module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.LinearAlgebra.RootSystem.IsValuedIn | {
"line": 130,
"column": 4
} | {
"line": 130,
"column": 42
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_4\nN : Type u_5\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : Module R M\ninst✝⁸ : AddCommGroup N\ninst✝⁷ : Module R N\nP : RootPairing ι R M N\nS : Type u_6\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra S R\nT : Type u_7\ninst✝⁴ : CommRing T\ninst✝³ : Algebr... | use algebraMap T S (P.pairingIn T i j) | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.LinearAlgebra.RootSystem.Finite.CanonicalBilinear | {
"line": 274,
"column": 2
} | {
"line": 274,
"column": 58
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹³ : CommRing R\ninst✝¹² : AddCommGroup M\ninst✝¹¹ : Module R M\ninst✝¹⁰ : AddCommGroup N\ninst✝⁹ : Module R N\nP : RootPairing ι R M N\nS : Type u_5\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra S R\ninst✝⁶ : FaithfulSMul S R\ninst✝⁵ : Module S M\n... | rw [hc, mul_comm, mul_smul, rootFormIn_self_smul_coroot] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.RootSystem.Finite.CanonicalBilinear | {
"line": 401,
"column": 4
} | {
"line": 402,
"column": 35
} | [
{
"pp": "case inr.inr\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : AddCommGroup M\ninst✝¹⁴ : Module R M\ninst✝¹³ : AddCommGroup N\ninst✝¹² : Module R N\nP : RootPairing ι R M N\nS : Type u_5\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : LinearOrder S\ninst✝⁹ : IsStrictOrderedRing ... | rw [le_iff_eq_or_lt, le_iff_eq_or_lt, or_iff_right hij, or_iff_right hji]
exact P.pairingIn_lt_zero_iff S | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.Finite.CanonicalBilinear | {
"line": 401,
"column": 4
} | {
"line": 402,
"column": 35
} | [
{
"pp": "case inr.inr\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : AddCommGroup M\ninst✝¹⁴ : Module R M\ninst✝¹³ : AddCommGroup N\ninst✝¹² : Module R N\nP : RootPairing ι R M N\nS : Type u_5\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : LinearOrder S\ninst✝⁹ : IsStrictOrderedRing ... | rw [le_iff_eq_or_lt, le_iff_eq_or_lt, or_iff_right hij, or_iff_right hji]
exact P.pairingIn_lt_zero_iff S | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.QuadraticForm.Prod | {
"line": 329,
"column": 8
} | {
"line": 329,
"column": 76
} | [
{
"pp": "case h.e'_4\nι : Type u_1\nR : Type u_2\nMᵢ : ι → Type u_8\ninst✝⁷ : CommSemiring R\ninst✝⁶ : (i : ι) → AddCommMonoid (Mᵢ i)\ninst✝⁵ : (i : ι) → Module R (Mᵢ i)\nP : Type u_10\ninst✝⁴ : Fintype ι\ninst✝³ : AddCommMonoid P\ninst✝² : PartialOrder P\ninst✝¹ : IsOrderedAddMonoid P\ninst✝ : Module R P\nQ : ... | Finset.sum_eq_single_of_mem i (Finset.mem_univ _) fun j _ hji => ?_, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Weights.RootSystem | {
"line": 255,
"column": 8
} | {
"line": 255,
"column": 42
} | [
{
"pp": "case neg.a\nK : Type u_1\nL : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : CharZero K\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra K L\ninst✝³ : IsKilling K L\ninst✝² : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nα β β' : Weight K (↥H) L\nn : ℤ\... | ← chainTopCoeff_add_chainBotCoeff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Weights.RootSystem | {
"line": 255,
"column": 43
} | {
"line": 255,
"column": 77
} | [
{
"pp": "case neg.a\nK : Type u_1\nL : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : CharZero K\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra K L\ninst✝³ : IsKilling K L\ninst✝² : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nα β β' : Weight K (↥H) L\nn : ℤ\... | ← chainTopCoeff_add_chainBotCoeff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Weights.RootSystem | {
"line": 348,
"column": 43
} | {
"line": 348,
"column": 53
} | [
{
"pp": "case inr.inr.refine_2\nK : Type u_1\nL : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : CharZero K\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra K L\ninst✝³ : IsKilling K L\ninst✝² : FiniteDimensional K L\nH✝ : LieSubalgebra K L\ninst✝¹ : H✝.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H✝) L\nα β : Weight K (↥H... | ← add_smul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.QuadraticForm.Dual | {
"line": 182,
"column": 91
} | {
"line": 183,
"column": 45
} | [
{
"pp": "ι : Type u_4\nR : Type u_5\nM : Type u_6\ninst✝⁴ : CommRing R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nB : LinearMap.BilinForm R M\nhB : (BilinMap.toQuadraticMap B).PosDef\nf : Module.Dual R M\nv : ι → M\nhp : ∀ (i : ι), 0 < f (v i)\nhn : Pai... | by
simpa [hx, hy, map_neg, Finset.mul_sum] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RepresentationTheory.Basic | {
"line": 481,
"column": 4
} | {
"line": 481,
"column": 15
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : Group G\nH : Type u_4\ninst✝ : MulAction G H\ng : G\nf : H →₀ k\nh : H\nh' : H := g⁻¹ • h\n⊢ Function.Injective fun x ↦ g • x",
"usedConstants": []
}
] | intro h₁ h₂ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Order.Interval.Set.OrdConnectedLinear | {
"line": 65,
"column": 4
} | {
"line": 65,
"column": 86
} | [
{
"pp": "case refine_2\nα : Type u_1\nI : Set α\ninst✝¹ : LinearOrder α\ninst✝ : LocallyFiniteOrder α\nh' : ∀ x ∈ I, ∀ y ∈ I, Ioo x y ⊆ Iᶜ → Ioo x y = ∅\nx : α\nhx : x ∈ I\ny : α\nhy : y ∈ I\nhxy : x < y\nz : α\nhz : z ∈ Ioo x y\nhz' : z ∉ I\nx' : α\nhx' : x ≤ x'\nhx'' : Maximal (fun x_1 ↦ x_1 ∈ I ∩ Icc x z) x'... | have hxz : x' < z := lt_of_le_of_ne hx''.1.2.2 (ne_of_mem_of_not_mem hx''.1.1 hz') | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas | {
"line": 164,
"column": 4
} | {
"line": 164,
"column": 90
} | [
{
"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\ninst✝³ : Finite ι\ninst✝² : CharZero R\ninst✝¹ : P.IsCrystallographic\ni j : ι\ninst✝ : IsDomain R\nB : P.Inv... | exact mul_right_cancel₀ (B.ne_zero j) (len_eq ▸ B.pairing_mul_eq_pairing_mul_swap j i) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.RootSystem.Base | {
"line": 254,
"column": 2
} | {
"line": 266,
"column": 39
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝ : CharZero R\ni j : ι\nhi : i ∈ b.support\nhj : j ∈ b.support\n⊢ P.root i - P.root j ∉ rang... | rcases eq_or_ne j i with rfl | hij
