module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.Lie.CartanExists | {
"line": 339,
"column": 51
} | {
"line": 339,
"column": 70
} | {
"line": 340,
"column": 2
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra K L\ninst✝ : Module.Finite K L\nhLK : ↑(finrank K L) ≤ #K\nU : LieSubalgebra K L\nx : L\nhxU : x ∈ U\ny : L\nhyU : y ∈ U\nEx : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩\nEy : ↑{x | ∃ y ∈ U, engel K y = x} :=... | [] | by cases n <;> simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.Rank | {
"line": 467,
"column": 45
} | {
"line": 467,
"column": 62
} | {
"line": 467,
"column": 63
} | [
{
"pp": "case mp\nm : Type um\nn : Type un\nR : Type uR\ninst✝⁴ : Fintype n\ninst✝³ : Fintype m\ninst✝² : Field R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nA : Matrix m n R\nx : n → R\nh : x ᵥ* Aᵀ ⬝ᵥ A *ᵥ x = 0\n⊢ A *ᵥ x = 0",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
... | [
"case mp\nm : Type um\nn : Type un\nR : Type uR\ninst✝⁴ : Fintype n\ninst✝³ : Fintype m\ninst✝² : Field R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nA : Matrix m n R\nx : n → R\nh : A *ᵥ x ⬝ᵥ A *ᵥ x = 0\n⊢ A *ᵥ x = 0"
] | vecMul_transpose, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Basis | {
"line": 291,
"column": 2
} | {
"line": 291,
"column": 7
} | {
"line": 293,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nL : Type u_3\ninst✝⁴ : Finite ι\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\nb : Basis ι R L\ninst✝ : Fintype ι\nb₁ : Module.Basis ι R ↥b.cartan :=\n (Basis.span ⋯).map (LinearEquiv.ofEq b.cartan.toSubmodule (Submodule.span R (range b.h)) ⋯).symm\nb₂ :... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Lie.Classical | {
"line": 284,
"column": 2
} | {
"line": 285,
"column": 31
} | {
"line": 286,
"column": 2
} | [
{
"pp": "l : Type u_4\nR : Type u₂\ninst✝³ : DecidableEq l\ninst✝² : CommRing R\ninst✝¹ : Fintype l\ninst✝ : Invertible 2\n⊢ PD l R * ⅟2 • (PD l R)ᵀ = 1",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Matrix.fromBlocks",
"Matrix.smul... | [
"l : Type u_4\nR : Type u₂\ninst✝³ : DecidableEq l\ninst✝² : CommRing R\ninst✝¹ : Fintype l\ninst✝ : Invertible 2\n⊢ fromBlocks (1 * ⅟2 • 1ᵀ + -1 * ⅟2 • (-1)ᵀ) (1 * ⅟2 • 1ᵀ + -1 * ⅟2 • 1ᵀ) (1 * ⅟2 • 1ᵀ + 1 * ⅟2 • (-1)ᵀ)\n (1 * ⅟2 • 1ᵀ + 1 * ⅟2 • 1ᵀ) =\n 1"
] | rw [PD, Matrix.fromBlocks_transpose, Matrix.fromBlocks_smul,
Matrix.fromBlocks_multiply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Lie.Classical | {
"line": 293,
"column": 2
} | {
"line": 298,
"column": 5
} | {
"line": 300,
"column": 0
} | [
{
"pp": "n : Type u_1\np : Type u_2\nq : Type u_3\nl : Type u_4\nR : Type u₂\ninst✝⁸ : DecidableEq p\ninst✝⁷ : DecidableEq q\ninst✝⁶ : DecidableEq l\ninst✝⁵ : CommRing R\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype p\ninst✝² : Fintype q\ninst✝¹ : Fintype l\ninst✝ : Invertible 2\n⊢ ↥(typeD l R) ≃ₗ⁅R⁆ ↥(so' l l R)",... | [] | apply (skewAdjointMatricesLieSubalgebraEquiv (JD l R) (PD l R) (by infer_instance)).trans
apply LieEquiv.ofEq
ext A
rw [jd_transform, ← val_unitOfInvertible (2 : R), ← Units.smul_def, LieSubalgebra.mem_coe,
mem_skewAdjointMatricesLieSubalgebra_unit_smul]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Classical | {
"line": 293,
"column": 2
} | {
"line": 298,
"column": 5
} | {
"line": 300,
"column": 0
} | [
{
"pp": "n : Type u_1\np : Type u_2\nq : Type u_3\nl : Type u_4\nR : Type u₂\ninst✝⁸ : DecidableEq p\ninst✝⁷ : DecidableEq q\ninst✝⁶ : DecidableEq l\ninst✝⁵ : CommRing R\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype p\ninst✝² : Fintype q\ninst✝¹ : Fintype l\ninst✝ : Invertible 2\n⊢ ↥(typeD l R) ≃ₗ⁅R⁆ ↥(so' l l R)",... | [] | apply (skewAdjointMatricesLieSubalgebraEquiv (JD l R) (PD l R) (by infer_instance)).trans
apply LieEquiv.ofEq
ext A
rw [jd_transform, ← val_unitOfInvertible (2 : R), ← Units.smul_def, LieSubalgebra.mem_coe,
mem_skewAdjointMatricesLieSubalgebra_unit_smul]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Free | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 33
} | {
"line": 89,
"column": 33
} | [
{
"pp": "R : Type u\nX : Type v\ninst✝ : CommRing R\na b c : lib R X\nh : Rel R X b c\n⊢ Rel R X (a + b) (a + c)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"FreeNonUnitalNonAssocAlgebra",
"congrArg",
"CommSemiring.toSemiring",
"id",
"Distr... | [
"R : Type u\nX : Type v\ninst✝ : CommRing R\na b c : lib R X\nh : Rel R X b c\n⊢ Rel R X (b + a) (c + a)"
] | rw [add_comm _ b, add_comm _ c] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Lie.Derivation.BaseChange | {
"line": 43,
"column": 22
} | {
"line": 49,
"column": 41
} | {
"line": 49,
"column": 42
} | [
{
"pp": "R : 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\nd : Derivation R A A\nx y : A ⊗[R] L\n⊢ { toFun := ⇑(LinearMap.rTensor L ↑d), map_add' := ⋯, map_smul' := ⋯ } ⁅x, y⁆ =\n ⁅x, { toFun := ⇑(LinearMap.rT... | [] | by
simp only [LinearMap.coe_mk, AddHom.coe_mk]
refine x.induction_on (by simp) (fun _ l ↦ ?_) (fun _ _ h1 h2 ↦ ?_)
· refine y.induction_on (by simp) (fun _ l' ↦ ?_) (fun _ _ h1 h2 ↦ ?_)
· simp [← lie_skew l' l, -lie_skew, add_tmul, tmul_neg]
· simp [h1, h2, sub_add_sub_comm]
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.LieTheorem | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 9
} | {
"line": 70,
"column": 2
} | [
{
"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 ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Lie.SemiDirect | {
"line": 151,
"column": 26
} | {
"line": 151,
"column": 31
} | {
"line": 153,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : LieRing K\ninst✝² : LieAlgebra R K\nL : Type u_3\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nψ : L →ₗ⁅R⁆ LieDerivation R K K\nx : K\ny : L\n⊢ { left := x, right := y } ∈ (inl ψ).range ↔\n { left := x, right := y } ∈ LieIdeal.toLieSubalgebra ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.GroupTheory.Perm.Cycle.Concrete | {
"line": 141,
"column": 93
} | {
"line": 148,
"column": 12
} | {
"line": 150,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\ns : Cycle α\nh : s.Subsingleton\n⊢ s.formPerm ⋯ = 1",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"Nat.instIsOrderedAddMonoid",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
"AddLeftCancelSemigrou... | [] | by
obtain ⟨s⟩ := s
simp only [formPerm_coe, mk_eq_coe]
simp only [length_subsingleton_iff, length_coe, mk_eq_coe] at h
obtain - | ⟨hd, tl⟩ := s
· simp
· simp only [length_eq_zero_iff, add_le_iff_nonpos_left, List.length, nonpos_iff_eq_zero] at h
simp [h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Perm.Cycle.Concrete | {
"line": 292,
"column": 6
} | {
"line": 292,
"column": 45
} | {
"line": 294,
"column": 6
} | [
{
