module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 811,
"column": 2
} | {
"line": 812,
"column": 74
} | {
"line": 814,
"column": 0
} | [
{
"pp": "R : 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 • ⊤\nhf : ∀ {m : ℕ}, Submodule.factorPow I N ⋯ ∘ₗ f (m + 1) = f m\nm n :... | [] | ext
simp [extend, ← factorPow_comp_eq_of_factorPow_comp_succ_eq ha f hf hle] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 223,
"column": 2
} | {
"line": 223,
"column": 78
} | {
"line": 224,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x\nH : Associates.mk fx.prod = Associates.mk x\n⊢... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x\nH : Associates.mk fx.prod = Associates.mk x\n⊢ (Multiset.r... | rw [← H, ← Associates.prod_mk, Associates.mk_pow, ← Multiset.prod_replicate] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Valuation.ValuationRing | {
"line": 214,
"column": 6
} | {
"line": 214,
"column": 40
} | {
"line": 215,
"column": 6
} | [
{
"pp": "case inl\nA : Type u\ninst✝⁵ : CommRing A\nK : Type v\ninst✝⁴ : Field K\ninst✝³ : Algebra A K\ninst✝² : IsDomain A\ninst✝¹ : ValuationRing A\ninst✝ : IsFractionRing A K\nxa ya : A\nhya : ya ∈ nonZeroDivisors A\nxb yb : A\nhyb : yb ∈ nonZeroDivisors A\nthis✝ : (algebraMap A K) ya ≠ 0\nthis : (algebraMap... | [
"A : Type u\ninst✝⁵ : CommRing A\nK : Type v\ninst✝⁴ : Field K\ninst✝³ : Algebra A K\ninst✝² : IsDomain A\ninst✝¹ : ValuationRing A\ninst✝ : IsFractionRing A K\nxa ya : A\nhya : ya ∈ nonZeroDivisors A\nxb yb : A\nhyb : yb ∈ nonZeroDivisors A\nthis✝ : (algebraMap A K) ya ≠ 0\nthis : (algebraMap A K) yb ≠ 0\nc : A\nh... | apply le_trans _ (le_max_left _ _) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 328,
"column": 2
} | {
"line": 328,
"column": 78
} | {
"line": 329,
"column": 2
} | [
{
"pp": "case h\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsDiscreteValuationRing R\nx : R\nhx : x ≠ 0\nϖ : R\nhirr : Irreducible ϖ\nthis : WfDvdMonoid R\nfx : Multiset R\nhfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x\nH : Associates.mk fx.prod = Associates.mk x\n⊢ Associates.mk ... | [
"case h\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsDiscreteValuationRing R\nx : R\nhx : x ≠ 0\nϖ : R\nhirr : Irreducible ϖ\nthis : WfDvdMonoid R\nfx : Multiset R\nhfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x\nH : Associates.mk fx.prod = Associates.mk x\n⊢ (Multiset.map Associates.m... | rw [← H, ← Associates.prod_mk, Associates.mk_pow, ← Multiset.prod_replicate] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 376,
"column": 54
} | {
"line": 388,
"column": 38
} | {
"line": 390,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsDiscreteValuationRing R\np q : R\nhp : Irreducible p\nhq : Irreducible q\nu v : Rˣ\nm n : ℕ\nh : ↑u * p ^ m = ↑v * q ^ n\n⊢ m = n",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Multiset.prod_replicate",
... | [] | by
have key : Associated (Multiset.replicate m p).prod (Multiset.replicate n q).prod := by
rw [Multiset.prod_replicate, Multiset.prod_replicate, Associated]
refine ⟨u * v⁻¹, ?_⟩
simp only [Units.val_mul]
rw [mul_left_comm, ← mul_assoc, h, mul_right_comm, Units.mul_inv, one_mul]
have := by
refine... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Valuation.Basic | {
"line": 323,
"column": 22
} | {
"line": 323,
"column": 43
} | {
"line": 323,
"column": 43
} | [
{
"pp": "R : Type u_3\nΓ₀ : Type u_4\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommMonoidWithZero Γ₀\nv : Valuation R Γ₀\nx y : R\nh : v y < v x\n⊢ v (x + -y) = v x",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AddGroupWithOne.toAddGroup",
"congrArg",
"Add... | [
"R : Type u_3\nΓ₀ : Type u_4\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommMonoidWithZero Γ₀\nv : Valuation R Γ₀\nx y : R\nh : v y < v x\n⊢ v x = v x",
"case h\nR : Type u_3\nΓ₀ : Type u_4\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommMonoidWithZero Γ₀\nv : Valuation R Γ₀\nx y : R\nh : v y < v x\n⊢ v (-y) < v x"
] | map_add_eq_of_lt_left | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.IsPrincipal | {
"line": 142,
"column": 59
} | {
"line": 144,
"column": 35
} | {
"line": 146,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nx y : Associates ↥R⁰\n⊢ ↑↑((associatesNonZeroDivisorsEquivIsPrincipal R) (x * y)) =\n ↑↑((associatesNonZeroDivisorsEquivIsPrincipal R) x) * ↑↑((associatesNonZeroDivisorsEquivIsPrincipal R) y)",
"ppTerm": "?m.25",
"assigned": true,
"u... | [] | by
simp_rw [associatesNonZeroDivisorsEquivIsPrincipal_coe, map_mul, Submonoid.coe_mul,
associatesEquivIsPrincipal_mul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Ideal.IsPrincipal | {
"line": 168,
"column": 21
} | {
"line": 168,
"column": 76
} | {
"line": 169,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nI : Ideal R\nhprinc : IsPrincipal I\nhI : I ≠ ⊥\nx : R := IsPrincipal.generator I\n⊢ x ≠ 0",
"ppTerm": "?m.87",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"Semiring.toModule",
"congrArg",
"... | [] | by rwa [Ne, ← IsPrincipal.eq_bot_iff_generator_eq_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.DiscreteValuationRing.TFAE | {
"line": 143,
"column": 6
} | {
"line": 143,
"column": 67
} | {
"line": 144,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsLocalRing R\ninst✝ : IsDedekindDomain R\nne_bot : ¬maximalIdeal R = ⊥\na : R\nha₁ : a ∈ maximalIdeal R\nha₂ : a ≠ 0\nhle : Ideal.span {a} ≤ maximalIdeal R\nthis✝ : (Ideal.span {a}).radical = maximalIdeal R\nthis : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}\nn... | [] | exact ⟨k, le_maximalIdeal h ⟨_, ⟨_, hm, rfl⟩, hk'⟩, hk'.symm⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Flat.TorsionFree | {
"line": 96,
"column": 2
} | {
"line": 121,
"column": 8
} | {
"line": 123,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\n⊢ Flat R M ↔ torsion R M = ⊥",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Ideal.isoBaseOfIsPrincipal",
"LinearMap.id",
"Eq.... | [] | refine ⟨fun _ ↦ torsion_eq_bot, ?_⟩
-- now assume R is a Bezout domain and M is a torsionfree R-module
intro htors
-- we need to show that if I is an ideal of R then the natural map I ⊗ M → M is injective
rw [iff_lift_lsmul_comp_subtype_injective]
rintro I hFG
-- If I = 0 this is obvious because I ⊗ M is a ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Flat.TorsionFree | {
"line": 96,
"column": 2
} | {
"line": 121,
"column": 8
} | {
"line": 123,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\n⊢ Flat R M ↔ torsion R M = ⊥",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Ideal.isoBaseOfIsPrincipal",
"LinearMap.id",
"Eq.... | [] | refine ⟨fun _ ↦ torsion_eq_bot, ?_⟩
-- now assume R is a Bezout domain and M is a torsionfree R-module
intro htors
-- we need to show that if I is an ideal of R then the natural map I ⊗ M → M is injective
rw [iff_lift_lsmul_comp_subtype_injective]
rintro I hFG
-- If I = 0 this is obvious because I ⊗ M is a ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Jacobson.Ring | {
"line": 538,
"column": 2
} | {
"line": 538,
"column": 82
} | {
"line": 540,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nP : Ideal R[X]\nhP : P.IsMaximal\ninst✝ : IsJacobsonRing R\nP' : Ideal R := comap C P\nthis✝ : P'.IsPrime\nf : R[X] →+* (R ⧸ P')[X] := mapRingHom (Ideal.Quotient.mk P')\nhf : Function.Surjective ⇑f\nhPJ : P = comap f (Ideal.map f P)\nthis : (Ideal.map (mapRingHom (Ide... | [] | rwa [Quotient.eq_zero_iff_mem, mem_comap, hPJ, mem_comap, coe_mapRingHom, map_C] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.LinearAlgebra.RootSystem.BaseChange | {