· simpa only [sub_self, mem_range, not_exists] using fun k ↦ P.ne_zero k
classical
let f : ι → ℤ := fun k ↦ if k = i then 1 else if k = j then -1 else 0
have hf : ∑ k ∈ b.support, f k • P.root k = P.root i - P.root j := by
have : {i, j} ⊆ b.support := by aesop (add simp Fi... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.Base | {
"line": 254,
"column": 2
} | {
"line": 266,
"column": 39
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝ : CharZero R\ni j : ι\nhi : i ∈ b.support\nhj : j ∈ b.support\n⊢ P.root i - P.root j ∉ rang... | rcases eq_or_ne j i with rfl | hij
· simpa only [sub_self, mem_range, not_exists] using fun k ↦ P.ne_zero k
classical
let f : ι → ℤ := fun k ↦ if k = i then 1 else if k = j then -1 else 0
have hf : ∑ k ∈ b.support, f k • P.root k = P.root i - P.root j := by
have : {i, j} ⊆ b.support := by aesop (add simp Fi... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.Base | {
"line": 346,
"column": 6
} | {
"line": 346,
"column": 31
} | [
{
"pp": "case inr\nι : 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\nb : P.Base\ni : ι\nthis :\n ∀ m ∈ AddSubmonoid.closure (⇑P.root '' ↑b.support),\n ∃ f, Function.... | exact ⟨f, hf, Or.inr hf'⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 103,
"column": 6
} | {
"line": 104,
"column": 86
} | [
{
"pp": "case a.inl\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁷ : Finite ι\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : CommRing R\ninst✝³ : Module R M\ninst✝² : Module R N\nP : RootPairing ι R M N\ninst✝¹ : P.IsRootSystem\ninst✝ : Nontrivial R\ns : Set ι\nhli : LinearIndepOn ... | simpa [SetLike.mem_coe, ← Submodule.mem_toAddSubgroup, span_int_eq_addSubgroupClosure,
← image_eq_range] using AddSubgroup.le_closure_toAddSubmonoid (P.root '' s) hi | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 103,
"column": 6
} | {
"line": 104,
"column": 86
} | [
{
"pp": "case a.inr\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁷ : Finite ι\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : CommRing R\ninst✝³ : Module R M\ninst✝² : Module R N\nP : RootPairing ι R M N\ninst✝¹ : P.IsRootSystem\ninst✝ : Nontrivial R\ns : Set ι\nhli : LinearIndepOn ... | simpa [SetLike.mem_coe, ← Submodule.mem_toAddSubgroup, span_int_eq_addSubgroupClosure,
← image_eq_range] using AddSubgroup.le_closure_toAddSubmonoid (P.root '' s) hi | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.RootSystem.Base | {
"line": 402,
"column": 8
} | {
"line": 402,
"column": 22
} | [
{
"pp": "case neg\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝ : CharZero R\ni : ι\nf : ι → ℤ\nhf₀ : Function.support f ⊆ ↑b.support\nhf₂ : P.ro... | simpa using hj | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.RootSystem.Base | {
"line": 402,
"column": 8
} | {
"line": 402,
"column": 22
} | [
{
"pp": "case neg\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝ : CharZero R\ni : ι\nf : ι → ℤ\nhf₀ : Function.support f ⊆ ↑b.support\nhf₂ : P.ro... | simpa using hj | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.Base | {
"line": 402,
"column": 8
} | {
"line": 402,
"column": 22
} | [
{
"pp": "case neg\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝ : CharZero R\ni : ι\nf : ι → ℤ\nhf₀ : Function.support f ⊆ ↑b.support\nhf₂ : P.ro... | simpa using hj | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.Base | {
"line": 409,
"column": 8
} | {
"line": 409,
"column": 22
} | [
{
"pp": "case neg\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝ : CharZero R\ni : ι\nf : ι → ℤ\nhf₀ : Function.support f ⊆ ↑b.support\nhf₂ : P.ro... | simpa using hj | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.RootSystem.Base | {
"line": 409,
"column": 8
} | {
"line": 409,
"column": 22
} | [
{
"pp": "case neg\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝ : CharZero R\ni : ι\nf : ι → ℤ\nhf₀ : Function.support f ⊆ ↑b.support\nhf₂ : P.ro... | simpa using hj | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.Base | {
"line": 409,
"column": 8
} | {
"line": 409,
"column": 22
} | [
{
"pp": "case neg\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\nb : P.Base\ninst✝ : CharZero R\ni : ι\nf : ι → ℤ\nhf₀ : Function.support f ⊆ ↑b.support\nhf₂ : P.ro... | simpa using hj | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 148,
"column": 63
} | {
"line": 148,
"column": 70
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Finite ι\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Field R\ninst✝⁴ : CharZero R\ninst✝³ : Module R M\ninst✝² : Module R N\nP : RootPairing ι R M N\ninst✝¹ : P.IsRootSystem\ninst✝ : P.IsCrystallographic\nf : M →+ ℚ\nhf : ... | h_span, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.RootSystem.BaseExists | {
"line": 235,
"column": 17
} | {
"line": 235,
"column": 27
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Field R\ninst✝⁵ : CharZero R\ninst✝⁴ : Module R M\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : P.IsRootSystem\ninst✝¹ : P.IsCrystallographic\ninst✝ : P.IsRedu... | ← add_smul | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.RootSystem.CartanMatrix | {
"line": 285,
"column": 43
} | {
"line": 285,
"column": 61
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁵ : CommRing R\ninst✝¹⁴ : AddCommGroup M\ninst✝¹³ : Module R M\ninst✝¹² : AddCommGroup N\ninst✝¹¹ : Module R N\ninst✝¹⁰ : CharZero R\ninst✝⁹ : IsDomain R\ninst✝⁸ : Finite ι\nι₂ : Type u_6\nM₂ : Type u_7\nN₂ : Type u_8\ninst✝⁷ : AddCommGroup ... | simpa using this j | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.RootSystem.CartanMatrix | {
"line": 285,
"column": 43
} | {
"line": 285,
"column": 61
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁵ : CommRing R\ninst✝¹⁴ : AddCommGroup M\ninst✝¹³ : Module R M\ninst✝¹² : AddCommGroup N\ninst✝¹¹ : Module R N\ninst✝¹⁰ : CharZero R\ninst✝⁹ : IsDomain R\ninst✝⁸ : Finite ι\nι₂ : Type u_6\nM₂ : Type u_7\nN₂ : Type u_8\ninst✝⁷ : AddCommGroup ... | simpa using this j | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.CartanMatrix | {
"line": 285,
"column": 43
} | {
"line": 285,
"column": 61
