module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.LinearAlgebra.RootSystem.Defs | {
"line": 377,
"column": 2
} | {
"line": 377,
"column": 72
} | {
"line": 378,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nP : RootPairing ι R M N\ni j k : ι\n⊢ P.pairing j ((P.reflectionPerm i) k) = P.pairing ((P.reflectionPerm i) j) k",
"ppTerm": "?m.4... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nP : RootPairing ι R M N\ni j k : ι\n⊢ (P.toLinearMap (P.root j)) ((P.coreflection i) (P.coroot k)) =\n (P.toLinearMap ((P.reflection i) (P.root ... | simp only [pairing, root', coroot_reflectionPerm, root_reflectionPerm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 529,
"column": 6
} | {
"line": 529,
"column": 17
} | {
"line": 529,
"column": 18
} | [
{
"pp": "case refine_1\nK : 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.... | [
"case refine_1\nK : 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.IsZero\nx : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Reflection | {
"line": 137,
"column": 48
} | {
"line": 137,
"column": 59
} | {
"line": 137,
"column": 60
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx : M\nf : Dual R M\nh : f x = 2\np : Submodule R M\nhx : x ∈ p\nthis : ∀ y ∈ p, (reflection h) y ∈ p\ny : M\nhy : y ∈ p\n⊢ y ∈ Submodule.comap (↑(reflection h)) p",
"ppTerm": "?m.60",
"assigned": true... | [
"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx : M\nf : Dual R M\nh : f x = 2\np : Submodule R M\nhx : x ∈ p\nthis : ∀ y ∈ p, (reflection h) y ∈ p\ny : M\nhy : y ∈ p\n⊢ (reflection h) y ∈ p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.Defs | {
"line": 419,
"column": 46
} | {
"line": 419,
"column": 57
} | {
"line": 419,
"column": 58
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nP : RootPairing ι R M N\ni : ι\ninst✝¹ : NeZero 2\ninst✝ : IsDomain R\nthis : IsReflexive R M\ncontra : i = -i\n⊢ P.root i = -P.root i... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nP : RootPairing ι R M N\ni : ι\ninst✝¹ : NeZero 2\ninst✝ : IsDomain R\nthis : IsReflexive R M\ncontra : i = -i\n⊢ P.root i = -P.root i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Reflection | {
"line": 139,
"column": 2
} | {
"line": 139,
"column": 61
} | {
"line": 139,
"column": 62
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx : M\nf : Dual R M\nh : f x = 2\np : Submodule R M\nhx : x ∈ p\ny : M\nhy : y ∈ p\n⊢ (reflection h) y ∈ p",
"ppTerm": "?m.71",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule... | [
"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx : M\nf : Dual R M\nh : f x = 2\np : Submodule R M\nhx : x ∈ p\ny : M\nhy : y ∈ p\n⊢ f y • x ∈ p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.Defs | {
"line": 460,
"column": 41
} | {
"line": 460,
"column": 52
} | {
"line": 460,
"column": 53
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nP : RootPairing ι R M N\ninst✝¹ : Finite ι\ninst✝ : IsAddTorsionFree N\ni j : ι\nt : R\nh : P.root j = t • P.root i\n⊢ t * P.pairing i... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nP : RootPairing ι R M N\ninst✝¹ : Finite ι\ninst✝ : IsAddTorsionFree N\ni j : ι\nt : R\nh : P.root j = t • P.root i\n⊢ t * P.pairing i j = 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 549,
"column": 43
} | {
"line": 549,
"column": 54
} | {
"line": 549,
"column": 55
} | [
{
"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 β : Weight K (↥H) L\nhyp : coroot α = coro... | [
"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 β : Weight K (↥H) L\nhyp : coroot α = coroot β\nhα : ¬... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Reflection | {
"line": 152,
"column": 4
} | {
"line": 152,
"column": 15
} | {
"line": 152,
"column": 16
} | [
{
"pp": "case refine_1\nR : Type u_1\nM : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nx : M\nf : Dual R M\ninst✝² : IsDomain R\ninst✝¹ : NeZero 2\ninst✝ : IsTorsionFree R M\nh : f x = 2\np : Submodule R M\nhp : Disjoint p (R ∙ x)\ny : M\nhy : y ∈ p\nhx : x ≠ 0\nh' : y - f y • x ... | [
"case refine_1\nR : Type u_1\nM : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nx : M\nf : Dual R M\ninst✝² : IsDomain R\ninst✝¹ : NeZero 2\ninst✝ : IsTorsionFree R M\nh : f x = 2\np : Submodule R M\nhp : Disjoint p (R ∙ x)\ny : M\nhy : y ∈ p\nhx : x ≠ 0\nh' : y - f y • x ∈ p\n⊢ f y •... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.Defs | {
"line": 494,
"column": 2
} | {
"line": 494,
"column": 57
} | {
"line": 494,
"column": 58
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nP : RootPairing ι R M N\ni j k : ι\nhij : P.reflectionPerm i = P.reflectionPerm j\nh : (P.reflection i) (P.root k) = (P.reflection j) (... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nP : RootPairing ι R M N\ni j k : ι\nhij : P.reflectionPerm i = P.reflectionPerm j\nh : (P.reflection i) (P.root k) = (P.reflection j) (P.root k)\n⊢... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.Defs | {
"line": 500,
"column": 2
} | {
"line": 500,
"column": 61
} | {
"line": 500,
"column": 62