"line": 80,
"column": 23
} | {
"line": 83,
"column": 65
} | {
"line": 84,
"column": 2
} | [
{
"pp": "ι : Type u_1\nL : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹² : Field L\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : AddCommGroup N\ninst✝⁹ : Module L M\ninst✝⁸ : Module L N\nP : RootPairing ι L M N\nK : Type u_5\ninst✝⁷ : Field K\ninst✝⁶ : Algebra K L\ninst✝⁵ : Module K M\ninst✝⁴ : Module K N\ninst✝³ : IsSc... | [] | by
have : algebraMap K L 2 = 2 := by
rw [← Int.cast_two (R := K), ← Int.cast_two (R := L), map_intCast]
exact FaithfulSMul.algebraMap_injective K L <| by simp [this] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.RootSystem.RootPositive | {
"line": 84,
"column": 60
} | {
"line": 88,
"column": 50
} | {
"line": 90,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_4\nN : Type u_5\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\nB : P.InvariantForm\ni j : ι\ninst✝¹ : IsDomain R\ninst✝ : NeZero 2\n⊢ (B.form (P.root i)) (P.root j) = 0 ↔ P... | [] | by
calc B.form (P.root i) (P.root j) = 0
↔ 2 * B.form (P.root i) (P.root j) = 0 := by simp [two_ne_zero]
_ ↔ P.pairing i j * B.form (P.root j) (P.root j) = 0 := by rw [B.two_mul_apply_root_root i j]
_ ↔ P.pairing i j = 0 := by simp [B.ne_zero j] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.RootSystem.RootPositive | {
"line": 199,
"column": 4
} | {
"line": 199,
"column": 73
} | {
"line": 200,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nS : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝¹² : CommRing S\ninst✝¹¹ : LinearOrder S\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Algebra S R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nP : RootPairing ι R M N\ninst✝⁴ : P.IsValuedIn S\n... | [] | simpa [map_ofNat] using B.toInvariantForm.two_mul_apply_root_root i j | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.RootSystem.Finite.Nondegenerate | {
"line": 152,
"column": 2
} | {
"line": 154,
"column": 36
} | {
"line": 155,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁶ : Fintype ι\ninst✝¹⁵ : AddCommGroup M\ninst✝¹⁴ : AddCommGroup N\ninst✝¹³ : CommRing R\ninst✝¹² : Module R M\ninst✝¹¹ : Module R N\nP : RootPairing ι R M N\nS : Type u_5\ninst✝¹⁰ : CommRing S\ninst✝⁹ : IsDomain R\ninst✝⁸ : IsDomain S\ninst✝... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁶ : Fintype ι\ninst✝¹⁵ : AddCommGroup M\ninst✝¹⁴ : AddCommGroup N\ninst✝¹³ : CommRing R\ninst✝¹² : Module R M\ninst✝¹¹ : Module R N\nP : RootPairing ι R M N\nS : Type u_5\ninst✝¹⁰ : CommRing S\ninst✝⁹ : IsDomain R\ninst✝⁸ : IsDomain S\ninst✝⁷ : Algebra ... | refine LinearMap.finrank_le_of_isSMulRegular (P.corootSpan S)
(LinearMap.range (M₂ := N) (P.PolarizationIn S))
(smul_right_injective N h_ne) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.RootSystem.Finite.Nondegenerate | {
"line": 158,
"column": 2
} | {
"line": 158,
"column": 44
} | {
"line": 159,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁶ : Fintype ι\ninst✝¹⁵ : AddCommGroup M\ninst✝¹⁴ : AddCommGroup N\ninst✝¹³ : CommRing R\ninst✝¹² : Module R M\ninst✝¹¹ : Module R N\nP : RootPairing ι R M N\nS : Type u_5\ninst✝¹⁰ : CommRing S\ninst✝⁹ : IsDomain R\ninst✝⁸ : IsDomain S\ninst✝... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁶ : Fintype ι\ninst✝¹⁵ : AddCommGroup M\ninst✝¹⁴ : AddCommGroup N\ninst✝¹³ : CommRing R\ninst✝¹² : Module R M\ninst✝¹¹ : Module R N\nP : RootPairing ι R M N\nS : Type u_5\ninst✝¹⁰ : CommRing S\ninst✝⁹ : IsDomain R\ninst✝⁸ : IsDomain S\ninst✝⁷ : Algebra ... | simp_rw [smul_smul, mul_comm, ← smul_smul] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.LinearAlgebra.QuadraticForm.Dual | {
"line": 176,
"column": 4
} | {
"line": 176,
"column": 80
} | {
"line": 177,
"column": 4
} | [
{
"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... | [
"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 : Pairwise fun i ... | simp_rw [hx, hy, neg_smul, Finset.sum_neg_distrib, ← add_eq_zero_iff_eq_neg] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Algebra.Lie.Weights.RootSystem | {
"line": 145,
"column": 16
} | {
"line": 145,
"column": 33
} | {
"line": 145,
"column": 33
} | [
{
"pp": "case neg\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) L\nhα : ¬α.IsZ... | [
"case neg\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) L\nhα : ¬α.IsZero\n⊢ ¬(cha... | ← Nat.succ_le_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Weights.RootSystem | {
"line": 147,
"column": 2
} | {
"line": 148,
"column": 59
} | {
"line": 149,
"column": 2
} | [
{
"pp": "case neg\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) L\nhα : ¬α.IsZ... | [
"case neg\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) L\nhα : ¬α.IsZero\ne : (ch... | apply genWeightSpace_nsmul_add_ne_bot_of_le α β
(Nat.sub_le (chainTopCoeff α β) (chainLength α β).succ) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Lie.Weights.RootSystem | {
"line": 217,
"column": 4
} | {
"line": 217,
"column": 19
} | {
"line": 218,
"column": 4
} | [
{
"pp": "case mpr\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) L\nhα : α.IsNo... | [
"case mpr\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) L\nhα : α.IsNonZero\nn : ℤ... | rintro ⟨h₁, h₂⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.Lie.Weights.RootSystem | {
"line": 251,
"column": 42
} | {
"line": 258,
"column": 31
} | {
"line": 260,
"column": 0
} | [
{
"pp": "K : 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\nhβ' : ⇑β' =... | [] | by
by_cases hα : α.IsZero
· rw [chainLength_of_isZero _ _ hα, chainLength_of_isZero _ _ hα]
· apply Nat.cast_injective (R := ℤ)
rw [← chainTopCoeff_add_chainBotCoeff, ← chainTopCoeff_add_chainBotCoeff,
Nat.cast_add, Nat.cast_add, chainTopCoeff_of_eq_zsmul_add α β hα β' n hβ',
chainBotCoeff_of_eq_z... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.Weights.RootSystem | {
"line": 295,
"column": 88
} | {
"line": 297,
"column": 34
} | {
"line": 299,
"column": 0
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα : Weight K (↥H) L\ninst✝ : Nontrivial L\n... | [] | by
rw [← chainBotCoeff_add_chainTopCoeff, chainTopCoeff_zero_right α hα,
chainBotCoeff_zero_right α hα] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RepresentationTheory.Basic | {
"line": 172,
"column": 2
} | {
"line": 173,
"column": 6
} | {
"line": 175,
"column": 0
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝³ : CommSemiring k\ninst✝² : Monoid G\ninst✝¹ : AddCommMonoid V\ninst✝ : Module k V\nρ : Representation k G V\nr : k\nx : V\n⊢ ρ.asModuleEquiv.symm (r • x) = (algebraMap k k[G]) r • ρ.asModuleEquiv.symm x",
"ppTerm": "?m.65",
"assigned": true,
... | [] | rw [LinearEquiv.symm_apply_eq]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RepresentationTheory.Basic | {
"line": 172,
"column": 2
} | {
"line": 173,
"column": 6
} | {
"line": 175,
"column": 0
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝³ : CommSemiring k\ninst✝² : Monoid G\ninst✝¹ : AddCommMonoid V\ninst✝ : Module k V\nρ : Representation k G V\nr : k\nx : V\n⊢ ρ.asModuleEquiv.symm (r • x) = (algebraMap k k[G]) r • ρ.asModuleEquiv.symm x",
"ppTerm": "?m.65",