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁵ : CommRing R\ninst✝¹⁴ : AddCommGroup M\ninst✝¹³ : Module R M\ninst✝¹² : AddCommGroup N\ninst✝¹¹ : Module R N\ninst✝¹⁰ : CharZero R\ninst✝⁹ : IsDomain R\ninst✝⁸ : Finite ι\nι₂ : Type u_6\nM₂ : Type u_7\nN₂ : Type u_8\ninst✝⁷ : AddCommGroup ... | simpa using this j | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.Factorial.BigOperators | {
"line": 51,
"column": 20
} | {
"line": 51,
"column": 33
} | [
{
"pp": "n k : ℕ\n⊢ n.ascFactorial (k + 1) = ∏ i ∈ range (k + 1), (n + i)",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"Nat.ascFactorial",
"id",
"instMulNat",
"instOfNatNat",
"Finset.prod",
"Finset.range",
"instHAdd",
"HAdd.hAdd",
... | ascFactorial, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.Factorial.SuperFactorial | {
"line": 82,
"column": 85
} | {
"line": 83,
"column": 61
} | [
{
"pp": "n : ℕ\n⊢ sf 4 * n = (∏ i ∈ range (2 * n), (2 * i + 1)!) ^ 2 * 2 ^ (2 * n) * (2 * n)!",
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"HMul.hMul",
"Nat.superFactorial",
"congrArg",
"Nat.instMonoid",
"mul_assoc",
"id",
"instMulNat",
"instO... | by
rw [← superFactorial_two_mul, ← mul_assoc, Nat.mul_two] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 306,
"column": 6
} | {
"line": 306,
"column": 44
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nn : ℕ\nk : ℤ\n⊢ ↑(eval k (descPochhammer ℤ n)) = eval (↑k) (descPochhammer R n)",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Polynomial.eval",
"Algebra.algebraMap",
"congrArg",
"descPochhammer",
"id",
"descPochhammer_map",
... | ← descPochhammer_map (algebraMap ℤ R), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 482,
"column": 6
} | {
"line": 482,
"column": 26
} | [
{
"pp": "case inl\nS : Type u_1\ninst✝² : Ring S\ninst✝¹ : PartialOrder S\ninst✝ : IsStrictOrderedRing S\nn : ℕ\ns : S\nh : ↑n - 1 ≤ s\nheq : ↑n - 1 = s\n⊢ 0 ≤ eval (↑n - 1 - ↑n + 1) (ascPochhammer S n)",
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
"NegZeroClass.toNeg",
"AddGroup... | sub_sub_cancel_left, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Lagrange | {
"line": 266,
"column": 4
} | {
"line": 267,
"column": 18
} | [
{
"pp": "case refine_3\nF : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns : Finset ι\nv : ι → F\nhvs : Set.InjOn v ↑s\nhs : s.Nonempty\ni : ι\nhi : i ∈ s\n⊢ eval (v i) (∑ j ∈ s, Lagrange.basis s v j) = eval (v i) 1",
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
"Non... | rw [eval_finsetSum, eval_one, ← add_sum_erase _ _ hi, eval_basis_self hvs hi,
add_eq_left] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Lagrange | {
"line": 322,
"column": 2
} | {
"line": 322,
"column": 74
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns : Finset ι\ni : ι\nv r : ι → F\nhvs : Set.InjOn v ↑s\nhi : i ∈ s\n⊢ eval (v i) (C (r i) * Lagrange.basis s v i) + ∑ x ∈ s.erase i, eval (v i) (C (r x) * Lagrange.basis s v x) = r i",
"usedConstants": [
"Eq.mpr",
"Pol... | simp_rw [eval_mul, eval_C, eval_basis_self hvs hi, mul_one, add_eq_left] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.LinearAlgebra.Lagrange | {
"line": 336,
"column": 89
} | {
"line": 343,
"column": 54
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns : Finset ι\nv r : ι → F\nhvs : Set.InjOn v ↑s\n⊢ ((interpolate s v) r).degree < ↑(#s)",
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"Pi.Function.module",
"lt_of_le_of_lt",
"Nat.instMulZeroC... | by
rw [Nat.cast_withBot]
rcases eq_empty_or_nonempty s with (rfl | h)
· rw [interpolate_empty, degree_zero, card_empty]
exact WithBot.bot_lt_coe _
· refine lt_of_le_of_lt (degree_interpolate_le _ hvs) ?_
rw [Nat.cast_withBot, WithBot.coe_lt_coe]
exact Nat.sub_lt (Nonempty.card_pos h) zero_lt_one | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.EngelSubalgebra | {
"line": 160,
"column": 2
} | {
"line": 160,
"column": 7
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\ninst✝ : IsNoetherian R L\nH : LieSubalgebra R L\nh : ∀ x ∈ H, H ≤ engel R x\nx : ↥H\nK : ℕ →o Submodule R ↥H := { toFun := fun n ↦ LinearMap.ker ((ad R ↥H) x ^ n), monotone' := ⋯ }\nn : ℕ\nhn : ∀ (m : ℕ), n ≤ ... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.LinearAlgebra.Lagrange | {
"line": 511,
"column": 2
} | {
"line": 511,
"column": 58
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns : Finset ι\nv : ι → F\nhvs : Set.InjOn v ↑s\nP : F[X]\ndeg : ℕ\nhdeg : ↑deg = P.degree\nhP : #s = deg + 1\nhdegree : P.degree = ↑(#s - 1)\n⊢ P.coeff (#s - 1) = ∑ i ∈ s, eval (v i) P / ∏ j ∈ s.erase i, (v i - v j)",
"usedConstant... | exact coeff_eq_sum hvs (by rw [hdegree]; norm_cast; lia) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous | {
"line": 436,
"column": 46
} | {
"line": 444,
"column": 34
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝¹ : CommSemiring R\nσ : Type u_3\ninst✝ : AddCommMonoid M\nw : σ → M\nm : M\np : MvPolynomial σ R\nhp : IsWeightedHomogeneous w p m\n⊢ (weightedHomogeneousComponent w m) p = p",
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"NonAssocSemi... | by
classical
ext x
rw [coeff_weightedHomogeneousComponent]
by_cases zero_coeff : coeff x p = 0
· split_ifs
· rfl
rw [zero_coeff]
· rw [hp zero_coeff, if_pos rfl] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous | {
"line": 478,
"column": 2
} | {
"line": 478,
"column": 53
} | [
{
"pp": "case h\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\nσ : Type u_3\ninst✝¹ : AddCommMonoid M\nw : σ → M\ninst✝ : DecidableEq M\nn : M\np : MvPolynomial σ R\nc : σ →₀ ℕ\n⊢ c ∈ ((weightedHomogeneousComponent w n) p).support ↔ c ∈ {c ∈ p.support | (weight w) c = n}",
"usedConstants": [
"F... | simp [coeff_weightedHomogeneousComponent, And.comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous | {
"line": 636,
"column": 4
} | {
"line": 646,
"column": 45
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\nσ : Type u_3\nw : σ → M\ninst✝¹ : AddCommMonoid M\ninst✝ : DecidableEq M\nφ : MvPolynomial σ R\n⊢ (DirectSum.coeAddMonoidHom (weightedHomogeneousSubmodule R w)) (decompose' R w φ) = φ",
"usedConstants": [
"Eq.mpr",
"DirectSum.coeAddMo... | classical
conv_rhs => rw [← sum_weightedHomogeneousComponent w φ]
rw [← DirectSum.sum_support_of (decompose' R w φ)]
simp only [DirectSum.coeAddMonoidHom_of, map_sum,
finsum_eq_sum _ (weightedHomogeneousComponent_finsupp φ)]
apply Finset.sum_congr _ (fun m _ ↦ by rw [decompose'_apply])
ext m
... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous | {
