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.RingTheory.Derivation.Lie | {
"line": 85,
"column": 18
} | {
"line": 85,
"column": 26
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\nA : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra R A\nD1 D2 : Derivation R A A\na : A\nA' : Type u_3\ninst✝³ : CommRing A'\ninst✝² : Algebra R A'\ninst✝¹ : Algebra A A'\ninst✝ : IsScalarTower R A A'\n⊢ ∀ (c : R) {x : Derivation R A' A' × Derivation R A A},\n x ∈... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Derivation.Lie | {
"line": 85,
"column": 18
} | {
"line": 85,
"column": 26
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\nA : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra R A\nD1 D2 : Derivation R A A\na : A\nA' : Type u_3\ninst✝³ : CommRing A'\ninst✝² : Algebra R A'\ninst✝¹ : Algebra A A'\ninst✝ : IsScalarTower R A A'\n⊢ ∀ (c : R) {x : Derivation R A' A' × Derivation R A A},\n x ∈... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Derivation.Lie | {
"line": 85,
"column": 18
} | {
"line": 85,
"column": 26
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\nA : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra R A\nD1 D2 : Derivation R A A\na : A\nA' : Type u_3\ninst✝³ : CommRing A'\ninst✝² : Algebra R A'\ninst✝¹ : Algebra A A'\ninst✝ : IsScalarTower R A A'\n⊢ ∀ (c : R) {x : Derivation R A' A' × Derivation R A A},\n x ∈... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Classical | {
"line": 275,
"column": 2
} | {
"line": 278,
"column": 17
} | [
{
"pp": "l : Type u_4\nR : Type u₂\ninst✝² : DecidableEq l\ninst✝¹ : CommRing R\ninst✝ : Fintype l\n⊢ (PD l R)ᵀ * JD l R * PD l R = 2 • S l R",
"usedConstants": [
"neg_add_rev",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"add_neg_cancel",
"NegZeroClass.toNeg",
"Matrix.from... | have h : (PD l R)ᵀ * JD l R = Matrix.fromBlocks 1 1 1 (-1) := by
simp [PD, JD, Matrix.fromBlocks_transpose, Matrix.fromBlocks_multiply]
rw [h, PD, s_as_blocks, Matrix.fromBlocks_multiply, Matrix.fromBlocks_smul]
simp [two_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Classical | {
"line": 275,
"column": 2
} | {
"line": 278,
"column": 17
} | [
{
"pp": "l : Type u_4\nR : Type u₂\ninst✝² : DecidableEq l\ninst✝¹ : CommRing R\ninst✝ : Fintype l\n⊢ (PD l R)ᵀ * JD l R * PD l R = 2 • S l R",
"usedConstants": [
"neg_add_rev",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"add_neg_cancel",
"NegZeroClass.toNeg",
"Matrix.from... | have h : (PD l R)ᵀ * JD l R = Matrix.fromBlocks 1 1 1 (-1) := by
simp [PD, JD, Matrix.fromBlocks_transpose, Matrix.fromBlocks_multiply]
rw [h, PD, s_as_blocks, Matrix.fromBlocks_multiply, Matrix.fromBlocks_smul]
simp [two_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.DirectSum | {
"line": 208,
"column": 30
} | {
"line": 208,
"column": 45
} | [
{
"pp": "case inl\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), ... | toAddMonoid_of, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Lie.Derivation.BaseChange | {
"line": 54,
"column": 4
} | {
"line": 54,
"column": 12
} | [
{
"pp": "case H\nR : Type u_1\ninst✝⁴ : CommRing R\nA : Type u_2\ninst✝³ : CommRing A\ninst✝² : Algebra R A\nL : Type u_3\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nx✝³ x✝² : Derivation R A A\nz x✝¹ x✝ : A ⊗[R] L\nhx :\n { toFun := ⇑(LinearMap.rTensor L ↑⁅x✝³, x✝²⁆), map_add' := ⋯, map_smul' := ⋯, leibniz' :... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Lie.Free | {
"line": 253,
"column": 67
} | {
"line": 254,
"column": 69
} | [
{
"pp": "R : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\nx : X\n⊢ ((lift R) f) (of R x) = f x",
"usedConstants": [
"LieHom",
"Eq.mpr",
"Equiv.instEquivLike",
"FreeLieAlgebra.instLieRing",
"FreeLieAlgebra.of_comp_li... | by
rw [← @Function.comp_apply _ _ _ (lift R f) (of R) x, of_comp_lift] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Perm.Cycle.Concrete | {
"line": 170,
"column": 54
} | {
"line": 170,
"column": 62
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nx : α\na✝ : List α\nh : Nodup (Quot.mk (⇑(IsRotated.setoid α)) a✝)\nhx : x ∈ Quot.mk (⇑(IsRotated.setoid α)) a✝\n⊢ x ∈ a✝",
"usedConstants": [
"List.IsRotated.setoid",
"Membership.mem",
"Eq.mp",
"id",
"List",
"List.instMembers... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.GroupTheory.Perm.Cycle.Concrete | {
"line": 170,
"column": 54
} | {
"line": 170,
"column": 62
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nx : α\na✝ : List α\nh : Nodup (Quot.mk (⇑(IsRotated.setoid α)) a✝)\nhx : x ∈ Quot.mk (⇑(IsRotated.setoid α)) a✝\n⊢ x ∈ a✝",
"usedConstants": [
"List.IsRotated.setoid",
"Membership.mem",
"Eq.mp",
"id",
"List",
"List.instMembers... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Perm.Cycle.Concrete | {
"line": 170,
"column": 54
} | {
"line": 170,
"column": 62
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nx : α\na✝ : List α\nh : Nodup (Quot.mk (⇑(IsRotated.setoid α)) a✝)\nhx : x ∈ Quot.mk (⇑(IsRotated.setoid α)) a✝\n⊢ x ∈ a✝",
"usedConstants": [
"List.IsRotated.setoid",
"Membership.mem",
"Eq.mp",
"id",
"List",
"List.instMembers... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Perm.Cycle.Concrete | {
"line": 258,
"column": 2
} | {
"line": 258,
"column": 90
} | [
{
"pp": "case neg.succ.succ\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx : α\nhx : x ∈ (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : ℕ\nhn✝ : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : ℕ\nhm✝ : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cyc... | have hm' : ¬orderOf (p.cycleOf x) ∣ m.succ := Nat.not_dvd_of_pos_of_lt m.zero_lt_succ hm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.GroupTheory.Perm.Cycle.Concrete | {