"assigned": true,
... | [] | rw [LinearEquiv.symm_apply_eq]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RepresentationTheory.Basic | {
"line": 178,
"column": 2
} | {
"line": 179,
"column": 6
} | {
"line": 181,
"column": 0
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝³ : CommSemiring k\ninst✝² : Monoid G\ninst✝¹ : AddCommMonoid V\ninst✝ : Module k V\nρ : Representation k G V\ng : G\nx : V\n⊢ ρ.asModuleEquiv.symm ((ρ g) x) = (of k G) g • ρ.asModuleEquiv.symm x",
"ppTerm": "?m.61",
"assigned": true,
"usedCons... | [] | rw [LinearEquiv.symm_apply_eq]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RepresentationTheory.Basic | {
"line": 178,
"column": 2
} | {
"line": 179,
"column": 6
} | {
"line": 181,
"column": 0
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝³ : CommSemiring k\ninst✝² : Monoid G\ninst✝¹ : AddCommMonoid V\ninst✝ : Module k V\nρ : Representation k G V\ng : G\nx : V\n⊢ ρ.asModuleEquiv.symm ((ρ g) x) = (of k G) g • ρ.asModuleEquiv.symm x",
"ppTerm": "?m.61",
"assigned": true,
"usedCons... | [] | rw [LinearEquiv.symm_apply_eq]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Weights.RootSystem | {
"line": 346,
"column": 26
} | {
"line": 346,
"column": 58
} | {
"line": 346,
"column": 59
} | [
{
"pp": "case inr.inr.refine_1\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... | [
"case inr.inr.refine_1\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✝) L\nhα : α... | chainBotCoeff_add_chainTopCoeff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RepresentationTheory.Basic | {
"line": 342,
"column": 17
} | {
"line": 346,
"column": 89
} | {
"line": 348,
"column": 0
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝⁴ : Semiring k\ninst✝³ : Group G\ninst✝² : AddCommMonoid V\ninst✝¹ : Module k V\nρ : Representation k G V\nS : Subgroup G\ninst✝ : IsTrivial (MonoidHom.comp ρ S.subtype)\ng h : G\nhgh : ↑g = ↑h\n⊢ ρ g = ρ h",
"ppTerm": "?m.29",
"assigned": true,
... | [] | by
ext x
apply (ρ.apply_bijective g⁻¹).1
simpa [← Module.End.mul_apply, ← map_mul, -isTrivial_def] using
(congr($(isTrivial_def (ρ.comp S.subtype) ⟨g⁻¹ * h, QuotientGroup.eq.1 hgh⟩) x)).symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.RootSystem.Hom | {
"line": 232,
"column": 2
} | {
"line": 233,
"column": 12
} | {
"line": 234,
"column": 2
} | [
{
"pp": "case coweightMap\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\nf g : P.End\nhfg : (coweightHom P) f = (coweightHom P) g\nx : N\n⊢ f.coweightMap x = g.cowei... | [
"case indexEquiv\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\nf g : P.End\nhfg : (coweightHom P) f = (coweightHom P) g\nx : ι\n⊢ f.indexEquiv x = g.indexEquiv x"
] | · dsimp [coweightHom] at hfg
simp_all | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.RootSystem.Chain | {
"line": 53,
"column": 4
} | {
"line": 53,
"column": 17
} | {
"line": 53,
"column": 18
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Finite ι\ninst✝⁷ : CommRing R\ninst✝⁶ : CharZero R\ninst✝⁵ : IsDomain 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\ninst✝ : P.IsCrystallographic\ni j : ι\nh : ∀ (s ... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Finite ι\ninst✝⁷ : CommRing R\ninst✝⁶ : CharZero R\ninst✝⁵ : IsDomain 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\ninst✝ : P.IsCrystallographic\ni j : ι\nh : ∀ (s t : ℤ), s • ... | intro ⟨z, hz⟩ | Lean.Elab.Tactic.evalIntro | null |
Mathlib.LinearAlgebra.RootSystem.Base | {
"line": 105,
"column": 2
} | {
"line": 108,
"column": 10
} | {
"line": 110,
"column": 0
} | [
{
"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✝ : Nontrivial R\ni j : ι\nhi : i ∈ b.support\nhj : j ∈ b.support\nhij : i ≠ j\n⊢ P.root i ≠ ... | [] | intro contra
have := linearIndepOn_iff'.mp b.linearIndepOn_root ({i, j} : Finset ι) 1
(by simp [Set.insert_subset_iff, hi, hj]) (by simp [Finset.sum_pair hij, contra])
simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.Base | {
"line": 105,
"column": 2
} | {
"line": 108,
"column": 10
} | {
"line": 110,
"column": 0
} | [
{
"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✝ : Nontrivial R\ni j : ι\nhi : i ∈ b.support\nhj : j ∈ b.support\nhij : i ≠ j\n⊢ P.root i ≠ ... | [] | intro contra
have := linearIndepOn_iff'.mp b.linearIndepOn_root ({i, j} : Finset ι) 1
(by simp [Set.insert_subset_iff, hi, hj]) (by simp [Finset.sum_pair hij, contra])
simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.Chain | {
"line": 261,
"column": 4
} | {
"line": 261,
"column": 45
} | {
"line": 262,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Finite ι\ninst✝⁷ : CommRing R\ninst✝⁶ : CharZero R\ninst✝⁵ : IsDomain 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\ninst✝ : P.IsCrystallographic\ni j : ι\nthis : In... | [] | rw [P.chainCoeff_reflectionPerm_left_aux] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.RootSystem.Chain | {
"line": 272,
"column": 4
} | {
"line": 272,
"column": 45
} | {
"line": 273,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Finite ι\ninst✝⁷ : CommRing R\ninst✝⁶ : CharZero R\ninst✝⁵ : IsDomain 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\ninst✝ : P.IsCrystallographic\ni j : ι\nthis : In... | [] | rw [P.chainCoeff_reflectionPerm_left_aux] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas | {
"line": 348,
"column": 4
} | {
"line": 348,
"column": 52
} | {
"line": 349,
"column": 2
} | [
{
"pp": "case pos\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Module R N\nP : RootPairing ι R M N\ninst✝⁵ : Finite ι\ninst✝⁴ : CharZero R\ninst✝³ : P.IsCrystallographic\ninst✝² : IsDomain R\ninst✝¹... | [] | rwa [h i j, mul_left_inj' (B.ne_zero j)] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.LinearAlgebra.Matrix.BilinearForm | {
"line": 356,
"column": 2
} | {
"line": 356,
"column": 49
} | {
"line": 358,
"column": 0
} | [
{
"pp": "R₁ : Type u_1\ninst✝² : CommSemiring R₁\nn : Type u_5\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nM : Matrix n n R₁\n⊢ (toBilin' M).IsSymm ↔ M.IsSymm",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Pi.Function.module",
"Algebra.to_smulCommClass",
"Semiring.toMod... | [] | simp [← M.toBilin'.isSymm_toMatrix'_iff_isSymm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Matrix.BilinearForm | {
"line": 356,
"column": 2
} | {
"line": 356,
"column": 49
} | {
"line": 358,
"column": 0
} | [
{
"pp": "R₁ : Type u_1\ninst✝² : CommSemiring R₁\nn : Type u_5\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nM : Matrix n n R₁\n⊢ (toBilin' M).IsSymm ↔ M.IsSymm",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Pi.Function.module",
"Algebra.to_smulCommClass",
"Semiring.toMod... | [] | simp [← M.toBilin'.isSymm_toMatrix'_iff_isSymm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.BilinearForm | {
"line": 356,
"column": 2
} | {
"line": 356,
"column": 49
} | {
"line": 358,
"column": 0
} | [
{
"pp": "R₁ : Type u_1\ninst✝² : CommSemiring R₁\nn : Type u_5\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nM : Matrix n n R₁\n⊢ (toBilin' M).IsSymm ↔ M.IsSymm",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Pi.Function.module",
"Algebra.to_smulCommClass",
"Semiring.toMod... | [] | simp [← M.toBilin'.isSymm_toMatrix'_iff_isSymm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 554,
"column": 18
} | {
"line": 554,
"column": 96
} | {
"line": 554,
"column": 96