"line": 636,
"column": 4
} | {
"line": 646,
"column": 45
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\nσ : Type u_3\nw : σ → M\ninst✝¹ : AddCommMonoid M\ninst✝ : DecidableEq M\nφ : MvPolynomial σ R\n⊢ (DirectSum.coeAddMonoidHom (weightedHomogeneousSubmodule R w)) (decompose' R w φ) = φ",
"usedConstants": [
"Eq.mpr",
"DirectSum.coeAddMo... | classical
conv_rhs => rw [← sum_weightedHomogeneousComponent w φ]
rw [← DirectSum.sum_support_of (decompose' R w φ)]
simp only [DirectSum.coeAddMonoidHom_of, map_sum,
finsum_eq_sum _ (weightedHomogeneousComponent_finsupp φ)]
apply Finset.sum_congr _ (fun m _ ↦ by rw [decompose'_apply])
ext m
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous | {
"line": 636,
"column": 4
} | {
"line": 646,
"column": 45
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\nσ : Type u_3\nw : σ → M\ninst✝¹ : AddCommMonoid M\ninst✝ : DecidableEq M\nφ : MvPolynomial σ R\n⊢ (DirectSum.coeAddMonoidHom (weightedHomogeneousSubmodule R w)) (decompose' R w φ) = φ",
"usedConstants": [
"Eq.mpr",
"DirectSum.coeAddMo... | classical
conv_rhs => rw [← sum_weightedHomogeneousComponent w φ]
rw [← DirectSum.sum_support_of (decompose' R w φ)]
simp only [DirectSum.coeAddMonoidHom_of, map_sum,
finsum_eq_sum _ (weightedHomogeneousComponent_finsupp φ)]
apply Finset.sum_congr _ (fun m _ ↦ by rw [decompose'_apply])
ext m
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.CartanCriterion | {
"line": 211,
"column": 2
} | {
"line": 214,
"column": 52
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CharZero R\ninst✝⁴ : IsDomain R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : IsNoetherian R L\ninst✝ : Module.Free R L\nI : LieIdeal R L\nDI : LieIdeal R L := ⁅I, I⁆\nh : ∀ x ∈ I, ∀ y ∈ DI, ((killingForm R L) x) y = 0\nDDI : LieIdeal R... | have tf_eq_zero : LieModule.traceForm R DI L = 0 := by
ext ⟨x, hx⟩ ⟨y, hy⟩
change killingForm R L x y = 0
exact h x (LieSubmodule.lie_le_left I I hx) y hy | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.MvPolynomial.Homogeneous | {
"line": 357,
"column": 2
} | {
"line": 357,
"column": 37
} | [
{
"pp": "R : Type u_3\ninst✝ : CommSemiring R\nN : ℕ\nφ : MvPolynomial (Fin (N + 1)) R\nn : ℕ\nhφ : φ.IsHomogeneous n\ni j : ℕ\nh : i + j = n\nd : Fin N →₀ ℕ\nhd : coeff d (((finSuccEquiv R N) φ).coeff i) ≠ 0\n⊢ (weight 1) d = j",
"usedConstants": [
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semi... | rw [finSuccEquiv_coeff_coeff] at hd | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Eigenspace.Zero | {
"line": 74,
"column": 4
} | {
"line": 74,
"column": 9
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : IsDomain R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Module.Finite R M\ninst✝¹ : Free R M\ninst✝ : IsNoetherian R M\nφ : End R M\ntfae_1_to_2 : IsNilpotent φ → charpoly φ = X ^ finrank R M\ntfae_2_to_3 : charpoly φ = X ^ finrank R ... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.LinearAlgebra.Eigenspace.Zero | {
"line": 71,
"column": 2
} | {
"line": 79,
"column": 40
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : IsDomain R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Module.Finite R M\ninst✝¹ : Free R M\ninst✝ : IsNoetherian R M\nφ : End R M\ntfae_1_to_2 : IsNilpotent φ → charpoly φ = X ^ finrank R M\ntfae_2_to_3 : charpoly φ = X ^ finrank R ... | tfae_have 3 → 1
| h => by
obtain ⟨n, hn⟩ := Filter.eventually_atTop.mp <| φ.eventually_iSup_ker_pow_eq
use n
ext x
rw [zero_apply, ← mem_ker, ← hn n le_rfl]
obtain ⟨k, hk⟩ := h x
rw [← mem_ker] at hk
exact Submodule.mem_iSup_of_mem _ hk | Mathlib.Tactic.TFAE._aux_Mathlib_Tactic_TFAE___macroRules_Mathlib_Tactic_TFAE_tfaeHave_1 | Mathlib.Tactic.TFAE.tfaeHave |
Mathlib.RingTheory.MvPolynomial.Homogeneous | {
"line": 475,
"column": 2
} | {
"line": 475,
"column": 77
} | [
{
"pp": "R : Type u_5\nσ : Type u_6\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nhnR : ↑n ≤ #R\nk : ℕ\nf : Fin k → σ\nhf : Function.Injective f\nF : MvPolynomial (Fin k) R\nhF✝ : ((rename f) F).IsHomogeneous n\nh : (rename f) F ≠ 0\nhF₀ : F ≠ 0\nhF : F.IsHomogeneous n\n⊢ ∃ r, (eval r) ((rename f) F) ≠ 0",
... | obtain ⟨r, hr⟩ := exists_eval_ne_zero_of_totalDegree_le_card_aux hF hF₀ hnR | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.MvPolynomial.Homogeneous | {
"line": 617,
"column": 77
} | {
"line": 619,
"column": 5
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝ : CommSemiring R\n⊢ DirectSum.Decomposition.decompose' = fun φ ↦\n (DirectSum.mk (fun i ↦ ↥(homogeneousSubmodule σ R i)) (Finset.image (⇑degree) φ.support)) fun m ↦\n ⟨(homogeneousComponent ↑m) φ, ⋯⟩",
"usedConstants": [
"Finsupp.instAddZeroClass",
... | by
rw [degree_eq_weight_one]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Eigenspace.Zero | {
"line": 154,
"column": 4
} | {
"line": 154,
"column": 9
} | [
{
"pp": "K : Type u_2\nM : Type u_3\ninst✝³ : Field K\ninst✝² : AddCommGroup M\ninst✝¹ : Module K M\ninst✝ : Module.Finite K M\nφ : End K M\nV : Submodule K M := φ.maxGenEigenspace 0\nhV : V = ⨆ n, ker (φ ^ n)\nW : Submodule K M := ⨅ n, range (φ ^ n)\nhVW : IsCompl V W\nx : M\nn : ℕ\nhx : (φ ^ n) x = 0\n⊢ ∃ k, ... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Algebra.Module.LinearMap.Polynomial | {
"line": 163,
"column": 74
} | {
"line": 165,
"column": 73
} | [
{
"pp": "R : Type u_1\nM₁ : Type u_2\nM₂ : Type u_3\nι₁ : Type u_4\nι₂ : Type u_5\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M₁\ninst✝⁵ : AddCommGroup M₂\ninst✝⁴ : Module R M₁\ninst✝³ : Module R M₂\ninst✝² : Fintype ι₁\ninst✝¹ : Finite ι₂\ninst✝ : DecidableEq ι₁\nb₁ : Basis ι₁ R M₁\nb₂ : Basis ι₂ R M₂\nf : M₁ ... | by
rw [toMvPolynomial, Matrix.toMvPolynomial_eval_eq_apply,
← LinearMap.toMatrix_mulVec_repr b₁ b₂, LinearEquiv.apply_symm_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Eigenspace.Zero | {
"line": 185,
"column": 4
} | {
"line": 185,
"column": 37
} | [
{
"pp": "case h.a\nK : Type u_2\nM : Type u_3\ninst✝³ : Field K\ninst✝² : AddCommGroup M\ninst✝¹ : Module K M\ninst✝ : Module.Finite K M\nφ : End K M\nV : Submodule K M := φ.maxGenEigenspace 0\nhV : V = ⨆ n, ker (φ ^ n)\nW : Submodule K M := ⨅ n, range (φ ^ n)\nhVW : IsCompl V W\nhφV : ∀ x ∈ V, φ x ∈ V\nhφW : ∀... | suffices x.1 ∈ V from ⟨this, x.2⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.LinearAlgebra.Eigenspace.Zero | {