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nP : RootPairing ι R M N\ni j k : ι\nhij : P.reflectionPerm i = P.reflectionPerm j\nh : (P.coreflection i) (P.coroot k) = (P.coreflectio... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nP : RootPairing ι R M N\ni j k : ι\nhij : P.reflectionPerm i = P.reflectionPerm j\nh : (P.coreflection i) (P.coroot k) = (P.coreflection j) (P.coro... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Reflection | {
"line": 153,
"column": 29
} | {
"line": 153,
"column": 40
} | {
"line": 153,
"column": 41
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nx : M\nf : Dual R M\ninst✝² : IsDomain R\ninst✝¹ : NeZero 2\ninst✝ : IsTorsionFree R M\nh : f x = 2\np : Submodule R M\nhp : Disjoint p (R ∙ x)\nh' : p ≤ LinearMap.ker f\ny : M\nhy : y ∈ p\n⊢ f y = 0",
"p... | [
"R : Type u_1\nM : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nx : M\nf : Dual R M\ninst✝² : IsDomain R\ninst✝¹ : NeZero 2\ninst✝ : IsTorsionFree R M\nh : f x = 2\np : Submodule R M\nhp : Disjoint p (R ∙ x)\nh' : p ≤ LinearMap.ker f\ny : M\nhy : y ∈ p\n⊢ f y = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 534,
"column": 35
} | {
"line": 553,
"column": 33
} | {
"line": 555,
"column": 0
} | [
{
"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 β : Weight K (↥H) L\n⊢ coroot α = coroot β... | [] | by
refine ⟨fun hyp ↦ ?_, fun h ↦ by rw [h]⟩
if hα : α.IsZero then
have hβ : β.IsZero := by
rw [← coroot_eq_zero_iff] at hα ⊢
rwa [← hyp]
ext
simp [hα.eq, hβ.eq]
else
have hβ : β.IsNonZero := by
contrapose hα
simp only [← coroot_eq_zero_iff] at hα ⊢
rwa [hyp]
have ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 560,
"column": 4
} | {
"line": 560,
"column": 15
} | {
"line": 560,
"column": 16
} | [
{
"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 : L... | [
"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 : L\nheα : e ∈ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Reflection | {
"line": 202,
"column": 6
} | {
"line": 202,
"column": 17
} | {
"line": 202,
"column": 18
} | [
{
"pp": "R : 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\nih :\n ((reflection hf * reflection hg) ^ m) z =\n z +\n (Polynomial.eval t (S R ((↑m - 2) / 2)) *\n... | [
"R : 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\nih :\n ((reflection hf * reflection hg) ^ m) z =\n z +\n (Polynomial.eval t (S R ((↑m - 2) / 2)) *\n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.Defs | {
"line": 576,
"column": 58
} | {
"line": 576,
"column": 84
} | {
"line": 576,
"column": 85
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nP : RootPairing ι R M N\ni j : ι\nh : P.IsOrthogonal i j\nx✝ : M\n⊢ P.pairing i j = 0 ∧ P.pairing j i = 0",
"ppTerm": "?m.88",
... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nP : RootPairing ι R M N\ni j : ι\nh : P.IsOrthogonal i j\nx✝ : M\n⊢ P.pairing i j = 0 ∧ P.pairing j i = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.Defs | {
"line": 625,
"column": 14
} | {
"line": 625,
"column": 29
} | {
"line": 625,
"column": 30
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nP : RootPairing ι R M N\ni j : ι\nh : (P.reflectionPerm i) j = j\n⊢ P.pairing j i • P.root i = 0",
"ppTerm": "?m.50",
"assigned... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nP : RootPairing ι R M N\ni j : ι\nh : (P.reflectionPerm i) j = j\n⊢ P.pairing j i • P.root i = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.Defs | {
"line": 636,
"column": 2
} | {
"line": 636,
"column": 60
} | {
"line": 637,
"column": 4
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\nP : RootPairing ι R M N\ni✝ j✝ : ι\ninst✝² : NeZero 2\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\ni j : ι\nh : P.pairing i j = 0\... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\nP : RootPairing ι R M N\ni✝ j✝ : ι\ninst✝² : NeZero 2\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\ni j : ι\nh : P.pairing i j = 0\n⊢ P.pairing... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.RootSystem.Defs | {
"line": 674,
"column": 31
} | {
"line": 674,
"column": 59
} | {
"line": 676,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ni✝ j✝ : ι\nι₂ : Type u_5\nM₂ : Type u_6\nN₂ : Type u_7\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝... | [] | by simp [coreflection_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 587,
"column": 23
} | {
"line": 587,
"column": 59
} | {
"line": 587,
"column": 60
} | [
{
"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... | [
"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 : L\nht : I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.BaseChange | {
"line": 58,
"column": 34
} | {
"line": 58,
"column": 45
} | {
"line": 58,
"column": 46
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nL : Type u_3\ninst✝³ : Finite m\ninst✝² : Fintype n\ninst✝¹ : DecidableEq m\ninst✝ : Field L\ne : m ≃ n\nK : Subfield L\nA : Matrix m n L\nB : Matrix n m L\nhAB : A * B = 1\nh_mem : ∀ (i : n) (j : m), B i j ∈ K\ni : m\nj : n\n⊢ Bᵀ * Aᵀ = 1",
"ppTerm": "?m.31",
"assig... | [
