ninst✝² : CommSemiring F\ninst✝¹ : Algebra F K\ninst✝ : IsScalarTower F E K\nL : Subalgebra F K\nι : Type u_4\nbL : Basis ι F ↥L\n⊢ toSubmodule L = span F (Set.range fun i ↦ ↑... | [] | simpa only [← L.range_val, Submodule.map_span, Submodule.map_top, ← Set.range_comp]
using! congr_arg (Submodule.map (L.val : L →ₗ[F] K)) bL.span_eq.symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.TensorProduct.Finite | {
"line": 113,
"column": 73
} | {
"line": 113,
"column": 84
} | {
"line": 113,
"column": 85
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Module.Finite R M\ninst✝ : Nontrivial M\nn : ℕ\nf : Fin n → M\nhf : Submodule.span R (Set.range f) = ⊤\ns : Set ℕ := {m | Submodule.span R (f '' Fin.val ⁻¹' Set.Iio m) ≠ ⊤}\nx : ℕ\nhx : x ∈ s\ne : n ... | [
"R : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Module.Finite R M\ninst✝ : Nontrivial M\nn : ℕ\nf : Fin n → M\nhf : Submodule.span R (Set.range f) = ⊤\ns : Set ℕ := {m | Submodule.span R (f '' Fin.val ⁻¹' Set.Iio m) ≠ ⊤}\nx : ℕ\nhx : x ∈ s\ne : n ≤ x\ny : Fin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.TensorProduct.Finite | {
"line": 135,
"column": 27
} | {
"line": 135,
"column": 38
} | {
"line": 135,
"column": 39
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Module.Finite R M\ninst✝ : Nontrivial M\nN : Submodule R M\nhN : N ≠ ⊤\nx : M ⧸ N\nhx : ⊤ = R ∙ x\nf : (R ⧸ (LinearMap.toSpanSingleton R (M ⧸ N) x).ker) ≃ₗ[R] M ⧸ N :=\n (LinearMap.toSpanSingleton R... | [
"R : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Module.Finite R M\ninst✝ : Nontrivial M\nN : Submodule R M\nhN : N ≠ ⊤\nx : M ⧸ N\nhx : ⊤ = R ∙ x\nf : (R ⧸ (LinearMap.toSpanSingleton R (M ⧸ N) x).ker) ≃ₗ[R] M ⧸ N :=\n (LinearMap.toSpanSingleton R (M ⧸ N) x).... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.Pointwise | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 13
} | {
"line": 133,
"column": 14
} | [
{
"pp": "S : Type u_1\nα : Type u_2\ninst✝² : InvolutiveStar α\ninst✝¹ : SetLike S α\ninst✝ : StarMemClass S α\ns : S\nx✝ : α\n⊢ x✝ ∈ star ↑s ↔ x✝ ∈ ↑s",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"Set.star",
"congrArg",
... | [
"S : Type u_1\nα : Type u_2\ninst✝² : InvolutiveStar α\ninst✝¹ : SetLike S α\ninst✝ : StarMemClass S α\ns : S\nx✝ : α\n⊢ star x✝ ∈ s ↔ x✝ ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.StarProjection | {
"line": 96,
"column": 4
} | {
"line": 96,
"column": 68
} | {
"line": 96,
"column": 69
} | [
{
"pp": "R : Type u_1\np q : R\ninst✝¹ : NonUnitalNonAssocSemiring R\ninst✝ : StarRing R\nhp : IsStarProjection p\nhq : IsStarProjection q\nhpq : p * q = 0\n⊢ q * p = 0",
"ppTerm": "?m.56",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\np q : R\ninst✝¹ : NonUnitalNonAssocSemiring R\ninst✝ : StarRing R\nhp : IsStarProjection p\nhq : IsStarProjection q\nhpq : p * q = 0\n⊢ q * p = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.StarProjection | {
"line": 112,
"column": 8
} | {
"line": 112,
"column": 72
} | {
"line": 112,
"column": 73
} | [
{
"pp": "R : Type u_1\np q : R\ninst✝¹ : NonUnitalNonAssocRing R\ninst✝ : StarRing R\nhp : IsStarProjection p\nhq : IsStarProjection q\nhpq : p * q = p\n⊢ q * p = p",
"ppTerm": "?m.36",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\np q : R\ninst✝¹ : NonUnitalNonAssocRing R\ninst✝ : StarRing R\nhp : IsStarProjection p\nhq : IsStarProjection q\nhpq : p * q = p\n⊢ q * p = p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.StarProjection | {
"line": 118,
"column": 6
} | {
"line": 118,
"column": 70
} | {
"line": 118,
"column": 71
} | [
{
"pp": "R : Type u_1\np q : R\ninst✝¹ : NonUnitalNonAssocRing R\ninst✝ : StarRing R\nhp : IsStarProjection p\nhq : IsStarProjection q\nhqp : q * p = p\n⊢ p * q = p",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\np q : R\ninst✝¹ : NonUnitalNonAssocRing R\ninst✝ : StarRing R\nhp : IsStarProjection p\nhq : IsStarProjection q\nhqp : q * p = p\n⊢ p * q = p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.Center | {
"line": 22,
"column": 4
} | {
"line": 22,
"column": 42
} | {
"line": 22,
"column": 43
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Mul R\ninst✝ : StarMul R\na : R\nha : a ∈ center R\nb c : R\n⊢ star a * (b * c) = star a * b * c",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝¹ : Mul R\ninst✝ : StarMul R\na : R\nha : a ∈ center R\nb c : R\n⊢ star a * (b * c) = star a * b * c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.Center | {
"line": 24,
"column": 4
} | {
"line": 24,
"column": 42
} | {
"line": 24,
"column": 43
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Mul R\ninst✝ : StarMul R\na : R\nha : a ∈ center R\nb c : R\n⊢ b * c * star a = b * (c * star a)",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝¹ : Mul R\ninst✝ : StarMul R\na : R\nha : a ∈ center R\nb c : R\n⊢ b * c * star a = b * (c * star a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.Center | {
"line": 40,
"column": 49
} | {
"line": 40,
"column": 60
} | {
"line": 40,
"column": 61
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Mul R\ninst✝ : StarMul R\na : R\ns : Set R\nh : ∀ a ∈ s, star a ∈ s\nha : a ∈ s.centralizer\ny : R\nhy : y ∈ s\n⊢ y * star a = star a * y",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝¹ : Mul R\ninst✝ : StarMul R\na : R\ns : Set R\nh : ∀ a ∈ s, star a ∈ s\nha : a ∈ s.centralizer\ny : R\nhy : y ∈ s\n⊢ y * star a = star a * y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Spectrum.Basic | {
"line": 165,
"column": 16
} | {
"line": 165,
"column": 45
} | {
"line": 165,
"column": 46
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝³ : CommSemiring R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : Subsingleton A\na : A\n⊢ (resolventSet R a)ᶜ = ∅",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Compl.compl",
"Set.univ",
"id",
... | [
"R : Type u\nA : Type v\ninst✝³ : CommSemiring R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : Subsingleton A\na : A\n⊢ univᶜ = ∅"
] | resolventSet_of_subsingleton, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Algebra.Spectrum.Basic | {
"line": 177,
"column": 12
} | {
"line": 177,
"column": 44
} | {
"line": 177,
"column": 45
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nr : Rˣ\ns : R\na : A\nh : ¬IsUnit (s • 1 - a)\nhu : IsUnit (r⁻¹ • (s • 1 - a))\n⊢ IsUnit (s • 1 - a)",