"line": 440,
"column": 4
} | {
"line": 440,
"column": 24
} | [
{
"pp": "case intro.refine_2\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : DecidableEq α\nf : Perm α\nhf : f.IsCycle\nval✝ : Fintype α\nx : α\nhx : f x ≠ x\nhy : ∀ ⦃y : α⦄, f y ≠ y → f.SameCycle x y\n⊢ ∀ (y : Cycle α), (fun s ↦ ∃ (h : s.Nodup), s.formPerm h = f) y → y = ↑(f.toList x)",
"usedConstants": [
... | rintro ⟨l⟩ ⟨hn, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.Lie.LieTheorem | {
"line": 139,
"column": 30
} | {
"line": 139,
"column": 38
} | [
{
"pp": "R : 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 ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Lie.LieTheorem | {
"line": 139,
"column": 30
} | {
"line": 139,
"column": 38
} | [
{
"pp": "R : 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 ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.LieTheorem | {
"line": 139,
"column": 30
} | {
"line": 139,
"column": 38
} | [
{
"pp": "R : 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 ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.LieTheorem | {
"line": 146,
"column": 39
} | {
"line": 146,
"column": 52
} | [
{
"pp": "R : 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 ... | by simp [hv'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Matrix.Cartan | {
"line": 285,
"column": 2
} | {
"line": 288,
"column": 17
} | [
{
"pp": "case mpr\nι : Type u_1\ninst✝ : LinearOrder ι\nA : Matrix ι ι ℤ\nhA : A.IsSymm\n⊢ (∀ ⦃i j : ι⦄, j < i → A i j = 0 ∨ A i j = -1) → A.IsSimplyLaced",
"usedConstants": [
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder",
"SemilatticeInf.toPartialOrder",
"Eq.mp",
... | · intro h i j hij
obtain hij | hij := hij.lt_or_gt
· simpa only [hA.apply i j] using h hij
· exact h hij | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Lie.Extension | {
"line": 337,
"column": 46
} | {
"line": 337,
"column": 72
} | [
{
"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 ... | LieEquiv.apply_symm_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Extension | {
"line": 337,
"column": 73
} | {
"line": 337,
"column": 99
} | [
{
"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 ... | LieEquiv.apply_symm_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.LieTheorem | {
"line": 201,
"column": 4
} | {
"line": 201,
"column": 58
} | [
{
"pp": "case h.refine_2\nk : 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... | simpa [ne_eq, LieSubmodule.mk_eq_zero] using hvc.right | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Lie.LieTheorem | {
"line": 201,
"column": 4
} | {
"line": 201,
"column": 58
} | [
{
"pp": "case h.refine_2\nk : 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... | simpa [ne_eq, LieSubmodule.mk_eq_zero] using hvc.right | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.LieTheorem | {
"line": 201,
"column": 4
} | {
"line": 201,
"column": 58
} | [
{
"pp": "case h.refine_2\nk : 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... | simpa [ne_eq, LieSubmodule.mk_eq_zero] using hvc.right | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.LinearRecurrence | {
"line": 165,
"column": 4
} | {
"line": 165,
"column": 60
} | [
{
"pp": "case mp\nR : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : u ∈ E.solSpace\nhv : v ∈ E.solSpace\n⊢ E.basis.repr ⟨u, hu⟩ = E.basis.repr ⟨v, hv⟩ → Set.EqOn u v ↑(range E.order)",
"usedConstants": [
"Finsupp.instFunLike",
"Pi.Function.module",
"Submodule"... | exact fun h n hn ↦ congr($h ⟨n, Finset.mem_range.mp hn⟩) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.LinearRecurrence | {
"line": 165,
"column": 4
} | {
"line": 165,
"column": 60
} | [
{
"pp": "case mp\nR : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : u ∈ E.solSpace\nhv : v ∈ E.solSpace\n⊢ E.basis.repr ⟨u, hu⟩ = E.basis.repr ⟨v, hv⟩ → Set.EqOn u v ↑(range E.order)",
"usedConstants": [
"Finsupp.instFunLike",
"Pi.Function.module",
"Submodule"... | exact fun h n hn ↦ congr($h ⟨n, Finset.mem_range.mp hn⟩) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.LinearRecurrence | {
"line": 165,
"column": 4
} | {
"line": 165,
"column": 60
} | [
{
"pp": "case mp\nR : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : u ∈ E.solSpace\nhv : v ∈ E.solSpace\n⊢ E.basis.repr ⟨u, hu⟩ = E.basis.repr ⟨v, hv⟩ → Set.EqOn u v ↑(range E.order)",
"usedConstants": [
"Finsupp.instFunLike",
"Pi.Function.module",
"Submodule"... | exact fun h n hn ↦ congr($h ⟨n, Finset.mem_range.mp hn⟩) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.LinearRecurrence | {
"line": 220,
"column": 15
} | {
"line": 220,
"column": 35
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nE : LinearRecurrence R\na✝ : Nontrivial R\n⊢ ((monomial E.order) 1).coeff E.order - (∑ i, (monomial ↑i) (E.coeffs i)).coeff E.order = 1",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"NonUnitalCommRing.toNonUnitalNonAsso... | coeff_monomial_same, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Basic | {
"line": 115,
"column": 2
} | {
"line": 120,
"column": 78
} | [
{
"pp": "A : Type u_2\ninst✝¹ : CommRing A\ninst✝ : IsDomain A\nh : IsDedekindDomainInv A\n⊢ IsNoetherianRing A",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"FractionRing.field",
"Semiring.toModule",
"OreLocalization.instAlgebra",
"congrArg",
"CommSemiring.toSemiring",... | let := h.commGroupWithZero (K := FractionRing A)
refine isNoetherianRing_iff.mpr ⟨fun I : Ideal A => ?_⟩
by_cases hI : I = ⊥