"line": 194,
"column": 2
} | {
"line": 194,
"column": 7
} | [
{
"pp": "K : Type u_2\nM : Type u_3\ninst✝³ : Field K\ninst✝² : AddCommGroup M\ninst✝¹ : Module K M\ninst✝ : Module.Finite K M\nφ : End K M\nV : Submodule K M := φ.maxGenEigenspace 0\nhV : V = ⨆ n, ker (φ ^ n)\nW : Submodule K M := ⨅ n, range (φ ^ n)\nhVW : IsCompl V W\nhφV : ∀ x ∈ V, φ x ∈ V\nhφW : ∀ x ∈ W, φ ... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.RingTheory.Derivation.Lie | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 44
} | [
{
"pp": "case mp\nR : Type u_1\ninst✝⁶ : CommRing R\nA : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra R A\nA' : Type u_3\ninst✝³ : CommRing A'\ninst✝² : Algebra R A'\ninst✝¹ : Algebra A A'\ninst✝ : IsScalarTower R A A'\nx : Derivation R A' A' × Derivation R A A\n⊢ x ∈ couple R A A' → ⇑x.1 ∘ ⇑(Algebra.ofId A ... | · intro hx; ext a; exact congrArg (· a) hx | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Derivation.Lie | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 44
} | [
{
"pp": "case mpr\nR : Type u_1\ninst✝⁶ : CommRing R\nA : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra R A\nA' : Type u_3\ninst✝³ : CommRing A'\ninst✝² : Algebra R A'\ninst✝¹ : Algebra A A'\ninst✝ : IsScalarTower R A A'\nx : Derivation R A' A' × Derivation R A A\n⊢ ⇑x.1 ∘ ⇑(Algebra.ofId A A') = ⇑(Algebra.ofI... | · intro hx; ext a; exact congrArg (· a) hx | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Lie.CartanExists | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 34
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra K L\ninst✝ : Module.Finite K L\nhLK : ↑(finrank K L) ≤ #K\nU : LieSubalgebra K L\nx : L\nhxU : x ∈ U\ny : L\nhyU : y ∈ U\nEx : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩\nEy : ↑{x | ∃ y ∈ U, engel K y = x} :=... | obtain rfl | hx₀ := eq_or_ne x 0 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Lie.CartanExists | {
"line": 286,
"column": 14
} | {
"line": 286,
"column": 16
} | [
{
"pp": "case h.refine_2\nK : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra K L\ninst✝ : Module.Finite K L\nhLK : ↑(finrank K L) ≤ #K\nU : LieSubalgebra K L\nx : L\nhxU : x ∈ U\ny : L\nhyU : y ∈ U\nEx : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩\nEy : ↑{x | ∃ y ∈ U, ... | hα | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.Lie.DirectSum | {
"line": 211,
"column": 34
} | {
"line": 211,
"column": 49
} | [
{
"pp": "case inr\nR : Type u\nι : Type v\ninst✝⁵ : CommRing R\nL : ι → Type w\ninst✝⁴ : (i : ι) → LieRing (L i)\ninst✝³ : (i : ι) → LieAlgebra R (L i)\ninst✝² : DecidableEq ι\nL' : Type w₁\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : (i : ι) → L i →ₗ⁅R⁆ L'\nhf : Pairwise fun i j ↦ ∀ (x : L i) (y : L j), ... | hf hij.symm x y | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Lie.CartanExists | {
"line": 300,
"column": 10
} | {
"line": 300,
"column": 12
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra K L\ninst✝ : Module.Finite K L\nhLK : ↑(finrank K L) ≤ #K\nU : LieSubalgebra K L\nx : L\nhxU : x ∈ U\ny : L\nhyU : y ∈ U\nEx : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩\nEy : ↑{x | ∃ y ∈ U, engel K y = x} :=... | hα | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.Lie.CartanExists | {
"line": 325,
"column": 4
} | {
"line": 325,
"column": 9
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra K L\ninst✝ : Module.Finite K L\nhLK : ↑(finrank K L) ≤ #K\nU : LieSubalgebra K L\nx : L\nhxU : x ∈ U\ny : L\nhyU : y ∈ U\nEx : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩\nEy : ↑{x | ∃ y ∈ U, engel K y = x} :=... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Data.Matrix.Cartan | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 53
} | [
{
"pp": "n : ℕ\ni j : Fin n\nh : i ≠ j\n⊢ B n i j ≤ 0",
"usedConstants": [
"_private.Mathlib.Data.Matrix.Cartan.0.CartanMatrix.B_off_diag_nonpos._proof_1_5",
"Eq.mpr",
"_private.Mathlib.Data.Matrix.Cartan.0.CartanMatrix.B_off_diag_nonpos._proof_1_2",
"congrArg",
"_private.Mathl... | simp only [B, Matrix.of_apply]; split_ifs <;> omega | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Matrix.Cartan | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 53
} | [
{
"pp": "n : ℕ\ni j : Fin n\nh : i ≠ j\n⊢ B n i j ≤ 0",
"usedConstants": [
"_private.Mathlib.Data.Matrix.Cartan.0.CartanMatrix.B_off_diag_nonpos._proof_1_5",
"Eq.mpr",
"_private.Mathlib.Data.Matrix.Cartan.0.CartanMatrix.B_off_diag_nonpos._proof_1_2",
"congrArg",
"_private.Mathl... | simp only [B, Matrix.of_apply]; split_ifs <;> omega | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.LieTheorem | {
"line": 186,
"column": 4
} | {
"line": 186,
"column": 64
} | [
{
"pp": "k : Type u_1\ninst✝¹⁰ : Field k\nL : Type u_2\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra k L\nV : Type u_3\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module k V\ninst✝⁵ : LieRingModule L V\ninst✝⁴ : LieModule k L V\ninst✝³ : CharZero k\ninst✝² : Module.Finite k V\ninst✝¹ : IsTriangularizable k L V\nA : LieIdeal ... | simpa [W, weightSpaceOfIsLieTower, mem_weightSpace] using hv | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Lie.Extension | {
"line": 446,
"column": 22
} | {
"line": 446,
"column": 70
} | [
{
"pp": "R : Type u_1\nL : Type u_3\nM : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\ninst✝² : LieRing M\ninst✝¹ : LieAlgebra R M\ninst✝ : IsLieAbelian M\nE : Extension R M L\ns₁ s₂ : L →ₗ[R] E.L\nhs₁ : LeftInverse ⇑E.proj ⇑s₁\nhs₂ : LeftInverse ⇑E.proj ⇑s₂\nx y : L\ns : L → E.L\n... | rw [LieHom.mem_ker, map_sub, sub_eq_zero, h, hs] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Lie.Extension | {
"line": 446,
"column": 22
} | {
"line": 446,
"column": 70
} | [
{
"pp": "R : Type u_1\nL : Type u_3\nM : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\ninst✝² : LieRing M\ninst✝¹ : LieAlgebra R M\ninst✝ : IsLieAbelian M\nE : Extension R M L\ns₁ s₂ : L →ₗ[R] E.L\nhs₁ : LeftInverse ⇑E.proj ⇑s₁\nhs₂ : LeftInverse ⇑E.proj ⇑s₂\nx y : L\ns : L → E.L\n... | rw [LieHom.mem_ker, map_sub, sub_eq_zero, h, hs] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Extension | {
"line": 446,
"column": 22
} | {
"line": 446,
"column": 70
} | [
{