"ppTerm": "?m.102",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nr : Rˣ\ns : R\na : A\nh : ¬IsUnit (s • 1 - a)\nhu : IsUnit (r⁻¹ • (s • 1 - a))\n⊢ IsUnit (s • 1 - a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Spectrum.Basic | {
"line": 181,
"column": 6
} | {
"line": 181,
"column": 62
} | {
"line": 181,
"column": 63
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nr : Rˣ\ns : R\na : A\nh : s ∉ σ a\n⊢ IsUnit (r • ↑ₐ (r⁻¹ • s) - a)",
"ppTerm": "?m.144",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Units.... | [
"R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nr : Rˣ\ns : R\na : A\nh : s ∉ σ a\n⊢ IsUnit (s • 1 - a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Spectrum.Basic | {
"line": 188,
"column": 2
} | {
"line": 188,
"column": 63
} | {
"line": 189,
"column": 4
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nr : Rˣ\na : A\n⊢ r • resolvent a ↑r = resolvent (r⁻¹ • a) 1",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Units.val",
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHSMul",... | [
"R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nr : Rˣ\na : A\n⊢ ↑r • resolvent a ↑r = resolvent (↑r⁻¹ • a) 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Spectrum.Basic | {
"line": 210,
"column": 2
} | {
"line": 210,
"column": 72
} | {
"line": 210,
"column": 73
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : Aˣ\n⊢ 0 ∈ resolventSet R ↑a",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"Algebra.algebraMap",
"spectrum",
"AddGroupWithOne.toAddGroup... | [
"R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : Aˣ\n⊢ IsUnit ↑a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Spectrum.Basic | {
"line": 258,
"column": 2
} | {
"line": 258,
"column": 13
} | {
"line": 258,
"column": 14
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na b : A\nr : R\n⊢ r ∈ {r | IsUnit r} ∩ σ (a * b) ↔ r ∈ {r | IsUnit r} ∩ σ (b * a)",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"spectrum",
"cong... | [
"R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na b : A\nr : R\n⊢ IsUnit r → (r ∈ σ (a * b) ↔ r ∈ σ (b * a))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Spectrum.Basic | {
"line": 267,
"column": 4
} | {
"line": 268,
"column": 32
} | {
"line": 268,
"column": 33
} | [
{
"pp": "case refine_1\nR : Type u\nA : Type v\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Ring A\ninst✝³ : Algebra R A\ninst✝² : InvolutiveStar R\ninst✝¹ : StarRing A\ninst✝ : StarModule R A\nr : R\na : A\nh : star r ∈ resolventSet R a\n⊢ r ∈ resolventSet R (star a)",
"ppTerm": "?refine_1",
"assigned": true,
... | [
"case refine_1\nR : Type u\nA : Type v\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Ring A\ninst✝³ : Algebra R A\ninst✝² : InvolutiveStar R\ninst✝¹ : StarRing A\ninst✝ : StarModule R A\nr : R\na : A\nh : star r ∈ resolventSet R a\n⊢ IsUnit (r • 1 - star a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Spectrum.Basic | {
"line": 267,
"column": 4
} | {
"line": 268,
"column": 32
} | {
"line": 268,
"column": 33
} | [
{
"pp": "case refine_2\nR : Type u\nA : Type v\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Ring A\ninst✝³ : Algebra R A\ninst✝² : InvolutiveStar R\ninst✝¹ : StarRing A\ninst✝ : StarModule R A\nr : R\na : A\nh : r ∈ resolventSet R (star a)\n⊢ star r ∈ resolventSet R a",
"ppTerm": "?refine_2",
"assigned": true,
... | [
"case refine_2\nR : Type u\nA : Type v\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Ring A\ninst✝³ : Algebra R A\ninst✝² : InvolutiveStar R\ninst✝¹ : StarRing A\ninst✝ : StarModule R A\nr : R\na : A\nh : r ∈ resolventSet R (star a)\n⊢ IsUnit (star r • 1 - a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Spectrum.Basic | {
"line": 313,
"column": 2
} | {
"line": 313,
"column": 36
} | {
"line": 313,
"column": 37
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : A\nr : R\n⊢ σ a - {r} = σ (a - ↑ₐ r)",
"ppTerm": "?m.29",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\nA : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : A\nr : R\n⊢ σ a - {r} = σ (a - ↑ₐ r)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Spectrum.Basic | {
"line": 372,
"column": 4
} | {
"line": 372,
"column": 35
} | {
"line": 372,
"column": 36
} | [
{
"pp": "case inl\n𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : Nontrivial A\na : A\nha : (σ a).Nonempty\n⊢ σ (0 • a) = 0 • σ a",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"instSMulOfMul",
"spe... | [
"case inl\n𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : Nontrivial A\na : A\nha : (σ a).Nonempty\n⊢ {0} = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Spectrum.Basic | {
"line": 435,
"column": 4
} | {
"line": 435,
"column": 83
} | {
"line": 436,
"column": 6
} | [
{
"pp": "F : Type u_1\nR : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Ring A\ninst✝⁴ : Ring B\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\ninst✝¹ : EquivLike F A B\ninst✝ : AlgEquivClass F R A B\nf : F\na : A\n⊢ spectrum R a ⊆ spectrum R (f a)",
"ppTerm": "?m.37",
"assigned"... | [
"F : Type u_1\nR : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Ring A\ninst✝⁴ : Ring B\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\ninst✝¹ : EquivLike F A B\ninst✝ : AlgEquivClass F R A B\nf : F\na : A\n⊢ spectrum R a ⊆ spectrum R (f a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Spectrum.Basic | {
"line": 453,
"column": 2
} | {
"line": 453,
"column": 52
} | {
"line": 453,
"column": 53
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na✝ : A\nu✝ : Aˣ\na : A\nu : Aˣ\nμ : R\nhμ : IsUnit ((algebraMap R A) μ - a)\n⊢ IsUnit ((algebraMap R A) μ - ↑u * a * ↑u⁻¹)",
"ppTerm": "?m.100",
"assigned": false,
"usedConstants": [],
"usedFVars"... | [
"R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na✝ : A\nu✝ : Aˣ\na : A\nu : Aˣ\nμ : R\nhμ : IsUnit ((algebraMap R A) μ - a)\n⊢ IsUnit ((algebraMap R A) μ - ↑u * a * ↑u⁻¹)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Spectrum.Basic | {
"line": 459,
"column": 2
} | {
"line": 459,
"column": 13
} | {
"line": 459,
"column": 14