· rw [hI]; apply Submodule.fg_bot
have hI : (I : FractionalIdeal A⁰ (FractionRing A)) ≠ 0 := coeIdeal_ne_zero.mpr hI
exact I.fg_of_isUnit (IsFractionRing.injective A (FractionRing A)) h... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DedekindDomain.Ideal.Basic | {
"line": 115,
"column": 2
} | {
"line": 120,
"column": 78
} | [
{
"pp": "A : Type u_2\ninst✝¹ : CommRing A\ninst✝ : IsDomain A\nh : IsDedekindDomainInv A\n⊢ IsNoetherianRing A",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"FractionRing.field",
"Semiring.toModule",
"OreLocalization.instAlgebra",
"congrArg",
"CommSemiring.toSemiring",... | let := h.commGroupWithZero (K := FractionRing A)
refine isNoetherianRing_iff.mpr ⟨fun I : Ideal A => ?_⟩
by_cases hI : I = ⊥
· rw [hI]; apply Submodule.fg_bot
have hI : (I : FractionalIdeal A⁰ (FractionRing A)) ≠ 0 := coeIdeal_ne_zero.mpr hI
exact I.fg_of_isUnit (IsFractionRing.injective A (FractionRing A)) h... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Weights.IsSimple | {
"line": 315,
"column": 4
} | {
"line": 315,
"column": 69
} | [
{
"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... | let γ : Weight K H L := ⟨χ.toLinear - α.toLinear, h_minus_ne_bot⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 571,
"column": 2
} | {
"line": 572,
"column": 5
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝² : CommRing P\ninst✝¹ : Algebra R P\ninst✝ : IsLocalization S P\nx : P\n⊢ ↑(spanSingleton S x) = R ∙ x",
"usedConstants": [
"Eq.mpr",
"Submodule",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Fracti... | rw [spanSingleton]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 571,
"column": 2
} | {
"line": 572,
"column": 5
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝² : CommRing P\ninst✝¹ : Algebra R P\ninst✝ : IsLocalization S P\nx : P\n⊢ ↑(spanSingleton S x) = R ∙ x",
"usedConstants": [
"Eq.mpr",
"Submodule",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Fracti... | rw [spanSingleton]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 681,
"column": 4
} | {
"line": 681,
"column": 13
} | [
{
"pp": "case mpr\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\nz : R\nh : z • (IsLocalization.map P' (RingHom.id R) ⋯) x ∈... | use z • x | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Algebra.Lie.Weights.IsSimple | {
"line": 362,
"column": 18
} | {
"line": 362,
"column": 24
} | [
{
"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... | hj_val | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 1009,
"column": 6
} | {
"line": 1009,
"column": 21
} | [
{
"pp": "R : Type u_5\nS : Type u_6\nK : Type u_7\nL : Type u_8\ninst✝⁹ : CommRing R\ninst✝⁸ : IsDomain R\ninst✝⁷ : CommRing S\ninst✝⁶ : IsDomain S\ninst✝⁵ : CommRing K\ninst✝⁴ : CommRing L\ninst✝³ : Algebra R K\ninst✝² : Algebra S L\ninst✝¹ : IsFractionRing R K\ninst✝ : IsFractionRing S L\nf : R ≃+* S\nx : K\n... | SetLike.ext_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Finiteness.Cofinite | {
"line": 94,
"column": 48
} | {
"line": 99,
"column": 29
} | [
{
"pp": "R : Type u_1\ninst✝³ : Ring R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsNoetherianRing R\ns : Finset (Submodule R M)\nhs : ∀ S ∈ s, S.CoFG\n⊢ (sInf ↑s).CoFG",
"usedConstants": [
"Eq.mpr",
"Submodule",
"Finset.coe_empty",
"CompleteLattice.toLattic... | classical
induction s using Finset.induction with
| empty => simp
| insert w s hws hs' =>
simp only [Finset.mem_insert, forall_eq_or_imp, Finset.coe_insert, sInf_insert] at *
exact hs.1.inf (hs' hs.2) | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.RingTheory.Finiteness.Cofinite | {
"line": 94,
"column": 48
} | {
"line": 99,
"column": 29
} | [
{
"pp": "R : Type u_1\ninst✝³ : Ring R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsNoetherianRing R\ns : Finset (Submodule R M)\nhs : ∀ S ∈ s, S.CoFG\n⊢ (sInf ↑s).CoFG",
"usedConstants": [
"Eq.mpr",
"Submodule",
"Finset.coe_empty",
"CompleteLattice.toLattic... | classical
induction s using Finset.induction with
| empty => simp
| insert w s hws hs' =>
simp only [Finset.mem_insert, forall_eq_or_imp, Finset.coe_insert, sInf_insert] at *
exact hs.1.inf (hs' hs.2) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Finiteness.Cofinite | {
"line": 94,
"column": 48
} | {
"line": 99,
"column": 29
} | [
{
"pp": "R : Type u_1\ninst✝³ : Ring R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsNoetherianRing R\ns : Finset (Submodule R M)\nhs : ∀ S ∈ s, S.CoFG\n⊢ (sInf ↑s).CoFG",
"usedConstants": [
"Eq.mpr",
"Submodule",
"Finset.coe_empty",
"CompleteLattice.toLattic... | classical
induction s using Finset.induction with
| empty => simp
| insert w s hws hs' =>
simp only [Finset.mem_insert, forall_eq_or_imp, Finset.coe_insert, sInf_insert] at *
exact hs.1.inf (hs' hs.2) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.LinearMap.FiniteRange | {
"line": 194,
"column": 23
} | {
"line": 194,
"column": 31
} | [
{
"pp": "K : Type u_1\nV : Type u_2\nV' : Type u_3\nV₂ : Type u_4\nV₂' : Type u_5\nV₃ : Type u_6\ninst✝⁶ : CommRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : AddCommGroup V₃\ninst✝ : Module K V₃\na✝ b✝ : V →ₗ[K] V₂\nhu : a✝ ∈ {u | u.HasNoetherianRa... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Module.LinearMap.FiniteRange | {
"line": 194,
"column": 23
} | {
"line": 194,
"column": 31
} | [
{
"pp": "K : Type u_1\nV : Type u_2\nV' : Type u_3\nV₂ : Type u_4\nV₂' : Type u_5\nV₃ : Type u_6\ninst✝⁶ : CommRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : AddCommGroup V₃\ninst✝ : Module K V₃\na✝ b✝ : V →ₗ[K] V₂\nhu : a✝ ∈ {u | u.HasNoetherianRa... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.LinearMap.FiniteRange | {