"pp": "case pos.zero\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nf : Perm α\nx y : α\nh : f.SameCycle x y\nhx : x ∈ f.support\nleft✝ : 0 < (f.cycleOf x).support.card\nhy : (f ^ 0) x = y\n⊢ f.toList x ~r f.toList y",
"ppTerm": "?pos.zero✝",
"assigned": true,
"usedConstants": [
"... | [
"case pos.zero\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nf : Perm α\nx y : α\nh : f.SameCycle x y\nhx : x ∈ f.support\nleft✝ : 0 < (f.cycleOf x).support.card\nhy : x = y\n⊢ f.toList x ~r f.toList y"
] | simp only [coe_one, id, pow_zero] at hy | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.Perm.Cycle.Concrete | {
"line": 421,
"column": 6
} | {
"line": 421,
"column": 31
} | {
"line": 422,
"column": 6
} | [
{
"pp": "case mk\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx✝ : α\nval✝ : Cycle α\ns : List α\nhn✝ : Cycle.Nodup (Quot.mk (⇑(IsRotated.setoid α)) s)\nht : Cycle.Nontrivial (Quot.mk (⇑(IsRotated.setoid α)) s)\nx : α\nhl : 2 ≤ s.length\nhn : s.Nodup\nhx : x ∈ s\n⊢ ↑(s.formPerm.toList x... | [
"case mk\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx✝ : α\nval✝ : Cycle α\ns : List α\nhn✝ : Cycle.Nodup (Quot.mk (⇑(IsRotated.setoid α)) s)\nht : Cycle.Nontrivial (Quot.mk (⇑(IsRotated.setoid α)) s)\nx : α\nhl : 2 ≤ s.length\nhn : s.Nodup\nhx : x ∈ s\n⊢ (IsRotated.setoid α) (s.formPerm.... | refine Quotient.sound' ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Lie.LieTheorem | {
"line": 174,
"column": 77
} | {
"line": 174,
"column": 82
} | {
"line": 174,
"column": 82
} | [
{
"pp": "k : Type u_1\ninst✝¹⁰ : Field k\nL : Type u_2\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra k L\nV : Type u_3\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module k V\ninst✝⁵ : LieRingModule L V\ninst✝⁴ : LieModule k L V\ninst✝³ : CharZero k\ninst✝² : Module.Finite k V\ninst✝¹ : IsTriangularizable k L V\nA : LieIdeal ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Lie.LieTheorem | {
"line": 174,
"column": 77
} | {
"line": 174,
"column": 82
} | {
"line": 174,
"column": 82
} | [
{
"pp": "k : Type u_1\ninst✝¹⁰ : Field k\nL : Type u_2\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra k L\nV : Type u_3\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module k V\ninst✝⁵ : LieRingModule L V\ninst✝⁴ : LieModule k L V\ninst✝³ : CharZero k\ninst✝² : Module.Finite k V\ninst✝¹ : IsTriangularizable k L V\nA : LieIdeal ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.LieTheorem | {
"line": 174,
"column": 77
} | {
"line": 174,
"column": 82
} | {
"line": 174,
"column": 82
} | [
{
"pp": "k : Type u_1\ninst✝¹⁰ : Field k\nL : Type u_2\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra k L\nV : Type u_3\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module k V\ninst✝⁵ : LieRingModule L V\ninst✝⁴ : LieModule k L V\ninst✝³ : CharZero k\ninst✝² : Module.Finite k V\ninst✝¹ : IsTriangularizable k L V\nA : LieIdeal ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.Cartan | {
"line": 293,
"column": 2
} | {
"line": 293,
"column": 7
} | {
"line": 295,
"column": 0
} | [
{
"pp": "ι : Type u_1\nA : Matrix ι ι ℤ\n⊢ (∀ (b a : ι), a ≠ b → Aᵀ a b = 0 ∨ Aᵀ a b = -1) ↔ ∀ ⦃i j : ι⦄, i ≠ j → A i j = 0 ∨ A i j = -1",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"False",
"congrArg",
"False.elim",
"Eq.mp",
"not_true_eq_false",
"id"... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.LieRinehartAlgebra.Subalgebra | {
"line": 70,
"column": 22
} | {
"line": 70,
"column": 27
} | {
"line": 71,
"column": 2
} | [
{
"pp": "A : Type u_1\nL : Type u_2\ninst✝² : CommRing A\ninst✝¹ : LieRing L\ninst✝ : Module A L\nL' : LieRinehartSubalgebra A L\nx y z : ↥L'\n⊢ ⟨⁅↑(x + y), ↑z⁆, ⋯⟩ = ⟨⁅↑x, ↑z⁆, ⋯⟩ + ⟨⁅↑y, ↑z⁆, ⋯⟩",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subtype.mk.congr_simp"... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.LieRinehartAlgebra.Subalgebra | {
"line": 70,
"column": 22
} | {
"line": 70,
"column": 27
} | {
"line": 71,
"column": 2
} | [
{
"pp": "A : Type u_1\nL : Type u_2\ninst✝² : CommRing A\ninst✝¹ : LieRing L\ninst✝ : Module A L\nL' : LieRinehartSubalgebra A L\nx y z : ↥L'\n⊢ ⟨⁅↑(x + y), ↑z⁆, ⋯⟩ = ⟨⁅↑x, ↑z⁆, ⋯⟩ + ⟨⁅↑y, ↑z⁆, ⋯⟩",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subtype.mk.congr_simp"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.LieRinehartAlgebra.Subalgebra | {
"line": 70,
"column": 22
} | {
"line": 70,
"column": 27
} | {
"line": 71,
"column": 2
} | [
{
"pp": "A : Type u_1\nL : Type u_2\ninst✝² : CommRing A\ninst✝¹ : LieRing L\ninst✝ : Module A L\nL' : LieRinehartSubalgebra A L\nx y z : ↥L'\n⊢ ⟨⁅↑(x + y), ↑z⁆, ⋯⟩ = ⟨⁅↑x, ↑z⁆, ⋯⟩ + ⟨⁅↑y, ↑z⁆, ⋯⟩",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subtype.mk.congr_simp"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.LieRinehartAlgebra.Subalgebra | {
"line": 69,
"column": 22
} | {
"line": 69,
"column": 27
} | {
"line": 70,
"column": 2
} | [
{
"pp": "A : Type u_1\nL : Type u_2\ninst✝² : CommRing A\ninst✝¹ : LieRing L\ninst✝ : Module A L\nL' : LieRinehartSubalgebra A L\nx y z : ↥L'\n⊢ ⟨⁅↑x, ↑(y + z)⁆, ⋯⟩ = ⟨⁅↑x, ↑y⁆, ⋯⟩ + ⟨⁅↑x, ↑z⁆, ⋯⟩",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subtype.mk.congr_simp"... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.LieRinehartAlgebra.Subalgebra | {
"line": 69,
"column": 22
} | {
"line": 69,
"column": 27
} | {
"line": 70,
"column": 2
} | [
{
"pp": "A : Type u_1\nL : Type u_2\ninst✝² : CommRing A\ninst✝¹ : LieRing L\ninst✝ : Module A L\nL' : LieRinehartSubalgebra A L\nx y z : ↥L'\n⊢ ⟨⁅↑x, ↑(y + z)⁆, ⋯⟩ = ⟨⁅↑x, ↑y⁆, ⋯⟩ + ⟨⁅↑x, ↑z⁆, ⋯⟩",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subtype.mk.congr_simp"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.LieRinehartAlgebra.Subalgebra | {
"line": 69,
"column": 22
} | {
"line": 69,
"column": 27
} | {
"line": 70,
"column": 2
} | [
{
"pp": "A : Type u_1\nL : Type u_2\ninst✝² : CommRing A\ninst✝¹ : LieRing L\ninst✝ : Module A L\nL' : LieRinehartSubalgebra A L\nx y z : ↥L'\n⊢ ⟨⁅↑x, ↑(y + z)⁆, ⋯⟩ = ⟨⁅↑x, ↑y⁆, ⋯⟩ + ⟨⁅↑x, ↑z⁆, ⋯⟩",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subtype.mk.congr_simp"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.LieRinehartAlgebra.Subalgebra | {
"line": 71,
"column": 19
} | {
"line": 71,
"column": 24
} | {
"line": 72,
"column": 2
} | [
{
"pp": "A : Type u_1\nL : Type u_2\ninst✝² : CommRing A\ninst✝¹ : LieRing L\ninst✝ : Module A L\nL' : LieRinehartSubalgebra A L\nx : ↥L'\n⊢ ⟨⁅↑x, ↑x⁆, ⋯⟩ = 0",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subtype.mk.congr_simp",
"LieRing.toAddCommGroup",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.LieRinehartAlgebra.Subalgebra | {
"line": 71,
"column": 19
} | {
"line": 71,
"column": 24
} | {
"line": 72,
"column": 2
} | [
{
"pp": "A : Type u_1\nL : Type u_2\ninst✝² : CommRing A\ninst✝¹ : LieRing L\ninst✝ : Module A L\nL' : LieRinehartSubalgebra A L\nx : ↥L'\n⊢ ⟨⁅↑x, ↑x⁆, ⋯⟩ = 0",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subtype.mk.congr_simp",