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : A\nu : Aˣ\n⊢ spectrum R (↑u⁻¹ * a * ↑u) = spectrum R a",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : A\nu : Aˣ\n⊢ spectrum R (↑u⁻¹ * a * ↑u) = spectrum R a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.TensorProduct.Basic | {
"line": 374,
"column": 16
} | {
"line": 374,
"column": 39
} | {
"line": 374,
"column": 40
} | [
{
"pp": "case tmul\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nC : Type u_3\ninst✝ : Semiring C\nf g : A ⊗[R] B →+* C\nh₁ : f.comp includeLeftRingHom = g.comp includeLeftRingHom\nh₂ : f.comp includeRight.t... | [
"case tmul\nR : Type uR\nA : Type uA\nB : Type uB\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nC : Type u_3\ninst✝ : Semiring C\nf g : A ⊗[R] B →+* C\nh₁ : f.comp includeLeftRingHom = g.comp includeLeftRingHom\nh₂ : f.comp includeRight.toRingHom = g... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.TensorProduct.Basic | {
"line": 513,
"column": 23
} | {
"line": 513,
"column": 34
} | {
"line": 513,
"column": 35
} | [
{
"pp": "R : Type uR\nR' : Type u_1\nS : Type uS\nT : Type u_2\nA : Type uA\nB : Type uB\nC : Type uC\nD : Type uD\nE : Type uE\nF : Type uF\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : CommSemiring B\ninst✝ : Algebra R B\na : A\nb : B\nx x✝¹ x✝ : A ⊗[R] B\nh₁ : a • b • x✝¹ = b ... | [
"R : Type uR\nR' : Type u_1\nS : Type uS\nT : Type u_2\nA : Type uA\nB : Type uB\nC : Type uC\nD : Type uD\nE : Type uE\nF : Type uF\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : CommSemiring B\ninst✝ : Algebra R B\na : A\nb : B\nx x✝¹ x✝ : A ⊗[R] B\nh₁ : a • b • x✝¹ = b • a • x✝¹\nh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Unitization | {
"line": 517,
"column": 6
} | {
"line": 517,
"column": 40
} | {
"line": 518,
"column": 6
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : Semiring R\ninst✝¹ : NonUnitalNonAssocSemiring A\ninst✝ : Module R A\nx✝² x✝¹ x✝ : Unitization R A\n⊢ (x✝² * (x✝¹ + x✝)).toProd.2 = (x✝² * x✝¹ + x✝² * x✝).toProd.2",
"ppTerm": "?m.83",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass"... | [
"R : Type u_1\nA : Type u_2\ninst✝² : Semiring R\ninst✝¹ : NonUnitalNonAssocSemiring A\ninst✝ : Module R A\nx✝² x✝¹ x✝ : Unitization R A\n⊢ x✝².toProd.1 • x✝¹.toProd.2 + x✝².toProd.1 • x✝.toProd.2 +\n (x✝¹.toProd.1 • x✝².toProd.2 + x✝.toProd.1 • x✝².toProd.2) +\n (x✝².toProd.2 * x✝¹.toProd.2 + x✝².toPro... | simp [smul_add, add_smul, mul_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Algebra.Spectrum.Quasispectrum | {
"line": 106,
"column": 2
} | {
"line": 106,
"column": 30
} | {
"line": 106,
"column": 31
} | [
{
"pp": "R : Type u_1\ninst✝ : NonUnitalSemiring R\nu : (PreQuasiregular R)ˣ\n⊢ (↑u⁻¹).val + (↑u).val + (↑u).val * (↑u⁻¹).val = 0",
"ppTerm": "?m.33",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : NonUnitalSemiring R\nu : (PreQuasiregular R)ˣ\n⊢ (↑u⁻¹).val + (↑u).val + (↑u).val * (↑u⁻¹).val = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Spectrum.Quasispectrum | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 30
} | {
"line": 111,
"column": 31
} | [
{
"pp": "R : Type u_1\ninst✝ : NonUnitalSemiring R\nu : (PreQuasiregular R)ˣ\n⊢ (↑u).val + (↑u⁻¹).val + (↑u⁻¹).val * (↑u).val = 0",
"ppTerm": "?m.33",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : NonUnitalSemiring R\nu : (PreQuasiregular R)ˣ\n⊢ (↑u).val + (↑u⁻¹).val + (↑u⁻¹).val * (↑u).val = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Spectrum.Quasispectrum | {
"line": 128,
"column": 4
} | {
"line": 128,
"column": 36
} | {
"line": 128,
"column": 37
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : (Unitization R A)ˣ\nhx : (↑x).toProd.1 = 1\n⊢ x⁻¹ ∈ {x | (↑x).toProd.1 = 1}",
"ppTerm": "?m.52",
"assigned": true,
"usedCons... | [
"R : Type u_1\nA : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : (Unitization R A)ˣ\nhx : (↑x).toProd.1 = 1\n⊢ (↑x⁻¹).toProd.1 = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Spectrum.Quasispectrum | {
"line": 151,
"column": 8
} | {
"line": 151,
"column": 36
} | {
"line": 151,
"column": 37
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : ↥(unitsFstOne R A)\n⊢ PreQuasiregular.equiv.symm (PreQuasiregular.equiv (↑↑x).toProd.2 * PreQuasiregular.equiv (↑↑x⁻¹).toProd.2) =\n ... | [
"R : Type u_1\nA : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : ↥(unitsFstOne R A)\n⊢ (↑(↑x)⁻¹).toProd.2 + (↑↑x).toProd.2 + (↑↑x).toProd.2 * (↑(↑x)⁻¹).toProd.2 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Spectrum.Quasispectrum | {
"line": 153,
"column": 8
} | {
"line": 153,
"column": 36
} | {
"line": 153,
"column": 37
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : ↥(unitsFstOne R A)\n⊢ PreQuasiregular.equiv.symm (PreQuasiregular.equiv (↑↑x⁻¹).toProd.2 * PreQuasiregular.equiv (↑↑x).toProd.2) =\n ... | [
"R : Type u_1\nA : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : ↥(unitsFstOne R A)\n⊢ (↑↑x).toProd.2 + (↑(↑x)⁻¹).toProd.2 + (↑(↑x)⁻¹).toProd.2 * (↑↑x).toProd.2 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Spectrum.Quasispectrum | {
"line": 171,
"column": 47
} | {
"line": 171,
"column": 58
} | {
"line": 171,
"column": 59
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : ↥(unitsFstOne R A)\n⊢ ↑↑((fun x ↦\n ⟨{ val := 1 + ↑(PreQuasiregular.equiv.symm ↑x), inv := 1 + ↑(PreQuasiregular.equiv.symm ↑... | [
"R : Type u_1\nA : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : ↥(unitsFstOne R A)\n⊢ 1 + ↑(↑↑x).toProd.2 = ↑↑x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Spectrum.Quasispectrum | {
"line": 202,
"column": 91
} | {
"line": 204,
"column": 54
} | {
"line": 206,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : NonUnitalSemiring R\nx : R\n⊢ IsQuasiregular x ↔ IsUnit (equiv x)",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Units.val",
"PreQuasiregular.equiv",
"Equiv.apply_symm_apply",
"Equiv.instEquivLike",
"congrArg",
"Function.I... | [] | by
simp only [IsQuasiregular, IsUnit, Equiv.apply_symm_apply,
← PreQuasiregular.equiv (R := R).injective.eq_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Algebra.Spectrum.Quasispectrum | {
"line": 213,
"column": 17
} | {
"line": 213,
"column": 28
} | {
"line": 213,
"column": 29
} | [
{
"pp": "F : Type u_1\nR : Type u_2\nS : Type u_3\ninst✝³ : NonUnitalSemiring R\ninst✝² : NonUnitalSemiring S\ninst✝¹ : FunLike F R S\ninst✝ : NonUnitalRingHomClass F R S\nf : F\nx y : R\nhy₁ : y + x + x * y = 0\nhy₂ : x + y + y * x = 0\n⊢ f y + f x + f x * f y = 0 ∧ f x + f y + f y * f x = 0",