"line": 194,
"column": 23
} | {
"line": 194,
"column": 31
} | [
{
"pp": "K : Type u_1\nV : Type u_2\nV' : Type u_3\nV₂ : Type u_4\nV₂' : Type u_5\nV₃ : Type u_6\ninst✝⁶ : CommRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : AddCommGroup V₃\ninst✝ : Module K V₃\na✝ b✝ : V →ₗ[K] V₂\nhu : a✝ ∈ {u | u.HasNoetherianRa... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.LinearMap.FiniteRange | {
"line": 196,
"column": 23
} | {
"line": 196,
"column": 31
} | [
{
"pp": "K : Type u_1\nV : Type u_2\nV' : Type u_3\nV₂ : Type u_4\nV₂' : Type u_5\nV₃ : Type u_6\ninst✝⁶ : CommRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : AddCommGroup V₃\ninst✝ : Module K V₃\nc : K\nhu : V →ₗ[K] V₂\n⊢ hu ∈ {u | u.HasNoetherianR... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Module.LinearMap.FiniteRange | {
"line": 196,
"column": 23
} | {
"line": 196,
"column": 31
} | [
{
"pp": "K : Type u_1\nV : Type u_2\nV' : Type u_3\nV₂ : Type u_4\nV₂' : Type u_5\nV₃ : Type u_6\ninst✝⁶ : CommRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : AddCommGroup V₃\ninst✝ : Module K V₃\nc : K\nhu : V →ₗ[K] V₂\n⊢ hu ∈ {u | u.HasNoetherianR... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.LinearMap.FiniteRange | {
"line": 196,
"column": 23
} | {
"line": 196,
"column": 31
} | [
{
"pp": "K : Type u_1\nV : Type u_2\nV' : Type u_3\nV₂ : Type u_4\nV₂' : Type u_5\nV₃ : Type u_6\ninst✝⁶ : CommRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : AddCommGroup V₃\ninst✝ : Module K V₃\nc : K\nhu : V →ₗ[K] V₂\n⊢ hu ∈ {u | u.HasNoetherianR... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 110,
"column": 2
} | {
"line": 111,
"column": 59
} | [
{
"pp": "A : Type u_2\ninst✝¹ : CommRing A\ninst✝ : IsDedekindDomain A\nP : Ideal A\nhP : P ≠ ⊥\nh : P.IsPrime\n⊢ Prime P",
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"Semiring.toModule",
"HMul.hMul",
"IsScalarTower.right",
"MulZeroClass.toMul",
"congrArg",
"Comm... | refine ⟨hP, mt isUnit_iff.mp h.ne_top, fun I J hIJ => ?_⟩
simpa only [dvd_iff_le] using h.mul_le.mp (le_of_dvd hIJ) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 110,
"column": 2
} | {
"line": 111,
"column": 59
} | [
{
"pp": "A : Type u_2\ninst✝¹ : CommRing A\ninst✝ : IsDedekindDomain A\nP : Ideal A\nhP : P ≠ ⊥\nh : P.IsPrime\n⊢ Prime P",
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"Semiring.toModule",
"HMul.hMul",
"IsScalarTower.right",
"MulZeroClass.toMul",
"congrArg",
"Comm... | refine ⟨hP, mt isUnit_iff.mp h.ne_top, fun I J hIJ => ?_⟩
simpa only [dvd_iff_le] using h.mul_le.mp (le_of_dvd hIJ) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.Presentation.Cokernel | {
"line": 92,
"column": 6
} | {
"line": 93,
"column": 11
} | [
{
"pp": "case inl\nA : Type u\ninst✝⁶ : Ring A\nM₁ : Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\ninst✝⁵ : AddCommGroup M₁\ninst✝⁴ : Module A M₁\ninst✝³ : AddCommGroup M₂\ninst✝² : Module A M₂\ninst✝¹ : AddCommGroup M₃\ninst✝ : Module A M₃\npres₂ : Presentation A M₂\nf : M₁ →ₗ[A] M₂\nι : Type w₁\ng₁ : ι → M₁\ndata : pr... | erw [pres₂.linearCombination_var_relation]
dsimp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.Presentation.Cokernel | {
"line": 92,
"column": 6
} | {
"line": 93,
"column": 11
} | [
{
"pp": "case inl\nA : Type u\ninst✝⁶ : Ring A\nM₁ : Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\ninst✝⁵ : AddCommGroup M₁\ninst✝⁴ : Module A M₁\ninst✝³ : AddCommGroup M₂\ninst✝² : Module A M₂\ninst✝¹ : AddCommGroup M₃\ninst✝ : Module A M₃\npres₂ : Presentation A M₂\nf : M₁ →ₗ[A] M₂\nι : Type w₁\ng₁ : ι → M₁\ndata : pr... | erw [pres₂.linearCombination_var_relation]
dsimp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 432,
"column": 4
} | {
"line": 432,
"column": 12
} | [
{
"pp": "case pos\nT : Type u_4\ninst✝¹ : CommRing T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhJ : Irreducible J\nn : ℕ\nhn : ↑n ≤ emultiplicity J I\nhI : I = ⊥\n⊢ J ^ n ⊔ I = J ^ n",
"usedConstants": [
"Semiring.toModule",
"IsScalarTower.right",
"congrArg",
"CommSemiring.toSemiri... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 432,
"column": 4
} | {
"line": 432,
"column": 12
} | [
{
"pp": "case pos\nT : Type u_4\ninst✝¹ : CommRing T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhJ : Irreducible J\nn : ℕ\nhn : ↑n ≤ emultiplicity J I\nhI : I = ⊥\n⊢ J ^ n ⊔ I = J ^ n",
"usedConstants": [
"Semiring.toModule",
"IsScalarTower.right",
"congrArg",
"CommSemiring.toSemiri... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 432,
"column": 4
} | {
"line": 432,
"column": 12
} | [
{
"pp": "case pos\nT : Type u_4\ninst✝¹ : CommRing T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhJ : Irreducible J\nn : ℕ\nhn : ↑n ≤ emultiplicity J I\nhI : I = ⊥\n⊢ J ^ n ⊔ I = J ^ n",
"usedConstants": [
"Semiring.toModule",
"IsScalarTower.right",
"congrArg",
"CommSemiring.toSemiri... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 553,
"column": 6
} | {
"line": 553,
"column": 81
} | [
{
"pp": "case map_units.left\nR : Type u_3\nM : Type u_4\nN : Type u_5\nN'✝ : Type ?u.147698\ninst✝¹⁴ : CommRing R\ninst✝¹³ : AddCommGroup M\ninst✝¹² : Module R M\ninst✝¹¹ : AddCommGroup N\ninst✝¹⁰ : Module R N\ninst✝⁹ : AddCommGroup N'✝\ninst✝⁸ : Module R N'✝\nS : Submonoid R\nf✝ : N →ₗ[R] N'✝\ninst✝⁷ : IsLoca... | exact fun _ _ e ↦ LinearMap.ext fun m ↦ this.left (LinearMap.congr_fun e m) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 553,
"column": 6
} | {
"line": 553,
"column": 81