"LieRing.toAddCommGroup",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.LieRinehartAlgebra.Subalgebra | {
"line": 71,
"column": 19
} | {
"line": 71,
"column": 24
} | {
"line": 72,
"column": 2
} | [
{
"pp": "A : Type u_1\nL : Type u_2\ninst✝² : CommRing A\ninst✝¹ : LieRing L\ninst✝ : Module A L\nL' : LieRinehartSubalgebra A L\nx : ↥L'\n⊢ ⟨⁅↑x, ↑x⁆, ⋯⟩ = 0",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subtype.mk.congr_simp",
"LieRing.toAddCommGroup",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.LieRinehartAlgebra.Subalgebra | {
"line": 72,
"column": 26
} | {
"line": 72,
"column": 31
} | {
"line": 74,
"column": 0
} | [
{
"pp": "A : Type u_1\nL : Type u_2\ninst✝² : CommRing A\ninst✝¹ : LieRing L\ninst✝ : Module A L\nL' : LieRinehartSubalgebra A L\nx y z : ↥L'\n⊢ ⟨⁅↑x, ↑⟨⁅↑y, ↑z⁆, ⋯⟩⁆, ⋯⟩ = ⟨⁅↑⟨⁅↑x, ↑y⁆, ⋯⟩, ↑z⁆, ⋯⟩ + ⟨⁅↑y, ↑⟨⁅↑x, ↑z⁆, ⋯⟩⁆, ⋯⟩",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Subtype.... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.LieRinehartAlgebra.Subalgebra | {
"line": 72,
"column": 26
} | {
"line": 72,
"column": 31
} | {
"line": 74,
"column": 0
} | [
{
"pp": "A : Type u_1\nL : Type u_2\ninst✝² : CommRing A\ninst✝¹ : LieRing L\ninst✝ : Module A L\nL' : LieRinehartSubalgebra A L\nx y z : ↥L'\n⊢ ⟨⁅↑x, ↑⟨⁅↑y, ↑z⁆, ⋯⟩⁆, ⋯⟩ = ⟨⁅↑⟨⁅↑x, ↑y⁆, ⋯⟩, ↑z⁆, ⋯⟩ + ⟨⁅↑y, ↑⟨⁅↑x, ↑z⁆, ⋯⟩⁆, ⋯⟩",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Subtype.... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.LieRinehartAlgebra.Subalgebra | {
"line": 72,
"column": 26
} | {
"line": 72,
"column": 31
} | {
"line": 74,
"column": 0
} | [
{
"pp": "A : Type u_1\nL : Type u_2\ninst✝² : CommRing A\ninst✝¹ : LieRing L\ninst✝ : Module A L\nL' : LieRinehartSubalgebra A L\nx y z : ↥L'\n⊢ ⟨⁅↑x, ↑⟨⁅↑y, ↑z⁆, ⋯⟩⁆, ⋯⟩ = ⟨⁅↑⟨⁅↑x, ↑y⁆, ⋯⟩, ↑z⁆, ⋯⟩ + ⟨⁅↑y, ↑⟨⁅↑x, ↑z⁆, ⋯⟩⁆, ⋯⟩",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Subtype.... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.LieRinehartAlgebra.Subalgebra | {
"line": 141,
"column": 86
} | {
"line": 145,
"column": 9
} | {
"line": 147,
"column": 0
} | [
{
"pp": "A : Type u_1\nL : Type u_2\ninst✝² : CommRing A\ninst✝¹ : LieRing L\ninst✝ : Module A L\n⊢ Function.Injective toSubmodule",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"LieRinehartSubalgebra.coe_set_eq",
"LieRing.toAddCommGroup",
... | [] | by
intro L₁' L₂' h
rw [SetLike.ext'_iff] at h
rw [← coe_set_eq]
exact h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.LieRinehartAlgebra.Subalgebra | {
"line": 185,
"column": 17
} | {
"line": 185,
"column": 22
} | {
"line": 187,
"column": 0
} | [
{
"pp": "A : Type u_1\nL : Type u_2\ninst✝⁸ : CommRing A\ninst✝⁷ : LieRing L\ninst✝⁶ : Module A L\nL' : LieRinehartSubalgebra A L\ninst✝⁵ : LieRingModule L A\ninst✝⁴ : LieRinehartRing A L\nR : Type u_3\ninst✝³ : CommRing R\ninst✝² : Algebra R A\ninst✝¹ : LieAlgebra R L\ninst✝ : LieRinehartAlgebra R A L\n⊢ ∀ (t ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.LieRinehartAlgebra.Subalgebra | {
"line": 185,
"column": 17
} | {
"line": 185,
"column": 22
} | {
"line": 187,
"column": 0
} | [
{
"pp": "A : Type u_1\nL : Type u_2\ninst✝⁸ : CommRing A\ninst✝⁷ : LieRing L\ninst✝⁶ : Module A L\nL' : LieRinehartSubalgebra A L\ninst✝⁵ : LieRingModule L A\ninst✝⁴ : LieRinehartRing A L\nR : Type u_3\ninst✝³ : CommRing R\ninst✝² : Algebra R A\ninst✝¹ : LieAlgebra R L\ninst✝ : LieRinehartAlgebra R A L\n⊢ ∀ (t ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.LieRinehartAlgebra.Subalgebra | {
"line": 185,
"column": 17
} | {
"line": 185,
"column": 22
} | {
"line": 187,
"column": 0
} | [
{
"pp": "A : Type u_1\nL : Type u_2\ninst✝⁸ : CommRing A\ninst✝⁷ : LieRing L\ninst✝⁶ : Module A L\nL' : LieRinehartSubalgebra A L\ninst✝⁵ : LieRingModule L A\ninst✝⁴ : LieRinehartRing A L\nR : Type u_3\ninst✝³ : CommRing R\ninst✝² : Algebra R A\ninst✝¹ : LieAlgebra R L\ninst✝ : LieRinehartAlgebra R A L\n⊢ ∀ (t ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.LieTheorem | {
"line": 243,
"column": 2
} | {
"line": 243,
"column": 32
} | {
"line": 244,
"column": 2
} | [
{
"pp": "k : Type u_1\ninst✝¹¹ : Field k\nL : Type u_2\ninst✝¹⁰ : LieRing L\ninst✝⁹ : LieAlgebra k L\nV : Type u_3\ninst✝⁸ : AddCommGroup V\ninst✝⁷ : Module k V\ninst✝⁶ : LieRingModule L V\ninst✝⁵ : LieModule k L V\ninst✝⁴ : CharZero k\ninst✝³ : Module.Finite k V\ninst✝² : Nontrivial V\ninst✝¹ : IsSolvable L\ni... | [
"k : Type u_1\ninst✝¹¹ : Field k\nL : Type u_2\ninst✝¹⁰ : LieRing L\ninst✝⁹ : LieAlgebra k L\nV : Type u_3\ninst✝⁸ : AddCommGroup V\ninst✝⁷ : Module k V\ninst✝⁶ : LieRingModule L V\ninst✝⁵ : LieModule k L V\ninst✝⁴ : CharZero k\ninst✝³ : Module.Finite k V\ninst✝² : Nontrivial V\ninst✝¹ : IsSolvable L\ninst✝ : IsTri... | let imL := (toEnd k L V).range | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Algebra.Module.Bimodule | {
"line": 87,
"column": 49
} | {
"line": 87,
"column": 97
} | {
"line": 87,
"column": 97
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\nM : Type u_4\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : Module R M\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Module A M\ninst✝⁵ : Module B M\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\ninst✝² : IsScalarTower R A M\ninst✝¹ : IsScal... | [] | simp only [zero_smul, SetLike.mem_coe, zero_mem] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Module.Bimodule | {
"line": 87,
"column": 49
} | {
"line": 87,
"column": 97
} | {
"line": 87,
"column": 97
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\nM : Type u_4\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : Module R M\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Module A M\ninst✝⁵ : Module B M\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\ninst✝² : IsScalarTower R A M\ninst✝¹ : IsScal... | [] | simp only [zero_smul, SetLike.mem_coe, zero_mem] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.Bimodule | {
"line": 87,
"column": 49
} | {
"line": 87,
"column": 97
} | {
"line": 87,
"column": 97
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\nM : Type u_4\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : Module R M\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Module A M\ninst✝⁵ : Module B M\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\ninst✝² : IsScalarTower R A M\ninst✝¹ : IsScal... | [] | simp only [zero_smul, SetLike.mem_coe, zero_mem] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.FractionalIdeal.Basic | {
"line": 570,
"column": 2
} | {
"line": 571,
"column": 57
} | {
"line": 573,
"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\nI J : Ideal R\n⊢ ↑(I * J) = ↑I * ↑J",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsLocalization.coeSubmodule_mul",
"Submodule",
"Semiri... | [] | simp only [mul_def]
exact coeToSubmodule_injective (coeSubmodule_mul _ _ _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.FractionalIdeal.Basic | {