"m : Type u_1\nn : Type u_2\nL : Type u_3\ninst✝³ : Finite m\ninst✝² : Fintype n\ninst✝¹ : DecidableEq m\ninst✝ : Field L\ne : m ≃ n\nK : Subfield L\nA : Matrix m n L\nB : Matrix n m L\nhAB : A * B = 1\nh_mem : ∀ (i : n) (j : m), B i j ∈ K\ni : m\nj : n\n⊢ Bᵀ * Aᵀ = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 595,
"column": 31
} | {
"line": 595,
"column": 70
} | {
"line": 595,
"column": 71
} | [
{
"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 :... | [
"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 : L\nheα : e ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.PerfectPairing.Restrict | {
"line": 55,
"column": 6
} | {
"line": 55,
"column": 17
} | {
"line": 55,
"column": 18
} | [
{
"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'... | [
"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'\ninst✝ : Mo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.PerfectPairing.Restrict | {
"line": 56,
"column": 42
} | {
"line": 56,
"column": 58
} | {
"line": 56,
"column": 59
} | [
{
"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'... | [
"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'\ninst✝ : Mo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.PerfectPairing.Restrict | {
"line": 57,
"column": 4
} | {
"line": 58,
"column": 11
} | {
"line": 58,
"column": 12
} | [
{
"pp": "case refine_1\nR : 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 : ... | [
"case refine_1\nR : 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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Reflection | {
"line": 212,
"column": 4
} | {
"line": 213,
"column": 52
} | {
"line": 214,
"column": 4
} | [
{
"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... | [
"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)) - Polynom... | 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
} | {
"line": 68,
"column": 18
} | [
{
"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'... | [
"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'\ninst✝ : Mo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 605,
"column": 4
} | {
"line": 605,
"column": 22
} | {
"line": 605,
"column": 23
} | [
{
"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 :... | [
"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 : L\nheα : e ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.PerfectPairing.Restrict | {
"line": 103,
"column": 6
} | {
"line": 103,
"column": 21
} | {
"line": 103,
"column": 22
} | [
{
"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... | [
"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\ninst✝¹⁰ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.PerfectPairing.Restrict | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 13
} | {
"line": 109,
"column": 14
} | [
{
"pp": "R : 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\ninst✝¹⁰ ... | [
"R : 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\ninst✝¹⁰ : Algebra S ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 644,
"column": 67
} | {
"line": 644,
"column": 78
} | {
"line": 644,
"column": 79
} | [
{
"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... | [
"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 : L\nt : Is... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 644,
"column": 64
} | {
"line": 644,
"column": 90
} | {
"line": 644,
"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.LinearAlgebra.PerfectPairing.Restrict | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 13
} | {
"line": 125,
"column": 14
} | [
{
"pp": "R : 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\ninst✝¹⁰ ... | [
"R : 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\ninst✝¹⁰ : Algebra S ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 646,
"column": 17
} | {
"line": 646,
"column": 28
} | {
"line": 646,
"column": 29
} | [
{
"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... | [
"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 : L\nt : Is... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 648,
"column": 67
} | {
"line": 648,
"column": 78
} | {
"line": 648,
"column": 79
} | [
{
"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... | [
"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 : L\nt : Is... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 650,
"column": 17
} | {
"line": 650,
"column": 28
} | {
"line": 650,
"column": 29
} | [
{
"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... | [
"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 : L\nt : Is... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.PerfectPairing.Restrict | {
"line": 199,
"column": 23
} | {
"line": 199,
"column": 38
} | {
"line": 199,
"column": 39
} | [
{
"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... | [
"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\ninst✝ : p.IsP... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.PerfectPairing.Restrict | {
"line": 199,
"column": 59
} | {
"line": 199,
"column": 75
} | {
"line": 199,
"column": 76
} | [
{
"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... | [