} | [
{
"pp": "case map_units.left\nR : Type u_3\nM : Type u_4\nN : Type u_5\nN'✝ : Type ?u.147698\ninst✝¹⁴ : CommRing R\ninst✝¹³ : AddCommGroup M\ninst✝¹² : Module R M\ninst✝¹¹ : AddCommGroup N\ninst✝¹⁰ : Module R N\ninst✝⁹ : AddCommGroup N'✝\ninst✝⁸ : Module R N'✝\nS : Submonoid R\nf✝ : N →ₗ[R] N'✝\ninst✝⁷ : IsLoca... | exact fun _ _ e ↦ LinearMap.ext fun m ↦ this.left (LinearMap.congr_fun e m) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 553,
"column": 6
} | {
"line": 553,
"column": 81
} | [
{
"pp": "case map_units.left\nR : Type u_3\nM : Type u_4\nN : Type u_5\nN'✝ : Type ?u.147698\ninst✝¹⁴ : CommRing R\ninst✝¹³ : AddCommGroup M\ninst✝¹² : Module R M\ninst✝¹¹ : AddCommGroup N\ninst✝¹⁰ : Module R N\ninst✝⁹ : AddCommGroup N'✝\ninst✝⁸ : Module R N'✝\nS : Submonoid R\nf✝ : N →ₗ[R] N'✝\ninst✝⁷ : IsLoca... | exact fun _ _ e ↦ LinearMap.ext fun m ↦ this.left (LinearMap.congr_fun e m) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 823,
"column": 2
} | {
"line": 823,
"column": 63
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nI J : Ideal R\nhJ : J.IsPrime\nhJ₀ : J ≠ ⊥\nhI : Associates.mk I ≠ 0\nhJ' : Irreducible (Associates.mk J)\n⊢ (Associates.mk J).count (Associates.mk I).factors = Multiset.count J (normalizedFactors I)",
"usedConstants": [
"UniqueFa... | apply (count_normalizedFactors_eq (p := J) (x := I) _ _).symm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 838,
"column": 2
} | {
"line": 838,
"column": 88
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\na a₀ x : R\nn : ℕ\nhx : Prime x\nha : ¬x ∣ a\nheq : a₀ = x ^ n * a\nhx0 : x ≠ 0\n⊢ (Associates.mk (span {x})).count (Associates.mk (span {a₀})).factors = n",
"usedConstants": [
"UniqueFactorizationMonoid.normalizedFactors",
... | rw [count_associates_factors_eq, UniqueFactorizationMonoid.count_normalizedFactors_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 1237,
"column": 4
} | {
"line": 1238,
"column": 45
} | [
{
"pp": "case neg\nA : Type u_4\ninst✝⁶ : CommRing A\np : Ideal A\nhpm : p.IsMaximal\nB : Type u_5\ninst✝⁵ : CommRing B\ninst✝⁴ : IsDedekindDomain B\ninst✝³ : Algebra A B\ninst✝² : IsDomain A\ninst✝¹ : IsTorsionFree A B\ninst✝ : Algebra.IsIntegral A B\nhpb : ¬p = ⊥\n⊢ (p.primesOver B).Finite",
"usedConstant... | rw [← coe_primesOverFinset hpb B]
exact (primesOverFinset p B).finite_toSet | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 1237,
"column": 4
} | {
"line": 1238,
"column": 45
} | [
{
"pp": "case neg\nA : Type u_4\ninst✝⁶ : CommRing A\np : Ideal A\nhpm : p.IsMaximal\nB : Type u_5\ninst✝⁵ : CommRing B\ninst✝⁴ : IsDedekindDomain B\ninst✝³ : Algebra A B\ninst✝² : IsDomain A\ninst✝¹ : IsTorsionFree A B\ninst✝ : Algebra.IsIntegral A B\nhpb : ¬p = ⊥\n⊢ (p.primesOver B).Finite",
"usedConstant... | rw [← coe_primesOverFinset hpb B]
exact (primesOverFinset p B).finite_toSet | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.PDeriv | {
"line": 157,
"column": 19
} | {
"line": 157,
"column": 27
} | [
{
"pp": "case add\nR : Type u\ninst✝ : CommSemiring R\nσ : Type u_1\nι : Type u_2\ni : σ\np q : MvPolynomial (σ ⊕ ι) R\na✝¹ : (pderiv i) ((sumToIter R σ ι) p) = (sumToIter R σ ι) ((pderiv (Sum.inl i)) p)\na✝ : (pderiv i) ((sumToIter R σ ι) q) = (sumToIter R σ ι) ((pderiv (Sum.inl i)) q)\n⊢ (pderiv i) ((sumToIte... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.MvPolynomial.PDeriv | {
"line": 157,
"column": 19
} | {
"line": 157,
"column": 27
} | [
{
"pp": "case add\nR : Type u\ninst✝ : CommSemiring R\nσ : Type u_1\nι : Type u_2\ni : σ\np q : MvPolynomial (σ ⊕ ι) R\na✝¹ : (pderiv i) ((sumToIter R σ ι) p) = (sumToIter R σ ι) ((pderiv (Sum.inl i)) p)\na✝ : (pderiv i) ((sumToIter R σ ι) q) = (sumToIter R σ ι) ((pderiv (Sum.inl i)) q)\n⊢ (pderiv i) ((sumToIte... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.PDeriv | {
"line": 157,
"column": 19
} | {
"line": 157,
"column": 27
} | [
{
"pp": "case add\nR : Type u\ninst✝ : CommSemiring R\nσ : Type u_1\nι : Type u_2\ni : σ\np q : MvPolynomial (σ ⊕ ι) R\na✝¹ : (pderiv i) ((sumToIter R σ ι) p) = (sumToIter R σ ι) ((pderiv (Sum.inl i)) p)\na✝ : (pderiv i) ((sumToIter R σ ι) q) = (sumToIter R σ ι) ((pderiv (Sum.inl i)) q)\n⊢ (pderiv i) ((sumToIte... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Kaehler.Polynomial | {
"line": 48,
"column": 4
} | {
"line": 48,
"column": 12
} | [
{
"pp": "case add\nR : Type u\ninst✝ : CommRing R\nσ : Type u_1\nf✝ g✝ : MvPolynomial σ R →₀ MvPolynomial σ R\na✝¹ :\n (Finsupp.linearCombination (MvPolynomial σ R) fun x ↦ (D R (MvPolynomial σ R)) (MvPolynomial.X x))\n ((MvPolynomial.mkDerivation R fun x ↦ Finsupp.single x 1).liftKaehlerDifferential.toFu... | | add => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.Kaehler.Polynomial | {
"line": 39,
"column": 4
} | {
"line": 39,
"column": 12
} | [
{
"pp": "case add\nR : Type u\ninst✝ : CommRing R\nσ : Type u_1\nf✝ g✝ : σ →₀ MvPolynomial σ R\na✝¹ :\n (MvPolynomial.mkDerivation R fun x ↦ Finsupp.single x 1).liftKaehlerDifferential.toFun\n ((Finsupp.linearCombination (MvPolynomial σ R) fun x ↦ (D R (MvPolynomial σ R)) (MvPolynomial.X x)) f✝) =\n f✝... | | add => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.Kaehler.Polynomial | {
"line": 90,
"column": 98
} | {
"line": 91,
"column": 86
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nσ : Type u_1\ni : σ\nx : MvPolynomial σ R\n⊢ (mvPolynomialBasis R σ).repr.symm (Finsupp.single i x) = x • (D R (MvPolynomial σ R)) (MvPolynomial.X i)",
"usedConstants": [
"Derivation",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Nat.instMulZeroClass... | by