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : DivisionRing β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : CauSeq β abv\nhf : ¬f.LimZero\nε : α\nε0 : ε > 0\nK : α\nK0 : K > 0\ni : ℕ\nH : ∀ j ≥ i, K ≤ abv (↑f j)\nj : ℕ\nij : j ≥ i\n⊢ abv ... | [] | by simpa [(abv_pos abv).1 (lt_of_lt_of_le K0 (H _ ij)), abv_zero abv] using ε0 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 558,
"column": 18
} | {
"line": 558,
"column": 96
} | {
"line": 558,
"column": 96
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : DivisionRing β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : CauSeq β abv\nhf : ¬f.LimZero\nε : α\nε0 : ε > 0\nK : α\nK0 : K > 0\ni : ℕ\nH : ∀ j ≥ i, K ≤ abv (↑f j)\nj : ℕ\nij : j ≥ i\n⊢ abv ... | [] | by simpa [(abv_pos abv).1 (lt_of_lt_of_le K0 (H _ ij)), abv_zero abv] using ε0 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 833,
"column": 2
} | {
"line": 833,
"column": 22
} | {
"line": 834,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na b c : CauSeq α abs\nha : a ≤ c\nhb : b ≤ c\n⊢ a ⊔ b ≤ c",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"CauSeq.instLTAbs._proof_1",
"CauSeq.instLTAbs",
"AddGroupWithOne.toAd... | [
"case inl\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na b c : CauSeq α abs\nhb : b ≤ c\nha : a < c\n⊢ a ⊔ b ≤ c",
"case inr\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na b c : CauSeq α abs\nhb : b ≤ c\nha : a ≈ c\n⊢ a ⊔ b ≤ c"... | obtain ha | ha := ha | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 839,
"column": 4
} | {
"line": 839,
"column": 52
} | {
"line": 840,
"column": 4
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na b c : CauSeq α abs\nhb : b ≤ c\nha : a ≈ c\n⊢ a ⊔ b ≤ c",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"AddGroupWithOne.toAddGroup",
"abs",
"IsAbsoluteValue.abs_isA... | [
"case inr\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na b c : CauSeq α abs\nha : a ≈ c\nhb : b ≤ a\n⊢ a ⊔ b ≤ c"
] | replace hb := le_of_le_of_eq hb (Setoid.symm ha) | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.Data.Real.Basic | {
"line": 551,
"column": 2
} | {
"line": 551,
"column": 60
} | {
"line": 553,
"column": 0
} | [
{
"pp": "f : CauSeq ℚ abs\nx : ℝ\ni : ℕ\nH : ∀ j ≥ i, ↑(↑f j) ≤ x\n⊢ -x ≤ mk (-f)",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"Real.le_mk_of_forall_le",
"Real.instLE",
"Real",
"DivisionRing.toRatC... | [] | exact le_mk_of_forall_le ⟨i, fun j ij => by simp [H _ ij]⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.Matrix.Vec | {
"line": 51,
"column": 64
} | {
"line": 52,
"column": 76
} | {
"line": 54,
"column": 0
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_3\nA B : Matrix m n R\n⊢ A.vec = B.vec ↔ A = B",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matrix",
"forall_comm",
"id",
"Prod.mk",
"_private.Mathlib.LinearAlgebra.Matrix.Ve... | [] | by
simp_rw [← Matrix.ext_iff, funext_iff, Prod.forall, @forall_comm m n, vec] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.Hermitian | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 36
} | {
"line": 112,
"column": 2
} | [
{
"pp": "α : Type u_1\nm : Type u_3\nn : Type u_4\ninst✝ : Star α\nA : Matrix n n α\nf : n ≃ m\n⊢ ((reindex f f) A).IsHermitian ↔ A.IsHermitian",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Equiv.instEquivLike",
"Matrix",
"Equiv",
"Matrix.IsHermitian.reindex",
... | [
"α : Type u_1\nm : Type u_3\nn : Type u_4\ninst✝ : Star α\nA : Matrix n n α\nf : n ≃ m\nh : ((reindex f f) A).IsHermitian\n⊢ A.IsHermitian"
] | refine ⟨fun h ↦ ?_, (·.reindex f)⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.Matrix.ZPow | {
"line": 194,
"column": 86
} | {
"line": 199,
"column": 27
} | {
"line": 201,
"column": 0
} | [
{
"pp": "n' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA B : M\nh : Commute A B\nm : ℤ\n⊢ Commute A (B ^ m)",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"False",
"Nat.instMulZeroClass",
... | [] | by
rcases nonsing_inv_cancel_or_zero B with (⟨hB, _⟩ | hB)
· refine SemiconjBy.zpow_right ?_ ?_ h _ <;> exact isUnit_det_of_left_inverse hB
· cases m
· simpa using h.pow_right _
· simp [← inv_pow', hB] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 173,
"column": 62
} | {
"line": 176,
"column": 84
} | {
"line": 178,
"column": 0
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_3\ninst✝² : Ring R\ninst✝¹ : PartialOrder R\ninst✝ : StarRing R\nM : Matrix n n R\nhM : M.PosDef\ne : m → n\nhe : Function.Injective e\n⊢ (M.submatrix e e).PosDef",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"add_mul... | [] | by
refine ⟨hM.1.submatrix _, fun x hx ↦ ?_⟩
simpa [Finsupp.sum_mapDomain_index, add_mul, mul_add] using
hM.2 <| Finsupp.mapDomain_injective he |>.ne_iff' Finsupp.mapDomain_zero |>.2 hx | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 194,
"column": 4
} | {
"line": 199,
"column": 95
} | {
"line": 201,
"column": 0
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝⁵ : Ring R\ninst✝⁴ : PartialOrder R\ninst✝³ : StarRing R\ninst✝² : StarOrderedRing R\ninst✝¹ : DecidableEq n\ninst✝ : NoZeroDivisors R\nd : n → R\nh : ∀ (i : n), 0 < d i\nx : n →₀ R\nhx : x ≠ 0\n⊢ 0 < x.sum fun i xi ↦ x.sum fun j xj ↦ star xi * diagonal d i j * xj",
... | [] | refine Finsupp.sum_pos' (fun _ _ ↦ Finsupp.sum_nonneg ?_) ?_
· simp +contextual [diagonal, apply_ite, star_left_conjugate_nonneg (h _).le]
obtain ⟨i, hxi⟩ := by simpa [Finsupp.ext_iff] using hx
refine ⟨i, ?_, Finsupp.sum_pos' ?_ ⟨i, ?_, ?_⟩⟩ <;> simp +contextual [diagonal,
apply_ite, star_left_conjuga... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 194,
"column": 4
} | {
"line": 199,
"column": 95
} | {
"line": 201,
"column": 0
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝⁵ : Ring R\ninst✝⁴ : PartialOrder R\ninst✝³ : StarRing R\ninst✝² : StarOrderedRing R\ninst✝¹ : DecidableEq n\ninst✝ : NoZeroDivisors R\nd : n → R\nh : ∀ (i : n), 0 < d i\nx : n →₀ R\nhx : x ≠ 0\n⊢ 0 < x.sum fun i xi ↦ x.sum fun j xj ↦ star xi * diagonal d i j * xj",
... | [] | refine Finsupp.sum_pos' (fun _ _ ↦ Finsupp.sum_nonneg ?_) ?_
· simp +contextual [diagonal, apply_ite, star_left_conjugate_nonneg (h _).le]
obtain ⟨i, hxi⟩ := by simpa [Finsupp.ext_iff] using hx
refine ⟨i, ?_, Finsupp.sum_pos' ?_ ⟨i, ?_, ?_⟩⟩ <;> simp +contextual [diagonal,
apply_ite, star_left_conjuga... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 156,
"column": 46
} | {
"line": 156,
"column": 67
} | {
"line": 156,
"column": 67
} | [
{
"pp": "n : ℕ\n⊢ 1 = n.ascFactorial 0",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"congrArg",
"Nat.ascFactorial_zero",
"Nat.ascFactorial",
"id",
"instOfNatNat",
"AddCommMonoidWithOne.t... | [
"n : ℕ\n⊢ 1 = 1"
] | Nat.ascFactorial_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Vandermonde | {
"line": 139,
"column": 2
} | {
"line": 143,
"column": 43
} | {
"line": 145,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\nv w : Fin (n + 1) → R\ni : Fin (n + 1)\nhw : w i = 0\n⊢ projVandermonde v w i = Pi.single (last n) (v i ^ n)",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instNeZeroNatHAdd_1",
"MulOne.toOne",
"LE.le.eq_... | [] | ext j
obtain rfl | hlt := j.le_last.eq_or_lt
· simp [projVandermonde_apply]
rw [projVandermonde_apply, Pi.single_eq_of_ne hlt.ne, hw, zero_pow, mul_zero]
simpa [Nat.sub_eq_zero_iff_le] using! hlt | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Vandermonde | {