"line": 570,
"column": 2
} | {
"line": 571,
"column": 57
} | {
"line": 573,
"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\nI J : Ideal R\n⊢ ↑(I * J) = ↑I * ↑J",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsLocalization.coeSubmodule_mul",
"Submodule",
"Semiri... | [] | simp only [mul_def]
exact coeToSubmodule_injective (coeSubmodule_mul _ _ _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.ChainOfDivisors | {
"line": 425,
"column": 95
} | {
"line": 427,
"column": 63
} | {
"line": 427,
"column": 63
} | [
{
"pp": "M : Type u_1\ninst✝⁶ : CommMonoidWithZero M\ninst✝⁵ : IsCancelMulZero M\nN : Type u_2\ninst✝⁴ : CommMonoidWithZero N\ninst✝³ : Subsingleton Mˣ\ninst✝² : Subsingleton Nˣ\ninst✝¹ : UniqueFactorizationMonoid M\ninst✝ : UniqueFactorizationMonoid N\nm p : M\nn : N\nhm : m ≠ 0\nhn : n ≠ 0\nhp : p ∈ normalize... | [] | by
simp only [dvd_of_mem_normalizedFactors hp, associatesEquivOfUniqueUnits_symm_apply,
associatesEquivOfUniqueUnits_apply, out_mk, normalize_eq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 362,
"column": 6
} | {
"line": 362,
"column": 15
} | {
"line": 362,
"column": 15
} | [
{
"pp": "R₁ : Type u_3\ninst✝² : CommRing R₁\nK : Type u_4\ninst✝¹ : Field K\ninst✝ : Algebra R₁ K\nI J : FractionalIdeal R₁⁰ K\nh : I * J = 1\nhI : I = 0\n⊢ 0 = I * J",
"ppTerm": "?m.91",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
... | [] | simp [hI] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.DedekindDomain.Ideal.Basic | {
"line": 360,
"column": 4
} | {
"line": 364,
"column": 43
} | {
"line": 366,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nK✝ : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K✝\ninst✝² : Algebra A K✝\ninst✝¹ : IsFractionRing A K✝\ninst✝ : IsDedekindDomain A\nI✝ : Ideal A\nI : { x // 0 < x }\nJ K : Ideal A\ne : (fun x1 x2 ↦ x1 ≤ x2) ((fun x y ↦ ↑x * y) I J) ((fun x y ↦ ↑x * y... | [] | rwa [← FractionalIdeal.coeIdeal_le_coeIdeal (FractionRing A),
← mul_le_mul_iff_right₀ (α := FractionalIdeal A⁰ (FractionRing A)) (a := I.1)
(by simpa [pos_iff_ne_zero] using I.2.ne'),
← FractionalIdeal.coeIdeal_mul, ← FractionalIdeal.coeIdeal_mul,
FractionalIdeal.coeIdeal_le_coeIdeal] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 130,
"column": 73
} | {
"line": 135,
"column": 55
} | {
"line": 137,
"column": 0
} | [
{
"pp": "A : Type u_2\ninst✝¹ : CommRing A\ninst✝ : IsDedekindDomain A\na : A\n⊢ Prime (span {a}) ↔ Prime a",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"not_iff_not",
"Eq.mpr",
"False",
"Semiring.toModule",
"congrArg",
"CommSemiring.toSemiring",
... | [] | by
rcases eq_or_ne a 0 with rfl | ha
· rw [Set.singleton_zero, span_zero, ← zero_eq_bot, ← not_iff_not]
simp only [not_prime_zero, not_false_eq_true]
· have ha' : span {a} ≠ ⊥ := by simpa only [ne_eq, span_singleton_eq_bot] using ha
rw [prime_iff_isPrime ha', span_singleton_prime ha] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 675,
"column": 4
} | {
"line": 675,
"column": 66
} | {
"line": 676,
"column": 4
} | [
{
"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 x' : P\nhx' : x' ∈ spanSingleton S x\nh : (IsLocalization.map P' (RingHom.id ... | obtain ⟨x', hx', rfl⟩ := (mem_canonicalEquiv_apply _ _ _).mp h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 268,
"column": 4
} | {
"line": 268,
"column": 13
} | {
"line": 269,
"column": 2
} | [
{
"pp": "case pos\nA : Type u_2\ninst✝² : CommRing A\ninst✝¹ : IsDedekindDomain A\nI : Ideal A\nι : Type u_4\ninst✝ : Nonempty ι\nJ : ι → Ideal A\nhI : I = 0\n⊢ I * ⨅ i, J i = ⨅ i, I * J i",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Submodule",
"iInf",
"Semiring.toMo... | [] | simp [hI] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 268,
"column": 4
} | {
"line": 268,
"column": 13
} | {
"line": 269,
"column": 2
} | [
{
"pp": "case pos\nA : Type u_2\ninst✝² : CommRing A\ninst✝¹ : IsDedekindDomain A\nI : Ideal A\nι : Type u_4\ninst✝ : Nonempty ι\nJ : ι → Ideal A\nhI : I = 0\n⊢ I * ⨅ i, J i = ⨅ i, I * J i",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Submodule",
"iInf",
"Semiring.toMo... | [] | simp [hI] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 268,
"column": 4
} | {
"line": 268,
"column": 13
} | {
"line": 269,
"column": 2
} | [
{
"pp": "case pos\nA : Type u_2\ninst✝² : CommRing A\ninst✝¹ : IsDedekindDomain A\nI : Ideal A\nι : Type u_4\ninst✝ : Nonempty ι\nJ : ι → Ideal A\nhI : I = 0\n⊢ I * ⨅ i, J i = ⨅ i, I * J i",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Submodule",
"iInf",
"Semiring.toMo... | [] | simp [hI] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 825,
"column": 4
} | {
"line": 825,
"column": 64
} | {
"line": 826,
"column": 4
} | [
{
"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⊢ I ≤ J ^ (Associates.mk J).count (Associates.mk I).factors",
"ppTerm": "?m.71",
"assigned": true,
"usedConstants": [
... | [
"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 ^ (Associates.mk J).count (Associates.mk I).factors ∣ Associates.mk I"
] | rw [← dvd_iff_le, ← Associates.mk_dvd_mk, Associates.mk_pow] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 825,
"column": 4
} | {
"line": 825,
"column": 64
} | {
"line": 826,
"column": 4
} | [
{
"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⊢ ¬I ≤ J ^ ((Associates.mk J).count (Associates.mk I).factors + 1)",
"ppTerm": "?m.72",
"assigned": true,
"usedConstants"... | [
"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 ^ ((Associates.mk J).count (Associates.mk I).factors + 1) ∣ Associates.mk I"
] | rw [← dvd_iff_le, ← Associates.mk_dvd_mk, Associates.mk_pow] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 847,
"column": 4
} | {
"line": 847,
"column": 9
} | {
"line": 848,
"column": 2
} | [
{
"pp": "case hI\nR : 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⊢ ¬a₀ = 0",
"ppTerm": "?hI",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsDedekindDomain.toIsDomain",
"False",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Module.LinearMap.FiniteRange | {
"line": 121,
"column": 2
} | {
"line": 123,
"column": 16
} | {
"line": 125,
"column": 0
} | [
{
"pp": "K : Type u_1\nV : Type u_2\nV₂ : Type u_4\ninst✝⁵ : Ring K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\ninst✝ : IsNoetherianRing K\nu : V →ₗ[K] V₂\nh : u.HasFiniteRange\n⊢ u.HasNoetherianRange",
"ppTerm": "?m.33",
"assigned": true,
"usedCons... | [] | rw [HasNoetherianRange]
have := Finite.of_fg h.fg_range
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.LinearMap.FiniteRange | {
"line": 121,
"column": 2
} | {
"line": 123,
"column": 16
} | {
"line": 125,
"column": 0
} | [
{
"pp": "K : Type u_1\nV : Type u_2\nV₂ : Type u_4\ninst✝⁵ : Ring K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\ninst✝ : IsNoetherianRing K\nu : V →ₗ[K] V₂\nh : u.HasFiniteRange\n⊢ u.HasNoetherianRange",