apply (mvPolynomialBasis R σ).repr.injective; simp [LinearEquiv.map_smul, -map_smul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Extension.Presentation.Basic | {
"line": 450,
"column": 2
} | {
"line": 450,
"column": 24
} | [
{
"pp": "R : Type u\nS : Type v\nι : Type w\nσ : Type t\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nι' : Type u_1\nσ' : Type u_2\nT : Type u_3\ninst✝³ : CommRing T\ninst✝² : Algebra S T\nQ : Presentation S T ι' σ'\nP : Presentation R S ι σ\ninst✝¹ : Algebra R T\ninst✝ : IsScalarTower R S T\... | change (Q.aux P) _ = _ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.RingTheory.Extension.Basic | {
"line": 513,
"column": 4
} | {
"line": 514,
"column": 83
} | [
{
"pp": "case refine_1\nR : Type u\nS : Type v\ninst✝²² : CommRing R\ninst✝²¹ : CommRing S\ninst✝²⁰ : Algebra R S\nP : Extension R S\nR' : Type ?u.346085\nS' : Type ?u.346088\ninst✝¹⁹ : CommRing R'\ninst✝¹⁸ : CommRing S'\ninst✝¹⁷ : Algebra R' S'\nP' : Extension R' S'\nR'' : Type ?u.346212\nS'' : Type ?u.346215\... | refine Submodule.smul_induction_on' (p := fun x (hx : x ∈ P.ker * P.ker) ↦
(1 : S) ⊗ₜ[P.Ring] (⟨x, Ideal.mul_le_right hx⟩ : P.ker) = 0) (hx := hx) ?_ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Extension.Generators | {
"line": 485,
"column": 23
} | {
"line": 485,
"column": 50
} | [
{
"pp": "case add\nR : Type u\nS : Type v\nι : Type w\ninst✝¹⁷ : CommRing R\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\nP : Generators R S ι\nR' : Type u_1\nS' : Type u_2\nι' : Type u_3\ninst✝¹⁴ : CommRing R'\ninst✝¹³ : CommRing S'\ninst✝¹² : Algebra R' S'\nP' : Generators R' S' ι'\nR'' : Type u_4\nS'' : Type... | simp only [map_add, hx, hy] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Extension.Generators | {
"line": 485,
"column": 23
} | {
"line": 485,
"column": 50
} | [
{
"pp": "case add\nR : Type u\nS : Type v\nι : Type w\ninst✝¹⁷ : CommRing R\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\nP : Generators R S ι\nR' : Type u_1\nS' : Type u_2\nι' : Type u_3\ninst✝¹⁴ : CommRing R'\ninst✝¹³ : CommRing S'\ninst✝¹² : Algebra R' S'\nP' : Generators R' S' ι'\nR'' : Type u_4\nS'' : Type... | simp only [map_add, hx, hy] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Extension.Generators | {
"line": 485,
"column": 23
} | {
"line": 485,
"column": 50
} | [
{
"pp": "case add\nR : Type u\nS : Type v\nι : Type w\ninst✝¹⁷ : CommRing R\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\nP : Generators R S ι\nR' : Type u_1\nS' : Type u_2\nι' : Type u_3\ninst✝¹⁴ : CommRing R'\ninst✝¹³ : CommRing S'\ninst✝¹² : Algebra R' S'\nP' : Generators R' S' ι'\nR'' : Type u_4\nS'' : Type... | simp only [map_add, hx, hy] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Extension.Generators | {
"line": 509,
"column": 21
} | {
"line": 509,
"column": 48
} | [
{
"pp": "case add\nR : Type u\nS : Type v\nι : Type w\ninst✝¹⁹ : CommRing R\ninst✝¹⁸ : CommRing S\ninst✝¹⁷ : Algebra R S\nP : Generators R S ι\nR' : Type u_1\nS' : Type u_2\nι' : Type u_3\ninst✝¹⁶ : CommRing R'\ninst✝¹⁵ : CommRing S'\ninst✝¹⁴ : Algebra R' S'\nP' : Generators R' S' ι'\nR'' : Type u_4\nS'' : Type... | simp only [map_add, hx, hy] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Extension.Generators | {
"line": 509,
"column": 21
} | {
"line": 509,
"column": 48
} | [
{
"pp": "case add\nR : Type u\nS : Type v\nι : Type w\ninst✝¹⁹ : CommRing R\ninst✝¹⁸ : CommRing S\ninst✝¹⁷ : Algebra R S\nP : Generators R S ι\nR' : Type u_1\nS' : Type u_2\nι' : Type u_3\ninst✝¹⁶ : CommRing R'\ninst✝¹⁵ : CommRing S'\ninst✝¹⁴ : Algebra R' S'\nP' : Generators R' S' ι'\nR'' : Type u_4\nS'' : Type... | simp only [map_add, hx, hy] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Extension.Generators | {
"line": 509,
"column": 21
} | {
"line": 509,
"column": 48
} | [
{
"pp": "case add\nR : Type u\nS : Type v\nι : Type w\ninst✝¹⁹ : CommRing R\ninst✝¹⁸ : CommRing S\ninst✝¹⁷ : Algebra R S\nP : Generators R S ι\nR' : Type u_1\nS' : Type u_2\nι' : Type u_3\ninst✝¹⁶ : CommRing R'\ninst✝¹⁵ : CommRing S'\ninst✝¹⁴ : Algebra R' S'\nP' : Generators R' S' ι'\nR'' : Type u_4\nS'' : Type... | simp only [map_add, hx, hy] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Extension.Generators | {
"line": 637,
"column": 52
} | {
"line": 640,
"column": 63
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nI : Type u_8\nf : MvPolynomial I R →ₐ[R] S\nh : Function.Surjective ⇑f\n⊢ (ofAlgHom f h).ker = RingHom.ker f.toRingHom",
"usedConstants": [
"Algebra.Generators.ker",
"Finsupp.instAddZeroClass",
... | by
change RingHom.ker _ = _
congr
exact MvPolynomial.ringHom_ext (by simp) (by simp [ofAlgHom]) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Extension.Generators | {
"line": 695,
"column": 8
} | {
"line": 695,
"column": 16
} | [
{
"pp": "R : Type u\nS : Type v\nι : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nι' : Type u_3\nT : Type u_7\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nQ : Generators S T ι'\nP : Generators R S ι\nthis : DecidableEq (ι' →₀ ℕ) := Cla... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.TensorProduct.Vanishing | {
"line": 112,
"column": 49
} | {
"line": 112,
"column": 52
} | [
{
"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\nk : ℕ\na : ι → Fin k → R\ny : Fin k → N\nh₁ : ∀ (i : ι), n i = ∑ j, a i j • y j\nh₂ : ∀ (j : ... | h₂, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Extension.Cotangent.Basic | {
"line": 180,
"column": 2
} | {
"line": 180,
"column": 10
} | [
{