"line": 139,
"column": 2
} | {
"line": 143,
"column": 43
} | {
"line": 145,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\nv w : Fin (n + 1) → R\ni : Fin (n + 1)\nhw : w i = 0\n⊢ projVandermonde v w i = Pi.single (last n) (v i ^ n)",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instNeZeroNatHAdd_1",
"MulOne.toOne",
"LE.le.eq_... | [] | ext j
obtain rfl | hlt := j.le_last.eq_or_lt
· simp [projVandermonde_apply]
rw [projVandermonde_apply, Pi.single_eq_of_ne hlt.ne, hw, zero_pow, mul_zero]
simpa [Nat.sub_eq_zero_iff_le] using! hlt | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 425,
"column": 8
} | {
"line": 425,
"column": 31
} | {
"line": 425,
"column": 31
} | [
{
"pp": "case succ\nR : Type u\ninst✝ : Ring R\nk n : ℕ\nih : k < n → eval (-↑k) (ascPochhammer R n) = 0\nh : k < n + 1\n⊢ eval (-↑k) (ascPochhammer R (n + 1)) = 0",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
"NegZeroClass.toNeg",
"No... | [
"case succ\nR : Type u\ninst✝ : Ring R\nk n : ℕ\nih : k < n → eval (-↑k) (ascPochhammer R n) = 0\nh : k < n + 1\n⊢ eval (-↑k) (ascPochhammer R n) * (-↑k + ↑n) = 0"
] | ascPochhammer_succ_eval | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 427,
"column": 4
} | {
"line": 427,
"column": 19
} | {
"line": 428,
"column": 4
} | [
{
"pp": "case succ.inl\nR : Type u\ninst✝ : Ring R\nk n : ℕ\nih : k < n → eval (-↑k) (ascPochhammer R n) = 0\nh : k < n + 1\nhkn : k < n\n⊢ eval (-↑k) (ascPochhammer R n) * (-↑k + ↑n) = 0",
"ppTerm": "?succ.inl",
"assigned": true,
"usedConstants": [
"Polynomial.eval",
"NegZeroClass.toNeg... | [
"case succ.inr.inl\nR : Type u\ninst✝ : Ring R\nk : ℕ\nih : k < k → eval (-↑k) (ascPochhammer R k) = 0\nh : k < k + 1\n⊢ eval (-↑k) (ascPochhammer R k) * (-↑k + ↑k) = 0",
"case succ.inr.inr\nR : Type u\ninst✝ : Ring R\nk n : ℕ\nih : k < n → eval (-↑k) (ascPochhammer R n) = 0\nh : k < n + 1\nhkn : n < k\n⊢ eval (-... | · simp [ih hkn] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.RootSystem.CartanMatrix | {
"line": 344,
"column": 4
} | {
"line": 344,
"column": 43
} | {
"line": 345,
"column": 4
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝²² : CommRing R\ninst✝²¹ : AddCommGroup M\ninst✝²⁰ : Module R M\ninst✝¹⁹ : AddCommGroup N\ninst✝¹⁸ : Module R N\nS : Type u_5\ninst✝¹⁷ : CommRing S\ninst✝¹⁶ : Algebra S R\nP✝ : RootPairing ι R M N\ninst✝¹⁵ : P✝.IsValuedIn S\nb✝ : P✝.Base\nins... | [
"case refine_1\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝²² : CommRing R\ninst✝²¹ : AddCommGroup M\ninst✝²⁰ : Module R M\ninst✝¹⁹ : AddCommGroup N\ninst✝¹⁸ : Module R N\nS : Type u_5\ninst✝¹⁷ : CommRing S\ninst✝¹⁶ : Algebra S R\nP✝ : RootPairing ι R M N\ninst✝¹⁵ : P✝.IsValuedIn S\nb✝ : P✝.Base\n... | refine fun m ↦ ⟨fun h ↦ ?_, fun h ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.Lagrange | {
"line": 217,
"column": 2
} | {
"line": 218,
"column": 32
} | {
"line": 220,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\nv : ι → F\ni j : ι\nhij : i ≠ j\n⊢ Lagrange.basis {i, j} v j = basisDivisor (v j) (v i)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Finset.pair_comm",
"Eq.mpr",
"congrArg",
"Finset",
... | [] | rw [pair_comm]
exact basis_pair_left hij.symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Lagrange | {
"line": 217,
"column": 2
} | {
"line": 218,
"column": 32
} | {
"line": 220,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\nv : ι → F\ni j : ι\nhij : i ≠ j\n⊢ Lagrange.basis {i, j} v j = basisDivisor (v j) (v i)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Finset.pair_comm",
"Eq.mpr",
"congrArg",
"Finset",
... | [] | rw [pair_comm]
exact basis_pair_left hij.symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Lagrange | {
"line": 497,
"column": 81
} | {
"line": 501,
"column": 12
} | {
"line": 503,
"column": 0
} | [
{
"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]\nhP : P.degree < ↑(#s)\n⊢ P.coeff (#s - 1) = ∑ i ∈ s, eval (v i) P / ∏ j ∈ s.erase i, (v i - v j)",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Mat... | [] | by
nth_rewrite 1 [eq_interpolate hvs hP, interpolate_apply, finsetSum_coeff]
congr! with i hi
rw [coeff_C_mul, ← natDegree_basis hvs hi, ← leadingCoeff, leadingCoeff_basis hvs hi]
field_simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.MvPolynomial.Monad | {
"line": 306,
"column": 4
} | {
"line": 318,
"column": 18
} | {
"line": 319,
"column": 2
} | [
{
"pp": "case calc_1\nσ : Type u_1\nτ : Type u_2\nR : Type u_3\ninst✝¹ : CommSemiring R\ninst✝ : DecidableEq τ\nf : σ → MvPolynomial τ R\nφ : MvPolynomial σ R\n⊢ (φ.support.biUnion fun d ↦ (C (coeff d φ) * ∏ i ∈ d.support, f i ^ d i).vars) ≤\n φ.support.biUnion fun d ↦ d.support.biUnion fun i ↦ (f i).vars",
... | [] | apply Finset.biUnion_mono
intro d _hd
calc
vars (C (coeff d φ) * ∏ i ∈ d.support, f i ^ d i) ≤
(C (coeff d φ)).vars ∪ (∏ i ∈ d.support, f i ^ d i).vars :=
vars_mul _ _
_ ≤ (∏ i ∈ d.support, f i ^ d i).vars := by
simp only [Finset.empty_union, vars_C, Finset.le_iff_subset, F... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Monad | {
"line": 306,
"column": 4
} | {
"line": 318,
"column": 18
} | {
"line": 319,
"column": 2
} | [
{
"pp": "case calc_1\nσ : Type u_1\nτ : Type u_2\nR : Type u_3\ninst✝¹ : CommSemiring R\ninst✝ : DecidableEq τ\nf : σ → MvPolynomial τ R\nφ : MvPolynomial σ R\n⊢ (φ.support.biUnion fun d ↦ (C (coeff d φ) * ∏ i ∈ d.support, f i ^ d i).vars) ≤\n φ.support.biUnion fun d ↦ d.support.biUnion fun i ↦ (f i).vars",
... | [] | apply Finset.biUnion_mono
intro d _hd
calc
vars (C (coeff d φ) * ∏ i ∈ d.support, f i ^ d i) ≤
(C (coeff d φ)).vars ∪ (∏ i ∈ d.support, f i ^ d i).vars :=
vars_mul _ _
_ ≤ (∏ i ∈ d.support, f i ^ d i).vars := by
simp only [Finset.empty_union, vars_C, Finset.le_iff_subset, F... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.CartanCriterion | {
"line": 146,
"column": 46
} | {
"line": 146,
"column": 88
} | {
"line": 146,
"column": 89
} | [
{
"pp": "L : Type u_2\nM : Type u_3\ninst✝⁹ : LieRing L\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : LieRingModule L M\nK : Type u_4\ninst✝⁶ : Field K\ninst✝⁵ : CharZero K\ninst✝⁴ : IsAlgClosed K\ninst✝³ : LieAlgebra K L\ninst✝² : Module K M\ninst✝¹ : LieModule K L M\ninst✝ : FiniteDimensional K M\nh : traceForm K L M = ... | [
"L : Type u_2\nM : Type u_3\ninst✝⁹ : LieRing L\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : LieRingModule L M\nK : Type u_4\ninst✝⁶ : Field K\ninst✝⁵ : CharZero K\ninst✝⁴ : IsAlgClosed K\ninst✝³ : LieAlgebra K L\ninst✝² : Module K M\ninst✝¹ : LieModule K L M\ninst✝ : FiniteDimensional K M\nh : traceForm K L M = 0\nφ : L →ₗ⁅... | aeval_apply_of_mem_apply_eq_smul (h₁ i j), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Charpoly.ToMatrix | {
"line": 68,
"column": 2
} | {
"line": 69,
"column": 68
} | {
"line": 71,
"column": 0
} | [
{
"pp": "R : Type u_1\nM₁ : Type u_3\nM₂ : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M₁\ninst✝⁶ : Module R M₁\ninst✝⁵ : Module.Finite R M₁\ninst✝⁴ : Free R M₁\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : Module.Finite R M₂\ninst✝ : Free R M₂\nf₁ : M₁ →ₗ[R] M₁\nf₂ : M₂ →ₗ[R] M₂\nb₁ : Basi... | [] | rw [← charpoly_toMatrix f₁ b₁, ← charpoly_toMatrix f₂ b₂, ← charpoly_toMatrix (f₁.prodMap f₂) b,
toMatrix_prodMap b₁ b₂ f₁ f₂, Matrix.charpoly_fromBlocks_zero₁₂] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Finsupp.Weight | {