"ppTerm": "?... | [
"F : Type u_1\nR : Type u_2\nS : Type u_3\ninst✝³ : NonUnitalSemiring R\ninst✝² : NonUnitalSemiring S\ninst✝¹ : FunLike F R S\ninst✝ : NonUnitalRingHomClass F R S\nf : F\nx y : R\nhy₁ : y + x + x * y = 0\nhy₂ : x + y + y * x = 0\n⊢ f y + f x + f x * f y = 0 ∧ f x + f y + f y * f x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Spectrum.Quasispectrum | {
"line": 285,
"column": 2
} | {
"line": 285,
"column": 27
} | {
"line": 285,
"column": 28
} | [
{
"pp": "F : Type u_3\nR : Type u_4\nS : Type u_5\nA : Type u_6\nB : Type u_7\ninst✝¹² : CommSemiring R\ninst✝¹¹ : Semiring S\ninst✝¹⁰ : NonUnitalRing A\ninst✝⁹ : NonUnitalRing B\ninst✝⁸ : Module R S\ninst✝⁷ : Module S A\ninst✝⁶ : Module R A\ninst✝⁵ : Module S B\ninst✝⁴ : Module R B\ninst✝³ : IsScalarTower R S ... | [
"F : Type u_3\nR : Type u_4\nS : Type u_5\nA : Type u_6\nB : Type u_7\ninst✝¹² : CommSemiring R\ninst✝¹¹ : Semiring S\ninst✝¹⁰ : NonUnitalRing A\ninst✝⁹ : NonUnitalRing B\ninst✝⁸ : Module R S\ninst✝⁷ : Module S A\ninst✝⁶ : Module R A\ninst✝⁵ : Module S B\ninst✝⁴ : Module R B\ninst✝³ : IsScalarTower R S A\ninst✝² : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Spectrum.Quasispectrum | {
"line": 300,
"column": 13
} | {
"line": 300,
"column": 24
} | {
"line": 300,
"column": 25
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : NonUnitalRing A\ninst✝ : Module R A\na : A\nr : Rˣ\nhr : ¬IsQuasiregular (-(r⁻¹ • a))\nx✝ : IsUnit ↑r\n⊢ ¬IsQuasiregular (-(x✝.unit⁻¹ • a))",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mp... | [
"R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : NonUnitalRing A\ninst✝ : Module R A\na : A\nr : Rˣ\nhr : ¬IsQuasiregular (-(r⁻¹ • a))\nx✝ : IsUnit ↑r\n⊢ ¬IsQuasiregular (-(r⁻¹ • a))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Spectrum.Quasispectrum | {
"line": 333,
"column": 23
} | {
"line": 333,
"column": 34
} | {
"line": 333,
"column": 35
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalRing A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\na : A\ny : Unitization R A\nhy₁ : y + ↑a + ↑a * y = 0\nhy₂ : ↑a + y + y * ↑a = 0\n⊢ y.toProd.1 = 0",
"ppTerm": "?m.57",
"assigned": fals... | [
"R : Type u_1\nA : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalRing A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\na : A\ny : Unitization R A\nhy₁ : y + ↑a + ↑a * y = 0\nhy₂ : ↑a + y + y * ↑a = 0\n⊢ y.toProd.1 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Spectrum.Quasispectrum | {
"line": 342,
"column": 2
} | {
"line": 342,
"column": 34
} | {
"line": 342,
"column": 35
} | [
{
"pp": "R : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommRing S\ninst✝⁷ : Nontrivial S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Module S A\ninst✝³ : IsScalarTower S A A\ninst✝² : SMulCommClass S A A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\na : A\nu : (U... | [
"R : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommRing S\ninst✝⁷ : Nontrivial S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Module S A\ninst✝³ : IsScalarTower S A A\ninst✝² : SMulCommClass S A A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\na : A\nu : (Unitization S... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Spectrum.Quasispectrum | {
"line": 347,
"column": 19
} | {
"line": 347,
"column": 40
} | {
"line": 347,
"column": 41
} | [
{
"pp": "R : Type u_3\nA : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : NonUnitalRing A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\na : A\nr : R\nhr : ¬IsUnit r\nh : r ∈ resolventSet R ↑a\n⊢ IsUnit r",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"us... | [
"R : Type u_3\nA : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : NonUnitalRing A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\na : A\nr : R\nhr : ¬IsUnit r\nh : r ∈ resolventSet R ↑a\n⊢ IsUnit r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Pi.Units | {
"line": 44,
"column": 94
} | {
"line": 45,
"column": 80
} | {
"line": 46,
"column": 0
} | [
{
"pp": "ι : Type u_1\nM : ι → Type u_2\ninst✝ : (i : ι) → Monoid (M i)\nx : (i : ι) → M i\nhx : IsUnit x\ni : ι\n⊢ ↑hx.unit⁻¹ i = ↑⋯.unit⁻¹",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Monoid",
"Units.val",
"Eq.mpr",
"MulEquiv.instEquivLike",
"Inv",
... | [] | by
rw [← Units.inv_eq_val_inv, ← MulEquiv.val_inv_piUnits_apply]; congr; ext; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Algebra.Spectrum.Quasispectrum | {
"line": 373,
"column": 2
} | {
"line": 373,
"column": 13
} | {
"line": 373,
"column": 14
} | [
{
"pp": "R : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝⁸ : Semifield R\ninst✝⁷ : Field S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Module S A\ninst✝³ : IsScalarTower S A A\ninst✝² : SMulCommClass S A A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\na : A\n⊢ spectrum R ↑a ∪ {0} = spectrum R ... | [
"R : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝⁸ : Semifield R\ninst✝⁷ : Field S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Module S A\ninst✝³ : IsScalarTower S A A\ninst✝² : SMulCommClass S A A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\na : A\n⊢ 0 ∈ spectrum R ↑a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Spectrum.Quasispectrum | {
"line": 383,
"column": 4
} | {
"line": 384,
"column": 32
} | {
"line": 384,
"column": 33
} | [
{
"pp": "case e'_3\nR : Type u_3\nA : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : NonUnitalRing A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\na b : A\n⊢ quasispectrum R (a * b) ∩ {r | IsUnit r} = quasispectrum R (b * a) ∩ {r | IsUnit r}",
"ppTerm": "?e'_3",
"assigned": t... | [
"case e'_3\nR : Type u_3\nA : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : NonUnitalRing A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\na b : A\n⊢ {r | IsUnit r} ∩ spectrum R (↑a * ↑b) = {r | IsUnit r} ∩ spectrum R (↑b * ↑a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Spectrum.Quasispectrum | {
"line": 383,
"column": 2
} | {
"line": 384,
"column": 73
} | {
"line": 385,
"column": 2
} | [
{