"ppTerm": "?m.33",
"assigned": true,
"usedCons... | [] | rw [HasNoetherianRange]
have := Finite.of_fg h.fg_range
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 1180,
"column": 6
} | {
"line": 1180,
"column": 23
} | {
"line": 1180,
"column": 24
} | [
{
"pp": "A : Type u_4\ninst✝⁵ : CommRing A\np : Ideal A\nhpb : p ≠ ⊥\nhpm : p.IsMaximal\nB : Type u_5\ninst✝⁴ : CommRing B\ninst✝³ : IsDedekindDomain B\ninst✝² : Algebra A B\ninst✝¹ : IsDomain A\ninst✝ : IsTorsionFree A B\nP : Ideal B\n⊢ P ∈ primesOverFinset p B ↔ P ∈ p.primesOver B",
"ppTerm": "?m.30",
... | [
"A : Type u_4\ninst✝⁵ : CommRing A\np : Ideal A\nhpb : p ≠ ⊥\nhpm : p.IsMaximal\nB : Type u_5\ninst✝⁴ : CommRing B\ninst✝³ : IsDedekindDomain B\ninst✝² : Algebra A B\ninst✝¹ : IsDomain A\ninst✝ : IsTorsionFree A B\nP : Ideal B\n⊢ P ∈ ↑(primesOverFinset p B) ↔ P ∈ p.primesOver B"
] | ← Finset.mem_coe, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 1237,
"column": 4
} | {
"line": 1237,
"column": 45
} | {
"line": 1239,
"column": 0
} | [
{
"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⊢ (↑(primesOverFinset p B)).Finite",
"ppT... | [] | exact (primesOverFinset p B).finite_toSet | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 1268,
"column": 22
} | {
"line": 1268,
"column": 27
} | {
"line": 1268,
"column": 27
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : Algebra R A\ninst✝² : FaithfulSMul R A\ninst✝¹ : Algebra.IsIntegral R A\ninst✝ : Nontrivial (HeightOneSpectrum R)\nthis : IsDomain R\nf : HeightOneSpectrum A → HeightOneSpectrum R := fun p ↦ HeightOneSpe... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 1268,
"column": 22
} | {
"line": 1268,
"column": 27
} | {
"line": 1268,
"column": 27
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : Algebra R A\ninst✝² : FaithfulSMul R A\ninst✝¹ : Algebra.IsIntegral R A\ninst✝ : Nontrivial (HeightOneSpectrum R)\nthis : IsDomain R\nf : HeightOneSpectrum A → HeightOneSpectrum R := fun p ↦ HeightOneSpe... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 1268,
"column": 22
} | {
"line": 1268,
"column": 27
} | {
"line": 1268,
"column": 27
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : Algebra R A\ninst✝² : FaithfulSMul R A\ninst✝¹ : Algebra.IsIntegral R A\ninst✝ : Nontrivial (HeightOneSpectrum R)\nthis : IsDomain R\nf : HeightOneSpectrum A → HeightOneSpectrum R := fun p ↦ HeightOneSpe... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.LinearMap.Index | {
"line": 129,
"column": 31
} | {
"line": 130,
"column": 85
} | {
"line": 132,
"column": 0
} | [
{
"pp": "M : Type u_1\nN : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\nk : Type u_3\ninst✝² : Field k\ninst✝¹ : Module k M\ninst✝ : Module k N\nf : M →ₗ[k] N\nt : k\nht : t ≠ 0\n⊢ (t • f).index = f.index",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | by
rw [index_eq_finrank_sub, index_eq_finrank_sub, ker_smul _ _ ht, range_smul _ _ ht] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.Presentation.Cokernel | {
"line": 115,
"column": 24
} | {
"line": 115,
"column": 29
} | {
"line": 116,
"column": 2
} | [
{
"pp": "A : Type u\ninst✝⁸ : Ring A\nM₁ : Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\ninst✝⁷ : AddCommGroup M₁\ninst✝⁶ : Module A M₁\ninst✝⁵ : AddCommGroup M₂\ninst✝⁴ : Module A M₂\ninst✝³ : AddCommGroup M₃\ninst✝² : Module A M₃\npres₂ : Presentation A M₂\nf : M₁ →ₗ[A] M₂\nι : Type w₁\ng₁ : ι → M₁\ndata : pres₂.Coker... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Module.Presentation.Cokernel | {
"line": 115,
"column": 24
} | {
"line": 115,
"column": 29
} | {
"line": 116,
"column": 2
} | [
{
"pp": "A : Type u\ninst✝⁸ : Ring A\nM₁ : Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\ninst✝⁷ : AddCommGroup M₁\ninst✝⁶ : Module A M₁\ninst✝⁵ : AddCommGroup M₂\ninst✝⁴ : Module A M₂\ninst✝³ : AddCommGroup M₃\ninst✝² : Module A M₃\npres₂ : Presentation A M₂\nf : M₁ →ₗ[A] M₂\nι : Type w₁\ng₁ : ι → M₁\ndata : pres₂.Coker... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.Presentation.Cokernel | {
"line": 115,
"column": 24
} | {
"line": 115,
"column": 29
} | {
"line": 116,
"column": 2
} | [
{
"pp": "A : Type u\ninst✝⁸ : Ring A\nM₁ : Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\ninst✝⁷ : AddCommGroup M₁\ninst✝⁶ : Module A M₁\ninst✝⁵ : AddCommGroup M₂\ninst✝⁴ : Module A M₂\ninst✝³ : AddCommGroup M₃\ninst✝² : Module A M₃\npres₂ : Presentation A M₂\nf : M₁ →ₗ[A] M₂\nι : Type w₁\ng₁ : ι → M₁\ndata : pres₂.Coker... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 534,
"column": 6
} | {
"line": 534,
"column": 69
} | {
"line": 535,
"column": 6
} | [
{
"pp": "R : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : Module R M\ninst✝⁹ : AddCommGroup N\ninst✝⁸ : Module R N\nS : Submonoid R\nM' : Type u_1\ninst✝⁷ : AddCommGroup M'\ninst✝⁶ : Module R M'\nf : M →ₗ[R] M'\ninst✝⁵ : IsLocalizedModule S f\nN' : Type u_2\nin... | [
"R : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : Module R M\ninst✝⁹ : AddCommGroup N\ninst✝⁸ : Module R N\nS : Submonoid R\nM' : Type u_1\ninst✝⁷ : AddCommGroup M'\ninst✝⁶ : Module R M'\nf : M →ₗ[R] M'\ninst✝⁵ : IsLocalizedModule S f\nN' : Type u_2\ninst✝⁴ : AddCo... | apply IsLocalizedModule.ext S f (IsLocalizedModule.map_units g) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 579,
"column": 4
} | {
"line": 579,
"column": 67
} | {
"line": 580,
"column": 4
} | [
{
"pp": "case surj\nR : Type u_3\nM : Type u_4\nN : Type u_5\nN'✝ : Type ?u.18\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 : IsLocalizedModule S ... | [
"case surj\nR : Type u_3\nM : Type u_4\nN : Type u_5\nN'✝ : Type ?u.18\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 : IsLocalizedModule S f✝\nM' : Typ... | apply IsLocalizedModule.ext S f (IsLocalizedModule.map_units g) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Module.PID | {
"line": 281,
"column": 4
} | {
"line": 281,
"column": 27
} | {
"line": 281,
"column": 28
} | [
{
"pp": "R : Type u\ninst✝⁵ : CommRing R\ninst✝⁴ : IsPrincipalIdealRing R\nM : Type v\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsDomain R\ninst✝ : Module.Finite R M\nm : ℕ\nι : Type u\nw✝ : Fintype ι\np : ι → R\nirr : ∀ (i : ι), Irreducible (p i)\nn : ι → ℕ\ne : M ≃ₗ[R] (Fin m →₀ R) × ⨁ (i : ι), ... | [
"R : Type u\ninst✝⁵ : CommRing R\ninst✝⁴ : IsPrincipalIdealRing R\nM : Type v\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsDomain R\ninst✝ : Module.Finite R M\nm : ℕ\nι : Type u\nw✝ : Fintype ι\np : ι → R\nirr : ∀ (i : ι), Irreducible (p i)\nn : ι → ℕ\ne : M ≃ₗ[R] (Fin m →₀ R) × ⨁ (i : ι), R ⧸ R ∙ p i ... | e.symm.map_eq_zero_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Extension.Presentation.Basic | {
"line": 492,
"column": 4
} | {
"line": 492,
"column": 55
} | {