"pp": "R : Type u_1\nL : Type u_3\nM : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\ninst✝² : LieRing M\ninst✝¹ : LieAlgebra R M\ninst✝ : IsLieAbelian M\nE : Extension R M L\ns₁ s₂ : L →ₗ[R] E.L\nhs₁ : LeftInverse ⇑E.proj ⇑s₁\nhs₂ : LeftInverse ⇑E.proj ⇑s₂\nx y : L\ns : L → E.L\n... | rw [LieHom.mem_ker, map_sub, sub_eq_zero, h, hs] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Extension | {
"line": 447,
"column": 22
} | {
"line": 447,
"column": 70
} | [
{
"pp": "R : Type u_1\nL : Type u_3\nM : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\ninst✝² : LieRing M\ninst✝¹ : LieAlgebra R M\ninst✝ : IsLieAbelian M\nE : Extension R M L\ns₁ s₂ : L →ₗ[R] E.L\nhs₁ : LeftInverse ⇑E.proj ⇑s₁\nhs₂ : LeftInverse ⇑E.proj ⇑s₂\nx y : L\ns : L → E.L\n... | rw [LieHom.mem_ker, map_sub, sub_eq_zero, h, hs] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Lie.Extension | {
"line": 447,
"column": 22
} | {
"line": 447,
"column": 70
} | [
{
"pp": "R : Type u_1\nL : Type u_3\nM : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\ninst✝² : LieRing M\ninst✝¹ : LieAlgebra R M\ninst✝ : IsLieAbelian M\nE : Extension R M L\ns₁ s₂ : L →ₗ[R] E.L\nhs₁ : LeftInverse ⇑E.proj ⇑s₁\nhs₂ : LeftInverse ⇑E.proj ⇑s₂\nx y : L\ns : L → E.L\n... | rw [LieHom.mem_ker, map_sub, sub_eq_zero, h, hs] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Extension | {
"line": 447,
"column": 22
} | {
"line": 447,
"column": 70
} | [
{
"pp": "R : Type u_1\nL : Type u_3\nM : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\ninst✝² : LieRing M\ninst✝¹ : LieAlgebra R M\ninst✝ : IsLieAbelian M\nE : Extension R M L\ns₁ s₂ : L →ₗ[R] E.L\nhs₁ : LeftInverse ⇑E.proj ⇑s₁\nhs₂ : LeftInverse ⇑E.proj ⇑s₂\nx y : L\ns : L → E.L\n... | rw [LieHom.mem_ker, map_sub, sub_eq_zero, h, hs] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.ChainOfDivisors | {
"line": 234,
"column": 8
} | {
"line": 234,
"column": 78
} | [
{
"pp": "M : Type u_1\ninst✝³ : CommMonoidWithZero M\ninst✝² : IsCancelMulZero M\nN : Type u_2\ninst✝¹ : CommMonoidWithZero N\ninst✝ : IsCancelMulZero N\nm u : Associates M\nn : Associates N\nhu' : u ≤ m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\nhu : ↑(d ⟨u, hu'⟩) = 1\n⊢ u = ↑(d.symm ⟨↑(d ⟨u, hu'⟩), ⋯⟩)",
"usedCon... | simp only [Subtype.coe_eta, OrderIso.symm_apply_apply, Subtype.coe_mk] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.ChainOfDivisors | {
"line": 234,
"column": 8
} | {
"line": 234,
"column": 78
} | [
{
"pp": "M : Type u_1\ninst✝³ : CommMonoidWithZero M\ninst✝² : IsCancelMulZero M\nN : Type u_2\ninst✝¹ : CommMonoidWithZero N\ninst✝ : IsCancelMulZero N\nm u : Associates M\nn : Associates N\nhu' : u ≤ m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\nhu : ↑(d ⟨u, hu'⟩) = 1\n⊢ u = ↑(d.symm ⟨↑(d ⟨u, hu'⟩), ⋯⟩)",
"usedCon... | simp only [Subtype.coe_eta, OrderIso.symm_apply_apply, Subtype.coe_mk] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.ChainOfDivisors | {
"line": 234,
"column": 8
} | {
"line": 234,
"column": 78
} | [
{
"pp": "M : Type u_1\ninst✝³ : CommMonoidWithZero M\ninst✝² : IsCancelMulZero M\nN : Type u_2\ninst✝¹ : CommMonoidWithZero N\ninst✝ : IsCancelMulZero N\nm u : Associates M\nn : Associates N\nhu' : u ≤ m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\nhu : ↑(d ⟨u, hu'⟩) = 1\n⊢ u = ↑(d.symm ⟨↑(d ⟨u, hu'⟩), ⋯⟩)",
"usedCon... | simp only [Subtype.coe_eta, OrderIso.symm_apply_apply, Subtype.coe_mk] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.ChainOfDivisors | {
"line": 388,
"column": 4
} | {
"line": 390,
"column": 72
} | [
{
"pp": "M : Type u_1\ninst✝⁶ : CommMonoidWithZero M\ninst✝⁵ : IsCancelMulZero M\nN : Type u_2\ninst✝⁴ : CommMonoidWithZero N\ninst✝³ : Subsingleton Mˣ\ninst✝² : Subsingleton Nˣ\ninst✝¹ : UniqueFactorizationMonoid M\ninst✝ : UniqueFactorizationMonoid N\nm p : M\nn : N\nhm : m ≠ 0\nhn : n ≠ 0\nhp : p ∈ normalize... | obtain ⟨q, hq, hq'⟩ :=
exists_mem_normalizedFactors_of_dvd hn this.irreducible
(d ⟨p, by apply dvd_of_mem_normalizedFactors; convert! hp⟩).prop | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.FractionalIdeal.Basic | {
"line": 259,
"column": 31
} | {
"line": 259,
"column": 82
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nI : Ideal R\n⊢ coeSubmodule P I ≤ 1",
"usedConstants": [
"Submodule",
"Semiring.toModule",
"congrArg",
"CommSemiring.toSemiring",
"PartialOrder.toPreorder",
... | by simpa using coeSubmodule_mono P (le_top : I ≤ ⊤) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.FractionalIdeal.Basic | {
"line": 529,
"column": 8
} | {
"line": 529,
"column": 78
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nI J : Submodule R P\naI : R\nhaI : aI ∈ S\nhI : ∀ b ∈ I, IsInteger R (aI • b)\naJ : R\nhaJ : aJ ∈ S\nhJ : ∀ b ∈ J, IsInteger R (aJ • b)\nb : P\nhb : b ∈ I * J\nm : P\nhm : m ∈ I\nn... | rw [mul_smul, mul_comm m, ← smul_mul_assoc, ← hn', ← Algebra.smul_def] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.DedekindDomain.Ideal.Basic | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 73
} | [
{
"pp": "A : Type u_2\ninst✝¹ : CommRing A\ninst✝ : IsDomain A\nh : IsDedekindDomainInv A\nthis : CommGroupWithZero (FractionalIdeal A⁰ (FractionRing A)) := h.commGroupWithZero\n⊢ IsIntegrallyClosed A",
"usedConstants": [
"Iff.mpr",
"Algebra.algebraMap",
"OreLocalization.instAlgebra",
... | refine (isIntegrallyClosed_iff (FractionRing A)).mpr (fun {x hx} => ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 671,
"column": 2
} | {
"line": 683,
"column": 34
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\ninst✝³ : IsLocalization S P\nP' : Type u_5\ninst✝² : CommRing P'\ninst✝¹ : Algebra R P'\ninst✝ : IsLocalization S P'\nx : P\n⊢ (canonicalEquiv S P P') (spanSingleton S x) = spanSingleton S ((IsL... | apply SetLike.ext_iff.mpr
intro y
constructor <;> intro h
· rw [mem_spanSingleton]
obtain ⟨x', hx', rfl⟩ := (mem_canonicalEquiv_apply _ _ _).mp h
obtain ⟨z, rfl⟩ := (mem_spanSingleton _).mp hx'
use z
rw [IsLocalization.map_smul, RingHom.id_apply]
· rw [mem_canonicalEquiv_apply]
obtain ⟨z, rf... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 671,
"column": 2
} | {
"line": 683,
"column": 34
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\ninst✝³ : IsLocalization S P\nP' : Type u_5\ninst✝² : CommRing P'\ninst✝¹ : Algebra R P'\ninst✝ : IsLocalization S P'\nx : P\n⊢ (canonicalEquiv S P P') (spanSingleton S x) = spanSingleton S ((IsL... | apply SetLike.ext_iff.mpr