"pp": "case e'_3\nR : Type u_3\nA : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : NonUnitalRing A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\na b : A\n⊢ quasispectrum R (a * b) ∩ {r | IsUnit r} = quasispectrum R (b * a) ∩ {r | IsUnit r}",
"ppTerm": "?e'_3",
"assigned": t... | [
"case e'_4\nR : Type u_3\nA : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : NonUnitalRing A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\na b : A\n⊢ quasispectrum R (a * b) ∩ {r | IsUnit r}ᶜ = quasispectrum R (b * a) ∩ {r | IsUnit r}ᶜ"
] | · simpa [Set.inter_comm _ {r | IsUnit r}, Unitization.quasispectrum_eq_spectrum_inr,
Unitization.inr_mul] using spectrum.setOf_isUnit_inter_mul_comm _ _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Algebra.Spectrum.Quasispectrum | {
"line": 499,
"column": 4
} | {
"line": 499,
"column": 56
} | {
"line": 500,
"column": 6
} | [
{
"pp": "case refine_1\nR : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝⁸ : Semifield R\ninst✝⁷ : Field S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Module R A\ninst✝⁴ : Module S A\ninst✝³ : Algebra R S\na : A\nf : S → R\ninst✝² : IsScalarTower S A A\ninst✝¹ : SMulCommClass S A A\ninst✝ : IsScalarTower R S A\nh : Quasi... | [
"case refine_1\nR : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝⁸ : Semifield R\ninst✝⁷ : Field S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Module R A\ninst✝⁴ : Module S A\ninst✝³ : Algebra R S\na : A\nf : S → R\ninst✝² : IsScalarTower S A A\ninst✝¹ : SMulCommClass S A A\ninst✝ : IsScalarTower R S A\nh : QuasispectrumRest... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Spectrum.Quasispectrum | {
"line": 554,
"column": 6
} | {
"line": 554,
"column": 17
} | {
"line": 554,
"column": 18
} | [
{
"pp": "case inl\nR : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nh₁ : Function.LeftInverse f ⇑(algebraMap R S)\nh₂ : Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S a)\nhx... | [
"case inl\nR : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nh₁ : Function.LeftInverse f ⇑(algebraMap R S)\nh₂ : Set.RightInvOn f (⇑(algebraMap R S)) (spectrum S a)\nhx : 0 ∈ quasi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Spectrum.Quasispectrum | {
"line": 568,
"column": 6
} | {
"line": 568,
"column": 17
} | {
"line": 568,
"column": 18
} | [
{
"pp": "case inl\nR : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : spectrum S a ⊆ Set.range ⇑(algebraMap R S)\n⊢ (algebraMa... | [
"case inl\nR : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝⁵ : Semifield R\ninst✝⁴ : Semifield S\ninst✝³ : Ring A\ninst✝² : Algebra R S\ninst✝¹ : Algebra R A\ninst✝ : Algebra S A\na : A\nf : S → R\nhf : Function.LeftInverse f ⇑(algebraMap R S)\nh : spectrum S a ⊆ Set.range ⇑(algebraMap R S)\n⊢ f 0 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Spectrum.Quasispectrum | {
"line": 592,
"column": 4
} | {
"line": 592,
"column": 51
} | {
"line": 593,
"column": 6
} | [
{
"pp": "case refine_1\nR : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝⁶ : Semifield R\ninst✝⁵ : Semifield S\ninst✝⁴ : Ring A\ninst✝³ : Algebra R S\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\na : A\nf : S → R\ninst✝ : IsScalarTower R S A\nh : SpectrumRestricts a f\n⊢ ⇑(algebraMap R S) '' spectrum R a ⊆ spectru... | [
"case refine_1\nR : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝⁶ : Semifield R\ninst✝⁵ : Semifield S\ninst✝⁴ : Ring A\ninst✝³ : Algebra R S\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\na : A\nf : S → R\ninst✝ : IsScalarTower R S A\nh : SpectrumRestricts a f\n⊢ ⇑(algebraMap R S) '' spectrum R a ⊆ spectrum S a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.StrictPositivity | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 13
} | {
"line": 97,
"column": 14
} | [
{
"pp": "A : Type u_1\ninst✝³ : Semiring A\ninst✝² : StarRing A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nu a : A\nhu : IsUnit u\n⊢ IsStrictlyPositive (star u * a * u) ↔ IsStrictlyPositive a",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals":... | [
"A : Type u_1\ninst✝³ : Semiring A\ninst✝² : StarRing A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nu a : A\nhu : IsUnit u\n⊢ IsStrictlyPositive (star u * a * u) ↔ IsStrictlyPositive a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.TensorProduct.Maps | {
"line": 140,
"column": 4
} | {
"line": 140,
"column": 15
} | {
"line": 140,
"column": 16
} | [
{
"pp": "R : Type uR\nR' : Type u_1\nS : Type uS\nT : Type u_2\nA : Type uA\nB : Type uB\nC : Type uC\nD : Type uD\nE : Type uE\nF : Type uF\ninst✝¹³ : CommSemiring R\ninst✝¹² : CommSemiring S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Semiring A\ninst✝⁹ : Algebra R A\ninst✝⁸ : Algebra S A\ninst✝⁷ : IsScalarTower R S A\... | [
"R : Type uR\nR' : Type u_1\nS : Type uS\nT : Type u_2\nA : Type uA\nB : Type uB\nC : Type uC\nD : Type uD\nE : Type uE\nF : Type uF\ninst✝¹³ : CommSemiring R\ninst✝¹² : CommSemiring S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Semiring A\ninst✝⁹ : Algebra R A\ninst✝⁸ : Algebra S A\ninst✝⁷ : IsScalarTower R S A\ninst✝⁶ : Se... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Subalgebra.Centralizer | {
"line": 90,
"column": 4
} | {
"line": 91,
"column": 59
} | {
"line": 91,
"column": 60
} | [
{
"pp": "case mp\nR : Type u_1\ninst✝⁵ : CommSemiring R\nA : Type u_2\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_3\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nℬ : Module.Basis (Module.Free.ChooseBasisIndex R B) R B := Module.Free.chooseBasis R B\nb : Module.Free.C... | [
"case mp\nR : Type u_1\ninst✝⁵ : CommSemiring R\nA : Type u_2\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_3\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nℬ : Module.Basis (Module.Free.ChooseBasisIndex R B) R B := Module.Free.chooseBasis R B\nb : Module.Free.ChooseBasisIn... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Star.Basic | {
"line": 62,
"column": 29
} | {
"line": 62,
"column": 51
} | {
"line": 62,
"column": 52
} | [
{
"pp": "case add\nR : Type u_1\nA : Type u_2\ninst✝⁷ : Semiring R\ninst✝⁶ : StarRing R\ninst✝⁵ : NonUnitalSemiring A\ninst✝⁴ : StarRing A\ninst✝³ : Module R A\ninst✝² : StarModule R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nr : R\na : A\nha : a ∈ AddSubmonoid.closure (range fun s ↦ star s *... | [