"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\ninst✝ : p.IsP... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 716,
"column": 19
} | {
"line": 716,
"column": 30
} | {
"line": 716,
"column": 31
} | [
{
"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' ... | [
"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' e f : L\nht ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Span.TensorProduct | {
"line": 54,
"column": 2
} | {
"line": 58,
"column": 17
} | {
"line": 60,
"column": 0
} | [
{
"pp": "R : 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\n⊢ Surjective ⇑(tensorToSpan A p)",
"ppTerm": "?m.25",
"assigned":... | [] | intro v
obtain ⟨f, hf⟩ := (Finsupp.mem_span_iff_linearCombination _ _ _).mp v.property
use f.sum fun x a ↦ a ⊗ₜ x
rw [map_finsuppSum, Subtype.ext_iff, ← Submodule.subtype_apply, map_finsuppSum]
simpa using! hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Span.TensorProduct | {
"line": 54,
"column": 2
} | {
"line": 58,
"column": 17
} | {
"line": 60,
"column": 0
} | [
{
"pp": "R : 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\n⊢ Surjective ⇑(tensorToSpan A p)",
"ppTerm": "?m.25",
"assigned":... | [] | intro v
obtain ⟨f, hf⟩ := (Finsupp.mem_span_iff_linearCombination _ _ _).mp v.property
use f.sum fun x a ↦ a ⊗ₜ x
rw [map_finsuppSum, Subtype.ext_iff, ← Submodule.subtype_apply, map_finsuppSum]
simpa using! hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Span.TensorProduct | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 61
} | {
"line": 68,
"column": 2
} | [
{
"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... | [
"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 ↑p) →ₗ[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
} | {
"line": 72,
"column": 46
} | [
{
"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",
"ppTerm": "?m.39",
"assigned": fal... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.NumDen | {
"line": 77,
"column": 14
} | {
"line": 77,
"column": 53
} | {
"line": 77,
"column": 54
} | [
{
"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",
"ppTerm": "?m.39",
"assigned": fal... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.NumDen | {
"line": 82,
"column": 14
} | {
"line": 82,
"column": 44
} | {
"line": 82,
"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 : num A y * ↑(den A x) = num A x * ↑(den A y)\n⊢ x = y",
"ppTerm": "?m.49",
"assigned": false,
"usedCons... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 745,
"column": 33
} | {
"line": 745,
"column": 65
} | {
"line": 745,
"column": 66
} | [
{
"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.... | [
"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.root\n⊢ χ.Is... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 227,
"column": 49
} | {
"line": 241,
"column": 25
} | {
"line": 243,
"column": 0
} | [
{
"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)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Nat.cast_mul._simp_1",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"With... | [] | 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": 241,
"column": 2
} | {
"line": 241,
"column": 13
} | {
"line": 241,
"column": 14
} | [
{
"pp": "R : 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 ((↑m - 2) ... | [
"R : 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 z +\n (Polynomial.eval t (S R ((↑m - 2) / 2)) *\n (Polyn... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 296,
"column": 39
} | {
"line": 296,
"column": 50
} | {
"line": 296,
"column": 51
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\n⊢ U R (-1) = 0",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : CommRing R\n⊢ U R (-1) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 305,
"column": 2
} | {
"line": 305,
"column": 68
} | {
"line": 305,
"column": 69
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\n⊢ U R (-2) = -1",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : CommRing R\n⊢ U R (-2) = -1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.RationalRoot | {
"line": 106,
"column": 4
} | {
"line": 106,
"column": 15
} | {
"line": 106,
"column": 16
} | [
{
"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... | [
"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 < p.natDegree"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 320,
"column": 54
} | {
"line": 320,
"column": 75
} | {
"line": 320,
"column": 76
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn : ℤ\n⊢ U R (-n) = -U R (n - 2)",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : CommRing R\nn : ℤ\n⊢ U R (-n) = -U R (n - 2)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 324,
"column": 2
} | {
"line": 324,
"column": 40
} | {
"line": 324,
"column": 41
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn : ℤ\n⊢ U R (-n - 2) = -U R n",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.instNeg",
"Polynomial.Chebyshev.U",
"congrArg",
"CommSemiring.toSemiring",
"AddMonoid.toAddZeroClass",
"... | [
"R : Type u_1\ninst✝ : CommRing R\nn : ℤ\n⊢ U R (-n + -2) = -U R n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.RationalRoot | {