"pp": "case h.add\nR : Type u\nS : Type v\ninst✝²² : CommRing R\ninst✝²¹ : CommRing S\ninst✝²⁰ : Algebra R S\nP : Extension R S\nR' : Type u'\nS' : Type v'\ninst✝¹⁹ : CommRing R'\ninst✝¹⁸ : CommRing S'\ninst✝¹⁷ : Algebra R' S'\nP' : Extension R' S'\ninst✝¹⁶ : Algebra R R'\ninst✝¹⁵ : Algebra S S'\ninst✝¹⁴ : Al... | | add => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.Extension.Cotangent.Basic | {
"line": 186,
"column": 4
} | {
"line": 186,
"column": 12
} | [
{
"pp": "case h.tmul.add\nR : Type u\nS : Type v\ninst✝²² : CommRing R\ninst✝²¹ : CommRing S\ninst✝²⁰ : Algebra R S\nP : Extension R S\nR' : Type u'\nS' : Type v'\ninst✝¹⁹ : CommRing R'\ninst✝¹⁸ : CommRing S'\ninst✝¹⁷ : Algebra R' S'\nP' : Extension R' S'\ninst✝¹⁶ : Algebra R R'\ninst✝¹⁵ : Algebra S S'\ninst✝¹⁴... | | add => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.Extension.Cotangent.Basic | {
"line": 296,
"column": 2
} | {
"line": 296,
"column": 10
} | [
{
"pp": "case h.add\nR : Type u\nS : Type v\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\nP : Extension R S\nR' : Type u'\nS' : Type v'\ninst✝⁷ : CommRing R'\ninst✝⁶ : CommRing S'\ninst✝⁵ : Algebra R' S'\nP' : Extension R' S'\ninst✝⁴ : Algebra R R'\ninst✝³ : Algebra S S'\ninst✝² : Algebra R ... | | add => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.Extension.Cotangent.Basic | {
"line": 302,
"column": 4
} | {
"line": 302,
"column": 12
} | [
{
"pp": "case h.tmul.add\nR : Type u\nS : Type v\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\nP : Extension R S\nR' : Type u'\nS' : Type v'\ninst✝⁷ : CommRing R'\ninst✝⁶ : CommRing S'\ninst✝⁵ : Algebra R' S'\nP' : Extension R' S'\ninst✝⁴ : Algebra R R'\ninst✝³ : Algebra S S'\ninst✝² : Algeb... | | add => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Data.Fin.Parity | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 47
} | [
{
"pp": "n : ℕ\nk : Fin n\nhk : Even k\n⊢ Odd n ∨ Even ↑k",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Odd",
"id",
"Nat.not_even_iff_odd",
"Fin.val",
"imp_iff_not_or",
"Nat",
"Even",
"propext",
"instAddNat",
"Nat.instSemiring",
"Or... | rw [← Nat.not_even_iff_odd, ← imp_iff_not_or] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Fin.Parity | {
"line": 99,
"column": 2
} | {
"line": 99,
"column": 47
} | [
{
"pp": "n : ℕ\nk : Fin n\ninst✝ : NeZero n\nhk : Odd k\n⊢ Odd n ∨ Odd ↑k",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Odd",
"id",
"Nat.not_even_iff_odd",
"Fin.val",
"imp_iff_not_or",
"Nat",
"Even",
"propext",
"instAddNat",
"Nat.instSemi... | rw [← Nat.not_even_iff_odd, ← imp_iff_not_or] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Alternating.Uncurry.Fin | {
"line": 106,
"column": 4
} | {
"line": 106,
"column": 24
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nM₂ : Type u_3\nN : Type u_4\nN₂ : Type u_5\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup M₂\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : AddCommGroup N₂\ninst✝³ : Module R M\ninst✝² : Module R M₂\ninst✝¹ : Module R N\ninst✝ : Module R N₂\nn : ℕ\nf : M →ₗ[R] M [⋀... | intro v i j hvij hij | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.LinearAlgebra.TensorPower.Basic | {
"line": 56,
"column": 4
} | {
"line": 56,
"column": 12
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nιι : Type u_3\nι : ιι → Type u_4\nai : ιι\na b : ⨂[R] (x : ι ai), M\nh : (reindex R (fun x ↦ M) (Equiv.cast ⋯)) a = b\n⊢ ⟨ai, a⟩ = ⟨ai, b⟩",
"usedConstants": [
"PiTensorProduct.instModule",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.TensorPower.Basic | {
"line": 151,
"column": 65
} | {
"line": 151,
"column": 81
} | [
{
"pp": "case smul_tprod\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nn : ℕ\nr : R\na : Fin n → M\n⊢ r • (cast R M ⋯) (GradedMonoid.GMul.mul ((tprod R) Fin.elim0) ((tprod R) a)) = r • (tprod R) a",
"usedConstants": [
"PiTensorProduct.instModule",
... | tprod_mul_tprod, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.LinearDisjoint | {
"line": 382,
"column": 2
} | {
"line": 384,
"column": 78
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : Ring S\ninst✝¹ : Algebra R S\nM N : Submodule R S\nH : M.LinearDisjoint N\ninst✝ : Flat R ↥M\nκ : Type u_1\nι : Type u_2\nm : κ → ↥M\nn : ι → ↥N\nhm : Function.Injective ⇑(Finsupp.linearCombination R m)\nhn : Function.Injective ⇑(Finsupp.linearCombi... | have : i = Finsupp.linearCombination R fun i ↦ (m i.1).1 * (n i.2).1 := by
ext x
simp [i, i0, i1, i2, finsuppTensorFinsupp'_symm_single_eq_single_one_tmul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.LinearDisjoint | {
"line": 403,
"column": 2
} | {
"line": 405,
"column": 78
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : Ring S\ninst✝¹ : Algebra R S\nM N : Submodule R S\nH : M.LinearDisjoint N\ninst✝ : Flat R ↥N\nκ : Type u_1\nι : Type u_2\nm : κ → ↥M\nn : ι → ↥N\nhm : Function.Injective ⇑(Finsupp.linearCombination R m)\nhn : Function.Injective ⇑(Finsupp.linearCombi... | have : i = Finsupp.linearCombination R fun i ↦ (m i.1).1 * (n i.2).1 := by
ext x
simp [i, i0, i1, i2, finsuppTensorFinsupp'_symm_single_eq_single_one_tmul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.LocalRing.Module | {
"line": 210,
"column": 6
} | {
"line": 214,
"column": 72
} | [
{
"pp": "case refine_2\nR : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsLocalRing R\ninst✝ : FinitePresentation R M\nι : Type u_5\nv : ι → M\nhli : LinearIndependent k (⇑((TensorProduct.mk R k M) 1) ∘ v)\nhsp : Submodule.span k (Set.range (⇑((TensorProdu... | apply hi'.injective
rw [LinearMap.baseChange_eq_ltensor]