intro y
constructor <;> intro h
· rw [mem_spanSingleton]
obtain ⟨x', hx', rfl⟩ := (mem_canonicalEquiv_apply _ _ _).mp h
obtain ⟨z, rfl⟩ := (mem_spanSingleton _).mp hx'
use z
rw [IsLocalization.map_smul, RingHom.id_apply]
· rw [mem_canonicalEquiv_apply]
obtain ⟨z, rf... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 869,
"column": 2
} | {
"line": 869,
"column": 59
} | [
{
"pp": "case neg\nR₁ : Type u_3\ninst✝⁴ : CommRing R₁\nK : Type u_4\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nx : R₁\nI : FractionalIdeal R₁⁰ K\nhI : ∀ J ≤ I, (↑J).FG\nhx : ¬x = 0\nh_gx : (algebraMap R₁ K) x ≠ 0\nh_spanx : spanSingleton R₁⁰ ((algebraMap R₁ K) ... | rw [← div_spanSingleton, le_div_iff_mul_le h_spanx] at hJ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 88,
"column": 2
} | {
"line": 93,
"column": 81
} | [
{
"pp": "R : Type u\nM : Type u_1\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nfp : FinitePresentation R M\n⊢ ∃ n K x, K.FG",
"usedConstants": [
"Submodule.comap_equiv_eq_map_symm",
"Eq.mpr",
"Pi.Function.module",
"Submodule",
"RingHomSurjective.ids",
"S... | have ⟨ι, ⟨hι₁, hι₂⟩⟩ := fp
refine ⟨_, LinearMap.ker (linearCombination R Subtype.val ∘ₗ
(lcongr ι.equivFin (.refl ..) ≪≫ₗ linearEquivFunOnFinite R R _).symm.toLinearMap),
(LinearMap.quotKerEquivOfSurjective _ <| LinearMap.range_eq_top.mp ?_).symm, ?_⟩
· simpa [range_linearCombination] using hι₁
· simpa [L... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 88,
"column": 2
} | {
"line": 93,
"column": 81
} | [
{
"pp": "R : Type u\nM : Type u_1\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nfp : FinitePresentation R M\n⊢ ∃ n K x, K.FG",
"usedConstants": [
"Submodule.comap_equiv_eq_map_symm",
"Eq.mpr",
"Pi.Function.module",
"Submodule",
"RingHomSurjective.ids",
"S... | have ⟨ι, ⟨hι₁, hι₂⟩⟩ := fp
refine ⟨_, LinearMap.ker (linearCombination R Subtype.val ∘ₗ
(lcongr ι.equivFin (.refl ..) ≪≫ₗ linearEquivFunOnFinite R R _).symm.toLinearMap),
(LinearMap.quotKerEquivOfSurjective _ <| LinearMap.range_eq_top.mp ?_).symm, ?_⟩
· simpa [range_linearCombination] using hι₁
· simpa [L... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 170,
"column": 17
} | {
"line": 173,
"column": 14
} | [
{
"pp": "A : Type u_2\ninst✝¹ : CommRing A\ninst✝ : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\ne : ℕ\nhe : 2 ≤ e\n⊢ I ^ e < I",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Semiring.toModule",
"IsScalarTower.right",
"congrArg",
"CommSemiring.toSemiring",... | by
convert! I.pow_right_strictAnti hI0 hI1 he
dsimp only
rw [pow_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 254,
"column": 8
} | {
"line": 254,
"column": 74
} | [
{
"pp": "case refine_2.refine_3\nA : Type u_2\nK : Type u_3\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDedekindDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nJ : Ideal A\nhJ : J ≠ ⊤\nι : Type u_4\ns : Finset ι\nf : ι → K\nj : ι\nhjs : j ∈ s\nhjf : f j ≠ 0\nI : FractionalIdeal A⁰ K := spanFi... | intro x y _ _ hx hy; rw [mul_add]; exact Submodule.add_mem _ hx hy | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 254,
"column": 8
} | {
"line": 254,
"column": 74
} | [
{
"pp": "case refine_2.refine_3\nA : Type u_2\nK : Type u_3\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDedekindDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nJ : Ideal A\nhJ : J ≠ ⊤\nι : Type u_4\ns : Finset ι\nf : ι → K\nj : ι\nhjs : j ∈ s\nhjf : f j ≠ 0\nI : FractionalIdeal A⁰ K := spanFi... | intro x y _ _ hx hy; rw [mul_add]; exact Submodule.add_mem _ hx hy | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 409,
"column": 8
} | {
"line": 409,
"column": 45
} | [
{
"pp": "case a.left\nT : Type u_4\ninst✝¹ : CommRing T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nthis : (normalizedFactors I ∩ normalizedFactors J).prod ≠ 0\n⊢ normalizedFactors (normalizedFactors I ∩ normalizedFactors J).prod ≤ normalizedFactors I",
"usedConstants": [
"Uniq... | normalizedFactors_prod_inter_eq_inter | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 409,
"column": 8
} | {
"line": 409,
"column": 45
} | [
{
"pp": "case a.right\nT : Type u_4\ninst✝¹ : CommRing T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nthis : (normalizedFactors I ∩ normalizedFactors J).prod ≠ 0\n⊢ normalizedFactors (normalizedFactors I ∩ normalizedFactors J).prod ≤ normalizedFactors J",
"usedConstants": [
"Uni... | normalizedFactors_prod_inter_eq_inter | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 645,
"column": 64
} | {
"line": 645,
"column": 71
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nK : Type u_3\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Field K\ninst✝ : IsDedekindDomain A\nI : Ideal R\nJ : Ideal A\nf : R ⧸ I →+* A ⧸ J\nhf : Function.Surjective ⇑f\nX : Ideal R\nhX : X ∣ I\nY : Ideal R\nhY : Y ∣ I\nh : X ≤ Y\n⊢ X ≤ Y ⊔ RingHom.ker (Ideal.Quotien... | mk_ker, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 589,
"column": 4
} | {
"line": 592,
"column": 39
} | [
{
"pp": "R : Type u_3\nM : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\nS : Submonoid R\nM' : Type u_1\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M'\nf : M →ₗ[R] M'\ninst✝² : IsLocalizedModule S f\ninst✝¹ : Module.Finite R M\ninst✝ : Module.FinitePresentation R M'\nthis : IsLoc... | convert! show Function.Bijective LinearMap.id from Function.bijective_id
apply IsLocalizedModule.ext S f
· exact IsLocalizedModule.map_units f
· simp [IsLocalizedModule.map_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 589,
"column": 4
} | {
"line": 592,
"column": 39
} | [
{
"pp": "R : Type u_3\nM : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\nS : Submonoid R\nM' : Type u_1\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M'\nf : M →ₗ[R] M'\ninst✝² : IsLocalizedModule S f\ninst✝¹ : Module.Finite R M\ninst✝ : Module.FinitePresentation R M'\nthis : IsLoc... | convert! show Function.Bijective LinearMap.id from Function.bijective_id
apply IsLocalizedModule.ext S f
· exact IsLocalizedModule.map_units f
· simp [IsLocalizedModule.map_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.PDeriv | {
"line": 75,
"column": 4