"line": 125,
"column": 37
} | {
"line": 125,
"column": 59
} | {
"line": 125,
"column": 60
} | [
{
"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)",
"ppTer... | [
"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)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.RationalRoot | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 15
} | {
"line": 126,
"column": 16
} | [
{
"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 ... | [
"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 * inv ∣ p.co... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Reflection | {
"line": 273,
"column": 2
} | {
"line": 273,
"column": 13
} | {
"line": 273,
"column": 14
} | [
{
"pp": "R : 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 ((m - 2) /... | [
"R : 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 z +\n (Polynomial.eval t (S R ((m - 2) / 2)) *\n (Polyno... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Quotient.PowTransition | {
"line": 55,
"column": 4
} | {
"line": 55,
"column": 54
} | {
"line": 55,
"column": 55
} | [
{
"pp": "case 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)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.m... | [
"case 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⊢ r ∈ J"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 558,
"column": 2
} | {
"line": 558,
"column": 34
} | {
"line": 558,
"column": 35
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\n⊢ C R 2 = X ^ 2 - 2",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"pow_two",
"Monoid.toMulOneClass",
"congrArg",
"CommSemiring.toSemiring",
"Nat.instAtLeastTwoHAddOfNat",
"HS... | [
"R : Type u_1\ninst✝ : CommRing R\n⊢ C R 2 = X * X - 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Length | {
"line": 81,
"column": 40
} | {
"line": 81,
"column": 51
} | {
"line": 81,
"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 ... | [
"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 {(a, b) | a ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Reflection | {
"line": 367,
"column": 15
} | {
"line": 367,
"column": 38
} | {
"line": 367,
"column": 39
} | [
{
"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... | [
"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\nhg₁ : g x = 2\... | Module.End.mul_eq_comp, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Length | {
"line": 82,
"column": 41
} | {
"line": 82,
"column": 52
} | {
"line": 82,
"column": 53
} | [
{
"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 ... | [
"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 {(a, b) | a ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Length | {
"line": 83,
"column": 4
} | {
"line": 83,
"column": 15
} | {
"line": 83,
"column": 16
} | [
{
"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... | [
"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' : RelSeries {(a,... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Reflection | {
"line": 372,
"column": 4
} | {
"line": 372,
"column": 47
} | {
"line": 372,
"column": 48
} | [
{
"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... | [
"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\nhg₂ : Maps... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Length | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 46
} | {
"line": 154,
"column": 47
} | [
{
"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 < ⊤",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"instTopEN... | [
"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⊢ Module.length R ↥N < ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Length | {
"line": 162,
"column": 2
} | {
"line": 162,
"column": 69
} | {
"line": 162,
"column": 70
} | [
{
"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",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"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⊢ Order.height N < Order.height ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Length | {
"line": 183,
"column": 31
} | {
"line": 183,
"column": 61
} | {
"line": 183,
"column": 62
} | [
{
"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... | [
"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 : Function.Exact... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 663,
"column": 39
} | {
"line": 663,
"column": 50
} | {
"line": 663,
"column": 51
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\n⊢ S R (-1) = 0",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : CommRing R\n⊢ S R (-1) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 672,
"column": 2
} | {
"line": 672,
"column": 68
} | {
"line": 672,
"column": 69
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\n⊢ S R (-2) = -1",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : CommRing R\n⊢ S R (-2) = -1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Length | {
"line": 187,
"column": 14
} | {
"line": 187,
"column": 58
} | {
"line": 187,
"column": 59
} | [
{
"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... | [