erw [← LinearMap.comp_apply (i.lTensor k), ← LinearMap.lTensor_comp]
rw [(LinearMap.exact_subtype_ker_map i).linearMap_comp_eq_zero]
simp only [LinearMap.lTensor_zero, LinearMap.zero_apply, map_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.LocalRing.Module | {
"line": 210,
"column": 6
} | {
"line": 214,
"column": 72
} | [
{
"pp": "case refine_2\nR : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsLocalRing R\ninst✝ : FinitePresentation R M\nι : Type u_5\nv : ι → M\nhli : LinearIndependent k (⇑((TensorProduct.mk R k M) 1) ∘ v)\nhsp : Submodule.span k (Set.range (⇑((TensorProdu... | apply hi'.injective
rw [LinearMap.baseChange_eq_ltensor]
erw [← LinearMap.comp_apply (i.lTensor k), ← LinearMap.lTensor_comp]
rw [(LinearMap.exact_subtype_ker_map i).linearMap_comp_eq_zero]
simp only [LinearMap.lTensor_zero, LinearMap.zero_apply, map_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.MinimalPrime.Colon | {
"line": 60,
"column": 8
} | {
"line": 60,
"column": 28
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nN : Submodule R M\nI : Ideal R\nx : M\ninst✝ : IsNoetherianRing R\nhx : x ∉ N\nann : Ideal R := N.colon {x}\nhI : I ∈ ann.minimalPrimes\nkey : ∃ n, n ≠ 0 ∧ ∃ J, I ^ n * J ≤ ann ∧ ¬J ≤ I\nJ : Ideal R\nhJI... | mem_colon_singleton, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.MinimalPrime.Colon | {
"line": 83,
"column": 43
} | {
"line": 83,
"column": 63
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nN : Submodule R M\nI : Ideal R\nx : M\ninst✝ : IsNoetherianRing R\nhx : x ∉ N\nann : Ideal R := N.colon {x}\nhI : I ∈ ann.minimalPrimes\nkey : ∃ n, n ≠ 0 ∧ ∃ J, I ^ n * J ≤ ann ∧ ¬J ≤ I\nJ : Ideal R\nhJI... | mem_colon_singleton, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.AssociatedPrime.Basic | {
"line": 126,
"column": 34
} | {
"line": 126,
"column": 54
} | [
{
"pp": "case h\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Type u_2\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nM' : Type u_3\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nf : M →ₗ[R] M'\nhf : Function.Injective ⇑f\nx : M\nh : IsAssociatedPrime (⊥.colon {x}).radical M\nr : R\n⊢ (∃ n, r ^ n ∈ ⊥.colon {x}... | mem_colon_singleton, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Ideal.AssociatedPrime.Basic | {
"line": 153,
"column": 8
} | {
"line": 153,
"column": 28
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nH : IsNoetherianRing R\nx : M\nhx : x ≠ 0\ny : M\nh₃ : ∀ I ∈ {P | ⊥.colon {x} ≤ P ∧ P ≠ ⊤ ∧ ∃ y, P = ⊥.colon {y}}, ¬⊥.colon {y} < I\nl : ⊥.colon {x} ≤ ⊥.colon {y}\nh₁ : ⊥.colon {y} ≠ ⊤\na b : R\nhab : (a ... | mem_colon_singleton, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.AssociatedPrime.Basic | {
"line": 156,
"column": 4
} | {
"line": 156,
"column": 24
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nH : IsNoetherianRing R\nx : M\nhx : x ≠ 0\ny : M\nh₃ : ∀ I ∈ {P | ⊥.colon {x} ≤ P ∧ P ≠ ⊤ ∧ ∃ y, P = ⊥.colon {y}}, ¬⊥.colon {y} < I\nl : ⊥.colon {x} ≤ ⊥.colon {y}\nh₁ : ⊥.colon {y} ≠ ⊤\na b : R\nhab : (a ... | mem_colon_singleton, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.AssociatedPrime.Basic | {
"line": 181,
"column": 10
} | {
"line": 181,
"column": 30
} | [
{
"pp": "case h\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.Injective ⇑f\nhfg... | mem_colon_singleton, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.AssociatedPrime.Basic | {
"line": 184,
"column": 10
} | {
"line": 184,
"column": 30
} | [
{
"pp": "case pos.h.refine_2\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... | mem_colon_singleton, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.AssociatedPrime.Basic | {
"line": 194,
"column": 10
} | {
"line": 194,
"column": 30
} | [
{
"pp": "case neg.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... | mem_colon_singleton, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.AssociatedPrime.Basic | {
"line": 196,
"column": 10
} | {
"line": 196,
"column": 30
} | [
{
"pp": "case neg.h.refine_2\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... | mem_colon_singleton, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.AssociatedPrime.Finiteness | {
"line": 82,
"column": 54
} | {
"line": 82,
"column": 62
} | [
{
"pp": "A : Type u\ninst✝² : CommRing A\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module A M\nN₁ N₂ : Submodule A M\nf : ↥N₂ ⧸ N₁.submoduleOf N₂ →ₗ[A] M ⧸ N₁ := (N₁.submoduleOf N₂).mapQ N₁ N₂.subtype ⋯\nhf₁ : f.ker = ⊥\nhf₂ : f.range = map N₁.mkQ N₂\nx✝ : ∃ x, (⊥.colon {N₁.mkQ x}).IsPrime ∧ N₂ = N₁ ⊔ A ∙ x... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Ideal.AssociatedPrime.Finiteness | {
"line": 82,
"column": 54
} | {
"line": 82,
"column": 62
} | [
{
"pp": "A : Type u\ninst✝² : CommRing A\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module A M\nN₁ N₂ : Submodule A M\nf : ↥N₂ ⧸ N₁.submoduleOf N₂ →ₗ[A] M ⧸ N₁ := (N₁.submoduleOf N₂).mapQ N₁ N₂.subtype ⋯\nhf₁ : f.ker = ⊥\nhf₂ : f.range = map N₁.mkQ N₂\nx✝ : ∃ x, (⊥.colon {N₁.mkQ x}).IsPrime ∧ N₂ = N₁ ⊔ A ∙ x... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.AssociatedPrime.Finiteness | {
"line": 82,
"column": 54
} | {
"line": 82,
"column": 62
} | [
{
"pp": "A : Type u\ninst✝² : CommRing A\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module A M\nN₁ N₂ : Submodule A M\nf : ↥N₂ ⧸ N₁.submoduleOf N₂ →ₗ[A] M ⧸ N₁ := (N₁.submoduleOf N₂).mapQ N₁ N₂.subtype ⋯\nhf₁ : f.ker = ⊥\nhf₂ : f.range = map N₁.mkQ N₂\nx✝ : ∃ x, (⊥.colon {N₁.mkQ x}).IsPrime ∧ N₂ = N₁ ⊔ A ∙ x... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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