"line": 295,
"column": 4
} | {
"line": 295,
"column": 47
} | {
"line": 296,
"column": 4
} | [
{
"pp": "case succ\nσ : Type u_5\nf g : σ →₀ ℕ\nhgf : g ≤ f\nIH : degree g ≤ degree f → ∃ g_1 ≤ f, degree g_1 = degree g\nhn : degree g + 1 ≤ degree f\n⊢ ∃ g_1 ≤ f, degree g_1 = degree g + 1",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Finsupp.instCanonicallyOrderedAddOfAddLeftMo... | [
"case succ\nσ : Type u_5\ng f : σ →₀ ℕ\nhgf : g ≤ g + f\nIH : degree g ≤ degree (g + f) → ∃ g_1 ≤ g + f, degree g_1 = degree g\nhn : degree g + 1 ≤ degree (g + f)\n⊢ ∃ g_1 ≤ g + f, degree g_1 = degree g + 1"
] | obtain ⟨f, rfl⟩ := le_iff_exists_add.mp hgf | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Data.Finsupp.Weight | {
"line": 305,
"column": 2
} | {
"line": 305,
"column": 45
} | {
"line": 306,
"column": 2
} | [
{
"pp": "σ : Type u_5\ns t : Set ℕ\nf : σ →₀ ℕ\nn : ℕ\nhn : n ∈ t\ng : σ →₀ ℕ\nhgf : g ≤ f\nhm : degree g ∈ s\ne : degree g + n = degree f\n⊢ f ∈ ⇑degree ⁻¹' s + ⇑degree ⁻¹' t",
"ppTerm": "?m.108",
"assigned": true,
"usedConstants": [
"Finsupp.instCanonicallyOrderedAddOfAddLeftMono",
"Fi... | [
"σ : Type u_5\ns t : Set ℕ\nn : ℕ\nhn : n ∈ t\ng : σ →₀ ℕ\nhm : degree g ∈ s\nf : σ →₀ ℕ\nhgf : g ≤ g + f\ne : degree g + n = degree (g + f)\n⊢ g + f ∈ ⇑degree ⁻¹' s + ⇑degree ⁻¹' t"
] | obtain ⟨f, rfl⟩ := le_iff_exists_add.mp hgf | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Lie.CartanCriterion | {
"line": 188,
"column": 8
} | {
"line": 188,
"column": 32
} | {
"line": 188,
"column": 32
} | [
{
"pp": "R : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : LieRing L\ninst✝⁶ : LieAlgebra R L\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : LieRingModule L M\ninst✝³ : Module R M\ninst✝² : LieModule R L M\ninst✝¹ : IsNoetherian R M\ninst✝ : Free R M\nh : tra... | [
"R : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : LieRing L\ninst✝⁶ : LieAlgebra R L\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : LieRingModule L M\ninst✝³ : Module R M\ninst✝² : LieModule R L M\ninst✝¹ : IsNoetherian R M\ninst✝ : Free R M\nh : traceForm R L M... | derivedSeries_baseChange | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.CartanCriterion | {
"line": 216,
"column": 2
} | {
"line": 217,
"column": 71
} | {
"line": 218,
"column": 2
} | [
{
"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... | [
"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 L := ⁅DI, D... | have module_nilp : LieModule.IsNilpotent (derivedSeries R DI 1) L :=
LieModule.isNilpotent_derivedSeries_of_traceForm_eq_zero tf_eq_zero | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous | {
"line": 294,
"column": 2
} | {
"line": 297,
"column": 26
} | {
"line": 299,
"column": 0
} | [
{
"pp": "case a\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\nσ : Type u_3\ninst✝¹ : AddCommMonoid M\nφ : MvPolynomial σ R\nn : M\ninst✝ : SemilatticeSup M\nw : σ → M\nhφ : IsWeightedHomogeneous w φ n\nh : φ ≠ 0\n⊢ ↑n ≤ φ.support.sup fun s ↦ ↑((weight w) s)",
"ppTerm": "?a✝",
"assigned": true,
... | [] | · obtain ⟨d, hd⟩ : ∃ d, coeff d φ ≠ 0 := exists_coeff_ne_zero h
simp only [← hφ hd]
replace hd := Finsupp.mem_support_iff.mpr hd
apply Finset.le_sup hd | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Lie.CartanCriterion | {
"line": 239,
"column": 9
} | {
"line": 239,
"column": 70
} | {
"line": 239,
"column": 70
} | [
{
"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\nh : ∀ (x y : L), y ∈ derivedSeries R L 1 → ((killingForm R L) x) y = 0\nthis : IsSolvable ↥⊤\n⊢ IsSolvable L",
... | [
"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\nh : ∀ (x y : L), y ∈ derivedSeries R L 1 → ((killingForm R L) x) y = 0\nthis : IsSolvable ↥⊤\n⊢ IsSolvable ↥⊤"
] | ← solvable_iff_equiv_solvable LieSubalgebra.topEquiv (R := R) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous | {
"line": 317,
"column": 8
} | {
"line": 317,
"column": 16
} | {
"line": 317,
"column": 16
} | [
{
"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\nmotive : (p : MvPolynomial σ R) → IsWeightedHomogeneous w p m → Prop\nzero : motive 0 ⋯\nadd :\n ∀ (p q : MvPolynomial σ R) (hp : IsWeightedHomogeneous w p m) (hq : IsWeightedHomogeneous w q m... | [] | apply hx | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous | {
"line": 455,
"column": 4
} | {
"line": 457,
"column": 42
} | {
"line": 459,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\nM : Type u_2\ninst✝¹ : CommSemiring R\nσ : Type u_3\ninst✝ : AddCommMonoid M\nw : σ → M\nm n : M\np : MvPolynomial σ R\nhp : IsWeightedHomogeneous w p m\nhn : n ≠ m\nx : σ →₀ ℕ\nzero_coeff : ¬coeff x p = 0\n⊢ (if (weight w) x = n then coeff x p else 0) = coeff x 0",
"ppTerm"... | [] | rw [if_neg]
· rw [coeff_zero]
· rw [hp zero_coeff]; exact Ne.symm hn | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous | {
"line": 455,
"column": 4
} | {
"line": 457,
"column": 42
} | {
"line": 459,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\nM : Type u_2\ninst✝¹ : CommSemiring R\nσ : Type u_3\ninst✝ : AddCommMonoid M\nw : σ → M\nm n : M\np : MvPolynomial σ R\nhp : IsWeightedHomogeneous w p m\nhn : n ≠ m\nx : σ →₀ ℕ\nzero_coeff : ¬coeff x p = 0\n⊢ (if (weight w) x = n then coeff x p else 0) = coeff x 0",
"ppTerm"... | [] | rw [if_neg]
· rw [coeff_zero]
· rw [hp zero_coeff]; exact Ne.symm hn | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.LinearMap.Polynomial | {
"line": 416,
"column": 2
} | {
"line": 417,
"column": 55
} | {
"line": 418,
"column": 2
} | [
{
"pp": "R : Type u_1\nL : Type u_2\nM : Type u_3\nι : Type u_5\nι' : Type u_6\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup L\ninst✝⁸ : Module R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nφ : L →ₗ[R] End R M\ninst✝⁵ : Fintype ι\ninst✝⁴ : Fintype ι'\ninst✝³ : DecidableEq ι\ninst✝² : DecidableEq ι'\ninst✝¹ ... | [
"R : Type u_1\nL : Type u_2\nM : Type u_3\nι : Type u_5\nι' : Type u_6\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup L\ninst✝⁸ : Module R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nφ : L →ₗ[R] End R M\ninst✝⁵ : Fintype ι\ninst✝⁴ : Fintype ι'\ninst✝³ : DecidableEq ι\ninst✝² : DecidableEq ι'\ninst✝¹ : Free R M\n... | have : toMvPolynomial b' B φ = fun i ↦ (MvPolynomial.bind₁ g) (toMvPolynomial b B φ i) :=
funext <| toMvPolynomial_comp b' b B φ LinearMap.id | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Lie.Basis | {
"line": 370,
"column": 19
} | {
"line": 370,
"column": 67
} | {
"line": 370,
"column": 67
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nL : Type u_3\ninst✝⁶ : Finite ι\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\nb : Basis ι R L\ninst✝² : Fintype ι\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\nx u v : L\nhx✝ : u ∈ lieSpan R L (range b.e)\nhy✝ : v ∈ lieSpan R L (range b.e)\nhu : u ∈ ⨆ n, ⨆ (... | [
"ι : Type u_1\nR : Type u_2\nL : Type u_3\ninst✝⁶ : Finite ι\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\nb : Basis ι R L\ninst✝² : Fintype ι\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\nx u v : L\nhx✝ : u ∈ lieSpan R L (range b.e)\nhy✝ : v ∈ lieSpan R L (range b.e)\nhu : u ∈ ⨆ n, ⨆ (_ : n ≠ 0), ... | rw [iSup_subtype', iSup_subtype', ← e.iSup_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Lie.Basis | {
"line": 372,
"column": 19
} | {
"line": 372,
"column": 67
} | {
"line": 372,