"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 : Function.Exact... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 683,
"column": 4
} | {
"line": 685,
"column": 73
} | {
"line": 687,
"column": 0
} | [
{
"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)",
"ppTerm": "?neg_add_one",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.... | [] | 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
} | {
"line": 687,
"column": 0
} | [
{
"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)",
"ppTerm": "?neg_add_one",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.... | [] | 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.Length | {
"line": 188,
"column": 14
} | {
"line": 188,
"column": 60
} | {
"line": 188,
"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... | [
"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 : Function.Exact... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 687,
"column": 54
} | {
"line": 687,
"column": 75
} | {
"line": 687,
"column": 76
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn : ℤ\n⊢ S R (-n) = -S R (n - 2)",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : CommRing R\nn : ℤ\n⊢ S R (-n) = -S R (n - 2)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 691,
"column": 2
} | {
"line": 691,
"column": 40
} | {
"line": 691,
"column": 41
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn : ℤ\n⊢ S R (-n - 2) = -S R n",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.instNeg",
"congrArg",
"CommSemiring.toSemiring",
"AddMonoid.toAddZeroClass",
"sub_eq_add_neg",
"HSub.hSu... | [
"R : Type u_1\ninst✝ : CommRing R\nn : ℤ\n⊢ S R (-n + -2) = -S R n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Reflection | {
"line": 454,
"column": 31
} | {
"line": 454,
"column": 50
} | {
"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
} | {
"line": 456,
"column": 14
} | [
{
"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... | [
"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) (range ⇑r)\ni j ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 30
} | {
"line": 73,
"column": 31
} | [
{
"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 • ⊤]",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
"Eq.mp... | [
"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 ∈ I ^ n • ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 178,
"column": 18
} | {
"line": 178,
"column": 68
} | {
"line": 178,
"column": 69
} | [
{
"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",
"ppTerm": "?m.21",
"assigned": false,
"... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 248,
"column": 2
} | {
"line": 249,
"column": 20
} | {
"line": 250,
"column": 2
} | [
{
"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 • ⊤]",
... | [
"case succ\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 • ⊤]\nn : ℕ\n⊢ f (n + 1) ≡ f 1 [SMOD ⊥ ^ (n + 1) •... | · rw [pow_zero, Ideal.one_eq_top, top_smul]
exact SModEq.top | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 983,
"column": 2
} | {
"line": 984,
"column": 58
} | {
"line": 985,
"column": 2
} | [
{
"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))",
"ppTerm": "?m.122",
"assigned": true,
... | [
"R : Type u_1\ninst✝ : CommRing R\nn : ℤ\nk : ℕ\nx : R\nh :\n eval x ((1 - X ^ 2) * (⇑derivative)^[k + 2] (U R n)) =\n eval x\n ((2 * ↑k + 3) * X * (⇑derivative)^[k + 1] (U R n) - ((↑n + 1) ^ 2 - (↑k + 1) ^ 2) * (⇑derivative)^[k] (U R n))\n⊢ (1 - x ^ 2) * eval x ((⇑derivative)^[k + 2] (U R n)) =\n (2 * ... | 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.Valuation.Integers | {
"line": 228,
"column": 68
} | {
"line": 228,
"column": 79
} | {
"line": 228,
"column": 80
} | [
{
"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",
"ppT... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Integers | {
"line": 238,
"column": 68
} | {
"line": 238,
"column": 79
} | {
"line": 238,
"column": 80
} | [
{
"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",
"ppT... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Integers | {
"line": 256,
"column": 37
} | {
"line": 256,
"column": 48
} | {
"line": 256,
"column": 49
} | [
{
"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... | [
"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 ∈ ⇑v ∘ ⇑(al... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Integers | {
"line": 267,
"column": 4
} | {
"line": 268,
"column": 11
} | {
"line": 268,
"column": 12
} | [
{
"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... | [
"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 1, add_mem'... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Integers | {
"line": 270,
"column": 4
} | {
"line": 271,
"column": 11
} | {
"line": 271,
"column": 12
} | [
{
"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... | [
"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 1, add_mem'... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Integers | {
"line": 311,
"column": 4
} | {
"line": 311,
"column": 34
} | {
"line": 311,
"column": 35
} | [
{