"case add\nR : Type u_1\nA : Type u_2\ninst✝⁷ : Semiring R\ninst✝⁶ : StarRing R\ninst✝⁵ : NonUnitalSemiring A\ninst✝⁴ : StarRing A\ninst✝³ : Module R A\ninst✝² : StarModule R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nr : R\na : A\nha : a ∈ AddSubmonoid.closure (range fun s ↦ star s * s)\nr₁ r₂ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Star.Basic | {
"line": 66,
"column": 29
} | {
"line": 66,
"column": 51
} | {
"line": 66,
"column": 52
} | [
{
"pp": "case mem.add\nR : Type u_1\nA : Type u_2\ninst✝⁷ : Semiring R\ninst✝⁶ : StarRing R\ninst✝⁵ : NonUnitalSemiring A\ninst✝⁴ : StarRing A\ninst✝³ : Module R A\ninst✝² : StarModule R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nr✝ : R\na : A\nr : R\nhr : r ∈ range fun s ↦ star s * s\na₁ a₂ ... | [
"case mem.add\nR : Type u_1\nA : Type u_2\ninst✝⁷ : Semiring R\ninst✝⁶ : StarRing R\ninst✝⁵ : NonUnitalSemiring A\ninst✝⁴ : StarRing A\ninst✝³ : Module R A\ninst✝² : StarModule R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nr✝ : R\na : A\nr : R\nhr : r ∈ range fun s ↦ star s * s\na₁ a₂ : A\nhx✝ : a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Subalgebra.Centralizer | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 37
} | {
"line": 117,
"column": 6
} | [
{
"pp": "case e'_2\nR : Type u_1\ninst✝⁵ : CommSemiring R\nA : Type u_2\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_3\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set B\ninst✝ : Module.Free R A\neq1 :\n comap (↑(Algebra.TensorProduct.comm R A B)) (centralizer R (⇑includeLeft '' S)) =\n comap (... | [
"case e'_2\nR : Type u_1\ninst✝⁵ : CommSemiring R\nA : Type u_2\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_3\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set B\ninst✝ : Module.Free R A\neq1 :\n comap (↑(Algebra.TensorProduct.comm R A B)) (centralizer R (⇑includeLeft '' S)) =\n comap (↑(Algebra.Te... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Subalgebra.Centralizer | {
"line": 116,
"column": 4
} | {
"line": 118,
"column": 65
} | {
"line": 119,
"column": 2
} | [
{
"pp": "case e'_2\nR : Type u_1\ninst✝⁵ : CommSemiring R\nA : Type u_2\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_3\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set B\ninst✝ : Module.Free R A\neq1 :\n comap (↑(Algebra.TensorProduct.comm R A B)) (centralizer R (⇑includeLeft '' S)) =\n comap (... | [] | simpa [mem_centralizer_iff] using
⟨fun h b hb ↦ (Algebra.TensorProduct.comm R A B).symm.injective <| by aesop, fun h b hb ↦
(Algebra.TensorProduct.comm R A B).injective <| by aesop⟩ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Order.Star.Basic | {
"line": 155,
"column": 2
} | {
"line": 157,
"column": 32
} | {
"line": 159,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : NonUnitalSemiring R\ninst✝³ : PartialOrder R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\nx y : R\ninst✝ : IsCancelAdd R\n⊢ x < y ↔ ∃ p, p ≠ 0 ∧ p ∈ AddSubmonoid.closure (range fun s ↦ star s * s) ∧ y = x + p",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants"... | [] | rw [lt_iff_le_and_ne, and_comm, StarOrderedRing.le_iff, ← exists_and_left]
congr! 2 with p
simp +contextual [← and_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Star.Basic | {
"line": 155,
"column": 2
} | {
"line": 157,
"column": 32
} | {
"line": 159,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : NonUnitalSemiring R\ninst✝³ : PartialOrder R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\nx y : R\ninst✝ : IsCancelAdd R\n⊢ x < y ↔ ∃ p, p ≠ 0 ∧ p ∈ AddSubmonoid.closure (range fun s ↦ star s * s) ∧ y = x + p",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants"... | [] | rw [lt_iff_le_and_ne, and_comm, StarOrderedRing.le_iff, ← exists_and_left]
congr! 2 with p
simp +contextual [← and_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Star.Basic | {
"line": 170,
"column": 4
} | {
"line": 170,
"column": 73
} | {
"line": 170,
"column": 74
} | [
{
"pp": "R : Type u_1\ninst✝² : NonUnitalRing R\ninst✝¹ : PartialOrder R\ninst✝ : StarRing R\nh_add : ∀ {x y : R}, x ≤ y → ∀ (z : R), z + x ≤ z + y\nh_nonneg_iff : ∀ (x : R), 0 ≤ x ↔ x ∈ AddSubmonoid.closure (range fun s ↦ star s * s)\nx y : R\nthis : AddLeftMono R\n⊢ x ≤ y ↔ ∃ p ∈ AddSubmonoid.closure (range f... | [
"R : Type u_1\ninst✝² : NonUnitalRing R\ninst✝¹ : PartialOrder R\ninst✝ : StarRing R\nh_add : ∀ {x y : R}, x ≤ y → ∀ (z : R), z + x ≤ z + y\nh_nonneg_iff : ∀ (x : R), 0 ≤ x ↔ x ∈ AddSubmonoid.closure (range fun s ↦ star s * s)\nx y : R\nthis : AddLeftMono R\n⊢ x ≤ y ↔ y - x ∈ AddSubmonoid.closure (range fun s ↦ sta... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Star.Basic | {
"line": 184,
"column": 4
} | {
"line": 184,
"column": 48
} | {
"line": 184,
"column": 49
} | [
{
"pp": "R : Type u_1\ninst✝² : NonUnitalRing R\ninst✝¹ : PartialOrder R\ninst✝ : StarRing R\nh_add : ∀ {x y : R}, x ≤ y → ∀ (z : R), z + x ≤ z + y\nh_nonneg_iff : ∀ (x : R), 0 ≤ x ↔ ∃ s, x = star s * s\nthis : AddLeftMono R\n⊢ ∀ (x y : R), x ≤ y ↔ ∃ s, y = x + star s * s",
"ppTerm": "?m.49",
"assigned"... | [
"R : Type u_1\ninst✝² : NonUnitalRing R\ninst✝¹ : PartialOrder R\ninst✝ : StarRing R\nh_add : ∀ {x y : R}, x ≤ y → ∀ (z : R), z + x ≤ z + y\nh_nonneg_iff : ∀ (x : R), 0 ≤ x ↔ ∃ s, x = star s * s\nthis : AddLeftMono R\n⊢ ∀ (x y : R), x ≤ y ↔ ∃ s, y = x + star s * s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Star.Basic | {
"line": 220,
"column": 2
} | {
"line": 220,
"column": 30
} | {
"line": 220,
"column": 31
} | [
{
"pp": "R : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nr : R\n⊢ 0 ≤ r * star r",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nr : R\n⊢ 0 ≤ r * star r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Star.Basic | {
"line": 224,
"column": 2
} | {
"line": 224,
"column": 26
} | {
"line": 224,
"column": 27
} | [
{
"pp": "R : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\na : R\nha : IsSelfAdjoint a\n⊢ 0 ≤ a * a",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\na : R\nha : IsSelfAdjoint a\n⊢ 0 ≤ a * a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Subalgebra.Directed | {
"line": 44,
"column": 2
} | {
"line": 45,
"column": 47
} | {
"line": 45,
"column": 48