"column": 67
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nL : Type u_3\ninst✝⁶ : Finite ι\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\nb : Basis ι R L\ninst✝² : Fintype ι\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\nx u v : L\nhx✝ : u ∈ lieSpan R L (range b.e)\nhy✝ : v ∈ lieSpan R L (range b.e)\nhv : v ∈ ⨆ n, ⨆ (... | [
"ι : Type u_1\nR : Type u_2\nL : Type u_3\ninst✝⁶ : Finite ι\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\nb : Basis ι R L\ninst✝² : Fintype ι\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\nx u v : L\nhx✝ : u ∈ lieSpan R L (range b.e)\nhy✝ : v ∈ lieSpan R L (range b.e)\nhv : v ∈ ⨆ n, ⨆ (_ : n ≠ 0), ... | rw [iSup_subtype', iSup_subtype', ← e.iSup_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Lie.Basis | {
"line": 374,
"column": 4
} | {
"line": 374,
"column": 52
} | {
"line": 374,
"column": 52
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nL : Type u_3\ninst✝⁶ : Finite ι\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\nb : Basis ι R L\ninst✝² : Fintype ι\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\nx u v : L\nhx✝ : u ∈ lieSpan R L (range b.e)\nhy✝ : v ∈ lieSpan R L (range b.e)\ns : Set (↥b.carta... | [
"ι : Type u_1\nR : Type u_2\nL : Type u_3\ninst✝⁶ : Finite ι\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\nb : Basis ι R L\ninst✝² : Fintype ι\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\nx u v : L\nhx✝ : u ∈ lieSpan R L (range b.e)\nhy✝ : v ∈ lieSpan R L (range b.e)\ns : Set (↥b.cartan → R) := {χ... | rw [iSup_subtype', iSup_subtype', ← e.iSup_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Lie.Extension | {
"line": 330,
"column": 44
} | {
"line": 330,
"column": 69
} | {
"line": 330,
"column": 69
} | [
{
"pp": "R : Type u_1\nN : Type u_2\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\nx y : L\nm : M\nh : HasRightInverse ⇑E.proj := Surjective.hasRightInverse (proj_surjective ... | [] | by simp [h.choose_spec _] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.Extension | {
"line": 336,
"column": 46
} | {
"line": 336,
"column": 71
} | {
"line": 336,
"column": 71
} | [
{
"pp": "R : Type u_1\nN : Type u_2\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\nx y : L\nm : M\nh : HasRightInverse ⇑E.proj := Surjective.hasRightInverse (proj_surjective ... | [] | by simp [h.choose_spec _] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.Extension | {
"line": 363,
"column": 48
} | {
"line": 363,
"column": 73
} | {
"line": 363,
"column": 73
} | [
{
"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\nthis : LieRingModule L M := E.ringModuleOf\nh : HasRightInverse ⇑E.proj := Surjective.hasRightInverse (pr... | [] | by simp [h.choose_spec _] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.LieTheorem | {
"line": 141,
"column": 2
} | {
"line": 145,
"column": 39
} | {
"line": 146,
"column": 2
} | [
{
"pp": "case inr\nR : Type u_1\nL : Type u_2\nA : Type u_3\nV : Type u_4\ninst✝¹⁹ : CommRing R\ninst✝¹⁸ : IsPrincipalIdealRing R\ninst✝¹⁷ : IsDomain R\ninst✝¹⁶ : CharZero R\ninst✝¹⁵ : LieRing L\ninst✝¹⁴ : LieAlgebra R L\ninst✝¹³ : LieRing A\ninst✝¹² : LieAlgebra R A\ninst✝¹¹ : Bracket L A\ninst✝¹⁰ : Bracket A ... | [
"case inr\nR : Type u_1\nL : Type u_2\nA : Type u_3\nV : Type u_4\ninst✝¹⁹ : CommRing R\ninst✝¹⁸ : IsPrincipalIdealRing R\ninst✝¹⁷ : IsDomain R\ninst✝¹⁶ : CharZero R\ninst✝¹⁵ : LieRing L\ninst✝¹⁴ : LieAlgebra R L\ninst✝¹³ : LieRing A\ninst✝¹² : LieAlgebra R A\ninst✝¹¹ : Bracket L A\ninst✝¹⁰ : Bracket A L\ninst✝⁹ : ... | have hvU : v ∈ U := by
apply Submodule.mem_iSup_of_mem 1
apply Submodule.subset_span
use 0, zero_lt_one
rw [pow_zero, Module.End.one_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Lie.Weights.IsSimple | {
"line": 198,
"column": 4
} | {
"line": 198,
"column": 45
} | {
"line": 199,
"column": 4
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : CharZero K\ninst✝¹ : IsKilling K L\ninst✝ : IsTriangularizable K (↥H) L\nI : LieIdeal K L\nα : ↥LieSubalgebra.root\nh... | [
"K : Type u_1\nL : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : CharZero K\ninst✝¹ : IsKilling K L\ninst✝ : IsTriangularizable K (↥H) L\nI : LieIdeal K L\nα : ↥LieSubalgebra.root\nhα_span : Wei... | rw [LieIdeal.rootSpan, Submodule.span_le] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.FractionalIdeal.Inverse | {
"line": 154,
"column": 4
} | {
"line": 154,
"column": 12
} | {
"line": 154,
"column": 13
} | [
{
"pp": "case mp\nK : Type u_3\ninst✝⁵ : Field K\nR₁ : Type u_4\ninst✝⁴ : CommRing R₁\ninst✝³ : IsDomain R₁\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\nI : FractionalIdeal R₁⁰ K\ninst✝ : (↑I).IsPrincipal\n⊢ I * I⁻¹ = 1 → generator ↑I ≠ 0",
"ppTerm": "?mp",
"assigned": true,
"usedConstants"... | [
"case mp\nK : Type u_3\ninst✝⁵ : Field K\nR₁ : Type u_4\ninst✝⁴ : CommRing R₁\ninst✝³ : IsDomain R₁\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\nI : FractionalIdeal R₁⁰ K\ninst✝ : (↑I).IsPrincipal\nhI : I * I⁻¹ = 1\n⊢ generator ↑I ≠ 0"
] | intro hI | Lean.Elab.Tactic.evalIntro | null |
Mathlib.RingTheory.FractionalIdeal.Inverse | {
"line": 160,
"column": 4
} | {
"line": 160,
"column": 12
} | {
"line": 161,
"column": 4
} | [
{
"pp": "case mpr\nK : Type u_3\ninst✝⁵ : Field K\nR₁ : Type u_4\ninst✝⁴ : CommRing R₁\ninst✝³ : IsDomain R₁\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\nI : FractionalIdeal R₁⁰ K\ninst✝ : (↑I).IsPrincipal\nhg : generator ↑I ≠ 0\n⊢ spanSingleton R₁⁰ (generator ↑I) ≠ 0",
"ppTerm": "?mpr",
"assig... | [
"case mpr\nK : Type u_3\ninst✝⁵ : Field K\nR₁ : Type u_4\ninst✝⁴ : CommRing R₁\ninst✝³ : IsDomain R₁\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\nI : FractionalIdeal R₁⁰ K\ninst✝ : (↑I).IsPrincipal\nhg : generator ↑I ≠ 0\nhI : spanSingleton R₁⁰ (generator ↑I) = 0\n⊢ False"
] | intro hI | Lean.Elab.Tactic.evalIntro | null |
Mathlib.RingTheory.FractionalIdeal.Inverse | {
"line": 160,
"column": 4
} | {
"line": 160,
"column": 12
} | {
"line": 161,
"column": 4
} | [
{
"pp": "case mpr\nK : Type u_3\ninst✝⁵ : Field K\nR₁ : Type u_4\ninst✝⁴ : CommRing R₁\ninst✝³ : IsDomain R₁\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\nI : FractionalIdeal R₁⁰ K\ninst✝ : (↑I).IsPrincipal\nhg : generator ↑I ≠ 0\n⊢ spanSingleton R₁⁰ (generator ↑I) ≠ 0",
"ppTerm": "?mpr",
"assig... | [
"case mpr\nK : Type u_3\ninst✝⁵ : Field K\nR₁ : Type u_4\ninst✝⁴ : CommRing R₁\ninst✝³ : IsDomain R₁\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\nI : FractionalIdeal R₁⁰ K\ninst✝ : (↑I).IsPrincipal\nhg : generator ↑I ≠ 0\nhI : spanSingleton R₁⁰ (generator ↑I) = 0\n⊢ False"
] | intro hI | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 210,
"column": 2
} | {
"line": 211,
"column": 49
} | {
"line": 213,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\ninj : Function.Injective ⇑(algebraMap R P)\nI : Ideal R\nh : IsUnit ↑I\n⊢ I.FG",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"FractionalIdeal.fg_of_i... | [] | rw [← coeIdeal_fg S inj I]
exact FractionalIdeal.fg_of_isUnit (R := R) I h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 210,
"column": 2
} | {
"line": 211,
"column": 49
} | {
"line": 213,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\ninj : Function.Injective ⇑(algebraMap R P)\nI : Ideal R\nh : IsUnit ↑I\n⊢ I.FG",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"FractionalIdeal.fg_of_i... | [] | rw [← coeIdeal_fg S inj I]