"line": 494,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type v\nι : Type w\nσ : Type t\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP✝ P : Presentation R S ι σ\nι' : Type u_1\nσ' : Type u_2\ne : ι' ≃ ι\nf : σ' ≃ σ\nhf : Function.Bijective ⇑(rename ⇑e.symm)\n⊢ Ideal.comap (rename ⇑e.symm) (Ideal.map (rename ⇑e.symm) (Ideal.... | [] | simp [Ideal.comap_map_of_bijective _ hf, rename_id] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Module.Presentation.Differentials | {
"line": 104,
"column": 4
} | {
"line": 104,
"column": 9
} | {
"line": 105,
"column": 2
} | [
{
"pp": "R : Type u\nS : Type v\nι : Type w\nσ : Type t\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\npres : Presentation R S ι σ\nφ : (σ →₀ S) →ₗ[pres.Ring] pres.toExtension.Cotangent := { toFun := ⇑(hom₁ pres), map_add' := ⋯, map_smul' := ⋯ }\nh₁ : Function.Surjective ⇑Extension.Cotangent.mk... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Module.Presentation.Free | {
"line": 49,
"column": 22
} | {
"line": 49,
"column": 27
} | {
"line": 50,
"column": 2
} | [
{
"pp": "A : Type u\ninst✝³ : Ring A\nrelations : Relations A\nM : Type v\ninst✝² : AddCommGroup M\ninst✝¹ : Module A M\ninst✝ : IsEmpty relations.R\n⊢ ∀ {N : Type w} [inst : AddCommGroup N] [inst_1 : Module A N] (s : relations.Solution N),\n relations.solutionFinsupp.postcomp (Finsupp.linearCombination A s.... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Module.Presentation.Free | {
"line": 49,
"column": 22
} | {
"line": 49,
"column": 27
} | {
"line": 50,
"column": 2
} | [
{
"pp": "A : Type u\ninst✝³ : Ring A\nrelations : Relations A\nM : Type v\ninst✝² : AddCommGroup M\ninst✝¹ : Module A M\ninst✝ : IsEmpty relations.R\n⊢ ∀ {N : Type w} [inst : AddCommGroup N] [inst_1 : Module A N] (s : relations.Solution N),\n relations.solutionFinsupp.postcomp (Finsupp.linearCombination A s.... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.Presentation.Free | {
"line": 49,
"column": 22
} | {
"line": 49,
"column": 27
} | {
"line": 50,
"column": 2
} | [
{
"pp": "A : Type u\ninst✝³ : Ring A\nrelations : Relations A\nM : Type v\ninst✝² : AddCommGroup M\ninst✝¹ : Module A M\ninst✝ : IsEmpty relations.R\n⊢ ∀ {N : Type w} [inst : AddCommGroup N] [inst_1 : Module A N] (s : relations.Solution N),\n relations.solutionFinsupp.postcomp (Finsupp.linearCombination A s.... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.Presentation.Tensor | {
"line": 90,
"column": 24
} | {
"line": 90,
"column": 29
} | {
"line": 91,
"column": 2
} | [
{
"pp": "A : Type u\ninst✝⁶ : CommRing A\nM₁ : Type v₁\nM₂ : Type v₂\ninst✝⁵ : AddCommGroup M₁\ninst✝⁴ : AddCommGroup M₂\ninst✝³ : Module A M₁\ninst✝² : Module A M₂\nrelations₁ : Relations A\nrelations₂ : Relations A\nsolution₁ : relations₁.Solution M₁\nsolution₂ : relations₂.Solution M₂\nh₁ : solution₁.IsPrese... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Module.Presentation.Tensor | {
"line": 90,
"column": 24
} | {
"line": 90,
"column": 29
} | {
"line": 91,
"column": 2
} | [
{
"pp": "A : Type u\ninst✝⁶ : CommRing A\nM₁ : Type v₁\nM₂ : Type v₂\ninst✝⁵ : AddCommGroup M₁\ninst✝⁴ : AddCommGroup M₂\ninst✝³ : Module A M₁\ninst✝² : Module A M₂\nrelations₁ : Relations A\nrelations₂ : Relations A\nsolution₁ : relations₁.Solution M₁\nsolution₂ : relations₂.Solution M₂\nh₁ : solution₁.IsPrese... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.Presentation.Tensor | {
"line": 90,
"column": 24
} | {
"line": 90,
"column": 29
} | {
"line": 91,
"column": 2
} | [
{
"pp": "A : Type u\ninst✝⁶ : CommRing A\nM₁ : Type v₁\nM₂ : Type v₂\ninst✝⁵ : AddCommGroup M₁\ninst✝⁴ : AddCommGroup M₂\ninst✝³ : Module A M₁\ninst✝² : Module A M₂\nrelations₁ : Relations A\nrelations₂ : Relations A\nsolution₁ : relations₁.Solution M₁\nsolution₂ : relations₂.Solution M₂\nh₁ : solution₁.IsPrese... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Extension.Basic | {
"line": 546,
"column": 4
} | {
"line": 546,
"column": 91
} | {
"line": 547,
"column": 4
} | [
{
"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.16\nS' : Type ?u.18\ninst✝¹⁹ : CommRing R'\ninst✝¹⁸ : CommRing S'\ninst✝¹⁷ : Algebra R' S'\nP' : Extension R' S'\nR'' : Type ?u.29\nS'' : Type ?u.31\ninst✝¹⁶ : CommR... | [
"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.16\nS' : Type ?u.18\ninst✝¹⁹ : CommRing R'\ninst✝¹⁸ : CommRing S'\ninst✝¹⁷ : Algebra R' S'\nP' : Extension R' S'\nR'' : Type ?u.29\nS'' : Type ?u.31\ninst✝¹⁶ : CommRing R''\nins... | obtain ⟨x, rfl⟩ := TensorProduct.mk_surjective P.Ring P.ker S P.algebraMap_surjective x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Extension.Cotangent.Basic | {
"line": 90,
"column": 55
} | {
"line": 90,
"column": 60
} | {
"line": 90,
"column": 60
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹³ : CommRing R\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP✝ : Extension R S\nA✝ : Type u_1\ninst✝¹⁰ : CommRing A✝\ninst✝⁹ : Algebra S A✝\ninst✝⁸ : Algebra P✝.Ring A✝\ninst✝⁷ : IsScalarTower P✝.Ring S A✝\nP : Type u_2\nA : Type u_3\ninst✝⁶ : CommRing P\ninst✝⁵ : CommRin... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Extension.Cotangent.Basic | {
"line": 97,
"column": 75
} | {
"line": 97,
"column": 80
} | {
"line": 97,
"column": 80
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹³ : CommRing R\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP✝ : Extension R S\nA✝ : Type u_1\ninst✝¹⁰ : CommRing A✝\ninst✝⁹ : Algebra S A✝\ninst✝⁸ : Algebra P✝.Ring A✝\ninst✝⁷ : IsScalarTower P✝.Ring S A✝\nP : Type u_2\nA : Type u_3\ninst✝⁶ : CommRing P\ninst✝⁵ : CommRin... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.TensorProduct.Vanishing | {
"line": 244,
"column": 2
} | {
"line": 244,
"column": 69
} | {
"line": 245,
"column": 2
} | [
{
"pp": "case e'_2\nR : Type u_1\ninst✝⁴ : CommRing R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_3\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nhMN : ∀ {l : ℕ} {m : Fin l → M} {n : Fin l → N}, ∑ i, m i ⊗ₜ[R] n i = 0 → VanishesTrivially R m n\nM' : Submodule R M\ns : Finset (↥M' × N... | [
"case e'_1\nR : Type u_1\ninst✝⁴ : CommRing R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_3\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nhMN : ∀ {l : ℕ} {m : Fin l → M} {n : Fin l → N}, ∑ i, m i ⊗ₜ[R] n i = 0 → VanishesTrivially R m n\nM' : Submodule R M\ns : Finset (↥M' × N)\ne : Fin (... | convert! (injective_iff_map_eq_zero' _).mp (injective_subtype M') _ | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.RingTheory.Extension.Cotangent.Basic | {
"line": 274,
"column": 4
} | {
"line": 279,
"column": 29
} | {
"line": 281,
"column": 0
} | [
{
"pp": "case refine_2\nR : Type u\nS : Type v\ninst✝²³ : CommRing R\ninst✝²² : CommRing S\ninst✝²¹ : Algebra R S\nP : Extension R S\nR' : Type u'\nS' : Type v'\ninst✝²⁰ : CommRing R'\ninst✝¹⁹ : CommRing S'\ninst✝¹⁸ : Algebra R' S'\nP' : Extension R' S'\ninst✝¹⁷ : Algebra R R'\ninst✝¹⁶ : Algebra S S'\ninst✝¹⁵ :... | [] | intro x y