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\nι : Type u_4\ninst✝ : Nonempty ι\nS : ι → Subalgebra R A\nhS : ∀ (i : ι), IsMulCommutative ↥(S i)\ndir : Directed (fun x1 x2 ↦ x1 ≤ x2) S\n⊢ IsMulCommutative ↥(⨆ i, S i)",
"ppTerm": "?m.32",
"assigne... | [
"R : Type u_1\nA : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\nι : Type u_4\ninst✝ : Nonempty ι\nS : ι → Subalgebra R A\nhS : ∀ (i : ι), IsMulCommutative ↥(S i)\ndir : Directed (fun x1 x2 ↦ x1 ≤ x2) S\n⊢ ∀ (a : A) (x : ι), a ∈ ↑(S x) → ∀ (a_1 : A) (x : ι), a_1 ∈ ↑(S x) → a * a_1 = ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Star.Basic | {
"line": 246,
"column": 2
} | {
"line": 246,
"column": 30
} | {
"line": 246,
"column": 31
} | [
{
"pp": "R : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\na : R\nha : 0 ≤ a\nc : R\n⊢ 0 ≤ c * a * star c",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\na : R\nha : 0 ≤ a\nc : R\n⊢ 0 ≤ c * a * star c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Star.Basic | {
"line": 266,
"column": 2
} | {
"line": 266,
"column": 30
} | {
"line": 266,
"column": 31
} | [
{
"pp": "R : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\na b : R\nhab : a ≤ b\nc : R\n⊢ c * a * star c ≤ c * b * star c",
"ppTerm": "?m.29",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\na b : R\nhab : a ≤ b\nc : R\n⊢ c * a * star c ≤ c * b * star c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Star.Basic | {
"line": 270,
"column": 2
} | {
"line": 270,
"column": 31
} | {
"line": 270,
"column": 32
} | [
{
"pp": "R : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\na b : R\nhab : a ≤ b\nc : R\nhc : IsSelfAdjoint c\n⊢ c * a * c ≤ c * b * c",
"ppTerm": "?m.27",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": ... | [
"R : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\na b : R\nhab : a ≤ b\nc : R\nhc : IsSelfAdjoint c\n⊢ c * a * c ≤ c * b * c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Star.Basic | {
"line": 279,
"column": 8
} | {
"line": 279,
"column": 36
} | {
"line": 279,
"column": 37
} | [
{
"pp": "R : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nx y : R\nthis : ∀ (x y : R), x ≤ y → star x ≤ star y\n⊢ star x ≤ star y → x ≤ y",
"ppTerm": "?m.33",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoa... | [
"R : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nx y : R\nthis : ∀ (x y : R), x ≤ y → star x ≤ star y\n⊢ star x ≤ star y → x ≤ y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Star.Basic | {
"line": 292,
"column": 4
} | {
"line": 292,
"column": 36
} | {
"line": 292,
"column": 37
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nx y : R\nh : ¬x = y\n⊢ star x < star y ↔ x < y",
"ppTerm": "?neg✝",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case neg\nR : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nx y : R\nh : ¬x = y\n⊢ star x < star y ↔ x < y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Star.Basic | {
"line": 300,
"column": 2
} | {
"line": 300,
"column": 13
} | {
"line": 300,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nx : R\n⊢ 0 ≤ star x ↔ 0 ≤ x",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nx : R\n⊢ 0 ≤ star x ↔ 0 ≤ x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Star.Basic | {
"line": 304,
"column": 2
} | {
"line": 304,
"column": 13
} | {
"line": 304,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nx : R\n⊢ star x ≤ 0 ↔ x ≤ 0",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nx : R\n⊢ star x ≤ 0 ↔ x ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Star.Basic | {
"line": 308,
"column": 2
} | {
"line": 308,
"column": 13
} | {
"line": 308,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nx : R\n⊢ 0 < star x ↔ 0 < x",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nx : R\n⊢ 0 < star x ↔ 0 < x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Star.Basic | {
"line": 312,
"column": 2
} | {
"line": 312,
"column": 13
} | {
"line": 312,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nx : R\n⊢ star x < 0 ↔ x < 0",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nx : R\n⊢ star x < 0 ↔ x < 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Star.Basic | {
"line": 321,
"column": 2
} | {
"line": 321,
"column": 30
} | {
"line": 321,
"column": 31
} | [
{
"pp": "R : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\na b : R\nhab : a < b\nc : R\nhc : IsRegular c\n⊢ c * a * star c < c * b * star c",
"ppTerm": "?m.31",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGo... | [
"R : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\na b : R\nhab : a < b\nc : R\nhc : IsRegular c\n⊢ c * a * star c < c * b * star c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Star.Basic | {
"line": 325,
"column": 2
} | {
"line": 325,
"column": 39
} | {
"line": 325,
"column": 40
} | [
{
"pp": "R : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\na : R\nha : 0 < a\nc : R\nhc : IsRegular c\n⊢ 0 < star c * a * c",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\na : R\nha : 0 < a\nc : R\nhc : IsRegular c\n⊢ 0 < star c * a * c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Star.Basic | {
"line": 329,
"column": 2
} | {
"line": 329,
"column": 30
} | {
"line": 329,
"column": 31
} | [
{
"pp": "R : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\na : R\nha : 0 < a\nc : R\nhc : IsRegular c\n⊢ 0 < c * a * star c",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\na : R\nha : 0 < a\nc : R\nhc : IsRegular c\n⊢ 0 < c * a * star c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Star.Basic | {
"line": 336,
"column": 2
} | {
"line": 336,
"column": 13
} | {
"line": 336,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : NonUnitalSemiring R\ninst✝³ : PartialOrder R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\ninst✝ : Nontrivial R\nx : R\nhx : IsRegular x\n⊢ 0 < x * star x",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝⁴ : NonUnitalSemiring R\ninst✝³ : PartialOrder R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\ninst✝ : Nontrivial R\nx : R\nhx : IsRegular x\n⊢ 0 < x * star x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Star.Basic | {
"line": 357,
"column": 20
} | {
"line": 357,
"column": 31
} | {
"line": 357,
"column": 32