exact FractionalIdeal.fg_of_isUnit (R := R) I h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 495,
"column": 2
} | {
"line": 495,
"column": 10
} | {
"line": 496,
"column": 2
} | [
{
"pp": "K : Type u_4\nL : Type u_5\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : IsFractionRing K L\nI : FractionalIdeal K⁰ L\n⊢ ¬I = 0 → I = 1",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"nonZeroDivisors",
"Field.toSemifield",
"Field.toCommRing"... | [
"K : Type u_4\nL : Type u_5\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : IsFractionRing K L\nI : FractionalIdeal K⁰ L\nhI : ¬I = 0\n⊢ I = 1"
] | intro hI | Lean.Elab.Tactic.evalIntro | null |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 495,
"column": 2
} | {
"line": 495,
"column": 10
} | {
"line": 496,
"column": 2
} | [
{
"pp": "K : Type u_4\nL : Type u_5\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : IsFractionRing K L\nI : FractionalIdeal K⁰ L\n⊢ ¬I = 0 → I = 1",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"nonZeroDivisors",
"Field.toSemifield",
"Field.toCommRing"... | [
"K : Type u_4\nL : Type u_5\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : IsFractionRing K L\nI : FractionalIdeal K⁰ L\nhI : ¬I = 0\n⊢ I = 1"
] | intro hI | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 506,
"column": 4
} | {
"line": 506,
"column": 43
} | {
"line": 508,
"column": 0
} | [
{
"pp": "case mpr\nK : Type u_4\nL : Type u_5\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : IsFractionRing K L\nI : FractionalIdeal K⁰ L\nhI : ¬I = 0\nx y : K\ny_ne : y ≠ 0\ny_mem : (algebraMap K L) y ∈ I\n⊢ (x / y) • (algebraMap K L) y ∈ I",
"ppTerm": "?mpr",
"assigned": true,
... | [] | exact smul_mem (M := L) I (x / y) y_mem | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 674,
"column": 8
} | {
"line": 674,
"column": 25
} | {
"line": 674,
"column": 25
} | [
{
"pp": "case mp\nR : 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\ny : P'\nh : y ∈ (canonicalEquiv S P P') (spanSingleton S x)\... | [
"case mp\nR : 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\ny : P'\nh : y ∈ (canonicalEquiv S P P') (spanSingleton S x)\n⊢ ∃ z, z • ... | mem_spanSingleton | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 31
} | {
"line": 106,
"column": 2
} | [
{
"pp": "case refine_1\nA : Type u_2\ninst✝¹ : CommRing A\ninst✝ : IsDedekindDomain A\nh : Prime 1\n⊢ False",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"CommSemiring.toSemiring",
"CommSemiring.toCommMonoidWithZero",
"CommMonoidWithZero.t... | [] | exact h.not_unit isUnit_one | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 313,
"column": 6
} | {
"line": 313,
"column": 49
} | {
"line": 314,
"column": 4
} | [
{
"pp": "A : Type u_2\ninst✝¹ : CommRing A\ninst✝ : IsDedekindDomain A\nI J : Ideal A\nthis : NormalizedGCDMonoid (Ideal A) := UniqueFactorizationMonoid.toNormalizedGCDMonoid (Ideal A)\n⊢ gcd I J ∣ I ∧ gcd I J ∣ J",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
"Dvd.dvd",
"Semi... | [] | exact ⟨gcd_dvd_left _ _, gcd_dvd_right _ _⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Module.GradedModule | {
"line": 146,
"column": 20
} | {
"line": 146,
"column": 75
} | {
"line": 148,
"column": 0
} | [
{
"pp": "ιA : Type u_1\nιB : Type u_2\nA : ιA → Type u_3\nM : ιB → Type u_4\ninst✝⁷ : AddMonoid ιA\ninst✝⁶ : VAdd ιA ιB\ninst✝⁵ : (i : ιA) → AddCommMonoid (A i)\ninst✝⁴ : (i : ιB) → AddCommMonoid (M i)\ninst✝³ : DecidableEq ιA\ninst✝² : DecidableEq ιB\ninst✝¹ : GSemiring A\ninst✝ : Gmodule A M\nx : ⨁ (i : ιB), ... | [] | simp only [smul_def, map_zero, AddMonoidHom.zero_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Module.GradedModule | {
"line": 146,
"column": 20
} | {
"line": 146,
"column": 75
} | {
"line": 148,
"column": 0
} | [
{
"pp": "ιA : Type u_1\nιB : Type u_2\nA : ιA → Type u_3\nM : ιB → Type u_4\ninst✝⁷ : AddMonoid ιA\ninst✝⁶ : VAdd ιA ιB\ninst✝⁵ : (i : ιA) → AddCommMonoid (A i)\ninst✝⁴ : (i : ιB) → AddCommMonoid (M i)\ninst✝³ : DecidableEq ιA\ninst✝² : DecidableEq ιB\ninst✝¹ : GSemiring A\ninst✝ : Gmodule A M\nx : ⨁ (i : ιB), ... | [] | simp only [smul_def, map_zero, AddMonoidHom.zero_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.GradedModule | {
"line": 146,
"column": 20
} | {
"line": 146,
"column": 75
} | {
"line": 148,
"column": 0
} | [
{
"pp": "ιA : Type u_1\nιB : Type u_2\nA : ιA → Type u_3\nM : ιB → Type u_4\ninst✝⁷ : AddMonoid ιA\ninst✝⁶ : VAdd ιA ιB\ninst✝⁵ : (i : ιA) → AddCommMonoid (A i)\ninst✝⁴ : (i : ιB) → AddCommMonoid (M i)\ninst✝³ : DecidableEq ιA\ninst✝² : DecidableEq ιB\ninst✝¹ : GSemiring A\ninst✝ : Gmodule A M\nx : ⨁ (i : ιB), ... | [] | simp only [smul_def, map_zero, AddMonoidHom.zero_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 408,
"column": 63
} | {
"line": 408,
"column": 76
} | {
"line": 408,
"column": 76
} | [
{
"pp": "T : 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⊢ I ≠ 0",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"hI"
],
"usedGoals": []
}... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 408,
"column": 63
} | {
"line": 408,
"column": 76
} | {
"line": 408,
"column": 76
} | [
{
"pp": "T : 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⊢ J ≠ 0",
"ppTerm": "?m.85",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"hJ"
],
"usedGoals": []
}... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.Weights.IsSimple | {
"line": 347,
"column": 9
} | {
"line": 347,
"column": 29
} | {
"line": 347,
"column": 29
} | [
{
"pp": "K : 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 : FiniteDimensional K L\ninst✝² : IsKilling K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nq : Submodule K (Dual K ↥H)\nhq : ∀ (i : ↥Li... | [
"K : 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 : FiniteDimensional K L\ninst✝² : IsKilling K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nq : Submodule K (Dual K ↥H)\nhq : ∀ (i : ↥LieSubalgebra.... | LieSubmodule.mem_bot | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Weights.IsSimple | {
"line": 352,
"column": 9
} | {
"line": 352,
"column": 29
} | {
"line": 352,
"column": 29
} | [
{
"pp": "K : 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 : FiniteDimensional K L\ninst✝² : IsKilling K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nq : Submodule K (Dual K ↥H)\nhq : ∀ (i : ↥Li... | [
"K : 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 : FiniteDimensional K L\ninst✝² : IsKilling K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nq : Submodule K (Dual K ↥H)\nhq : ∀ (i : ↥LieSubalgebra.... | LieSubmodule.mem_bot | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 719,
"column": 4
} | {
"line": 719,
"column": 12
} | {
"line": 720,
"column": 4
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDedekindDomain A\nI : Ideal R\nJ : Ideal A\ninst✝ : IsDedekindDomain R\nf : R ⧸ I ≃+* A ⧸ J\nhJ : J ≠ ⊥\nL : Ideal R\nhL : L ∈ normalizedFactors I\n⊢ I ≠ ⊥",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants"... | [
"R : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDedekindDomain A\nI : Ideal R\nJ : Ideal A\ninst✝ : IsDedekindDomain R\nf : R ⧸ I ≃+* A ⧸ J\nhJ : J ≠ ⊥\nL : Ideal R\nhL : L ∈ normalizedFactors I\nhI : I = ⊥\n⊢ False"
] | intro hI | Lean.Elab.Tactic.evalIntro | null |
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