ext
simp only [LinearMap.coe_comp, LinearMap.coe_restrictScalars, Function.comp_apply,
Cotangent.val_mk, Cotangent.val_add, Cotangent.val_smul''', ← map_smul, ← map_add,
Ideal.toCotangent_eq]
exact Hom.sub_aux f g x y | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Extension.Cotangent.Basic | {
"line": 274,
"column": 4
} | {
"line": 279,
"column": 29
} | {
"line": 281,
"column": 0
} | [
{
"pp": "case refine_2\nR : Type u\nS : Type v\ninst✝²³ : CommRing R\ninst✝²² : CommRing S\ninst✝²¹ : Algebra R S\nP : Extension R S\nR' : Type u'\nS' : Type v'\ninst✝²⁰ : CommRing R'\ninst✝¹⁹ : CommRing S'\ninst✝¹⁸ : Algebra R' S'\nP' : Extension R' S'\ninst✝¹⁷ : Algebra R R'\ninst✝¹⁶ : Algebra S S'\ninst✝¹⁵ :... | [] | intro x y
ext
simp only [LinearMap.coe_comp, LinearMap.coe_restrictScalars, Function.comp_apply,
Cotangent.val_mk, Cotangent.val_add, Cotangent.val_smul''', ← map_smul, ← map_add,
Ideal.toCotangent_eq]
exact Hom.sub_aux f g x y | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Extension.Cotangent.Basic | {
"line": 556,
"column": 55
} | {
"line": 556,
"column": 63
} | {
"line": 556,
"column": 63
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : FinitePresentation R S\nP : Presentation R S (Fin (Presentation.ofFinitePresentationVars R S))\n (Fin (Presentation.ofFinitePresentationRels R S)) :=\n Presentation.ofFinitePresentation R S\n⊢ FiniteType R... | [
"R : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : FinitePresentation R S\nP : Presentation R S (Fin (Presentation.ofFinitePresentationVars R S))\n (Fin (Presentation.ofFinitePresentationRels R S)) :=\n Presentation.ofFinitePresentation R S\n⊢ FiniteType R (Presentati... | simp [P] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Extension.Cotangent.Basic | {
"line": 626,
"column": 55
} | {
"line": 626,
"column": 63
} | {
"line": 626,
"column": 63
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹² : CommRing R\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R S\nι : Type w\nι' : Type u_1\nP✝ : Generators R S ι\nS' : Type u_2\ninst✝⁹ : CommRing S'\ninst✝⁸ : Algebra R S'\nT : Type w\ninst✝⁷ : CommRing T\ninst✝⁶ : Algebra R T\ninst✝⁵ : Algebra S T\ninst✝⁴ : IsScalarTower R ... | [
"R : Type u\nS : Type v\ninst✝¹² : CommRing R\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R S\nι : Type w\nι' : Type u_1\nP✝ : Generators R S ι\nS' : Type u_2\ninst✝⁹ : CommRing S'\ninst✝⁸ : Algebra R S'\nT : Type w\ninst✝⁷ : CommRing T\ninst✝⁶ : Algebra R T\ninst✝⁵ : Algebra S T\ninst✝⁴ : IsScalarTower R S T\ninst✝³ ... | simp [P] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Support | {
"line": 121,
"column": 2
} | {
"line": 124,
"column": 79
} | {
"line": 126,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\n⊢ support R M = ∅ ↔ Subsingleton M",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Submonoid.powers_one",
"Mu... | [] | rw [← Set.subset_empty_iff, ← PrimeSpectrum.zeroLocus_singleton_one,
← LocalizedModule.subsingleton_iff_support_subset, LocalizedModule.subsingleton_iff,
subsingleton_iff_forall_eq 0]
simp only [Submonoid.powers_one, Submonoid.mem_bot, exists_eq_left, one_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Support | {
"line": 121,
"column": 2
} | {
"line": 124,
"column": 79
} | {
"line": 126,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\n⊢ support R M = ∅ ↔ Subsingleton M",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Submonoid.powers_one",
"Mu... | [] | rw [← Set.subset_empty_iff, ← PrimeSpectrum.zeroLocus_singleton_one,
← LocalizedModule.subsingleton_iff_support_subset, LocalizedModule.subsingleton_iff,
subsingleton_iff_forall_eq 0]
simp only [Submonoid.powers_one, Submonoid.mem_bot, exists_eq_left, one_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.SpanRankOperations | {
"line": 46,
"column": 2
} | {
"line": 46,
"column": 63
} | {
"line": 48,
"column": 0
} | [
{
"pp": "case hd\nR : Type u_1\nA : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\nM : Type u_3\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nN : Submodule R M\nfg : N.FG\n⊢ Cardinal.lift.{u_2, u_3} N.spanRank < Cardinal.aleph0",
"ppTerm": "?hd",
"assigned": true,
"usedCon... | [] | simp [Cardinal.lift_lt_aleph0, spanRank_finite_iff_fg.mpr fg] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Multilinear.TensorProduct | {
"line": 48,
"column": 4
} | {
"line": 50,
"column": 38
} | {
"line": 52,
"column": 0
} | [
{
"pp": "R : Type u_1\nι₁ : Type u_2\nι₂ : Type u_3\nι₃ : Type u_4\nι₄ : Type u_5\ninst✝⁶ : CommSemiring R\nN₁ : Type u_6\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : Module R N₁\nN₂ : Type u_7\ninst✝³ : AddCommMonoid N₂\ninst✝² : Module R N₂\nN : ι₁ ⊕ ι₂ → Type u_8\ninst✝¹ : (i : ι₁ ⊕ ι₂) → AddCommMonoid (N i)\ninst✝ ... | [] | rintro _ m (i₁ | i₂) p q
· letI := Classical.decEq ι₁; simp
· letI := Classical.decEq ι₂; simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Multilinear.TensorProduct | {
"line": 48,
"column": 4
} | {
"line": 50,
"column": 38
} | {
"line": 52,
"column": 0
} | [
{
"pp": "R : Type u_1\nι₁ : Type u_2\nι₂ : Type u_3\nι₃ : Type u_4\nι₄ : Type u_5\ninst✝⁶ : CommSemiring R\nN₁ : Type u_6\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : Module R N₁\nN₂ : Type u_7\ninst✝³ : AddCommMonoid N₂\ninst✝² : Module R N₂\nN : ι₁ ⊕ ι₂ → Type u_8\ninst✝¹ : (i : ι₁ ⊕ ι₂) → AddCommMonoid (N i)\ninst✝ ... | [] | rintro _ m (i₁ | i₂) p q
· letI := Classical.decEq ι₁; simp
· letI := Classical.decEq ι₂; simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Multilinear.TensorProduct | {
"line": 58,
"column": 8
} | {
"line": 58,
"column": 13
} | {
"line": 58,
"column": 13
} | [
{
"pp": "R : Type u_1\nι₁ : Type u_2\nι₂ : Type u_3\nι₃ : Type u_4\nι₄ : Type u_5\ninst✝⁶ : CommSemiring R\nN₁ : Type u_6\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : Module R N₁\nN₂ : Type u_7\ninst✝³ : AddCommMonoid N₂\ninst✝² : Module R N₂\nN : ι₁ ⊕ ι₂ → Type u_8\ninst✝¹ : (i : ι₁ ⊕ ι₂) → AddCommMonoid (N i)\ninst✝ ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Multilinear.TensorProduct | {
"line": 58,
"column": 8
} | {
"line": 58,
"column": 13
} | {
"line": 58,
"column": 13
} | [
{
"pp": "R : Type u_1\nι₁ : Type u_2\nι₂ : Type u_3\nι₃ : Type u_4\nι₄ : Type u_5\ninst✝⁶ : CommSemiring R\nN₁ : Type u_6\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : Module R N₁\nN₂ : Type u_7\ninst✝³ : AddCommMonoid N₂\ninst✝² : Module R N₂\nN : ι₁ ⊕ ι₂ → Type u_8\ninst✝¹ : (i : ι₁ ⊕ ι₂) → AddCommMonoid (N i)\ninst✝ ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Multilinear.TensorProduct | {
"line": 58,
"column": 8
} | {
"line": 58,
"column": 13
} | {
"line": 58,
"column": 13
} | [
{