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝³ : Semiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\n⊢ 0 ≤ 1",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\nA : Type u_2\ninst✝³ : Semiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\n⊢ 0 ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Star.Basic | {
"line": 361,
"column": 2
} | {
"line": 361,
"column": 13
} | {
"line": 361,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝³ : Semiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nx : R\n⊢ 1 ≤ star x ↔ 1 ≤ x",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝³ : Semiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nx : R\n⊢ 1 ≤ star x ↔ 1 ≤ x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Star.Basic | {
"line": 365,
"column": 2
} | {
"line": 365,
"column": 13
} | {
"line": 365,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝³ : Semiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nx : R\n⊢ star x ≤ 1 ↔ x ≤ 1",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝³ : Semiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nx : R\n⊢ star x ≤ 1 ↔ x ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Star.Basic | {
"line": 369,
"column": 2
} | {
"line": 369,
"column": 13
} | {
"line": 369,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝³ : Semiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nx : R\n⊢ 1 < star x ↔ 1 < x",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝³ : Semiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nx : R\n⊢ 1 < star x ↔ 1 < x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Star.Basic | {
"line": 373,
"column": 2
} | {
"line": 373,
"column": 13
} | {
"line": 373,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝³ : Semiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nx : R\n⊢ star x < 1 ↔ x < 1",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝³ : Semiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nx : R\n⊢ star x < 1 ↔ x < 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Star.Basic | {
"line": 383,
"column": 13
} | {
"line": 383,
"column": 38
} | {
"line": 383,
"column": 39
} | [
{
"pp": "R : Type u_1\ninst✝³ : Semiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nu x : R\nhu : IsUnit u\nh : 0 ≤ u * x * star u\nv : R\nhv : v * u = 1\n⊢ ?m.52",
"ppTerm": "?m.53",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"R : Type u_1\ninst✝³ : Semiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nu x : R\nhu : IsUnit u\nh : 0 ≤ u * x * star u\nv : R\nhv : v * u = 1\n⊢ ?m.52"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Star.Basic | {
"line": 388,
"column": 2
} | {
"line": 388,
"column": 13
} | {
"line": 388,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝³ : Semiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nu x : R\nhu : IsUnit u\n⊢ 0 ≤ star u * x * u ↔ 0 ≤ x",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝³ : Semiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nu x : R\nhu : IsUnit u\n⊢ 0 ≤ star u * x * u ↔ 0 ≤ x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Star.Basic | {
"line": 402,
"column": 4
} | {
"line": 402,
"column": 30
} | {
"line": 404,
"column": 0
} | [
{
"pp": "case e'_2\nR : Type u_1\nA : Type u_2\ninst✝³ : NonUnitalSemiring R\ninst✝² : StarRing R\ninst✝¹ : PartialOrder R\ninst✝ : StarOrderedRing R\nx y : Rᵐᵒᵖ\np : R\n⊢ unop y = unop x + p ↔ y = x + op p",
"ppTerm": "?e'_2",
"assigned": true,
"usedConstants": [
"congrArg",
"MulOpposit... | [] | · simp [← op_inj (α := R)] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Order.Star.Basic | {
"line": 458,
"column": 2
} | {
"line": 458,
"column": 17
} | {
"line": 460,
"column": 0
} | [
{
"pp": "case mem\nR : Type u_4\nS : Type u_5\ninst✝⁷ : NonUnitalSemiring R\ninst✝⁶ : PartialOrder R\ninst✝⁵ : StarRing R\ninst✝⁴ : StarOrderedRing R\ninst✝³ : NonUnitalSemiring S\ninst✝² : PartialOrder S\ninst✝¹ : StarRing S\ninst✝ : StarOrderedRing S\nf : R →⋆ₙ+* S\nx p : R\nhf : ∀ (r : R), f (star r) = star ... | [] | all_goals aesop | Lean.Elab.Tactic.evalAllGoals | Lean.Parser.Tactic.allGoals |
Mathlib.LinearAlgebra.Dimension.Constructions | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 32
} | {
"line": 82,
"column": 33
} | [
{
"pp": "case refine_2\nR : Type u\nM : Type v\nι : Type w\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ns t : Set ι\nf : ι → M\nhst : Disjoint s t\nh : LinearIndepOn R f (s ∪ t)\n⊢ Disjoint (span R (range fun x ↦ f ↑x)) (span R (f '' s)).mkQ.ker",
"ppTerm": "?refine_2",
"assigned": tru... | [
"case refine_2\nR : Type u\nM : Type v\nι : Type w\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ns t : Set ι\nf : ι → M\nhst : Disjoint s t\nh : LinearIndepOn R f (s ∪ t)\n⊢ Disjoint (span R (f '' t)) (span R (f '' s))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Star.Basic | {
"line": 532,
"column": 6
} | {
"line": 532,
"column": 17
} | {
"line": 532,
"column": 18
} | [
{
"pp": "case inr\nR✝ : Type u_1\nA : Type u_2\ninst✝⁷ : NonUnitalRing R✝\ninst✝⁶ : PartialOrder R✝\ninst✝⁵ : StarRing R✝\ninst✝⁴ : StarOrderedRing R✝\np q : R✝\nR : Type u_3\ninst✝³ : NonUnitalRing R\ninst✝² : LinearOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nr : R\nhr : 0 ≤ -r\n⊢ star r = r",
... | [
"case inr\nR✝ : Type u_1\nA : Type u_2\ninst✝⁷ : NonUnitalRing R✝\ninst✝⁶ : PartialOrder R✝\ninst✝⁵ : StarRing R✝\ninst✝⁴ : StarOrderedRing R✝\np q : R✝\nR : Type u_3\ninst✝³ : NonUnitalRing R\ninst✝² : LinearOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nr : R\nhr : 0 ≤ -r\n⊢ star r = r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.Finite | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 15
} | {
"line": 69,
"column": 16
} | [
{
"pp": "case mp\nR : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\na✝ : Nontrivial R\nx : M\nhx : ∀ (a : R), a ≠ 0 → a • x ≠ 0\nthis : LinearIndependent R fun x_1 ↦ x\n⊢ 1 ≤ Module.rank R M",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [
"case mp\nR : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\na✝ : Nontrivial R\nx : M\nhx : ∀ (a : R), a ≠ 0 → a • x ≠ 0\nthis : LinearIndependent R fun x_1 ↦ x\n⊢ 1 ≤ Module.rank R M"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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