} | {
"line": 75,
"column": 33
} | [
{
"pp": "case refine_1\nR : Type u\nσ : Type v\na : R\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\ni j : σ\nx✝ : j ∈ s.support\nhne : j ≠ i\n⊢ (fun a_1 b ↦ (monomial (s - single a_1 1)) (a * ↑b) * Pi.single i 1 a_1) j (s j) = 0",
"usedConstants": [
"Finsupp.instFunLike",
"Nat.instCanonicallyOrderedAdd",... | simp [Pi.single_eq_of_ne hne] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.MvPolynomial.PDeriv | {
"line": 75,
"column": 4
} | {
"line": 75,
"column": 33
} | [
{
"pp": "case refine_1\nR : Type u\nσ : Type v\na : R\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\ni j : σ\nx✝ : j ∈ s.support\nhne : j ≠ i\n⊢ (fun a_1 b ↦ (monomial (s - single a_1 1)) (a * ↑b) * Pi.single i 1 a_1) j (s j) = 0",
"usedConstants": [
"Finsupp.instFunLike",
"Nat.instCanonicallyOrderedAdd",... | simp [Pi.single_eq_of_ne hne] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.PDeriv | {
"line": 75,
"column": 4
} | {
"line": 75,
"column": 33
} | [
{
"pp": "case refine_1\nR : Type u\nσ : Type v\na : R\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\ni j : σ\nx✝ : j ∈ s.support\nhne : j ≠ i\n⊢ (fun a_1 b ↦ (monomial (s - single a_1 1)) (a * ↑b) * Pi.single i 1 a_1) j (s j) = 0",
"usedConstants": [
"Finsupp.instFunLike",
"Nat.instCanonicallyOrderedAdd",... | simp [Pi.single_eq_of_ne hne] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.TensorProduct.Vanishing | {
"line": 139,
"column": 2
} | {
"line": 140,
"column": 66
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\nM : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : Type u_3\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nι : Type u_4\ninst✝ : Fintype ι\nm : ι → M\nn : ι → N\nhm : span R (Set.range m) = ⊤\nhmn : ∑ i, m i ⊗ₜ[R] n i = 0\nG : (ι →₀ R) →ₗ[R] M := linearCo... | have exact_rTensor_ker_subtype : Exact (rTensor N (ker G).subtype) (rTensor N G) :=
rTensor_exact (M := ↥(ker G)) N exact_ker_subtype G_surjective | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.TensorProduct.Vanishing | {
"line": 254,
"column": 4
} | {
"line": 255,
"column": 59
} | [
{
"pp": "case mp\nR : Type u_1\ninst✝⁴ : CommRing R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_3\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\n⊢ (∀ {l : ℕ} {m : Fin l → M} {n : Fin l → N}, ∑ i, m i ⊗ₜ[R] n i = 0 → VanishesTrivially R m n) →\n ∀ (M' : Submodule R M), Injective ⇑(r... | intro h
exact rTensor_injective_of_forall_vanishesTrivially R h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.TensorProduct.Vanishing | {
"line": 254,
"column": 4
} | {
"line": 255,
"column": 59
} | [
{
"pp": "case mp\nR : Type u_1\ninst✝⁴ : CommRing R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_3\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\n⊢ (∀ {l : ℕ} {m : Fin l → M} {n : Fin l → N}, ∑ i, m i ⊗ₜ[R] n i = 0 → VanishesTrivially R m n) →\n ∀ (M' : Submodule R M), Injective ⇑(r... | intro h
exact rTensor_injective_of_forall_vanishesTrivially R h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Localization.Finiteness | {
"line": 74,
"column": 36
} | {
"line": 74,
"column": 44
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : CommSemiring S\nM : Submonoid R\nS' : Type u_4\ninst✝⁵ : CommSemiring S'\ninst✝⁴ : Algebra S S'\ninst✝³ : Algebra R S\ninst✝² : Algebra R S'\ninst✝¹ : IsScalarTower R S S'\ninst✝ : IsLocalization (Submonoid.map (algebraMap R S) M) S'\nx : S\... | g_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Localization.Finiteness | {
"line": 316,
"column": 2
} | {
"line": 317,
"column": 53
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nI : Ideal R\nt : Set R\nht : span t = ⊤\nH : ∀ (g : ↑t), (map (algebraMap R (Localization.Away ↑g)) I).FG\n⊢ Module.Finite R ↥I",
"usedConstants": [
"Semiring.toModule",
"instSMulOfMul",
"IsScalarTower.right",
"Algebra.algebraMap",
"... | let k (g : t) : I →ₗ[R] (I.map (algebraMap R (Localization.Away g.val))) :=
Algebra.idealMap I (S := Localization.Away g.val) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.LinearAlgebra.Multilinear.TensorProduct | {
"line": 43,
"column": 21
} | {
"line": 46,
"column": 38
} | [
{
"pp": "R : Type u_1\nι₁ : Type u_2\nι₂ : Type u_3\nι₃ : Type u_4\nι₄ : Type u_5\ninst✝⁶ : CommSemiring R\nN₁ : Type u_6\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : Module R N₁\nN₂ : Type u_7\ninst✝³ : AddCommMonoid N₂\ninst✝² : Module R N₂\nN : ι₁ ⊕ ι₂ → Type u_8\ninst✝¹ : (i : ι₁ ⊕ ι₂) → AddCommMonoid (N i)\ninst✝ ... | by
rintro _ _ (_ | _) _ _
· letI := Classical.decEq ι₁; simp
· letI := Classical.decEq ι₂; simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.LocalRing.Module | {
"line": 353,
"column": 6
} | {
"line": 355,
"column": 86
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R M\ninst✝⁴ : Module R N\ninst✝³ : IsLocalRing R\ninst✝² : Module.Finite R M\ninst✝¹ : Module.Finite R N\ninst✝ : Free R N\nl : M →ₗ[R] N\nh : Function.Injective ⇑(LinearMap.... | have : Function.Exact (LinearMap.ker l).subtype
(l.codRestrict (LinearMap.range l) (LinearMap.mem_range_self l)) := by
rw [LinearMap.exact_iff, LinearMap.ker_rangeRestrict, Submodule.range_subtype] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Ideal.AssociatedPrime.Basic | {
"line": 180,
"column": 6
} | {
"line": 180,
"column": 11
} | [
{
"pp": "case pos.h.refine_1\nR : Type u_1\ninst✝⁶ : CommSemiring R\nM : Type u_2\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nM' : Type u_3\ninst✝³ : AddCommMonoid M'\ninst✝² : Module R M'\nf : M →ₗ[R] M'\nM'' : Type u_4\ninst✝¹ : AddCommMonoid M''\ninst✝ : Module R M''\ng : M' →ₗ[R] M''\nhf : Function.Inje... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.LinearAlgebra.PiTensorProduct | {
"line": 399,
"column": 26
} | {
"line": 399,
"column": 64
} | [
{
"pp": "ι : Type u_1\nι₂ : Type u_2\nι₃ : Type u_3\nR : Type u_4\ninst✝⁸ : CommSemiring R\nR₁ : Type u_5\nR₂ : Type u_6\ns : ι → Type u_7\ninst✝⁷ : (i : ι) → AddCommMonoid (s i)\ninst✝⁶ : (i : ι) → Module R (s i)\nM : Type u_8\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nE : Type u_9\ninst✝³ : AddCommMonoid... | simp_rw [← smul_add, φ.map_update_add] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
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