"pp": "R : Type u_1\nι₁ : Type u_2\nι₂ : Type u_3\nι₃ : Type u_4\nι₄ : Type u_5\ninst✝⁶ : CommSemiring R\nN₁ : Type u_6\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : Module R N₁\nN₂ : Type u_7\ninst✝³ : AddCommMonoid N₂\ninst✝² : Module R N₂\nN : ι₁ ⊕ ι₂ → Type u_8\ninst✝¹ : (i : ι₁ ⊕ ι₂) → AddCommMonoid (N i)\ninst✝ ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Multilinear.TensorProduct | {
"line": 58,
"column": 19
} | {
"line": 58,
"column": 24
} | {
"line": 58,
"column": 24
} | [
{
"pp": "R : Type u_1\nι₁ : Type u_2\nι₂ : Type u_3\nι₃ : Type u_4\nι₄ : Type u_5\ninst✝⁶ : CommSemiring R\nN₁ : Type u_6\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : Module R N₁\nN₂ : Type u_7\ninst✝³ : AddCommMonoid N₂\ninst✝² : Module R N₂\nN : ι₁ ⊕ ι₂ → Type u_8\ninst✝¹ : (i : ι₁ ⊕ ι₂) → AddCommMonoid (N i)\ninst✝ ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Multilinear.TensorProduct | {
"line": 58,
"column": 19
} | {
"line": 58,
"column": 24
} | {
"line": 58,
"column": 24
} | [
{
"pp": "R : Type u_1\nι₁ : Type u_2\nι₂ : Type u_3\nι₃ : Type u_4\nι₄ : Type u_5\ninst✝⁶ : CommSemiring R\nN₁ : Type u_6\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : Module R N₁\nN₂ : Type u_7\ninst✝³ : AddCommMonoid N₂\ninst✝² : Module R N₂\nN : ι₁ ⊕ ι₂ → Type u_8\ninst✝¹ : (i : ι₁ ⊕ ι₂) → AddCommMonoid (N i)\ninst✝ ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Multilinear.TensorProduct | {
"line": 58,
"column": 19
} | {
"line": 58,
"column": 24
} | {
"line": 58,
"column": 24
} | [
{
"pp": "R : Type u_1\nι₁ : Type u_2\nι₂ : Type u_3\nι₃ : Type u_4\nι₄ : Type u_5\ninst✝⁶ : CommSemiring R\nN₁ : Type u_6\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : Module R N₁\nN₂ : Type u_7\ninst✝³ : AddCommMonoid N₂\ninst✝² : Module R N₂\nN : ι₁ ⊕ ι₂ → Type u_8\ninst✝¹ : (i : ι₁ ⊕ ι₂) → AddCommMonoid (N i)\ninst✝ ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Multilinear.TensorProduct | {
"line": 58,
"column": 30
} | {
"line": 58,
"column": 35
} | {
"line": 58,
"column": 35
} | [
{
"pp": "R : Type u_1\nι₁ : Type u_2\nι₂ : Type u_3\nι₃ : Type u_4\nι₄ : Type u_5\ninst✝⁶ : CommSemiring R\nN₁ : Type u_6\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : Module R N₁\nN₂ : Type u_7\ninst✝³ : AddCommMonoid N₂\ninst✝² : Module R N₂\nN : ι₁ ⊕ ι₂ → Type u_8\ninst✝¹ : (i : ι₁ ⊕ ι₂) → AddCommMonoid (N i)\ninst✝ ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Multilinear.TensorProduct | {
"line": 58,
"column": 30
} | {
"line": 58,
"column": 35
} | {
"line": 58,
"column": 35
} | [
{
"pp": "R : Type u_1\nι₁ : Type u_2\nι₂ : Type u_3\nι₃ : Type u_4\nι₄ : Type u_5\ninst✝⁶ : CommSemiring R\nN₁ : Type u_6\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : Module R N₁\nN₂ : Type u_7\ninst✝³ : AddCommMonoid N₂\ninst✝² : Module R N₂\nN : ι₁ ⊕ ι₂ → Type u_8\ninst✝¹ : (i : ι₁ ⊕ ι₂) → AddCommMonoid (N i)\ninst✝ ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Multilinear.TensorProduct | {
"line": 58,
"column": 30
} | {
"line": 58,
"column": 35
} | {
"line": 58,
"column": 35
} | [
{
"pp": "R : Type u_1\nι₁ : Type u_2\nι₂ : Type u_3\nι₃ : Type u_4\nι₄ : Type u_5\ninst✝⁶ : CommSemiring R\nN₁ : Type u_6\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : Module R N₁\nN₂ : Type u_7\ninst✝³ : AddCommMonoid N₂\ninst✝² : Module R N₂\nN : ι₁ ⊕ ι₂ → Type u_8\ninst✝¹ : (i : ι₁ ⊕ ι₂) → AddCommMonoid (N i)\ninst✝ ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Multilinear.TensorProduct | {
"line": 58,
"column": 41
} | {
"line": 58,
"column": 46
} | {
"line": 58,
"column": 46
} | [
{
"pp": "R : Type u_1\nι₁ : Type u_2\nι₂ : Type u_3\nι₃ : Type u_4\nι₄ : Type u_5\ninst✝⁶ : CommSemiring R\nN₁ : Type u_6\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : Module R N₁\nN₂ : Type u_7\ninst✝³ : AddCommMonoid N₂\ninst✝² : Module R N₂\nN : ι₁ ⊕ ι₂ → Type u_8\ninst✝¹ : (i : ι₁ ⊕ ι₂) → AddCommMonoid (N i)\ninst✝ ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Multilinear.TensorProduct | {
"line": 58,
"column": 41
} | {
"line": 58,
"column": 46
} | {
"line": 58,
"column": 46
} | [
{
"pp": "R : Type u_1\nι₁ : Type u_2\nι₂ : Type u_3\nι₃ : Type u_4\nι₄ : Type u_5\ninst✝⁶ : CommSemiring R\nN₁ : Type u_6\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : Module R N₁\nN₂ : Type u_7\ninst✝³ : AddCommMonoid N₂\ninst✝² : Module R N₂\nN : ι₁ ⊕ ι₂ → Type u_8\ninst✝¹ : (i : ι₁ ⊕ ι₂) → AddCommMonoid (N i)\ninst✝ ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Multilinear.TensorProduct | {
"line": 58,
"column": 41
} | {
"line": 58,
"column": 46
} | {
"line": 58,
"column": 46
} | [
{
"pp": "R : Type u_1\nι₁ : Type u_2\nι₂ : Type u_3\nι₃ : Type u_4\nι₄ : Type u_5\ninst✝⁶ : CommSemiring R\nN₁ : Type u_6\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : Module R N₁\nN₂ : Type u_7\ninst✝³ : AddCommMonoid N₂\ninst✝² : Module R N₂\nN : ι₁ ⊕ ι₂ → Type u_8\ninst✝¹ : (i : ι₁ ⊕ ι₂) → AddCommMonoid (N i)\ninst✝ ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Alternating.Uncurry.Fin | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 53
} | {
"line": 82,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R M\ninst✝ : Module R N\nn : ℕ\nf : M [⋀^Fin n]→ₗ[R] N\nv : Fin (n + 1) → M\nj : Fin (n + 1)\ni : Fin n\nhvij : v (j.succAbove i) = v j\nhij : j.succAbove i ≠ j\n⊢ (-1) ^ ↑(j... | [
"R : Type u_1\nM : Type u_2\nN : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R M\ninst✝ : Module R N\nm : ℕ\nf : M [⋀^Fin (m + 1)]→ₗ[R] N\nv : Fin (m + 1 + 1) → M\nj : Fin (m + 1 + 1)\ni : Fin (m + 1)\nhvij : v (j.succAbove i) = v j\nhij : j.succAbove i ≠ j\n⊢ (-... | obtain ⟨m, rfl⟩ : ∃ m, m + 1 = n := by simp [i.pos] | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.LinearAlgebra.Alternating.Uncurry.Fin | {
"line": 183,
"column": 4
} | {
"line": 183,
"column": 64
} | {
"line": 184,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R M\ninst✝ : Module R N\nn : ℕ\nf : M →ₗ[R] M →ₗ[R] M [⋀^Fin n]→ₗ[R] N\nv : Fin (n + 2) → M\ni j : Fin (n + 1)\nhj : i ≤ j\nH₁ : i.castSucc.removeNth v j = v j.succ\n⊢ j.succ... | [] | simp [Fin.removeNth_apply, Fin.succAbove_of_castSucc_lt, hj] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.TensorPower.Basic | {
"line": 148,
"column": 2
} | {
"line": 157,
"column": 57
} | {
"line": 159,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nn : ℕ\na : ⨂[R]^n M\n⊢ (cast R M ⋯) (GradedMonoid.GMul.mul GradedMonoid.GOne.one a) = a",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"PiTensorProduct.instModule",
"Eq.... | [] | rw [gMul_def, gOne_def]
induction a using PiTensorProduct.induction_on with
| smul_tprod r a =>
rw [TensorProduct.tmul_smul, map_smul, map_smul, ← gMul_def, tprod_mul_tprod, cast_tprod]
congr 2 with i
rw [Fin.elim0_append]
refine congr_arg a (Fin.ext ?_)
simp
| add x y hx hy =>
rw [TensorP... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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