"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",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Valuation.mem_l... | [
"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⊢ v ↑r * v x ≤ γ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Integers | {
"line": 317,
"column": 4
} | {
"line": 317,
"column": 34
} | {
"line": 317,
"column": 35
} | [
{
"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",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Units.val",
... | [
"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⊢ v ↑r * v x < ↑γ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Integers | {
"line": 353,
"column": 2
} | {
"line": 353,
"column": 55
} | {
"line": 353,
"column": 56
} | [
{
"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",
"ppTerm": "?inr",
"assigned": false,
"used... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 538,
"column": 4
} | {
"line": 538,
"column": 31
} | {
"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.Valuation.Integers | {
"line": 370,
"column": 6
} | {
"line": 370,
"column": 17
} | {
"line": 370,
"column": 18
} | [
{
"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",
"ppTerm": "?m.43",
"assigned": true,
"usedConstant... | [
"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⊢ v ↑r * v ↑x ≤ γ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Integers | {
"line": 377,
"column": 4
} | {
"line": 377,
"column": 34
} | {
"line": 377,
"column": 35
} | [
{
"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",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
... | [
"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 * v ↑x < ↑γ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 551,
"column": 2
} | {
"line": 551,
"column": 13
} | {
"line": 551,
"column": 14
} | [
{
"pp": "R : 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) = 0 n",
... | [
"R : 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⊢ ↑f m ∈ I ^ n • ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 608,
"column": 4
} | {
"line": 608,
"column": 33
} | {
"line": 608,
"column": 34
} | [
{
"pp": "case mp\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",
"ppTerm": "?mp",
"assigned": true,
"usedConstants... | [
"case mp\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⊢ x ∈ I ^ n • ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 615,
"column": 4
} | {
"line": 615,
"column": 29
} | {
"line": 615,
"column": 30
} | [
{
"pp": "case mpr\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 • ⊤]",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule... | [
"case mpr\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 ∈ I ^ n • ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.Integers | {
"line": 435,
"column": 35
} | {
"line": 435,
"column": 46
} | {
"line": 435,
"column": 47
} | [
{
"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",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
... | [
"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 / v ↑x ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 639,
"column": 4
} | {
"line": 639,
"column": 45
} | {
"line": 639,
"column": 46
} | [
{
"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\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 = ↑u... | [
"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\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⊢ Submodule.Quotient.mk a = Submod... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 691,
"column": 2
} | {
"line": 691,
"column": 55
} | {
"line": 691,
"column": 56
} | [
{
"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 • ⊤",
"ppTerm": "?refine_2",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"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 • ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 802,
"column": 4
} | {
"line": 802,
"column": 15
} | {
"line": 802,
"column": 16
} | [
{
"pp": "case 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} (... | [
"case 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} (x : M), (Sub... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 171,
"column": 4
} | {
"line": 171,
"column": 48
} | {
"line": 172,
"column": 2
} | [
{
"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",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
... | [] | simp [not_irreducible_one, pow_zero] at this | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 172,
"column": 4
} | {
"line": 172,
"column": 30
} | {
"line": 172,
"column": 31
} | [
{
"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",
"ppTerm": "?inr.inl",
"assigned": false,
"usedConstants": [],
... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Jacobson.Polynomial | {
"line": 37,
"column": 2
} | {
"line": 37,
"column": 18
} | {
"line": 37,
"column": 19
} | [
{
"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 ∈ ⊥",
"ppTerm": "?m.179",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"Semiring... | [
"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 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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